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Chapter 3: Number Relationships
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Chapter 3
Page 1
Number Relationships
Contents
OVERVIEW
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Curriculum across Grades 5 to 7: Number . . . . . . . . . . . . 2
Math Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Planning for Instruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Reading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Connections to Literature . . . . . . . . . . . . . . . . . . . . . . . 3
Connections to Other Math Strands . . . . . . . . . . . . . . . 3
Connections to Other Curricula . . . . . . . . . . . . . . . . . . 3
Connections to Home and Community . . . . . . . . . . . . 3
Chapter 3 Planning Chart . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 3 Assessment Summary . . . . . . . . . . . . . . . . . . . . 6
TEACHING NOTES
Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Getting Started: Banner Design . . . . . . . . . . . . . . . . . . . . . 9
Lesson 1: Identifying Factors . . . . . . . . . . . . . . . . . . . . . . 13
Lesson 2: Identifying Multiples . . . . . . . . . . . . . . . . . . . . 18
Curious Math: String Art . . . . . . . . . . . . . . . . . . . . . . . . . 22
Lesson 3: Prime and Composite Numbers . . . . . . . . . . . . 24
Math Game: Colouring Factors . . . . . . . . . . . . . . . . . . . . 29
Lesson 4: Identifying Factors by Dividing . . . . . . . . . . . . 31
Lesson 5: Creating Composite Numbers . . . . . . . . . . . . . 35
Mid-Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Lesson 6: Solving Problems Using an Organized List . . . . . 43
Lesson 7: Representing Integers . . . . . . . . . . . . . . . . . . . . 47
Curious Math: Countdown Clock . . . . . . . . . . . . . . . . . . 51
Lesson 8: Comparing and Ordering Integers . . . . . . . . . . 53
Lesson 9: Order of Operations . . . . . . . . . . . . . . . . . . . . . 58
Math Game: Four in a Row . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter Task: A Block Dropping Game . . . . . . . . . . . . . 69
Chapters 1–3 Cumulative Review . . . . . . . . . . . . . . . . . . 72
CHAPTER 3 BLACKLINE MASTERS
Family Letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Scaffolding for Getting Started . . . . . . . . . . . . . . . . . 75–76
Scaffolding for Lesson 2, Question 3 . . . . . . . . . . . . . . . . 77
String Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Mid-Chapter Review—Frequently Asked Questions . . . . 79
Four in a Row Game Board . . . . . . . . . . . . . . . . . . . . . . . 80
Calculation Cards . . . . . . . . . . . . . . . . . . . . . . . . . . . 81–82
Chapter Review—Frequently Asked Questions . . . . . . . . 83
Chapter 3 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84–86
Chapter 3 Task: A Block Dropping Game . . . . . . . . . 87–88
Answers for Chapter 3 Masters . . . . . . . . . . . . . . . . . 89–91
From Masters Booklet
Review of Essential Skills: Chapter 3 . . . . . . . . . . . . . . . . . 5
1 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 cm Grid Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
100 Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Number Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Initial Assessment Summary . . . . . . . . . . . . . . . . . . . . . . 57
Assessment Rubrics for Mathematical Processes . . . . . 58–61
Chapter Checklist: Chapter 3 . . . . . . . . . . . . . . . . . . . . . 64
Self-Assessment: Chapter 3 Lesson Goals . . . . . . . . . . . . . 75
Self-Assessment: Mathematical Processes . . . . . . . . . . . . . 84
Self-Assessment: What I Like . . . . . . . . . . . . . . . . . . . . . . 85
Self-Assessment: How I Learn . . . . . . . . . . . . . . . . . . . . . 85
Introduction
This chapter provides students with opportunities to use
their understanding of number relationships to identify
factors and multiples, to determine whether a number is
prime or composite, to compare and order integers, and to
use the rules for order of operations to calculate the value of
an expression. They will build upon the mental mathematics
strategies developed in Grade 5 to determine factors and
multiples.
Throughout the chapter, students use concrete and
pictorial models to help develop an understanding of new
concepts before attempting to use mental mathematics
strategies.
Copyright © 2010 Nelson Education Ltd.
Answers and Solutions
Answers to all numbered questions are provided in the
Student Book. Full solutions are provided in the Solutions
Manual. Selected answers are provided in the Teacher’s
Resource lesson notes.
Contents
1
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Curriculum across Grades 5 to 7: Number
The Grade 6 outcomes and achievement indicators listed below are addressed in this chapter.
When the outcome or indicator is the focus of a lesson or feature, the lesson number or feature is indicated in brackets.
Grade 5
Grade 6
Grade 7
Specific Outcomes
N3. Demonstrate an understanding of factors and multiples by
• determining multiples and factors of numbers less than 100
• identifying prime and composite numbers
• solving problems involving multiples. (1, 2, CM1, 3, MG1,
4, 5, 6)
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them. (2, CM1, 6)
• Determine all the whole-number factors of a given
number using arrays. (1, MG1)
• Identify the factors for a given number and explain the
strategy used, e.g., concrete or visual representations,
repeated division by prime numbers, or factor trees.
(1, MG1, 4, 5, 6)
• Provide an example of a prime number and explain why it
is a prime number. (3, 4, 5)
• Provide an example of a composite number and explain
why it is a composite number. (3, 4, 5)
• Sort a given set of numbers as prime and composite. (3)
• Solve a given problem involving factors or multiples.
(1, 2, CM1, 3, 6)
• Explain why 0 and 1 are neither prime nor composite. (3)
Specific Outcomes
N1. Determine and explain why a number is
divisible by 2, 3, 4, 5, 6, 8, 9, or 10, and
why a number cannot be divided by 0.
[C, R]
Strand: Number
General Outcome: Develop number sense.
Specific Outcome
N3. Apply mental mathematics strategies and number
properties, such as
• skip counting from a known fact
• using doubling or halving
• using patterns in the 9s facts
• using repeated doubling or halving
to determine answers for basic multiplication facts
to 81 and related division facts.
[C, CN, ME, R, V]
N6.
Demonstrate an understanding of
addition and subtraction of integers,
concretely, pictorially, and symbolically.
[C, CN, PS, R, V]
N7. Demonstrate an understanding of integers, concretely,
pictorially, and symbolically. (7, CM2, 8)
[C, CN, R, V]
Achievement Indicators
• Extend a given number line by adding numbers less than
zero and explain the pattern on each side of zero. (7, CM2)
• Place given integers on a number line and explain how
integers are ordered. (8)
• Describe contexts in which integers are used, e.g., on a
thermometer. (7, CM2)
• Compare two integers; represent their relationship using
the symbols <, >, and =, and verify using a number line. (8)
• Order given integers in ascending or descending order. (8)
N9. Explain and apply the order of operations, excluding
exponents, with and without technology (limited to whole
numbers). (9 MG2)
[CN, ME, PS, T]
Achievement Indicators
• Demonstrate and explain with examples why there is a
need to have a standardized order of operations. (9)
• Apply the order of operations to solve multi-step
problems with or without technology, e.g., computer,
calculator. (9, MG2)
Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization
Features: CM1 (Curious Math: String Art), MG1 (Math Game: Colouring Factors), CM2 (Curious Math: Countdown Clock), MG2 (Math Game: Four in a Row)
2
Chapter 3: Number Relationships
Copyright © 2010 Nelson Education Ltd.
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Page 3
Math Background
An understanding of number relationships is essential to
functioning in daily life. Students gain this understanding
by exploring factors, multiples, and integers directly. Students
also gain an intuitive understanding about numbers by relating
numbers to a variety of real-world contexts. For example,
students use reasoning to solve number problems in the real
world. In addition, visualizing number patterns and
relationships allows students to make connections and identify
number relationships, further developing number sense.
Throughout the chapter, students are encouraged to use
mental math to determine factors and multiples and to solve
complex expressions using the order of operations. It is
important for students to demonstrate computational
math skills as well as flexibility with numbers. Students
are encouraged to use reasoning to check their answers,
to analyze and evaluate their thinking, and to listen
and learn from the strategies of others.
See PRIME (Professional Resources and Instruction for
Mathematics Educators): Number and Operations by Marian
Small (Thomson Nelson, 2005) for additional math
background and teaching strategies.
Planning for Instruction
Problem Solving
In Lesson 6, students solve problems by using an organized
list. Students will also solve a variety of problems throughout
the chapter as they apply their understanding of factors,
multiples, and integers.
Assign a Problem of the Week from the selection below or
from your own collection.
1. A number has nine different factors. Two of its multiples
are 72 and 108. What is the number? (36: factors are 1, 2,
3, 4, 6, 9, 12, 18, and 36; 2 36 72; 3 36 108)
2. The temperature on Monday was 11 °C. The temperature
on Tuesday was 15 °C. The temperature on Wednesday
was 13 °C. On Thursday, it was colder than Monday but
warmer than Wednesday. What was the temperature on
Thursday? (12 °C: 13 12 11)
3. Maddy copied down a number sentence in math class,
but she forgot to write the brackets. Where should Maddy
place the brackets to make the number sentence true?
3 2 6 12 8 6 (Maddy should place the
brackets around the addition and subtraction.
3 (2 6) (12 8)
384
24 4
6)
Reading Strategies
The reading strategies highlighted in this chapter are
Monitoring Comprehension (Mid-Chapter Review) and
Finding Important Information (Lesson 6). To reinforce the
use of these strategies, you may apply them to other questions
throughout the lessons as opportunities present themselves.
Connections to Literature
Expand your classroom library or math centre with books
related to the math in this chapter. For example:
• Frasier, Debra. On the Day You Were Born. Harcourt
Children’s Books, 1991.
Copyright © 2010 Nelson Education Ltd.
• Merrill, Jean. The Toothpaste Millionaire. Houghton Mifflin,
2006.
• Murphy, Stuart. Less Than Zero. HarperTrophy, 2003.
Connections to Other Math Strands
Patterns and Algebra: In the Getting Started activity,
students will identify the pattern in a banner design. In
Lesson 2, students will use number patterns as a way to
identify multiples.
Shape and Space: In the Chapter Task, students will describe
how squares and rectangles with different dimensions can be
used to fill a large square.
Measurement: In Lesson 8, students will use their knowledge
of integers to compare and order temperatures.
Connections to Other Curricula
Art: In Curious Math: String Art, students will use a modified
version of string art to represent multiples of numbers.
Science: In Lesson 2, students will use multiples to determine
the years in which the comet Kojima will likely be visible
from Earth. In Lesson 8, students will compare and order
positive and negative temperatures. In Lesson 9, students will
use formulas to calculate heart rate and lung capacity.
Connections to Home and Community
• Have students use everyday situations to order and compare
numbers, identify factors and multiples, and use the order of
operations.
• Send home Family Letter p. 74, which contains suggestions
for a variety of activities related to the math in this chapter
that students can do at home.
• Have students complete the Nelson Math Focus 6 Workbook
pages for this chapter at home.
• Use the suggestions for at-home activities in Follow-Up and
Preparation for Next Class in various lessons.
Overview
3
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Chapter 3 Planning Chart
Key Concepts*
Key Principles
Number and Operations
• Numbers tell how many or how much.
• Classifying numbers provides information about the characteristics
of the numbers.
• There are different, but equivalent, representations for a number.
• Benchmark numbers are useful for relating and estimating numbers.
• A number can be described as the product of its factors.
• Describing a number as a multiple suggests thinking of it in terms of a unit
other than 1; for example, since 6 is a multiple of 3, it is two 3s.
• Knowing that a number is prime or composite gives you information about
how many factors it has, as well as about how it can be represented as an
array.
• Integers include the natural numbers and their opposites, as well as zero.
They describe amounts above, below, and including the zero benchmark.
• Integers can be compared by using their positions relative to the zero
benchmark.
• Order of operations rules are used to ensure that everyone reading an
expression interprets it the same way.
*PRIME (Professional Resources and Instruction for Math Educators): Number and
Operations by Marian Small (Thomson Nelson, 2005)
Student Book Section
Lesson Goal
Getting Started
Banner Designs
pp. 68–69 (TR pp. 9–12)
Activate knowledge about
number relationships.
Lesson 1
Identifying Factors
pp. 70–73 (TR pp. 13–17)
Identify factors to solve
problems.
Lesson 2
Identifying Multiples
pp. 74–76 (TR pp. 18–21)
Grade 6
Outcomes
Pacing
13 Days
Prerequisite Skills/Concepts
1 day
• Recall multiplication facts and related division facts to 81.
• Identify and extend number patterns.
N3
1 day
•
•
•
•
Identify multiples to solve
problems.
N3
1 day
• Identify factors of whole numbers.
• Extend a number pattern by multiplying or adding whole numbers.
Lesson 3
Prime and Composite Numbers
pp. 78–80 (TR pp. 24–28)
Identify prime and
composite numbers.
N3
1 day
• Identify factors and multiples of whole numbers.
Lesson 4
Identifying Factors by Dividing
pp. 82–84 (TR pp. 31–34)
Identify factors by dividing
composite numbers by
primes.
N3
1 day
• Identify prime and composite numbers.
• Identify factors of whole numbers.
Lesson 5
Creating Composite Numbers
p. 85 (TR pp. 35–38)
Multiply combinations of
factors to create composite
numbers.
N3
1 day
• Multiply and divide combinations of one-digit and two-digit numbers.
• Identify prime and composite numbers.
Lesson 6
Solving Problems Using an Organized
List, pp. 88–89 (TR pp. 43–46)
Use an organized list to
solve problems that involve
number relationships.
N3
1 day
• Identify factors and multiples of whole numbers.
• Identify prime and composite numbers.
Lesson 7
Representing Integers
pp. 90–92 (TR pp. 47–50)
Use integers to describe
situations.
N7
1 day
• Locate numbers on a number line.
Lesson 8
Comparing and Ordering Integers
pp. 94–97 (TR pp. 53–57)
Use a number line to
compare and order
integers.
N7
1 day
• Locate integers on a number line.
• Use the symbols <, >, and to compare numbers.
Lesson 9
Order of Operations
pp. 98–100 (TR pp. 58–61)
Apply the rules for order of
operations with whole
numbers.
N9
1 day
• Use mental math to add, subtract, multiply, and divide whole numbers.
Curious Math 1 p. 77 (TR pp. 22–23)
Math Game 1 p. 81 (TR pp. 29–30)
Mid-Chapter Review pp. 86–87 (TR pp. 39–42)
Curious Math 2 p. 93 (TR pp.51–52)
Math Game 2 p. 101 (TR pp. 62–63)
Chapter Review, pp. 102–104 (TR pp. 64–68)
Chapter Task, p. 105 (TR pp. 69–71)
Chapters 1–3 Cumulative Review pp. 106–107 (TR pp. 72–73)
4
Chapter 3: Number Relationships
Calculate products and quotients using mental math.
Divide a two-digit number by a one-digit number.
Understand the meaning of the term factor.
Use arrays to multiply and divide numbers.
3 days
Copyright © 2010 Nelson Education Ltd.
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Page 5
Chapter Goals
•
•
•
•
•
Identify prime numbers, composite numbers, factors, and multiples.
Determine the factors of a composite number.
Use an organized list to solve problems.
Represent, order, and compare integers.
Explain and apply the order of operations with whole numbers.
Materials
Masters
• pencil crayons
•
•
•
•
• Optional: counters
• Optional: linking
cubes
• Optional: 1 cm Grid Paper, Masters Booklet p. 22
• Optional: Chapter Checklist: Chapter 3, Masters Booklet p. 64
Mid-Chapter Review Questions 1 & 2
Chapter Review Questions 1, 2, & 3
Workbook, p. 17
• rulers
• Optional: counters
• Number Lines, Masters Booklet p. 33
• Optional: Scaffolding for Lesson 2, Question 3 p. 77
Mid-Chapter Review Questions 3 & 4
Chapter Review Questions 4 & 5
Workbook, p. 18
• counters
• 100 Chart, Masters Booklet p. 30
• 2 cm Grid Paper, Masters Booklet p. 23
Mid-Chapter Review Question 5
Chapter Review Questions 6 & 7
Workbook p. 19
2 cm Grid Paper, Masters Booklet p. 23
Optional: Scaffolding for Getting Started pp. 75–76
Optional: Review of Essential Skills: Chapter 3, Masters Booklet p. 5
Optional: Initial Assessment Summary, Masters Booklet p. 57
• number cards 40 to 50
• Optional: chart paper
and markers
Extra Practice in the Student Book and Workbook
Mid-Chapter Review Questions 6 & 7
Chapter Review Question 8
Workbook, p. 20
• Optional: 100 Chart, Masters Booklet p. 30
• Optional: 1 cm Grid Paper, Masters Booklet p. 23
Workbook p. 21
• Optional: 100 Chart, Masters Booklet p. 30
Chapter Review Question 9
Workbook, p. 22
• Number Lines, Masters Booklet p. 33
Chapter Review Question 10
Workbook p. 23
• Number Lines, Masters Booklet p. 33
Chapter Review Questions 11 & 12
Workbook, p. 24
• calculators
Chapter Review Questions 13 & 14
Workbook p. 25
For materials and masters for features, reviews, and the Chapter Task, see the TR section.
Workbook p. 26
Copyright © 2010 Nelson Education Ltd.
Overview
5
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Chapter 3 Assessment Summary
(also part of Assessment for Learning) are provided in Getting
Started. Summative assessment (Assessment of Learning)
opportunities are provided in the Mid-Chapter Review,
Chapter Review, and Chapter Task. Have students self-assess
their learning (Assessment as Learning) using one of the
self-assessment tools provided in the Masters Booklet.
These charts list references to the many assessment
opportunities in the chapter. Formative assessment
(Assessment for Learning) provides information about
students’ understanding of concepts and helps you adapt
instruction to students’ needs. A key question in each lesson
links to the lesson goal. Initial or diagnostic assessment ideas
Opportunities for Feedback: Assessment for Learning
Student Book Section
Chart
Lesson 1
Identifying Factors
pp. 70–73
TR p. 17
Lesson 2
Identifying Multiples pp. 74–76
TR p. 21
Curious Math
String Art p. 77
TR p. 23
Lesson 3
Prime and Composite Numbers
pp. 78–80
TR p. 28
Math Game
Colouring Factors p. 81
TR p. 30
Lesson 4
Identifying Factors by Dividing
pp. 82–84
TR p. 34
Lesson 5
Creating Composite Numbers p. 85
Mid-Chapter Review pp. 86–87
Key Question
Grade 6 Outcomes
Mathematical Process Focus for Key Question
N3. Demonstrate an understanding of
factors and multiples by
• determining multiples and factors of
numbers less than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Reasoning, Visualization
N3
Problem Solving, Visualization
N3
Problem Solving, Reasoning, Visualization
N3
Reasoning, Visualization
N3
Reasoning
4, written answer
N3
Reasoning
TR p. 38
entire exploration,
investigation
N3
Problem Solving, Reasoning
TR p. 41
1, model, written answer
N3
Visualization
2, short answer
N3
Reasoning
3, short answer
N3
Problem Solving
4, short answer, written answer
N3
Reasoning
5, short answer
N3
Problem Solving
6, short answer
N3
Reasoning, Visualization
7, written answer
N3
Reasoning
5, model, written answer
5, short answer,
written answer
4, written answer
Lesson 6
Solving Problems Using an
Organized List pp. 88–89
TR p. 46
6, written answer
N3
Problem Solving, Reasoning
Lesson 7
Representing Integers
pp. 90–92
TR p. 50
4, short answer, model
N7. Demonstrate an understanding of
integers, concretely, pictorially, and
symbolically.
[C, CN, R, V]
Reasoning, Visualization
Curious Math
Countdown Clock p. 93
TR p. 52
N7
Connections, Reasoning
Lesson 8
Comparing and Ordering Integers
pp. 94–97
TR p. 57
6, model, written answer
N7
Communication, Connection, Visualization
Lesson 9
Order of Operations
pp. 98–100
TR p. 61
4, short answer
N9. Explain and apply the order of
operations, excluding exponents, with and
without technology (limited to whole
numbers).
[CN, ME, PS, T]
Connections, Mental Mathematics and
Estimation, Problem Solving, Technology
Math Game
Countdown Clock p. 101
TR p. 63
N9
Mental Mathematics and Estimation
Mathematical Processes: C Communication, CN Connections, ME Mental Mathematics and Estimation, PS Problem Solving, R Reasoning, T Technology, V Visualization
6
Chapter 3: Number Relationships
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Assessment of Learning
Student Book Section
Chart
Question
Grade 6 Outcome
Mathematical Process Focus for Question
Mid-Chapter Review
pp. 86–87
TR pp. 41–42
1, model, written answer
N3
Visualization
2, short answer
N3
Reasoning
3, short answer
N3
Problem Solving
4, short answer, written answer
N3
Reasoning
5, short answer
N3
Problem Solving
6, short answer
N3
Visualization
7, written answer
N3
Reasoning
1, written answer
N3
Visualization
2, written answer
N3
Reasoning
3, 4, short answer
N3
Reasoning
5, written answer
N3
Problem Solving
6, short answer, written answer
N3
Reasoning
7, written answer
N3
Reasoning
8, short answer
N3
Reasoning
9, short answer
N3
Problem Solving, Reasoning
10, written answer
N7
Communication
11, written answer, model
N7
Visualization
12, short answer, model
N7
Communication, Visualization
13, short answer
N9
Mental Mathematics and Estimation
14, short answer, written answer
N9
Problem Solving
entire task, investigation
N3
Problem Solving, Reasoning, Visualization
Chapter Review
pp. 102–104
and
Chapter Test
(TR pp. 84–86)
Chapter Task
A Block Dropping Game, p. 105
TR pp. 66–68
TR p. 71
Assessment as Learning
Student Book Section
Student Self-Assessment Masters
Mid-Chapter Review
pp. 86–87
Chapter 3 Lesson Goals, Masters Booklet p. 75
Self-Assessment: Mathematical Processes, Masters Booklet p. 84
Self-Assessment: What I Like, Masters Booklet p. 85
Self-Assessment: How I Learn, Masters Booklet p. 85
Chapter Review
pp. 102–104
Chapter 3 Lesson Goals, Masters Booklet p. 75
Self-Assessment: Mathematical Processes, Masters Booklet p. 84
Self-Assessment: What I Like, Masters Booklet p. 85
Self-Assessment: How I Learn, Masters Booklet p. 85
Copyright © 2010 Nelson Education Ltd.
Overview
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STUDENT BOOK PAGES 66–67
Chapter Opener
Using the Chapter Opener
Draw students’ attention to the photograph on Student Book
pages 66 and 67. Tell students that the Craik Eco-Centre is
an energy-efficient building that uses renewable energy.
Together, read the opening task. Record and discuss students’
responses.
If students have trouble getting started, have them use 12
linking cubes and make as many different rectangular prisms
as they can. Encourage students to arrange prisms in multiple
layers, such as 2 ⫻ 2 ⫻ 3, as well as single layers, such as
1 ⫻ 2 ⫻ 6. Review how the length, width, and thickness
of a prism can be used to identify factors of 12. As students
assemble model walls with 36 linking cubes, encourage them
to build walls with layers as well. Students might identify
different numbers of walls depending on whether or not
they distinguish between the order of the dimensions. For
example, they might consider a 2 ⫻ 18 and an 18 ⫻ 2 wall
to be equivalent walls.
Sample Discourse
“Suppose a wall has a thickness of 1 cube. How many
different walls can you make with 36 cubes?”
• Five: 36 cubes long and 1 cube high, 18 cubes long and
2 cubes high, 12 cubes long and 3 cubes high, 9 cubes long
and 4 cubes high, and 6 cubes long and 6 cubes high
• If you know two factors that multiply together to make 36,
these factors represent the length and height of a wall.
“Suppose a wall has a thickness of 2 cubes. How many
different walls can you make with 36 cubes?”
• Three: 18 cubes long and 1 cube high, 9 cubes long and
2 cubes high, and 6 cubes long and 3 cubes high
“What other wall can you make with 36 linking cubes?
• I can make a wall 4 cubes thick, 3 cubes long, and
3 cubes high.
• I can make a wall 2 cubes thick, 9 cubes long, and
2 cubes high.
• I can make a wall 6 cubes thick, 3 cubes long, and
2 cubes high.
Read and discuss the five goals of the chapter. Ask students
to suggest different ways they can determine the factors of a
number. Have students record in their journals their thoughts
about one of the goals, using a prompt such as “Examples of
situations where I would need to identify the factors of a
number are….” At the end of the chapter, you can ask
students to complete the same prompt. Then they can
compare their ideas with the ones recorded at the beginning
of the chapter and reflect on what they have learned.
8
Chapter 3: Number Relationships
At this point, it would be appropriate to
• send home Family Letter p. 74
• ask students to look through the chapter and add math
word cards to your classroom word wall. Here are some
terms related to this chapter:
Family Letter p. 74
factor
product
multiple
prime number
composite number
integer
opposite integers
rules for order of operations
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STUDENT BOOK PAGES 68–69
Getting Started
Banner Designs
GOAL
Activate knowledge about number relationships.
PREREQUISITE SKILLS/CONCEPTS
• Recall multiplication facts and related division facts to 81.
• Identify and extend number patterns.
2 cm Grid Paper, Masters
Booklet p. 23
Optional: Scaffolding for
Getting Started p. 75–76
Optional: Review of
Essential Skills: Chapter 3,
Masters Booklet p. 5
Optional: Initial
Assessment Summary,
Masters Booklet p. 57
Preparation and Planning
Pacing
30–40 min Activity
10–20 min What Do You Think?
Materials
• pencil crayons
Masters
• 2 cm Grid Paper, Masters Booklet p. 23
• Optional: Scaffolding for Getting
Started pp. 75–76
• Optional: Review of Essential Skills:
Chapter 3, Masters Booklet p. 5
• Optional: Initial Assessment Summary,
Masters Booklet p. 57
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Math Background
The Getting Started activity helps students activate
knowledge of number relationships and principles learned
in earlier grades. Specifically, students will use number
patterns, skip counting, and multiplication to determine
multiples of two whole numbers. Students need a firm
understanding of multiplication and division facts to help
them identify both multiples and factors of whole
numbers.
Copyright © 2010 Nelson Education Ltd.
Getting Started: Banner Designs
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Using the Activity
(Whole Class/Pairs/Small Groups)
± 30–40 min
Use this activity to activate knowledge of factors and
multiples and number patterns and as an opportunity for
initial assessment.
Together, read about Daniel’s Heritage Day banner and
then read the central question on Student Book page 68.
Distribute grid paper to students. Have students work in pairs
or small groups to answer Prompts A to C. Students having
difficulty sketching may prefer writing the letter E for eagle
instead of drawing the symbol. Discuss the answers to these
prompts as a class. Have students work in groups to answer
Prompt D. Have volunteers share their banners with the class.
If extra support is required, guide these students and provide
copies of Scaffolding for Getting Started pp. 75–76.
horse
Answers to the Activity
A. For example,
10
Chapter 3: Number Relationships
horse
10
Every 10th square has a horse in a yellow square. So I
predict that the number of yellow squares with a horse in
100 squares is 100 ⫼ 10 = 10.
B. For example, I saw the pattern 6, 12, 18.
6
The pattern shows skip counting by 6s. So the next
square with an eagle should be the 24th square because
18 ⫹ 6 = 24.
C. For example, I can multiply 1, 2, and 3 by 6 to get
6 ⫻ 1 = 6, 6 ⫻ 2 = 12, and 6 ⫻ 3 = 18, which are
the numbers of the first three red squares that have an
eagle. So I can solve the equation
⫻ 6 = 30 to figure
out the number of red squares with an eagle.
I can divide by 6 to solve the problem. There are 30
squares and 30 ⫼ 6 = 5, so 5 red squares will have
an eagle.
D. For example, I’ll create a banner with 100 squares. I’ll
colour every second square yellow. Every fifth square will
have the symbol for a horse. I’ll figure out how many
yellow squares will have a horse.
12
18
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Using What Do You Think?
(Small Groups/Whole Class) ± 10–20 min
Use this anticipation guide to activate knowledge and
understanding of factors and multiples. Explain to students
that the statements involve math concepts or skills they will
learn about in the chapter—they are not expected to know
the answers at this point. Ask students to read the statements,
think about each one for a few seconds, and decide whether
they agree or disagree. Have volunteers explain the reasons
for their choices. Students can exchange their thoughts in
small groups, in groups where all agree or disagree, or in a
general class discussion. Tell students they can revisit their
ideas at the end of the chapter.
Possible Responses for What Do You Think?
Correct responses are indicated with an asterisk (*). Students
should be able to give correct responses by the end of the
chapter.
1. For example, agree. If you multiply 5 by 6, you get 30.
You can also multiply 1 and 30 to get 30, and there are
other factors of 30 too. So when you multiply two whole
numbers, the product has more than two factors.
*For example, disagree. When you multiply 1 by 1, you
get 1, and 1 is the only factor.
Copyright © 2010 Nelson Education Ltd.
2. For example, agree. The last digit is 0 so when you
multiply numbers like 10 and 20, you get a 0 in the ones
digit of the product.
*For example, disagree. 8 ⫻ 25 = 200 and neither factor
has 0 as the ones digit.
3. *For example, agree. If you extend the first pattern by
adding 5 and the second pattern by adding 7, you get
35 on both lists. Then if you keep adding 5 and 7, you
will get 70 as the next number on both lists. So if you
continue adding both 5 and 7, you will get lots of the
same numbers on both lists.
For example, disagree. The three numbers in each list are
different. One list of numbers goes up by 5s and the
other list goes up by 7s. So you will not get many of the
same numbers.
4. For example, agree. 3 has two factors, 1 and 3.
2 ⫻ 3 = 6. 6 has four factors: 1, 2, 3, and 6.
So multiplying 3 by 2 doubled the number of factors.
*For example, disagree. 4 has three factors: 1, 2, and 4;
2 ⫻ 4 = 8. 8 has four factors: 1, 2, 4 and 8. So when
you multiply 4 by 2, you do not get double the number
of factors.
Getting Started: Banner Designs
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Initial Assessment: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
Prompt B
• Students explain how to use a number pattern to predict the next red banner
square that will have an eagle symbol.
• Students may not recognize that every sixth square has both characteristics
(eagle, red) and cannot extend the pattern 6, 12, 18, … beyond 18. (See 3
below.)
Prompt C
• Students explain how to use a multiplication equation to figure out how many
red squares will have an eagle.
• Students may not be able to connect multiplication facts with determining the
number of red squares that will have an eagle. (See 4 below.)
Prompts C & D
• Students determine the number of coloured squares that will have a symbol
and explain their method.
• Students may not connect determining the number of squares with number
patterns or multiplication facts. (See 3 and 5 below.)
Differentiating Instruction: How you can respond
SUPPORTING STUDENTS WHO ARE ALMOST THERE
1. Use Scaffolding for Getting Started pp. 75–76.
2. Use Review of Essential Skills: Chapter 3, Masters Booklet p. 5 to
activate students’ skills.
3. Have students number the squares from left to right and note the numbers
of the coloured squares that have a symbol. Discuss the pattern in the
numbers (6, 12, 18, …) and discuss the strategies that students might use
to determine the next number in the pattern, for example, skip counting
by 6.
5. Remind students that in a multiplication fact, two factors are multiplied to
give a product. Help students understand that one of the factors is the
number of squares from one coloured square with a symbol to the next, and
the product is the total number of squares in the banner. The unknown factor
is the number of coloured squares with a symbol that will be in the banner.
For example, if there is a coloured square with a symbol every 5 squares and
a total of 50 squares, students can use the multiplication fact
⫻ 5 = 50
to calculate the number of coloured squares with a symbol that will be in
the banner.
4. Remind students that a multiplication fact is another way to represent skip
counting. For example, to complete the multiplication sentence
⫻ 6 = 18,
students can skip count by 6s until they reach 18, and count the number of
skips. There are three skips, so 3 ⫻ 6 = 18. Suggest students use the same
thinking for patterns that reach greater numbers.
SUPPORTING STUDENTS WHO ARE NOT READY
For this activity:
This chapter assumes that students are already comfortable identifying and
extending number patterns and calculating the missing factor in a multiplication
equation.
• You may want to focus on working with number patterns and eliminate
consideration of multiplication equations.
In some lessons, suggestions for adapting the lesson to deal with students who
are in a lower developmental phase can be found at the end of the
Opportunities for Feedback: Assessment for Learning chart.
12
Chapter 3: Number Relationships
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Identifying Factors
STUDENT BOOK PAGES 70–73
GOAL
Identify factors to solve problems.
PREREQUISITE SKILLS/CONCEPTS
•
•
•
•
Calculate products and quotients using mental math.
Divide a two-digit number by a one-digit number.
Understand the meaning of the term factor.
Use arrays to multiply and divide numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Determine all the whole-number factors of a given
number using arrays.
• Identify the factors and multiples for a given number
and explain the strategy used, e.g., concrete or visual
representations, repeated division by prime numbers, or
factor trees.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Pacing
5–10 min Introduction
15–20 min Teaching and Learning
20–30 min Consolidation
Materials
• Optional: counters
• Optional: linking cubes
Masters
• Optional: 1 cm Grid Paper, Masters
Booklet p. 22
• Optional: Chapter Checklist: Chapter 3,
Masters Booklet p. 64
Recommended
Practising Questions
Questions 3, 4, 5, 6, 7, 8, & 13
Key Question
Question 5
Extra Practice
Mid-Chapter Review Questions 1 & 2
Chapter Review Questions 1, 2, & 3
Workbook p. 17
Mathematical
Process Focus
R (Reasoning) and V (Visualization)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Copyright © 2010 Nelson Education Ltd.
Math Background
Students should be familiar with the relationship between
factors of a number and division of that number. For
example, 2 is a factor of 10 because the quotient (5) is a
whole number and the remainder is 0. To identify all of
the factors of a number and to help them visualize those
factors, students can use arrays.
An array is a pictorial or concrete model of a number
in which the rows and columns of the array represent
factors of the number. For example, a 4-by-5 array shows
that 4 and 5 are factors of 20 because the array has 4 rows,
5 columns, and a total of 20 elements.
As students use reasoning and mental math to identify
the factors of a number, they can show the factors in a
factor rainbow. A factor rainbow lists all of a number’s
factors in a row and pictorially links the factors that can
be multiplied together to result in that number. It is
important to list the factors systematically so none are
forgotten.
Optional: 1 cm Grid Paper,
Masters Booklet, p. 22
Optional: Chapter
Checklist: Chapter 3,
Masters Booklet p. 64
Lesson 1: Identifying Factors
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1
1 Introduction
2
(Small Groups/Whole Class)
± 5–10 min
Distribute 12 counters to each group. Have students form
the counters into an array. Alternatively, have them colour
arrays of 12 on grid paper. Ask volunteers to share their
arrays with the class. Try to elicit all of the possible arrays for
the number 12: 1-by-12, 2-by-6, and 3-by-4. Some students
may also suggest reversing the order of rows and columns, for
example, 12-by-1. Accept these answers but make sure
students realize that the factors are still the same.
3
4
5
Sample Discourse
“How did you decide how many counters would go in each
row of your array?”
• I tried to make rows that were all the same size without
having any counters left over. I then counted the number of
counters in each row to determine one factor.
• I chose a number of rows that is a factor of 12, and then put
the counters into that number of rows.
“ Can you make an array with five rows?”
• No, because there will be two counters left over.
• No, because 5 is not a factor of 12.
• No, because 5 does not divide evenly into 12.
6
7
2 Teaching and Learning
3
(Whole Class/Pairs) ± 15–20 min
Together, read about the Earth Day project and then read the
central question on Student Book page 70. Work through
Mai’s Arrays together. Students may represent the arrays with
symbols as Mai did, or they may use counters or grid paper.
Some students may use pairs of factors to identify two arrays
rather than one array.
Work through Jason’s Factor Rainbow with students to
show how to systematically record all the factors of 18. For
example, students may reverse the rows and columns to get
6 arrays for 18 seedlings: 1-by-18 and 18-by-1, 2-by-9 and
9-by-2, and 3-by-6 and 6-by-3. Tell students they can solve the
problem either way as long as they list the number of arrays
the same way for each number in the chart. They should also
note that the factors 1, 2, 3, 6, 9, and 18 remain the same.
Have students work in pairs to complete Prompts A to C.
When students have completed the activity, discuss the
answers as a class.
4
5
6
7
8
8
14
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Answers to Prompts
A. For example, I used a factor rainbow to record the
number of factors and the number of arrays for each
number of seedlings.
The factors of 25 are 1, 5, and 25. So 25 seedlings can be
planted in 2 arrays:
1-by-25, 5-by-5
5
25
The factors of 29 are 1 and 29. So 29 seedlings can be
planted in 1 array:
1-by-29
Grade 3
1
29
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. So
36 seedlings can be planted in 5 arrays:
1-by-36, 2-by-18,
3-by-12, 4-by-9,
6-by-6
Grade 4
1
2 3
Copyright © 2010 Nelson Education Ltd.
4
6
1 2
3 4 6
8 12 16 24 48
The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. So 56
seedlings can be planted in 4 arrays:
1-by-56, 2-by-28,
Grade 6
4-by-14, 7-by-8
Grade 2
1
The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
So 48 seedlings can be planted in 5 arrays:
1-by-48, 2-by-24,
Grade 5
3-by-16, 4-by-12,
6-by-8
9 12 18 36
1
2
4
7
8 14 28 56
B. For example, I chose Jason’s method because I can use
mental math to figure out the factors of a number. The
factor rainbow helps me keep track of the factors I have
figured out. I didn’t use Mai’s method because it takes
too long to draw all the arrays for each number.
C. Both 36 and 48 seedlings can be planted in 5 arrays.
Reflecting (Whole Class)
Here students compare and contrast arrays with factor rainbows
as methods for identifying the factors of a number. Students
also explain how they know when they have identified all of
the factors, using each method. Ensure students understand
that arrays can be used to identify factors, while factor
rainbows are primarily a method for recording the factors.
Students should also connect the dimensions of the arrays
with the factors listed in the factor rainbow.
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Answers to Reflecting Questions
D. For example, they are the same in that each of the
dimensions of Mai’s arrays matches a factor pair in Jason’s
factor rainbow. They are different because Mai has to
draw rectangles to list arrays while Jason uses mental
math to list factors.
E. For example, Mai drew arrays with 1, 2, and 3 rows. She
knew that 4 and 5 aren’t factors of 18, so she couldn’t
plant 18 seedlings in 4 or 5 rows. She knew that a 3-by-6
array can be arranged in either 3 rows of 6 or 6 rows of 3.
She knew that 7 and 8 aren’t factors of 18, so she
couldn’t plant 18 seedlings in 7 or 8 rows. She knew that
a 2-by-9 array can be arranged in either 2 rows of 9 or
9 rows of 2. So she knew that there are no other possible
arrays for 18 seedlings.
Jason’s factor rainbow shows he identified the matching
factors of 1 and 18, 2 and 9, and 3 and 6. He only had
to see if 4 or 5 is a factor because he had already figured
out factors of 18 that are 6 or greater. Because 4 and 5
are not factors, he knew he had identified all factors of 18.
1
2
3-by-8
4-by-6
b) For example, each number of rows and columns in an
array represents a factor of 24. So the factors of 24 are
1, 2, 3, 4, 6, 8, 12, and 24.
Closing (Whole Class)
Question 13 allows students to reflect on and consolidate
their learning for this lesson as they connect the numbers of
rows and columns in an array to the factors of the number.
Answer to Closing Question
13. For example, if you want to identify the factors of 26,
you can draw arrays.
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Students can use either arrays or factor rainbows to identify
the factors. Refer students to Mai’s and Jason’s methods for
guidance. Have counters and grid paper available for students
to use to model arrays.
5
The numbers of rows and columns of the arrays are the
factors of 26. So 1, 2, 13, and 26 are factors of 26.
You can also use mental math to identify the factors and
use a rainbow to help you keep track.
Practising (Individual)
6
These questions provide students with practice in using
arrays and factor rainbows to identify and record factors.
Provide counters or grid paper to students to help them
model the arrays.
6. Students should recognize that the number of coins can
only be divided by 1 and itself. In Lesson 3, students will
formalize this understanding as they learn about prime
and composite numbers.
7
8
1
2
13
26
Follow-Up and Preparation for Next Class
Have students follow up on the lesson at home using a group
of small items such as toothpicks. Suggest that students
arrange the group into an array. Using the array, students
should identify factors of the number used in the array.
Answers to Key Question
5. a)
1-by-24
2-by-12
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students use arrays and/or factor rainbows to identify the factors of
a number.
• Students may not identify all of the factors. (See Extra Support 1.)
Key Question 5 (Reasoning, Visualization)
• Students draw all of the possible arrays for the number 24 and explain how
the dimensions of the arrays relate to the factors of 24.
• Students may not connect the numbers of rows and columns with the factors
of the number. (See Extra Support 2.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Help students understand how they can use a factor rainbow to organize
their work. Have students write the numbers 1 to 16 in a row. Tell students
to look at each number in the row and use mental math or arrays to decide if
it is a factor of 16. If it is a factor, have students circle the number; if it is
not a factor, have students cross out the number. Finally, have students draw
arches to connect the numbers that can be multiplied together to give a
product of 16. For square numbers (16 = 4 ⫻ 4), suggest that students
simply draw an arch from the 4 to itself.
2. Have students use grid paper and shade in as many rectangles as possible
that have a total area of 24 grid squares. Then have students label each
rectangle with the number of rows and the number of columns that are
shaded, for example, “4-by-6.” Guide students to understand that “4-by-6”
means “4 multiplied by 6.” Since the area of the rectangle is 24, 4 and 6 are
factors of 24. Repeat the exercise, using counters in an array in place of the
grid paper, and guide students to connect the numbers of rows and columns
in the arrays with the factors of 24.
EXTRA CHALLENGE
• Challenge students to identify the number between 1 and 50 that can be
modelled with the greatest number of arrays. Encourage students to develop
strategies to help them eliminate some numbers, rather than drawing the
arrays for each number. For example, students might eliminate any number
that can only be drawn in an array with one row.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Provide students with an array and have them work together or individually to
identify the factors. Then ask students to create another array with the same
number of counters. This exercise will give students an opportunity to explore
factors and products without identifying all of the factors of a particular
number.
SUPPORTING LEARNING STYLE DIFFERENCES
• Kinesthetic learners will benefit from creating their arrays with counters
rather than just drawing them.
Copyright © 2010 Nelson Education Ltd.
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Identifying Multiples
STUDENT BOOK PAGES 74–76
GOAL
Identify multiples to solve problems.
PREREQUISITE SKILLS/CONCEPTS
• Identify factors of whole numbers.
• Extend a number pattern by multiplying or adding whole
numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
10–15 min Teaching and Learning
20–30 min Consolidation
Materials
• rulers
• Optional: counters
Masters
• Number Lines, Masters Booklet p. 33
• Optional: Scaffolding for Lesson 2,
Question 3 p. 77
Recommended
Practising Questions
Questions 2, 3, 5, 8, & 9
Key Question
Question 5
Extra Practice
Mid-Chapter Review Questions 3 & 4
Chapter Review Questions 4 & 5
Workbook p. 18
Mathematical
Process Focus
PS (Problem Solving) and V (Visualization)
Vocabulary/Symbols
multiple
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
18
Chapter 3: Number Relationships
Math Background
In previous grades, students have multiplied factors to
calculate a product. In this lesson, students will approach
multiplication from a different perspective as they
calculate multiples of a number using known
multiplication facts and skip counting. Students will
multiply a given number by sequential whole numbers to
build a list of multiples. For example, to build a list of
multiples of 6, students will multiply 6 by 1, 2, 3, 4, … to
get the multiples 6, 12, 18, 24, …. To use skip counting,
students will count in units of the given number. For
example, to build a list of multiples of 5, students will
count by 5s to get the multiples 5, 10, 15, 20, and so on.
Students use a number line to help them visualize the
pattern in the list of multiples. Students will apply these
skills in various problem-solving contexts.
Number Lines, Masters
Booklet p. 33
Optional: Scaffolding for
Lesson 2, Question 3 p. 77
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1
1 Introduction
2 Teaching and Learning
2
3
(Whole Class) ± 5–10 min
Briefly review some mental math strategies that students have
learned for multiplication. On the board, on a transparency,
or on an interactive whiteboard, write the following
multiplication expressions:
3
4⫻8
4
6⫻7
8⫻5
Ask volunteers to share their strategies for calculating each
product. Try to elicit a variety of strategies.
5
Sample Discourse
“How can you calculate the product of 4 and 8?”
• I used doubling. I know 2 ⫻ 8 = 16,
so 4 ⫻ 8 = 16 ⫹ 16, which is 32.
• I used doubling. I know 4 ⫻ 4 = 16,
so 4 ⫻ 8 = 16 ⫹ 16, which is 32.
“How can you calculate the product of 6 and 7?”
• I skip counted up. I know 6 ⫻ 6 = 36,
so 6 ⫻ 7 = 36 ⫹ 6, which is 42.
• I skip counted down. I know 7 ⫻ 7 = 49,
so 6 ⫻ 7 = 49 ⫺ 7, which is 42.
“How can you calculate the product of 8 and 5?”
• I used doubling. I know 2 ⫻ 5 = 10, so 4 ⫻ 5 = 10 ⫹ 10,
which is 20, and 8 ⫻ 5 is 20 ⫹ 20, or 40.
• I skip counted down. I know 10 ⫻ 5 = 50, so 9 ⫻ 5 =
50 ⫺ 5, which is 45, and 8 ⫻ 5 = 45 ⫺ 5, which is 40.
6
7
8
Copyright © 2010 Nelson Education Ltd.
(Whole Class/Small Groups)
± 10–15 min
Before reading, remind students that a comet is a small body
that orbits the Sun, and it is only visible from Earth at
certain points in its orbit. Comets that appear regularly are
referred to as periodic comets. Together, read about the
comets and then read the central question on Student Book
page 74. Have students set up Oleh’s List and retrace his
steps to show the first multiples of 7. Then direct them to
Léa’s Number Line. Tell students to use their rulers to draw
an open number line with two arrows. Ask them to point out
which number Léa starts with on the number line and how
she gets to the next number. When students have become
comfortable with Léa’s method, have them work through
Prompts A to C in small groups. You may want to discuss the
two methods as a group and have volunteers explain which
method they prefer.
4
5
6
7
8
Sample Discourse
“Which math operations did Oleh use in his method? How is
Oleh’s method different from Léa’s method?”
• Oleh used multiplication to determine the multiples of 7 and
addition to calculate the years the comet would be seen from
Earth. Léa only used addition to figure out the years after
2000 the comet would be seen.
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Answers to Reflecting Questions
D. For example, you create a multiple of 7 by multiplying
7 by a counting number. So any multiple is 7 times a
counting number and 7 must be a factor.
E. For example, any factor of 9 has to be 9 or less, so there
are only 9 possible numbers. But multiples of 9 are
created by continually adding 9s and you can add 9s
forever.
1
2
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Draw students’ attention to the Communication Tip. Ensure
that they are comfortable with the notation “…,” which is
called an ellipsis. If students require additional guidance, refer
them to Oleh’s and Léa’s methods in the example. You may
want to distribute number lines to students; however, students
do not need to use scaled number lines; rather, they can
sketch empty number lines.
5
6
Practising (Individual)
7
These questions give students opportunities to practise
calculating multiples. Students will also explain connections
between factors and multiples. Encourage students to use
mental math strategies in their calculations. Encourage
students to use number lines as visualization tools.
2. Ensure students understand that the “first five multiples”
can be calculated by multiplying by the first five
counting numbers, 1, 2, 3, 4, and 5, or by repeatedly
adding the number to itself until five multiples
are listed.
3. If extra support is required, guide these students and
provide copies of Scaffolding for Lesson 2,
Question 3 p. 77.
7. Students create lists of multiples of two numbers and
then identify the numbers that appear in both lists. In
later grades, students will formalize this understanding
as they learn about common multiples.
8
“Which method is easier for you to use? Explain.”
• Oleh’s method is easier because multiplying to determine the
multiples is faster than adding, and I only have to replace the
last digits of 2000 with the multiples of 7 to get the years.
• Léa’s method is easier because I like adding better than
multiplying.
Answers to Prompts
A. For example, I multiplied 7 by 3 to get 21.
B. For example, I added 7 to 2014 to get the year 2021.
C. For example, I listed the multiples of 7 until I got to 70.
I stopped at 70 because I know 2000 ⫹ 70 = 2070 is
past 2067.
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …
I added these multiples to 2000 to get these years in
which the comet will likely be seen from Earth: 2007,
2014, 2021, 2028, 2035, 2042, 2049, 2056, and 2063.
Reflecting (Whole Class)
Here students reflect on the relationship between factors and
multiples. Students should recognize that a multiple is the
product of a factor and a counting number.
20
Chapter 3: Number Relationships
Answers to Key Question
5. a) 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 plates
b) For example, 10 packages; Pauline needs to buy plates
for 80 people, and 80 plates are in 10 packages.
c) 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
d) For example, 7 packages; Pauline needs to buy at least
80 glasses, and 6 packages have 72 glasses, which is
too little, but 7 packages have 84 glasses, which is
enough.
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Closing (Whole Class)
Question 9 allows students to reflect on and consolidate their
learning for this lesson as they think about multiples of a
number.
Follow-Up and Preparation for Next Class
Have students list the multiples of 2 from 2 to 48. Challenge
students to explain which of the numbers they listed has the
most factors.
Answer to Closing Question
9. Disagree; for example, numbers like 1, 10, 19, and
28 are 9 apart but none are multiples of 9. The list
would have to start at 0, 9, or a multiple of 9 for the
numbers to be all multiples of 9.
Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students multiply by counting numbers and/or use skip counting to calculate
multiples.
• Students may have difficulty using mental math to calculate multiples. (See
Extra Support 1.)
Key Question 5 (Problem Solving, Visualization)
• Students use mental math to calculate multiples of 8 up to 80 and multiples
of 12 up to 120 and use their calculations to solve a problem.
• Students may have difficulty choosing the correct number of plates and cups.
(See Extra Support 2.)
• Students may have difficulty using mental math to calculate multiples. (See
Extra Support 1.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Discuss mental math strategies for multiplying by counting numbers:
Doubling: Students can multiply a known factor by 2. For example, since
2 ⫻ 6 = 12, then doubling the counting number will result in 4 ⫻ 6 = 24.
Doubling can be repeated. For example, 8 ⫻ 6 = 48.
2. Guide students to skip count by 8s using a 100 chart until they reach a
number between 70 and 80, circling each multiple of 8. Repeat with 12s,
circling each multiple of 12 with a different colour.
Skip counting: Students can skip count from a known factor. For example,
since 5 ⫻ 6 = 30, then 6 ⫻ 6 = 30 ⫹ 6, which is 36. Students can also
skip count down. For example, since 5 ⫻ 6 = 30, then 4 ⫻ 6 = 30 ⫺ 6,
which is 24.
EXTRA CHALLENGE
• Have students research and write a problem about an event that occurs every
number of years, for example, the Olympics or leap years. Then have students
exchange their problems with a partner and solve the problems.
b) Andrea’s 21st birthday is in the year 2016, and she wants to know if the
same year will have an Olympic Games. Which Olympic Games, if any, is
occurring that year?
Example:
a) The summer and winter Olympics both occur every four years. Calculate
the years for the next five Olympic Summer Games, starting with 2008.
Then calculate the years for the next five Olympic Winter Games, starting
with 2006.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may be able to determine the multiples but have difficulty
adding them to a first number, like to the year 2007. The addition component
might be eliminated for these students.
• Other students might have difficulty calculating multiples without concrete
support. Provide counters to help students create equal groups to determine
multiples.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may benefit from comparing visual representations of different
sets of multiples. For example, on a 100 chart, they can colour the multiples
of 6, 8, and 9 in different colours to see how the multiples of 9 are more
spread out than the multiples of 6 or 8.
Copyright © 2010 Nelson Education Ltd.
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Curious Math
String Art
STUDENT BOOK PAGE 77
PREREQUISITE SKILL/CONCEPT
• Identify multiples of whole numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Materials
• pencil crayons
• rulers
Masters
• String Art Circle p. 78
Mathematical
Process Focus
PS (Problem Solving), R (Reasoning), and
V (Visualization)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Math Background
String art, or curve stitching, is a technique that uses line
segments to produce apparent curves. Collectively, the
lines form an approximation of a curve. In this activity,
students will draw line segments of different lengths to
connect multiples of various numbers. The frame for the
string art consists of dots arranged in a circle and
numbered from 2 to 48. Students will use different
colours to draw the lines for various multiples. The
resulting design is a visual representation of factors and
multiples. Students will use reasoning to identify factors
based on the colours of lines joined at the number.
String Art p. 78
22
Chapter 3: Number Relationships
Using Curious Math
In this activity, students are exposed to another visual
representation of factors and multiples as they create and
interpret a string art design. Students will identify the
patterns formed in the string art and determine which
numbers should be connected with each colour. Encourage
students to explain how they are completing their project,
using the terms factors and multiples in their explanations.
Encourage students to see that all numbers connected by the
same colour string have at least one factor in common.
Students may draw conclusions about the numbers joined by
two or more colours.
Answers to Curious Math
1. For example, the multiples of 12 up to 48 are connected
by blue lines.
2. 12, 24, 36, 48
3. 12, 24, 36, 48
4. For example, if I use yellow to connect multiples of 4,
I predict there will be 4 lines at 12, 24, 36 and 48.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students draw lines to connect multiples of different numbers.
• Students may have difficulty calculating multiples of the different numbers.
(See Extra Support 1.)
• Students make the connection between lines that will be joined at 48 and the
factors of 48.
• Students may have difficulty identifying other numbers that can be connected
in lines that end at 48. (See Extra Support 2.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students talk about some of the mental math strategies that they can
use to calculate multiples of the different numbers. For example, students
might skip count by 3 to identify the multiples of 3, or connect all of the even
numbers to identify the multiples of 2.
2. Have students talk about what the numbers 2, 3, 4, and 12 have in common.
Students might mention that they are all factors of the same numbers, such
as 12 and 24. Guide students to understand that the lines connecting the
multiples of these numbers end at 48 because they are all factors of 48.
Discuss how students might find other factors of 48.
EXTRA CHALLENGE
• Challenge students to create string art with a different shape and a different
number of dots. For example, students may use a hexagon shape with a
number such as 36, and connect multiples of 2, 3, 4, 6, 9, 12, and 18.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may not be able to calculate factors and multiples using
mental math. Provide these students with 48 counters and have them form
groups or arrays to assist with their calculations.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may enjoy experimenting with different colours. Provide these
students with multiple copies of the String Art Circle blackline master and
encourage them to create a variety of designs.
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Prime and Composite
Numbers
STUDENT BOOK PAGES 78–80
GOAL
Identify prime and composite numbers.
PREREQUISITE SKILL/CONCEPT
• Identify factors and multiples of whole numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Provide an example of a prime number and explain
why it is a prime number.
• Provide an example of a composite number and
explain why it is a composite number.
• Sort a given set of numbers as prime and composite.
• Solve a given problem involving factors or multiples.
• Explain why 0 and 1 are neither prime nor composite.
Preparation and Planning
Pacing
5–10 min Introduction
20–25 min Teaching and Learning
15–25 min Consolidation
Materials
• counters
Masters
• 100 Chart, Masters Booklet p. 30
• 2 cm Grid Paper, Masters Booklet p. 23
Recommended
Practising Questions
Questions 2, 3, 4, & 8
Key Question
Question 4
Extra Practice
Mid-Chapter Review Question 5
Chapter Review Questions 6 & 7
Workbook p. 19
Mathematical
Process Focus
R (Reasoning) and V (Visualization)
Vocabulary/Symbols
prime number, composite number
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
24
Chapter 3: Number Relationships
Math Background
In Lesson 1, students used arrays to determine the factors
of numbers and investigated the different arrays in which
numbers could be arranged. In this lesson, students
formalize their understanding of prime and composite
numbers as they use reasoning to identify numbers that
can be arranged in only one array. By arranging counters
in arrays, students are able to visualize numbers that can
be arranged in only one row or column; these numbers
are prime, as their only factors are 1 and themselves.
Numbers that can be arranged in more than one array are
composite; each array represents two factors.
A 100 chart is used to identify prime and composite
numbers to 100, using a procedure called the Sieve of
Eratosthenes (er-uh-tos-thuh-neez), which was developed
and named for the ancient Greek mathematician
Eratosthenes. In this procedure, the smallest prime
number on the chart is circled and then each of its
multiples is crossed off. This is repeated until all of the
composite numbers have been crossed off, leaving only
the prime numbers.
100 Chart,
Masters Booklet p. 30
2 cm Grid Paper,
Masters Booklet p. 23
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1
1 Introduction
2 Teaching and Learning
2
3
(Whole Class/Small Groups)
± 5–10 min
Distribute various numbers of counters to each small group, and
have them arrange their counters into as many arrays as possible
with no counters left over. Ask a volunteer from each group to
say the number of counters they had and describe the different
arrays they were able to make. On the board, on a transparency,
or on an interactive whiteboard, record the number of counters
and the number of arrays for each group. As a class, talk about
what the rows and columns in an array represent.
3
4
5
Sample Discourse
“How can you use arrays to find the factors of a number?”
• The numbers of rows and columns in an array are factors of
the number.
• I can arrange counters in rows and columns to find the factors.
6
7
8
Copyright © 2010 Nelson Education Ltd.
(Whole Class/Small Groups)
± 20–25 min
Together, read about Robin’s batteries and then read the
central question on Student Book page 78. Distribute
10 counters to each small group and have students form
arrays for 2, 3, and 4 and relate these to the packages of
batteries. Have them continue to make as many arrays as
they can for the numbers 5 to 10. Talk about which numbers
can be arranged in only one row or column (2, 3, 5, and 7)
and which numbers can be arranged in more than one way
(4, 6, 8, 9, and 10). Draw students’ attention to the margin
definitions and ensure they understand the difference
between prime numbers and composite numbers.
Distribute 100 charts and work through Robin’s Chart on
page 79 together. Have students work through Prompts A to
D in groups. Talk about the answer to Prompt C as a class.
Students should realize that after they cross off the multiples
of 7, only prime numbers will remain in the chart. Ask
volunteers to share their solutions to Prompt D to ensure
that each group correctly identified the prime numbers to 50.
4
5
6
7
8
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B. For example, 5 is the next prime number because 4 is
composite and 5 has only two different factors, 1 and 5.
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
C. For example, when I crossed off multiples of 2, 3, and 5,
then 14, 28, 35, and 42 were crossed off, so only 49 is
left. It was not crossed off because 49 is a multiple of
7 but not a multiple of 2, 3, or 5.
D. The prime numbers to 50 are 2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, and 47.
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
Answers to Prompts
A. For example, you get every third number in the chart by
skip counting by 3. So every third number in the chart is
a multiple of 3 and can be divided by 3. Each multiple
of 3 greater than 3 has at least 1, 3, and the number itself
as a factor. So multiples of 3 greater than 3 have more
than two different factors and are composite numbers.
1
2
3
4
5
6
7
8
9
10
Reflecting (Whole Class)
Students explain how they used Robin’s Chart to identify all
of the prime numbers to 50.
Answers to Reflecting Questions
E. For example, each multiple of 11 has been crossed off as a
multiple of 2 or 3. All multiples of primes greater than 11
have either been crossed off.
F. 1 is the only number that isn’t circled or crossed off.
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
26
Chapter 3: Number Relationships
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1
2
3 Consolidation ± 15–25 min
Checking (Pairs)
4
Students complete their 100 chart as they apply the Sieve of
Eratosthenes to the numbers from 50 to 100. Discuss the
answer as a class to ensure that students have identified all of
the prime numbers.
5
Practising
(Individual)
6
These questions provide students with opportunities to
practise identifying prime and composite numbers in a
variety of contexts.
2. Students use reasoning to determine whether numbers
are prime or composite. You may want to make counters
available.
3. Students may have difficulty communicating their
answers for parts b) and c). Ask them how their answers
support the definitions of a prime number and a
composite number.
7
8
Answers to Key Question
4. a) For example, every number of candles that is prime
can be arranged in only one row or in one column.
I will be 12 next month. So when I am 13 or 17,
I can arrange the number of candles on a birthday
cake in only one array. For all other ages up to 18, I
can arrange the number of candles in more than one
array.
b) For example, I know prime numbers have only two
factors and one of the factors has to be 1. So you can
represent the numbers in only one row or one column.
Composite numbers have more than two factors so
you can arrange them in more than one array. So I
just had to identify the prime numbers from 12 to 18
to answer part a).
Copyright © 2010 Nelson Education Ltd.
Closing (Whole Class)
Question 6 allows students to reflect on and consolidate their
learning for this lesson as they explain the connection between
the number of arrays that can be used to represent a number
and whether that number is prime or composite.
Answer to Closing Question
6. For example, if you can arrange the counters in only one
array, the number of counters is a prime number. Seven
counters can only be arranged as a 1-by-7 array. So it is a
prime number.
1 row of 7
If you can arrange the counters in more than one
array, the number of counters is a composite number
because it means the number has more than two
different factors. Six counters can be arranged as 1-by-6
and 2-by-3 arrays. So it is a composite number.
1 row of 6
2 rows of 3
Follow-Up and Preparation for Next Class
Students can review how to determine factors of numbers
from 1 to 100 at home. They can use small objects such as
marbles or building blocks as counters to set up different
arrays of a particular number. Encourage students to explain
to a friend or family member what they are doing.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students identify prime numbers and composite numbers.
• Students may have difficulty crossing off multiples using a 100 chart to
identify prime numbers. (See Extra Support 1.)
Key Question 4 (Reasoning, Visualization)
• Students use their understanding of prime and composite numbers to identify
the ages from 12 to 18 for which the candles on a cake can be arranged in
only one array.
• Students may not realize that prime numbers can be arranged in only one
array and composite numbers can be arranged in more than one array. (See
Extra Support 2.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students practise identifying multiples of a number, such as 3, by skip
counting or by counting 123 123 123, etc.
2. Have students talk about what the rows and columns in an array represent.
Elicit from students that the number of rows and the number of columns are
the factors of a number; that is, if they multiply the number of rows by the
number of columns, the product will be the number. Distribute counters and
have students record the factors in the different arrays that they can build
for each number from 12 to 18. Talk about which numbers can only be
represented with one array (13 and 17) and talk about the factors for those
arrays (1 and 13; 1 and 17). Elicit from students that if a number can only be
represented by one array, its only factors are 1 and itself, and it is a prime
number (except for the number 1).
EXTRA CHALLENGE
• Have students create a game involving prime numbers and composite
numbers, using a pair of dice and a 100 chart. For example, a player rolls the
dice and determines the sum of the numbers on the dice. Then the player
crosses the number off the chart. If the sum is prime, the player can also
cross off the multiples of the number. The first player to cross off all the
numbers to 50 on his or her chart is the winner.
• Have students predict the number of factors of a product when two different
prime numbers are multiplied together.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Ask students to use counters to create equal groups to show all the multiples
(other than the number itself) of 2 from 2 to 50, all the multiples of 3 from 3 to
50, etc. Then have them mark off those products on a 100 chart. Explain that
the leftover numbers (other than 1) are the primes and the crossed-off
numbers are the composites.
SUPPORTING LEARNING STYLE DIFFERENCES
• Encourage students to use a variety of colours to complete their 100 charts.
For example, students can use a different colour to circle each prime number
and then use the same colour to shade in the squares that are multiples of
that prime number.
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Chapter 3: Number Relationships
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STUDENT BOOK PAGE 81
Math Game
Colouring Factors
PREREQUISITE SKILL/CONCEPT
• Identify factors of whole numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Determine all the whole-number factors of a given
number using arrays.
• Identify the factors for a given number and explain the
strategy used.
Preparation and Planning
Number of Players
2
Materials
• pencil crayons
• Optional: counters
Masters
• 100 Chart, Masters Booklet p. 30
Mathematical
Process Focus
R (Reasoning)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
100 Chart,
Masters Booklet p. 30
Math Background
This math game helps students consolidate their
understanding of factors, prime numbers, and composite
numbers. Students will apply their reasoning skills to
select the numbers with the fewest factors and to
identify the factors of the numbers selected by their
partners.
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Using the Math Game
Provide each pair of students with a 100 chart and pencil
crayons. Have students cut off the numbers from 51 to 100
on their 100 chart. The game of colouring factors can give
students an opportunity to apply what they have learned
about numbers and their factors. Make sure all students
understand the rules of the game. Allow time for students to
discuss the strategies they applied while playing the game.
When to Play
Students can play the game after they demonstrate an
understanding of identifying the factors of a number.
While understanding of prime and composite numbers is
not essential to playing the game, it will allow for more
sophisticated strategies.
Strategies
Have students discuss the strategies to colour a number. To
minimize the number of factors their opponent can colour,
students should choose prime numbers. To minimize their
opponent’s total score, students should choose numbers with
few factors. To maximize their own score, students should
choose large prime numbers.
Discuss
Ask students to share effective strategies with the rest of the
class to encourage students to learn from one another.
Opportunities for Feedback: Assessment for Learning
What you will see students doing
Proficient players
Less-proficient players
• Students use reasoning to identify numbers with as few factors as possible to
limit their opponents’ scores.
• Students may make poor choices about which numbers to colour. (See Extra
Support 1 and 3.)
• Students identify all the factors of the number selected for them to maximize
their scores.
• Students may skip over a factor of the chosen number in the chart. (See Extra
Support 2 and 3.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students work with a partner against whom they will not be playing
and give them a short planning period before the game begins. Encourage
students to talk about the numbers with the most factors and the numbers
with the fewest factors. Then have them talk about which numbers they
should choose when it is their turn to colour a number.
2. Remind students to think of the factor rainbows they have created for
numbers. They should be colouring all the factors of a number as if they
were creating a factor rainbow.
3. Have less-proficient students play the game with numbers from 1 to 20,
gradually working up to 50.
EXTRA CHALLENGE
• Have students play using the entire 100 chart, rather than just the numbers
from 1 to 50. This will make for a longer game that requires more complicated
calculations.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may have difficulty developing their own strategies. Provide
counters. Allow students to try to quickly rearrange the counters into arrays to
help them decide which numbers to colour or which numbers are factors of
the other player’s number.
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Identifying Factors
by Dividing
STUDENT BOOK PAGES 82–84
GOAL
Identify factors by dividing composite numbers by primes.
PREREQUISITE SKILLS/CONCEPTS
• Identify prime and composite numbers.
• Identify factors of whole numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Math Background
In this lesson, students use their reasoning ability to
identify a prime number that is a factor of a given
number. They will divide the given number by the prime
factor. This can be done by using repeated division or
factor trees. Both techniques help students to identify the
factors of the number, including factors that are prime.
Students do not need to complete the division or factor
tree, but they should try starting with a prime number
when they divide.
Achievement Indicator
• Identify the factors for a given number and explain
the strategy used, e.g., concrete or visual
representations, repeated division by prime numbers,
or factor trees.
Preparation and Planning
Pacing
5–10 min Introduction
15–25 min Teaching and Learning
20–25 min Consolidation
Materials
• number cards 40 to 50
Recommended
Practising Questions
Questions 2, 4, 6, & 7
Key Question
Question 4
Extra Practice
Mid-Chapter Review Questions 6 & 7
Chapter Review Question 8
Workbook p. 20
Mathematical
Process Focus
R (Reasoning)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Copyright © 2010 Nelson Education Ltd.
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1
1 Introduction
2
(Whole Class) ± 5–10 min
Remind students that they have written factor pairs for
numbers, for example, 3 ⫻ 6 for 18. To encourage students
to think of a number as the product of three factors, remind
students that 6 can be written as 2 ⫻ 3. So 18 can be written
as 3 ⫻ 2 ⫻ 3.
Next, ask students to write 24 as the product of three
factors other than 1. Have students share their answers with
the class. Students will see that there are various solutions.
For example, students may write 2 ⫻ 2 ⫻ 6 ⫽ 24 or
4 ⫻ 2 ⫻ 3 ⫽ 24.
If time permits, repeat the activity for 30 and 75.
Recall the definitions of prime and composite numbers
with students. Write the equation 24 ⫽ 3 ⫻ 2 ⫻ 4 on the
board, on a transparency, or on an interactive whiteboard.
3
4
5
6
7
Sample Discourse
“What are the two least prime numbers?”
• 2 and 3
“What is 24 divided by 2?”
• 24 divided by 2 is 12.
“What is 12 divided by 3?”
• 12 divided by 3 is 4.
“What type of numbers did you divide by each time, prime
or composite?
• Each time, I divided by a prime number, either 2 or 3.
8
32
Chapter 3: Number Relationships
2 Teaching and Learning
3
(Whole Class/Small Groups)
± 15–25 min
Together, read about Daniel and Léa’s card game with
composite numbers. Read the central question on Student
Book page 82. Work through Léa’s Repeated Division
together. Point out that Léa started by dividing by 5, but
she could have started with other prime numbers. Discuss
how to identify other factors of 45 from Léa’s division
(e.g., 9 and 15). Then have students repeat Léa’s process
starting with 45 and dividing by 3.
Then direct students to Daniel’s method. Tell students that
Daniel starts by dividing 40 by 2, and then continues to
divide by prime numbers. Have students work through
Daniel’s method with the number 45 to see if they get the
same results as Léa.
4
5
6
7
8
Sample Discourse
“Is there another pair of factors Daniel can start his factor
tree with other than 2 and 20?”
• He can start with 5 and 8, because 5 ⫻ 8 is 40.
“If Daniel uses 5 and 8, which number will he continue
to factor?”
• 8, because 5 is prime, but 8 is composite.
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1
2
3 Consolidation ± 20–25 min
Checking (Pairs)
4
Remind students to factor each number using repeated
division or factor trees before determining different factors
that are prime. Point out that they are not finding the
number of factors that are prime, rather they are finding the
number of different or distinct factors that are prime.
5
6
Practising (Individual)
These questions provide students with opportunities to
practise identifying the factors of a number. Remind students
that they can use other appropriate strategies besides the
factor tree.
4. a) Ask students how Manon got 32, and then have them
work backward to get the top number.
7
8
Answers to Key Question
4. a) 96; for example, she divided a number by 3 and got
32. So the number must be 3 ⫻ 32 ⫽ 96.
b) For example, once you get to 16, you can divide only
by 2.
c) For example, if I divide 16 by 2, I get another factor, 8.
Closing (Whole Class)
Answers to Prompts
A. 3, 5, 9, and 45
B. 2, 4, 5, 20, and 40
C. No; for example, 2 is the only prime number that is a
factor of the quotient 4 in his factor tree and he already
divided by 2.
D. 42: 3 points; 44: 2 points; 45: 2 points; 46: 2 points;
48: 2 points; 49: 1 point; 50: 2 points; 42 has the
highest score.
Reflecting (Whole Class)
Remind students that a factor of a number is any number
that can be divided into that number and leave no remainder.
Answers to Reflecting Questions
E. For example, 2 is prime and a factor of every even
number. So you score at least 1 point.
F. No. For example, when I kept dividing 32 by 2, I kept
getting numbers that I could divide by 2. So I only
scored 1 point.
Question 7 allows students to reflect on and consolidate their
learning for this lesson. Students should familiarize themselves
with using repeated division or factor trees to determine factors
of a composite number. Encourage students to determine the
factors of various composite numbers. Observe the number
of factors that are prime in each composite number.
Answer to Closing Question
7. For example, not always. 16 is greater than 12. But 2 is
the only prime number that is a factor of 16, while 2 and
3 are two different factors that are prime for 12.
Follow-Up and Preparation for Next Class
At home, students can practise factoring two-digit numbers
with the help of a parent or siblings. Students can present
their factor trees on poster paper and bring the poster to class
to display on the wall.
32
2
16
2
8
2
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students divide any composite number less than 100 by a prime number.
• Students may not know how to select a prime number to use as a divisor of a
composite number. (See Extra Support 1 or 2.)
Key Question 4 (Reasoning)
• Students use a factor tree to determine factors of a composite number.
• Students may not recognize when a number in the factor tree can be factored
further. (See Extra Support 3.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students look at the first 10 multiples of 2 and see what they notice.
They can talk about how that might help them recognize other multiples of 2.
Do the same with the first 10 multiples of 5.
2. Ensure students notice that multiples of 3 are 3 apart. For example, how do
they know that 3 is not a factor of 85? Since they know that 90 is a multiple
of 3, they could count back by 3s to see that 87 and 84 are multiples of 3.
Thus, 3 is a factor of 87 and 84, but not of 85.
3. Students might benefit from using square tiles to see if a particular number
of square tiles can be rearranged into a rectangle that is not 1 unit wide.
This would mean that some of the factors on the tree can be factored
further.
EXTRA CHALLENGE
• Mai conjectures that all even composite numbers will have more factors
that are prime (repeated or non-repeated) than odd composite numbers, if
both numbers have the same tens digit. Challenge students to explore
Mai’s conjecture. For example, 45 has more factors that are prime, (3, 3,
and 5) than 46 (2 and 23).
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Encourage students to use counters to model the factors of each number from
10 to 20. To determine the factors of 18, for example, a student may begin by
arranging 18 counters into an array of 6 rows of 3 counters each. Have them
write 6 and 3 as factors of 18. Next have them try to arrange 3 counters into
34
Chapter 3: Number Relationships
an array other than 1 row of 3 or 3 rows of 1 and determine that 3 is prime.
Since 6 counters can be arranged in an array of 2 rows of 3 counters each,
6 is a composite number with factors 2 and 3 that are prime. Thus, students
determine that 2 and 3 are factors of 18 that are prime.
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STUDENT BOOK PAGE 85
Creating Composite
Numbers
GOAL
Multiply combinations of factors to create composite
numbers.
PREREQUISITE SKILLS/CONCEPTS
• Multiply and divide combinations of one-digit and
two-digit numbers.
• Identify prime and composite numbers.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples
[PS, R, V]
Math Background
Students will begin this activity using reasoning to
determine the prime numbers from 1 to 50. Students have
already learned that prime numbers have only two factors
and that composite numbers have more than two factors.
Now they will explore the implications of that distinction
in the context of solving a problem. They will have the
opportunity to recognize that prime numbers can be
multiplied to make any composite number from 2 to 50.
Optional: 100 Chart,
Masters Booklet p. 30
Optional: 1 cm Grid Paper,
Masters Booklet p. 22
Achievement Indicators
• Identify the factors for a given number and explain the
strategy used.
• Provide an example of a prime number and explain
why it is a prime number.
• Provide an example of a composite number and
explain why it is a composite number.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
15–25 min Teaching and Learning
15–20 min Consolidation
Materials
• Optional: chart paper and markers
Masters
• Optional: 100 Chart, Masters Booklet p. 30
• Optional: 1 cm Grid Paper, Masters Booklet
p. 22
Key Question
entire exploration
Extra Practice
Workbook p. 21
Mathematical
Process Focus
PS (Problem Solving) and R (Reasoning)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Copyright © 2010 Nelson Education Ltd.
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1
2 Teaching and Learning
3
(Pairs/Small Groups) ± 15–25 min
With students, read about Oleh’s licorice-stretching machine
on Student Book page 85. Clarify that the buttons on Oleh’s
machine can be pressed more than once to stretch the
licorice. Ask them why a 1 button wouldn’t be needed on the
machine. Ensure they understand that pressing the 2, 3, and
5 buttons means 2 ⫻ 3 ⫻ 5 ⫽ 30, which produces a length
30 times as long as the original. Also, pressing the 2 button
three times means 2 ⫻ 2 ⫻ 2 ⫽ 8.
Read the central question and have students work in small
groups to answer it. Have available 100 charts (or at least the
numbers from 1 to 50); grid paper; chart paper; and markers.
Explain that students are to
• write out the main points in their solution on chart paper
• be prepared to communicate their solution process to the
rest of the class
• describe the buttons needed to stretch the licorice using
multiple stretches from 2 to 50
No one approach to the problem should be suggested.
Encourage students to choose their own methods. They must
take the information given and work toward a solution.
Circulate and observe students as they work.
4
5
6
7
8
1 Introduction
2
(Whole Class) ± 5–10 min
Review with students the various ways they can identify
factors of numbers. These include forming arrays of counters,
factor rainbows, repeated division, and factor trees.
3
Sample Discourse
“How would you identify the factors of 24?”
• I would start by listing 1 and 24 because 1 ⫻ 24 ⫽ 24.
I know 2 ⫻ 12 ⫽ 24 so 2 and 12 are factors. 3 ⫻ 8 ⫽ 24 so
3 and 8 are factors. 4 ⫻ 6 ⫽ 24 so 4 and 6 are factors. The
factors of 24 in order are 1, 2, 3, 4, 6, 8, 12, and 24.
“How can you use your list of factors of 24 to tell whether 24
is a prime or composite number?”
• It’s not a prime number because it has more than two different
factors.
“In your list of factors of 24, which factors are prime
numbers?”
• 2 and 3 are the only two prime numbers that are factors of 24.
“Can you multiply combinations of only 2 and 3 to get 24?”
• Yes; if you calculate 2 ⫻ 2 ⫻ 2 ⫻ 3, you get 24.
4
5
6
Sample Discourse
“What buttons can you include to stretch the licorice
12 times as long as the original licorice?”
• I can use a 6 and a 2.
• I can use a 12 and a 1.
• I can use a 2 and a 2 and a 3.
“How does the length of the licorice increase when you press
a 2 button or a 3 button three or four times?”
• If I press a 2 button three times, I have 2 ⫻ 2 ⫻ 2 ⫽ 8. If I
press a 2 button four times, I have 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫽ 16.
• If I press a 3 button three times, I have 3 ⫻ 3 ⫻ 3 ⫽ 27.
“What button do you need to stretch the licorice 17 times as
long as the original licorice?”
• I need a 17 button.
“Why do you only need buttons that are prime numbers?”
• I only need buttons that are prime numbers because pressing
prime-number buttons once will give the prime numbers needed
and pressing combinations of prime-number buttons will give
all the composite numbers needed.
7
8
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Possible Solutions
Sample Solution 1:
We used the numbers from 2 to 50 to keep track of buttons
that aren’t needed. Then we made a chart to show one way
the needed buttons can be used to stretch licorice from 2 to
50 times. The order of pressing the buttons does not matter.
2 Need
3 Need
4 Press 2 twice
5 Need
6 Press 2, then 3
7 Need
8 Press 2 three times
9 Press 3 twice
10 Press 2, then 5
11 Need
12 Press 2 twice, then 3 13 Need
14 Press 2, then 7
15 Press 3, then 5
17 Need
18 Press 2, then 3 twice 19 Need
16 Press 2 four times
Sample Solution 2:
We discovered that we can stretch licorice a composite
number of times from 2 to 50 by using only the primenumber buttons. We made a chart. The checkmarks show
what buttons you must push. Sometimes you have to press
the same button more than once.
Composite
Number
x
2
4
¸
6
¸
8
¸
22 Press 2, then 11
23 Need
24 Press 2 three times,
then 3
27 Press 3 three times
25 Press 5 twice
10
¸
28 Press 2 twice, then 7
12
¸
30 Press 2, then 3,
then 5
33 Press 3, then 11
31 Need
14
¸
36 Press 2 twice, then
3 twice
39 Press 3, then 13
37 Need
29 Need
32 Press 2 five times
35 Press 5, then 7
38 Press 2, then 19
41 Need
34 Press 2, then 17
40 Press 2 three times,
then 5
43 Need
42 Press 2, then 3,
then 7
44 Press 2 twice, then 11 45 Press 3 twice, then 5 46 Press 2, then 23
49 Press 7 twice
47 Need
48 Press 2 four times,
then 3
50 Press 2, then 5 twice
The only buttons we need to include are the prime numbers
from 2 to 47. You can stretch the licorice from 2 to 50 times
by pressing combinations of the prime-number buttons.
16
¸
18
¸
20
¸
22
¸
24
¸
x
17
x
19
x
23
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
27
28
¸
30
¸
32
¸
¸
¸
¸
¸
33
¸
¸
¸
¸
35
36
¸
38
¸
¸
¸
¸
¸
39
40
¸
42
¸
44
¸
46
¸
48
¸
¸
¸
¸
¸
¸
¸
45
¸
¸
¸
¸
49
Copyright © 2010 Nelson Education Ltd.
x
13
¸
25
50
x
11
¸
¸
21
34
x
7
¸
¸
15
26
x
5
¸
9
20 Press 2 twice, then 5 21 Press 3, then 7
26 Press 2, then 13
x
3
¸
¸
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1
2
3 Consolidation ± 15–20 min
Closing (Whole Class)
4
Provide an opportunity for students to share and communicate
about their work. Have students describe to the rest of the
class how they solved the problem, using chart paper as an
organizing tool for students to follow. Ask students to
comment on the approach presented. The presenters may
5
invite questions from other students and attempt to answer
the questions. Encourage students to identify similarities and
differences among their methods.
Follow-Up and Preparation for Next Class
Next class is the Mid-Chapter Review. Ask students to go
through Lessons 1 to 5 and note any questions or problems
they have.
6
Opportunities for Feedback: Assessment for Learning
7
What you will see students doing
When students understand
If students misunderstand
• Students use different combinations of prime numbers to calculate as many
products as possible. Students then use reasoning to discover that all the
prime numbers from 2 to 47 will yield every number needed.
• Students may not identify the combinations of prime numbers needed to form
the composite numbers. This may result in incomplete lists of numbers
needed or a list containing a mix of prime and composite numbers. (See Extra
Support 1.)
8
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Remind students that factors can be repeated when calculating a product.
Point out that this will allow them to use fewer buttons, but still arrive at
the correct product. For example, have students use repeated division of 16
to see that only the 2 button is needed. Help students see that whenever a
composite button is left, it could be replaced by other buttons, e.g., 6 by 2
and 3 or 10 by 2 and 5. They may use repeated division by primes or factor
trees to determine the prime numbers needed and use a chart to organize
their findings.
EXTRA CHALLENGE
• Oleh believes that his machine can also stretch the licorice for all composite
values from 51 to 100 times without adding any additional buttons.
Challenge students to show whether Oleh’s belief is correct or incorrect.
• Have students work in pairs to answer questions such as the following:
If the licorice-stretching machine works for 9, but not for 12, what button is
broken?
If the licorice-stretching machine works for 16, but not for 28, what button is
broken?
Challenge students to formulate similar questions for a partner to answer
and explain.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• For some students, the abstractness of the context may be a problem. Allow
these students to continue to explore the concept of a number being prime
using a more concrete model. For example, tell students that they are trying
to create paper strips of all the lengths from 2 cm to 50 cm using as few
strip lengths as possible. Have them use paper strips of lengths 2 cm, 3 cm,
5 cm, and 7 cm. Ask them to use each strip more than once, but as many
times as they want to try to make a total length. For example, three 2 cm
strips can be used to make 6 cm. They can record which lengths they are
able to make and explore why these are composite numbers (since they are
groups of another number).
SUPPORTING LEARNING STYLE DIFFERENCES
• Rather than presenting their work to the whole class, some students might
prefer presenting to a smaller group.
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Mid-Chapter Review
STUDENT BOOK PAGES 86–87
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
• Determine all the whole-number factors of a given
number using arrays.
• Identify the factors for a given number and explain the
strategy used, e.g., concrete or visual representations,
repeated division by prime numbers, or factor trees.
• Provide an example of a prime number and explain
why it is a prime number.
• Provide an example of a composite number and
explain why it is a composite number.
Reading Strategy
Monitoring Comprehension is a strategy that readers
use when what they are reading does not make sense.
Effective readers try several approaches to find meaning
when they have trouble understanding something they
are reading. Often they look at the context for clues to
figure out unknown words. In mathematics, students
might encounter new vocabulary or a challenging
procedure that affects comprehension. When this
occurs, students need to call on other known strategies
or a combination of strategies such as visualizing,
questioning, predicting, summarizing, inferring, and
rereading to regain comprehension.
Use a self-questioning strategy with the class. Have
students use key words from the first half of the chapter
and use a check mark to signify their level of
understanding for each term.
Key Word
Lots!
Some
Preparation and Planning
Materials
• Optional: counters
Masters
• Mid-Chapter Review—Frequently Asked
Questions p. 79
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Mid-Chapter Review—
Frequently Asked
Questions p. 79
Not Much
factor
factor rainbow
multiple
Have students share their responses in pairs and tell
each other what they know about each term. Use the
glossary at the back of the Student Book to check.
Students are prompted to use a Monitoring
Comprehension strategy in the Practice questions
of this Mid-Chapter Review.
Copyright © 2010 Nelson Education Ltd.
Mid-Chapter Review
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Frequently Asked Questions
(Whole Class)
Have students keep their Student Books closed. Write the
Frequently Asked Questions on Student Book page 86 on the
board, or use Mid-Chapter Review—Frequently Asked
Questions p. 79. (Distribute the master or display it using
an overhead transparency.) Use the discussion to draw out
what the class thinks is the best answer to each question.
Then have students compare the class answers with the
answers in the Student Book. Have students summarize the
answers in their own words as a way of reflecting on the
concepts. Students can refer to the answers to the Frequently
Asked Questions as they work through the Practice questions.
At this time, you can also discuss any other questions
related to Lessons 1 to 5 that students may have.
Practice (Individual)
Students should be able to complete all the questions in class.
For Question 5, encourage students to identify all the
possible two-digit numbers that can be spun by making a list
or chart, e.g., 22, 23, 24, 25, 32, 33, 34, 35, and so on.
Encourage students to identify which questions they found
easy and which more challenging. Ask them what they can
do to become more proficient at questions they found
challenging. The review questions are organized by lesson.
Students can go back to the lesson indicated to review the
concepts for the question.
Using the Mid-Chapter Review
This review provides an opportunity for students to monitor
their progress with the chapter skills and concepts (Assessment
as Learning), as well as for you to monitor the progress of the
class and see where re-teaching may be required (Assessment
for Learning). You may also use it to assess individual student
achievement (Assessment of Learning).
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Chapter 3: Number Relationships
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
Question 1 (Visualization)
• Students use arrays to identify the factors of 40 and connect the dimensions
of the arrays with the factors of 40.
Question 2 (Reasoning)
• Students calculate factors of numbers.
• Students may have difficulty identifying all of the possible arrays for 40. They
may not connect the numbers of rows and columns with the pairs of factors
that multiply to 40.
• Students may not identify all of the factors.
Question 3 (Problem Solving)
• Students use multiples to calculate the years after 2007 in which the
Women’s World Cup will occur.
Question 4 (Reasoning)
• Students calculate multiples of numbers and explain their reasoning clearly
and concisely.
Question 5 (Problem Solving)
• Students identify the prime numbers and composite numbers that can be
formed with the digits 2, 3, 4, and 5.
• Students may have difficulty identifying the numbers to multiply together or
the number by which to skip count to calculate the years in which the
Women’s World Cup will be played.
• Students may have difficulty identifying the numbers to multiply together or
the number by which to skip count. Students may arrive at correct answers
but not be able to explain their thinking.
• Students may be confused by the numbers in a new context (forming twodigit numbers using a spinner) and not recognize a simple problem in which
they must identify prime and composite numbers.
Question 6 (Reasoning, Visualization)
• Students identify factors from a factor tree.
• Students may not be able to interpret the factor tree.
Question 7 (Reasoning)
• Students identify three possible numbers that have three different prime
numbers as factors.
• Students may not be able to identify one or more numbers that have three
prime numbers as factors.
Differentiating Instruction: How you can respond
Refer to the Differentiating Instruction ideas in Lessons 1 to 5.
Assessment of Learning—What to look for in student work
Specific Outcome and Process Focus: N3 [V]
Question 1, written answer, model
• A veterinarian has 40 indoor dog kennels.
a) What arrays can she form with 40 kennels?
b) How can you use the arrays in part a) to identify all the factors of 40?
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• uses visual representations
insightfully to demonstrate a
thorough understanding of factors
• uses visual representations
meaningfully to demonstrate a
reasonable understanding of factors
• uses visual representations simply to
demonstrate a basic understanding of
factors
• uses visual representations poorly to
demonstrate an incomplete
understanding of factors
Question 2, short answer
• Identify the factors of each number.
a) 14
b) 45
Specific Outcome and Process Focus: N3 [R]
c) 54
d) 75
(Score 1 point for each correct answer for a total out of 4.)
(Continued on next page)
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Mid-Chapter Review
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Assessment of Learning—What to look for in student work
Specific Outcome and Process Focus: N3 [PS]
Question 3, short answer
• The Women’s World Cup of soccer is held every four years. The World Cup was played in China in 2007.
In what years will the five World Cups after China be played?
(Score 1 point for each correct year for a total out of 5.)
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• demonstrates an insightful
understanding of the problem
• demonstrates a complete
understanding of the problem
• demonstrates a basic understanding
of the problem
• demonstrates a limited
understanding of the problem
• develops a thorough plan for
solving the problem
• develops a workable plan for
solving the problem
• develops a basic plan for solving the
problem
• develops a minimal and/or flawed
plan for solving the problem
Specific Outcome and Process Focus: N3 [R]
Question 4, short answer, written answer
• Identify the first five multiples of each number. Explain what you did for one number.
a) 11
b) 22
c) 20
d) 35
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• chooses efficient and effective
strategies to identify multiples
• chooses workable and reasonable
strategies to identify multiples
• chooses partially appropriate and
workable strategies to identify
multiples
• chooses inappropriate and/or
unworkable strategies to identify
multiples
Specific Outcome and Process Focus: N3 [PS]
Question 5, short answer
• You can form a two-digit number by spinning the spinner twice. The first number spun is the tens digit.
The second number spun is the ones digit. How many more composite numbers than prime numbers can be spun?
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• demonstrates an insightful
understanding of the problem
• demonstrates a complete
understanding of the problem
• demonstrates a basic understanding
of the problem
• demonstrates a limited
understanding of the problem
• differentiates between relevant
and irrelevant information
• identifies relevant information
• identifies some relevant information
• has difficulty discerning relevant
from irrelevant information
• develops a thorough plan for
solving the problem
• develops a workable plan for
solving the problem
• develops a basic plan for solving the
problem
• develops a minimal and/or flawed
plan for solving the problem
• chooses an efficient and effective
strategy; may demonstrate
creativity and innovation in
his/her approach
• chooses an appropriate and
workable strategy
• chooses a simplistic and/or routine
strategy
• chooses an inappropriate or
unworkable strategy
Specific Outcome and Process Focus: N3 [R, V]
Question 6, short answer
• What factors of 48 can you identify from the factor tree at the left?
(Score 1 point for all factors listed for a total out of 5.)
Question 7, written answer
• Pablo found that his uncle’s age can be divided by three different prime numbers.
What are three possible ages for his uncle? Show your work.
Specific Outcome and Process Focus: N3 [R]
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• chooses efficient and effective
strategies to identify possible
ages
• chooses workable and reasonable
strategies to identify possible ages
• chooses partially appropriate and
workable strategies to identify
possible ages
• chooses inappropriate and/or
unworkable strategies to identify
possible ages
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Solving Problems Using
an Organized List
STUDENT BOOK PAGES 88–89
GOAL
Use an organized list to solve problems that involve
number relationships.
PREREQUISITE SKILLS/CONCEPTS
• Identify factors and multiples of whole numbers.
• Identify prime and composite numbers.
• Identify the factors for a given number and explain the
strategy used, e.g., concrete or visual representations,
repeated division by prime numbers, or factor trees.
• Solve a given problem involving factors or multiples.
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
Preparation and Planning
Pacing
5–10 min Introduction
15–20 min Teaching and Learning
20–30 min Consolidation
Masters
• Optional: 100 Chart, Masters Booklet p. 30
Recommended
Practising Questions
Questions 2, 6, & 7
Key Question
Question 6
Extra Practice
Chapter Review Question 9
Workbook p. 22
Mathematical
Process Focus
PS (Problem Solving) and R (Reasoning)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Optional: 100 Chart,
Masters Booklet p. 30
Copyright © 2010 Nelson Education Ltd.
Math Background
When there is more than one condition to be satisfied in
order to solve a problem, making an organized list is an
appropriate strategy. An organized list can be written to
satisfy the initial condition and then the list can be
narrowed down, based on additional conditions. This
problem-solving strategy allows students to reason that
no possible solution has been overlooked or eliminated in
error. The conditions that students will work with in this
lesson involve multiples, prime and composite numbers,
and factors. Students will apply what they have learned in
previous lessons about these concepts to arrive at a solution
to each problem. For example, students will identify a
multiple of two different numbers by listing multiples of
the first number and then identifying multiples of the
second number in the same list.
Reading Strategy
Finding Important Information is a reading strategy
that students use to focus their attention on useful
parts of the text and ignore irrelevant information. In
mathematics, students identify the question being asked,
decide the most relevant information needed to answer
the question, and categorize the rest of the information
as useful or not useful. Knowing essential information
makes problem solving manageable.
Students are prompted to use a Finding Important
Information strategy in Question 1. As you discuss the
problem with students, ask them to identify the facts
given in the problem. Then have them identify which
facts are not necessary for solving the problem. Ask them
to state in their own words what the problem asks them
to find out, and discuss strategies for solving the problem.
Lesson 6: Solving Problems Using an Organized List
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1
1 Introduction
2
(Whole Class) ± 5–10 min
2 Teaching and Learning
3
(Whole Class) ± 15–20 min
To prepare students for making organized lists, use skills
learned in previous lessons to play “What’s my number?” Tell
students that you are thinking of a number from 10 to 16
whose factors include 1, 2, 3, 4, and 6. Ask students to write
their answers on a piece of paper. When everyone is finished,
have students hold up their answers. Ask several students
what method they used to determine the number.
Together, read the information about cones for Sage’s jingle
dress and then read the central question on page 88 of the
Student Book. Discuss what information can be used to solve
the problem. Together, read Mai’s understanding of the
problem. Point out that Mai has stated what she needs to
determine and the conditions that must be met to answer the
question. Work through the rest of Mai’s Solution together.
Sample Discourse
“What method did you use to determine the number?”
• I tried each even number because 2 is a factor.
• I eliminated the prime numbers 11 and 13 first.
• I wrote down each number from 10 to 16 and tested to see
if it was the number.
“Why did you write the numbers down?”
• It was a good way to keep track of each number as I tested
whether or not it was the solution.
“How did you keep track of the numbers as you tested
them?”
• I crossed off the numbers that did not have all the factors.
“What is my number?”
• Your number is 12.
Repeat the activity with each clue below.
• I am thinking of a prime number between 20 and 28. (23)
• I am thinking of a number between 16 and 26 that is a
multiple of 9. (18)
Sample Discourse
“After Mai understands the information given in the
problem, how does she plan to solve the problem?”
• She plans to make a list of possible numbers of cones, starting
with multiples of 4 between 20 and 50.
• She lists the multiples of 4, starting with 24 and ending at 48.
“As Mai carries out her plan, she must consider more
information about the number of cones. What else does she
know about the number of cones?”
• The number of cones is a multiple of 3.
• The cones can be arranged in three equal rows with none
left over.
“Which multiples of 4 are also multiples of 3?”
• The numbers 24, 36, and 48 are also multiples of 3.
“How could you check that Mai’s answer of 24, 36, or
48 cones meets all the conditions given in the problem?”
• Each number is between 20 and 50, each number has 4 as
a factor, and each number has 3 as a factor.
3
4
5
6
7
8
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Chapter 3: Number Relationships
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5
6
7
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Reflecting (Whole Class)
Students reflect on why using an organized list is a good
problem-solving strategy for this problem.
Answer to Reflecting Question
A. For example, an organized list was a good strategy for
Mai to use because it allowed her to list all possible
answers. She could list numbers based on one of the
clues. Then she could use the other clues to eliminate
some of the numbers she listed for the first clue.
1
2
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Have students identify the conditions that must be satisfied
in the problem. Encourage students to restate these
conditions in their own words as Mai did in the Understand
part of her problem-solving plan. Have student pairs make a
plan to solve the problem and list the steps they will take to
carry out the plan. The plan they carry out must address all
information given about the number of cones, so it is
important that students identify that information correctly.
Remind students that an array represents a pair of factors of a
number.
5
6
7
Practising
(Individual)
8
Tell students that an organized list can give them a good start
to the problem. Suggest that they use the information in
each problem to list all the possible answers and then use
additional information to add to and/or narrow down the list.
2. Students should begin by listing the prime numbers
between 20 and 50.
6. Remind students that all two-digit numbers formed
using the spinner will yield numbers from 11 to 99
inclusive. None of the numbers will have 0 as the ones
digit, because there is no 0 on the spinner.
Answer to Key Question
6. Natalie. For example, use the problem-solving process.
Understand: Since the spinner contains the numbers
1 through 9 and each girl spins the spinner twice, it is
possible to create any two-digit number between 11 and
99 that doesn’t have 0 as the ones digit. I need to
determine even multiples of 7 and odd multiples of 9.
Make a Plan: I will list all the two-digit numbers
between 11 and 99 that are multiples of 7 and 9 and
don’t have 0 as the ones digit. Then, I will circle the
even multiples of 7 and odd multiples of 9.
Copyright © 2010 Nelson Education Ltd.
Carry Out the Plan: This is the list of two-digit numbers
between 11 and 99 that are multiples of 7 or multiples of
9 and don’t have 0 as the ones digit.
Multiples of 7: 14
21
28
35
42
49
56
63
77
Multiples of 9: 18
27
36
45
54
63
72
81
99
84
91
98
The circled numbers are the even multiples of 7 and the
odd multiples of 9.
Multiples of 7: 14
21
28
35
42
49
56
63
77
Multiples of 9: 18
27
36
45
54
63
72
81
99
84
91
98
There are six even multiples of 7 between 11 and 99 that
don’t have 0 as the ones digit, so Natalie has seven ways
to score 1 point. There are five odd multiples of 9 between
11 and 99, so Gwen has five ways to score 1 point.
Natalie has more ways to score 1 point.
Look Back: I checked all the circled numbers to see if
they match the conditions.
14 ⫽ 7 ⫻ 2
28 ⫽ 7 ⫻ 4
56 ⫽ 7 ⫻ 8
84 ⫽ 7 ⫻ 12
27 ⫽ 9 ⫻ 3
45 ⫽ 9 ⫻ 5
81 ⫽ 9 ⫻ 9
99 ⫽ 9 ⫻ 11
My solutions are reasonable.
42 ⫽ 7 ⫻ 6
98 ⫽ 7 ⫻ 14 63 ⫽ 9 ⫻ 7 Closing (Whole Class)
Question 7 allows students to reflect on and consolidate their
learning for this lesson. Ask students to remember to find
something the numbers 42, 45, and 48 have in common
before they begin writing the problem.
Answer to Closing Question
7. For example, I created this problem:
Shaun has between 40 and 50 model cars in his
collection.
The number of cars is a multiple of 3.
What are the possible numbers of cars in Shaun’s
collection?
I solved the problem by listing the multiples of 3 from
40 to 50: 42, 45, and 48. Shaun has 42, 45, or 48 cars in
his collection.
Follow-Up and Preparation for Next Class
Have students research the locations of the warmest
temperatures (in Celsius) in Canada using the Internet,
newspapers, or magazines. Tell them to organize their findings
in a table and bring it in for discussion in the next class.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students use an organized list as a problem-solving strategy.
• Students may not consider all the information given in the problem. (See Extra
Support 1.)
Key Question 6 (Problem Solving, Reasoning)
• Students make an organized list to determine each girl’s chances of scoring
1 point and then use reasoning to determine who has more chances to score
points.
• Students may not understand that each girl’s situation is an individual
problem to be solved first. Then the results must be compared to solve the
problem. (See Extra Support 2.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Point out to students that there is a connection between the information
they state in the Understand part of the problem-solving plan and the steps
taken to carry out the plan. Suggest that students restate each condition
from the problem in one column on their paper and write the step taken to
address that condition next to it.
2. Remind students that since both girls will spin the same spinner, the
possible range of numbers for each girl is the same. Tell students that after
they determine the possible range of numbers, they should consider each
girl’s ways of scoring in separate problems. Lastly, they should decide which
player has more ways to score points by comparing the results of each
separate problem they solved.
EXTRA CHALLENGE
• Challenge students to use an organized list to determine the possible
numbers of packages of hot dogs and numbers of packages of hot dog rolls
for a crowd of between 50 and 100 people. Hot dogs come in packages of
four and rolls come in packages of six. Plan to provide two hot dogs for each
person. Students should be prepared to explain their solution to the class.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may have difficulty making an organized list. Allow these
students to use a model to look for possible solutions. For example, to model
the multiples of 3 and multiples of 4 given in the opening question, students
can make a 3-by-4 array. One array has 12 counters.
Extend the array by repeating the process.
This array has 36 counters. Since 36 is between 20 and 50, and it’s still a
multiple of 3 and 4, 36 is another solution.
Ask students if 12 counters are a possible solution. Because 12 is less than
20, 12 is not a possible solution.
Have students extend the array by forming another 3-by-4 array and counting
the total number of counters. Emphasize that they can keep extending the
3-by-4 array because this array is already in multiples 3 and multiples of 4. This
will result in the following sets of arrays having multiples of 3 and multiples of 4.
Extend the array by repeating the process.
This array has 48 counters. Since 48 is between 20 and 50, and it’s still a
multiple of 3 and 4, 48 is another solution.
Students should realize that if they repeat the process one more time, the
total number of counters will exceed 50, which is not part of the solution. So
Sage could have 24, 36, or 48 metal cones for the jingle dress.
This array has 24 counters. Since 24 is between 20 and 50, and it’s still a
multiple of 3 and 4, 24 is one of the solutions.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may benefit from starting with a visual representation of
possible solutions. Have students work with a 100 chart and start each
problem by highlighting the range of possible solutions on the chart.
Suggest that students use different colours to circle possible numbers for
each condition given in the problem. Only those values that meet all the
conditions can be a solution to the problem.
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Representing Integers
STUDENT BOOK PAGES 90–92
GOAL
Use integers to describe situations.
PREREQUISITE SKILL/CONCEPT
• Locate numbers on a number line.
SPECIFIC OUTCOME
N7. Demonstrate an understanding of integers,
concretely, pictorially, and symbolically.
[C, CN, R, V]
Achievement Indicators
• Extend a given number line by adding numbers
less than zero and explain the pattern on each side
of zero.
• Describe contexts in which integers are used,
e.g., on a thermometer.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
10–15 min Teaching and Learning
20–30 min Consolidation
Masters
• Number Lines, Masters Booklet p. 33
Recommended
Practising Questions
Questions 3, 4, & 6
Key Question
Question 4
Extra Practice
Chapter Review Question 10
Workbook p. 23
Mathematical
Process Focus
R (Reasoning) and V (Visualization)
Vocabulary/Symbols
integer, opposite integer
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Copyright © 2010 Nelson Education Ltd.
Math Background
Students are familiar with the set of whole numbers,
which includes the counting numbers and zero. In this
lesson, students are introduced to the set of integers,
which includes positive and negative whole numbers and
zero. A number line is used as a visualization tool for the
set of integers. Positive integers are integers to the right of
zero on a number line. Negative integers are integers to
the left of zero on a number line.
The purpose of this lesson is to help students reason
and understand that many contexts exist where integers
are used, and to understand the relationship among
positive numbers, negative numbers, and zero. Students
will also be introduced to the concept of opposite integers,
or integers that are the same distance from zero, but on
opposite sides on a number line. For example, ⫹6 and –6
are opposite integers.
Although a positive (⫹) sign is not often used to denote
positive integers, students will use both the positive and
negative (⫺) signs throughout this lesson to solidify their
understanding. However, zero is never written with a
positive or negative sign. This point will be formally made
in the next lesson on temperatures.
Number Lines, Masters
Booklet p. 33
Lesson 7: Representing Integers
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1
1 Introduction
2
(Whole Class) ± 5–10 min
If students collected data about the warmest temperatures in
Canada in the follow-up to the previous lesson, invite them to
present that data. On the board, on a transparency, or on an
interactive whiteboard, draw a number line from 0 to 20.
Have students locate the position of one of their temperatures
on the number line. Make sure everyone in the class is using
temperatures in Celsius. Have students practise moving up
and down the number line; for example, have students locate
a number that is between two temperatures. Talk about how
students can use the number line to identify numbers.
3
4
5
Sample Discourse
“How can you identify a number that is between 10 and 15?”
• I can look for marked numbers between 10 and 15, such as
11, 12, 13, and 14.
• I can pick any number that is to the right of 10 and to the
left of 15.
6
7
2 Teaching and Learning
3
(Whole Class/Pairs) ± 10–15 min
Together, discuss the information and central question about
Jason’s cursor on Student Book page 90. Distribute number
lines or have students sketch number lines, and work through
Jason's Number Line together. Draw students’ attention to
the definition of integers. Ensure students understand that
positive integers are to the right of zero on the number line
and negative integers are to the left of zero.
Have students work through Prompts A to C in pairs, and
then discuss the answers as a class.
4
5
6
Answers to Prompts
A. –2
B. He pressed d six times to get to –6.
C.
7
8
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1
0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10
8
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Practising (Individual)
These questions provide students with opportunities to apply
their understanding of integers. Students will use reasoning
to identify integers. Remind students to use number lines to
help them visualize the relative positions of the integers.
4. Tell students to look for the integers between each pair,
but not including the pair.
Answers to Key Question
4. a) The number line shows the integers between –4 and
⫹4. The integers ⫺3, ⫺2, ⫺1, 0, ⫹1, ⫹2, and ⫹3
are between ⫺4 and ⫹4.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
b) The number line shows the integers between –3 and
0. The integers between ⫺3 and 0 are ⫺2 and ⫺1.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
c) The number line shows the integers between
–2 and –5. The integers between ⫺2 and ⫺5 are ⫺3
and ⫺4.
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
d) The number line shows that there are no integers
between 0 and –1.
Reflecting (Whole Class)
Draw students’ attention to the margin definition of opposite
integers. Talk about other examples of opposites related to
directions, such as east and west, right and left, and up and
down. Have students work through Prompt D individually
and then discuss the answer as a class.
Sample Discourse
“How do you know east and west are opposites?”
• Because I would go in one direction to go east and in the
opposite direction to go west.
“How do you know right and left are opposites?”
• For example, if two people stood in the centre of the room and
one walked to the right and the other walked to the left, they
would end up on opposite sides of the room.
Answer to Reflecting Question
D. For example, they are opposite integers because they are
both 4 units from 0, but in opposite directions. You can
use n to move the cursor 4 units from 0 to the right,
but you need to use m to move the cursor 4 units from
0 in the opposite direction.
1
2
3
Consolidation
± 20–30 min
Checking (Pairs)
4
Ask for volunteers to show solutions on the board and discuss
the solutions as a class. Provide students with number lines.
5
Copyright © 2010 Nelson Education Ltd.
6
7
–5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5
Closing (Whole Class)
Question 6 allows students to reflect on and consolidate their
learning for this lesson. Students will explain and interpret
the use of integers in a variety of real-life situations.
Encourage students to share their solutions, and try to elicit a
variety of examples.
Answer to Closing Question
6. For example, my birthday is on April 13. So I can write
–3 to represent 3 days before my birthday, or April 10,
and I can use ⫹3 to represent 3 days after my birthday,
or April 16.
For example, if a car is 10 km north, I can represent
the distance as ⫹10. If the car is 10 km south, I can
represent the distance as –10.
For example, if I take $5 out of the piggy bank, I can
write –5 to show that the amount in the piggy bank is
$5 less. But if I added $5, I can write ⫹5 to show that
the amount in the piggy bank is $5 more.
Follow-Up and Preparation for Next Class
Have students find additional examples of situations in their
daily lives that can be represented by integers. Have students
write down the examples in their notebooks and share with
their classmates. This will help to solidify their understanding
of integers.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students identify integers based on clues and number patterns.
• Students may have difficulty using clues to locate negative integers on a
number line. (See Extra Support 1.)
• Students interpret integers in the contexts of different situations.
• Students may be confused by the context and may not be able to interpret
positive and negative values. (See Extra Support 2.)
Key Question 4 (Reasoning, Visualization)
• Students identify the integers between two integers.
• Students may have difficulty locating the integers or may miss some of the
points in between as they mark the number line. (See Extra Support 3.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students begin by practising locating positive numbers on a number
line. Draw a number line from 0 to 10 on the board, on a transparency, or
on an interactive whiteboard. Present students with modified versions of the
clues from Question 3. For example:
a) It is the same distance from 6 as 4 is from 6.
b) It is between 3 and 5.
c) It is the next integer to the right of 2.
d) It is halfway between 0 and 10.
Talk about the strategies students used to identify the numbers described in
the clues. Point out that students can use the same strategies to work with
negative numbers.
2. Explain what 0 represents in each situation: in a rocket launch, 0 means the
time at which the rocket takes off; for days before and after your birthday,
0 means the day of your birthday; for kilometres from your town, 0 means the
location of your town; and for money taken from or added to a piggy bank, 0
means the number of dollars you started with. Then talk about what positive
numbers mean in each situation and what negative numbers mean in each
situation.
3. Write the numbers ⫹5 and ⫺5 on the board, on a transparency, or on an
interactive whiteboard. Make sure students understand that the digit
5 represents the distance from 0 on the number line, so both ⫹5 and ⫺5 are
5 units from 0. Next, point out that the negative (⫺) sign in front of a
number means that it is to the left of 0 on the number line, or less than 0;
the positive (⫹) sign in front of a number means that it is to the right of 0
on the number line, or greater than 0. Display a partially completed number
line on the board with –4, –2, ⫹2, and ⫹4 marked for students to copy.
Have students mark the missing integers between –4 and ⫹4. Check to
see that students remember to mark the 0.
EXTRA CHALLENGE
• Challenge students to compose puzzle questions involving integers for
classmates to solve. Puzzles may take the form of “What integer am I?” and
include clues as to where the integer may be found on a number line.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may have difficulty conceptualizing negative numbers.
Provide these students with additional examples of situations they may
encounter that can be represented with integers. For example, an elevator
at an office building might use “G” to represent the ground floor. The floor
numbers (1, 2, 3, and so on) are positive because you go up in the elevator to
reach them, and the parking levels (P1, P2, P3, and so on) are negative
because you go down in the elevator to reach them.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may benefit from using different colours to label the
positive and negative numbers on their number lines. This will help them
discriminate visually between values greater than and less than zero.
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Curious Math
Countdown Clock
STUDENT BOOK PAGE 93
PREREQUISITE SKILL/CONCEPT
• Locate numbers on a number line.
SPECIFIC OUTCOME
N7. Demonstrate an understanding of integers,
concretely, pictorially, and symbolically.
[C, CN, R, V]
Achievement Indicators
• Extend a given number line by adding numbers
less than zero and explain the pattern on each side
of zero.
• Describe contexts in which integers are used, e.g., on
a thermometer.
Preparation and Planning
Masters
• Optional: Number Lines, Masters
Booklet p. 33
Mathematical
Process Focus
CN (Connections) and R (Reasoning)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Math Background
A countdown clock is a real-life example of using integers
with which students are likely familiar. On a countdown
clock, the time of the event is assigned 0. Time before the
event has a negative value, and time after the event has a
positive value. Although many students will be familiar
with the idea of counting down to a date, most will have
done so from the perspective of today’s date rather from
the perspective of the event date; that is, they would have
considered the event to be, for example, 5 days in the
future, rather than considering today to be 5 days before
the event.
Connections to existing knowledge are made as students
work with positive and negative integers. Students will use
this reasoning to assign integer values to dates before and
after an event.
Optional: Number Lines,
Masters Booklet p. 33
Copyright © 2010 Nelson Education Ltd.
Using Curious Math
In Lesson 7, students were introduced to situations that can
be represented with integers. In this activity, students apply
their understanding in the context of a countdown clock.
Students can work through the questions individually. You
may want to provide copies of number lines, so that students
can visualize the relative positions of the days in the
countdown. Talk about when the countdown clock would
display a negative integer and when it would display a
positive integer.
Sample Discourse
“When would the countdown clock display a negative
integer?”
• It would display a negative integer before the school play
because the play is on day 0.
• It would display a negative integer on days before the play
because on a number line, those days would be to the left of
the play, which is on day 0.
“When would the countdown clock display a positive integer?”
• It would display a positive integer after the school play because
the play is on day 0.
• It would display a positive integer on days after the play
because on a number line, those days would be to the right of
the play, which is on day 0.
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Answers to Curious Math
1. For example, it represents 5 days before the opening
night of the school play.
2. –4, –3, –2, –1, 0
3. 0
4. ⫹3
5. For example, I chose my birthday on March 21. So
–3 would represent 3 days before my birthday, or March
18; 0 would represent the day of my birthday; ⫹3 would
represent 3 days after my birthday, or March 24.
Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students use integers to represent days before and after an event.
• Students may confuse positive and negative integers. (See Extra Support 1
and 2.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Help students relate the countdown clock to a calendar. If possible, show
students a calendar and ask them to assign a date to the school play, for
example, the 10th of the month. Talk about how different dates in the month
can be expressed in terms of the date of the play. For example, if today is
the 7th, you would need to subtract 3 from the date of the play to get today’s
date, so in integer terms, today has a value of ⫺3. If today is the 14th of the
month, you would need to add 4 to the date of the play to get today’s date,
so in integer terms, today has a value of ⫹4.
2. Students may be accustomed to thinking about an event from today’s
perspective, rather than from the perspective of the event. Help students
connect these two perspectives. Draw a number line on the board, on a
transparency, or on an interactive whiteboard. Mark “today” at 0 and
“school play” at ⫹5. Draw another number line below the first so that 0 on
the new line is aligned with ⫹5 on the old line. Discuss what integer would
represent “today” if the school play is 0. Students should see that “today” is
aligned with ⫺5 on the new line.
EXTRA CHALLENGE
• Have students create a timeline of recent and future events, assigning today as
0. Students can show the different events along the timeline, assigning the
dates integer values and drawing pictures to represent the events.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may have difficulty understanding the concept of why the
number of days is 5 units before or after the opening night. Instead of
focusing on real-life examples of negative integers, have students practise
labelling number lines. Have them put a counter or a small object at a
certain number and then tell them to move the object along the number
line to assigned positions. As they move, encourage them to count the units
out loud.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students will benefit from sketching a number line with the different
days and the event labelled on it. Students can use different colours to
indicate days before (negative integers) and after (positive integers) the
school play.
52
Chapter 3: Number Relationships
• Use masking tape or chalk to draw a number line across the classroom
floor. Some students will better understand the concept by walking along
the number line to an assigned position and figuring out the number of
units from the initial position.
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Comparing and Ordering
Integers
STUDENT BOOK PAGES 94–97
GOAL
Use a number line to compare and order integers.
PREREQUISITE SKILLS/CONCEPTS
• Locate integers on a number line.
• Use the symbols , , and to compare numbers.
SPECIFIC OUTCOME
N7. Demonstrate an understanding of integers,
concretely, pictorially, and symbolically.
[C, CN, R, V]
Achievement Indicators
• Place given integers on a number line and explain
how integers are ordered.
• Compare two integers; represent their relationship
using the symbols<,>, and ⴝ, and verify using a
number line.
• Order given integers in ascending or descending order.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
10–15 min Teaching and Learning
20–30 min Consolidation
Masters
• Number Lines, Masters Booklet p. 33
Recommended
Practising Questions
Questions 3, 5, 6, 7, 8, & 10
Key Question
Question 6
Extra Practice
Chapter Review Questions 11 & 12
Workbook p. 24
Mathematical
Process Focus
C (Communication), CN (Connections),
and V (Visualization)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Copyright © 2010 Nelson Education Ltd.
Math Background
In Lesson 7, students were introduced to situations that
can be represented with integers. In this lesson, students
build upon and expand their understanding as they
compare and order integers. Here are some key ideas
about comparing integers:
• Numbers become greater as you move to the right
along a number line, and smaller as you move to the
left along a number line.
• Positive numbers are greater than zero and negative
numbers are less than zero.
• Any positive number is greater than any negative
number.
Students will use numbers lines to help them visualize
the relative sizes of integers. Connections are formed
between positive and negative integers and relative size of
integer amounts. Students develop mathematical
communication skills as they explain their solutions.
Number Lines, Masters
Booklet p. 33
Lesson 8: Comparing and Ordering Integers
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1
1 Introduction
2
(Whole Class) ± 5–10 min
On the board, on a transparency, or on an interactive
whiteboard, draw a number line from 0 to 20 but label only
0 and 20. Have students locate different positive numbers on
the number line. Talk about how students can use a number
line to compare numbers. Write the following number
sentences on the board and ask volunteers to complete them
with or .
5 12
10 7
18 9
3 11
3
4
5
Sample Discourse
“In which direction do numbers increase on a number line?”
• Numbers increase as you move to the right.
“How do you know that 15 is greater than 10?”
• 15 is to the right of 10 on the number line.
• 10 is to the left of 15 on the number line.
6
7
8
54
Chapter 3: Number Relationships
2 Teaching and Learning
3
(Pairs/Whole Class) ± 10–15 min
Together, read about Léa’s report and then read the central
question on Student Book page 94. Work through Léa’s
Comparison together. Point out that Léa’s number line starts
at 40 and that 0 is on the right. Ask students if this setup
makes sense considering the temperatures she collected in her
chart. Distribute number lines and have students copy Léa’s
number line and mark the low temperature for Iqaluit and
Yellowknife before working through Prompts A to E in pairs.
When students have completed the activity, draw a large
number line on the board, on a transparency, or on an
interactive whiteboard. Have volunteers mark the high
temperatures on the number line (Prompt E) and describe
the strategies they used.
4
5
6
7
Answers to Prompts
A. For example, if the temperature shows a positive integer,
the temperature is above the freezing point of water. If the
temperature shows a negative integer, the temperature is
below the freezing point of water. If the integer is 0, the
temperature is 0 C or the freezing point of water.
B. For example, the temperature 31 C is the farthest to
the left of zero on the number line. So it is the coldest
temperature.
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C.
Regina
–21
Iqaluit
Yellowknife
Edmonton Victoria
–31
+1
–19
–40 °C –30 °C –20 °C –10 °C
0 °C +10 °C +20 °C
Whitehorse
–22
Winnipeg
Freezing point
–23
of water
22 C is to the left of 21C, so it is the colder
temperature.
D. For example, to order the temperatures from coldest to
warmest, I first picked the lowest temperature, which is
31 C. Then I chose temperatures in order that are to
the right of 31 C.
Yellowknife & Iqaluit: 31 C
Winnipeg: 23 C
Whitehorse: 22 C
Regina: 21 C
Edmonton: 19 C
Victoria: 1 C
Copyright © 2010 Nelson Education Ltd.
E. Yellowknife: 23 C
Iqaluit: 22 C
Whitehorse & Winnipeg: 13 C
Regina: 11 C
Edmonton: 8 C
Victoria: 7 C
Reflecting (Whole Class)
Here students form and articulate generalizations about
comparing a positive number with a negative number, a
positive number with a positive number, and a negative
number with a negative number.
Answers to Reflecting Questions
F. For example, positive temperatures are above 0 C, the
freezing point of water. So they can be shown on the
right side of zero on a number line. Negative
temperatures are below 0 C. They can be shown on the
left side of zero on a number line. So any positive
temperature is greater than any negative temperature.
Negative
Temperatures
–20 °C
–10 °C
Positive
Temperatures
0 °C
+10 °C
+20 °C
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G. For example, it is the same because temperatures to the
right are greater than temperatures to the left on a
number line.
+1 °C
–5
–4
–3
–2
–1
0
+1
+5 °C
+2
+3
+4
+5
5 is to the right of 1, so 5 C is warmer than 1 C.
–5 °C
–10 –9
–4
–3
–2
–1
0
+1
+2
+3
+4
–1 °C
–5 °C
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Encourage students to use a number line to help them
visualize the relative positions of the temperatures. You may
want to point out to students that Question 2 a) asks them to
order the temperatures from warmest to coldest, while part b)
asks them to order the temperatures from coldest to warmest.
5
6
Practising (Individual)
These questions provide students with practice in comparing
and ordering integers. Students can use various number lines
to help them visualize the relative values of the numbers.
9. c) There are some exceptions to the apparent
relationship between surface temperature and average
distance of planets from the Sun, such as Venus.
Students can do research to find the typical surface
temperatures for the planets that are not listed.
7
8
Chapter 3: Number Relationships
–8
–7
–6
–5
–4
–3
–2
–1
0
+5
25 is to the left of 1, so 5 C is colder than 1 C.
56
–10 °C
–1 °C
1
–5
Answer to Key Question
6. For example, 5 C is below the freezing point of water so
it is to the left of zero on a number line. Temperatures like
10 C are to the left of 5 C and are colder than 5 C.
Temperatures like 1 C are to the right of 5 C and are
warmer than 5 C. Positive temperatures are to the right
of zero, which is to the right of 5 C, so any positive
temperature is warmer than 5 C.
Closing (Whole Class)
Question 10 allows students to reflect on and consolidate
their learning for this lesson as they articulate the connection
between comparing temperatures and comparing integers.
Answer to Closing Question
10. For example, I can compare 10 and 5 by thinking of
the temperatures 10 C and 5 C. 10 C is colder
than 5 C so 10 5.
Follow-Up and Preparation for Next Class
Have students check the newspaper or the Internet for the
week’s forecasted temperatures and order them from coldest
to warmest. Encourage students to present their findings to
their friends or family members. They can elaborate their
presentation on a number line, compare how many degrees
(how many units) apart the temperatures are for certain days
by counting up or down on the number line, and so on.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students compare and order integers.
• Students may compare numbers without regard to the integer sign. (See Extra
Support 1.)
• Students may be unable to use a number line to compare integers. (See Extra
Support 2.)
Key Question 6 (Communication, Connections, Visualization)
• Students use a number line to explain how they can compare integers.
• Students may be unable to connect temperatures with a number line. (See
Extra Support 3.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Talk about what the negative and positive signs represent. Remind students
to first look at the integer sign to determine whether a number is positive or
negative, and then consider the digit to determine where to place the number
on a number line. Talk about positive and negative numbers relative to zero.
Ensure students understand that all negative numbers are less than zero and
all positive numbers are greater than zero.
2. On the board, on a transparency, or on an interactive whiteboard, use sticky
notes to place the numbers 3, 4, 8, 0, and 2 on a number line, with
the 2 and 8 interchanged. Ask students to find the mistake. Help them
see why 8 has to be the left of 2 since 8 is to the right of 2.
3. Draw a magnified basic thermometer on a large piece of paper cut out to the
thermometer’s size, and point out 0 C. Tell students that 0 C on the paper
thermometer represents the same as 0 on the number line. Take the paper
thermometer and place it sideways on the board. Ask volunteers to use the
thermometer to place different positive and negative temperatures on the
number line. Talk about which temperatures are greater than 5 C and
which are less than 5 C.
EXTRA CHALLENGE
• Have students develop a game in which they compare integers. Provide
dice of different colours if available; otherwise, have students use one set
of dice to represent positive integers and another set to represent negative
integers.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students would benefit from further opportunities to describe situations
involving negative numbers or locating negative integers on a number line
rather than comparing them.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students are able to communicate orally better than in writing. Allow
these students to work on the Practising questions in pairs. Partners can help
students clarify any concepts that are not clear.
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Order of Operations
STUDENT BOOK PAGES 98–100
GOAL
Apply the rules for order of operations with whole numbers.
PREREQUISITE SKILL/CONCEPT
• Use mental math to add, subtract, multiply, and divide
whole numbers.
SPECIFIC OUTCOME
N9. Explain and apply the order of operations, excluding
exponents, with and without technology (limited to
whole numbers).
[CN, ME, PS, T]
Achievement Indicators
• Demonstrate and explain with examples why there is
a need to have a standardized order of operations.
• Apply the order of operations to solve multi-step
problems with or without technology, e.g., computer,
calculator.
Preparation and Planning
Pacing
(allow 5 min for
previous homework)
5–10 min Introduction
10–15 min Teaching and Learning
20–30 min Consolidation
Materials
• calculators
Recommended
Practising Questions
Questions 2, 4, 5, 6, & 9
Key Question
Question 5
Extra Practice
Chapter Review Questions 13 & 14
Workbook p. 25
Mathematical
Process Focus
CN (Connections), ME (Mental Mathematics
and Estimation), PS (Problem Solving), and
T (Technology)
Vocabulary/Symbols
rules for order of operations
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
58
Chapter 3: Number Relationships
Math Background
In this lesson, students learn to use both mental math
skills and technology to calculate the answers to problems
involving many operations. The rules for order of
operations tell which operation should be performed first.
The purpose of the order of operations is to ensure that
the same answer is reached regardless of who performs the
calculations. When more than one operation appears in an
expression or equation, the operations must be performed
in the following order:
• Do the operations in brackets first.
• Then divide and multiply from left to right.
• Finally, add and subtract from left to right.
In this lesson, students demonstrate their understanding
of the connections among operations by applying the
rules for order of operations in a variety of problemsolving situations. Students check to see whether their
calculator follows the rules for order of operations.
Calculators may yield different results depending on
their type.
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1
1 Introduction
2
(Whole Class/Small Groups)
± 5–10 min
Write the following expression on the board, on a transparency,
or on an interactive whiteboard.
10 2 3 6 2
Have small groups of students calculate the value of the
expression. Ask volunteers to share their solutions on the
board. Discuss why some found different answers. Tell
students they will learn rules for doing calculations so that
everyone always gets the same answer.
3
4
5
Sample Discourse
“Which operation did you perform first”?
• I subtracted 10 2 because it is the first operation.
• I added 3 6 because addition is the easiest operation.
• I multiplied 2 3 because I knew it was equal to 6.
“What answer did you calculate?”
• I did the operations in order from left to right and calculated
an answer of 15.
• I did the subtraction and then the addition and calculated an
answer of 36.
6
7
2 Teaching and Learning
3
(Whole Class) ± 10–15 min
Together, read Oleh’s calculation to find his minimum
training heart rate and then read the central question on
Student Book page 98. Work through Oleh’s Solution
together. Draw students’ attention to the definition of rules
for order of operations and talk about how Oleh followed the
rules to calculate his minimum training heart rate.
4
5
Sample Discourse
“How do you know that Oleh followed the rules for order of
operations?”
• Oleh did the operations inside the brackets first, which is the
first step in the order of operations.
“Why did Oleh do the division last, even though division
comes before addition and subtraction according to the rules
of order of operation?”
• The addition and subtraction are inside the brackets, and
brackets come before division in the order of operations.
6
7
8
8
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Reflecting (Whole Class)
Here students reflect on how the rules for order of operations
affect the answer to a calculation. Draw students’ attention to
the Communication Tip. Explain to students that early
calculators, unlike modern ones, did not use the rules for
order of operations. If possible, have students enter Oleh’s
calculation into their calculators to demonstrate how they
can use brackets with a calculator.
For Prompt A, some students may notice that Oleh could
calculate the same answer if he ignored the brackets and
performed the operations from left to right. However, since
Oleh knows the rules for order of operations, if he ignored
the brackets he would likely perform the division first, which
would lead to an incorrect answer.
Answers to Reflecting Questions
A. For example, if he ignored the brackets in the formula
but used the rules for order of operations, he would
divide 72 by 2 first to get 36. Then he could do the rest
of the calculations in order:
220 12 36 208 36 244
So the answer would change from 140 beats each minute
to 244 beats each minute.
B. No, it didn’t matter which operation Oleh did in the
brackets for this calculation. For example, because the
operations are addition and subtraction. You can subtract
12 from 220 to get 208 and then add 72 to get 280.
Or you can add 72 to 220 to get 292 and then subtract
12 to get 280. The answer in the brackets is still 280.
1
2
3 Consolidation ± 20–30 min
Checking (Pairs)
4
Provide a sample calculation of different age and height for
Question 1. Ask volunteers to share their solutions to
Question 1a) with the class. Discuss different strategies that
students used to check the reasonableness of their answers.
5
Practising
(Individual)
6
These questions provide students opportunities to practise
applying the rules for order of operations in a variety of
problem-solving situations. Encourage students to use mental
math strategies to perform the calculations.
3. Provide calculators to students.
5. Note that there are many ways to use 4s to create each
number in part b).
7
8
Answers to Key Question
5. a) 4 4 4 4 4 1 4
54
1
(4 4) (4 4) 8 (4 4)
88
1
44 44 1
60
Chapter 3: Number Relationships
b) For example,
24444
3 (4 4 4) 4
4 (4 4) 4 4
5 (4 4 4) 4
Closing (Whole Class)
Question 9 allows students to reflect on and consolidate their
learning for this lesson. Ask for volunteers to share their skilltesting questions. Discuss why the different skill-testing
questions would or would not likely be solved correctly
without using the order of operations.
Answer to Closing Question
9. a) For example, some would answer mixed calculations
correctly if they can be done correctly from left to
right because most people calculate in that order:
6 3 5.
b) For example, 3 5 20 5 would probably be
done incorrectly if a person calculated in order from
left to right. 15 20 5 35 5, or 7. The
correct answer, however, is 19.
Follow-Up and Preparation for Next Class
Next class is the Chapter Review. Ask students to go through
Lessons 1 to 9 and note any questions or problems they have.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
When students understand
If students misunderstand
• Students understand and use the rules for order of operations to solve
problems with multiple operations.
• Students may perform the operations in the order in which they appear from
left to right. (See Extra Support 1.)
• Students identify expressions that do and do not need brackets to be solved
correctly.
• Students may not understand the purpose of brackets in an expression with
multiple operations. (See Extra Support 2.)
Key Question 5 (Connections, Problem Solving)
• Students use the rules for order of operations to show that different
expressions have a value of 1.
• Students may perform the operations in the order in which they appear from
left to right. (See Extra Support 1.)
• Students use the rules for order of operations to write expressions with
specific answers.
• Students may be unable to create alternative expressions as specified. (See
Extra Support 3.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students give examples of situations in which they need to follow rules,
such as traffic lights for pedestrians and rules for any game. Talk about how
rules help them know what to do and how to do it. Tell students that the
rules for order of operations need to be followed to make sure that everyone
solves the problem the same way and gets the same answer. You may want
to help them think of ways they can remember the order of operations.
3. Talk about the various ways 4s can be combined to get a value of 1, using
different operations and brackets. Have students try various combinations of
brackets and operations with four 4s to get a value of 2. They might begin by
guessing and testing, then use reasoning to get closer to the answer. In the
process, they might find expressions that have a value of 3, 4, or 5 instead.
Have them continue until they have an expression for each value.
2. Emphasize the role that brackets play in calculating an answer. Ensure
students understand that brackets indicate that they should perform an
operation first. Review the rules for order of operations and point out to
students that without brackets, multiplication and division are always
performed before addition and subtraction. Help students understand that
brackets are needed if the addition and subtraction are supposed to be done
first, but not if the multiplication and division are supposed to be done first.
EXTRA CHALLENGE
• Challenge students to use the digits from 1 to 5, as well as addition,
subtraction, multiplication, division, and brackets, to write as many
expressions with different answers as possible.
• Students may make up puzzles by creating a calculation and then erasing the
operations. Challenge other students to figure out the missing operations, as
in Question 8.
For example, (1 2) 5 (4 3) 15
SUPPORTING DEVELOPMENTAL DIFFERENCES
• For students uncomfortable with performing calculations with multiple
operations, have them practise solving simple expressions that have
parentheses, such as (3 1) 4. Students can then focus on solving the
expression in the parentheses first each time. This will allow students to
practise their mental math skills and solve problems with multiple steps.
SUPPORTING LEARNING STYLE DIFFERENCES
• Some students may find the task easier if different colours are associated
with and than with and .
Copyright © 2010 Nelson Education Ltd.
• Some students may benefit from devising a mnemonic device to help them
remember the order of operations.
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Math Game
Four in a Row
STUDENT BOOK PAGE 101
PREREQUISITE SKILL/CONCEPT
• Apply the rules for order of operations with whole
numbers.
SPECIFIC OUTCOME
N9. Explain and apply the order of operations, excluding
exponents, with and without technology (limited to whole
numbers).
[CN, ME, PS, T]
Achievement Indicator
• Apply the order of operations to solve multi-step
problems with or without technology.
Preparation and Planning
Number of Players
2
Materials
• coloured counters
Masters
• Four in a Row Game Board p. 80
• Calculation Cards pp. 81–82
Mathematical
Process Focus
ME (Mental Mathematics and Estimation)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Four in a Row Game
Board p. 80
Calculation Cards
pp. 81–82
Using the Math Game
In this game, students use mental math skills to calculate the
answers to expressions from Calculation Cards pp. 81–82.
On each turn, a student chooses a card and completes the
calculation. The student then places a coloured marker on
the game board square, using Four in a Row Game Board
p. 80, that coincides with their calculated answer. If the
student has four coloured markers in a row, column, or
diagonal, the student wins the game. Students may play until
all the cards are used up, with neither player winning.
When to Play
Students can play the game after they demonstrate an
understanding of how to use the order of operations.
Strategies
Have students keep track of which numbers they need to
form four counters in a row. They may then quickly estimate
a calculation by performing the multiplication and division
operations mentally before choosing a card.
Discuss
After the game, ask students to share any successful strategies
they used to win the game. You may also ask students to
share experiences as they applied the order of operations
using mental math.
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Opportunities for Feedback: Assessment for Learning
What you will see students doing
Proficient players
Less-proficient players
• Students use reasoning and mental math to perform calculations using the
order of operations.
• Students may be unable to correctly simplify an expression using the order of
operations. (See Extra Support 1.)
Differentiating Instruction: How you can respond
EXTRA SUPPORT
1. Have students work in teams of two to verify whether a calculation was
performed correctly. Display a visual showing the order of operations for
students to refer to. Encourage students to discuss which cards are more
likely to yield the desired answer.
EXTRA CHALLENGE
• Have students play using the same cards, but in case of not getting the
desired answer, have them add more operations to the calculation card
in order to end up with the answer they need to form four in a row. To
maintain a competitive element, have students work within a given
time limit.
SUPPORTING DEVELOPMENTAL DIFFERENCES
• Some students may have difficulty calculating on the spot. Allow students
to work with simpler cards that include only two operations, such as
multiplication and subtraction, or division and addition.
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STUDENT BOOK PAGES 102–104
Chapter Review
SPECIFIC OUTCOMES
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicators
• Identify multiples for a given number and explain the
strategy used to identify them.
• Determine all the whole-number factors of a given
number using arrays.
• Identify the factors for a given number and explain
the strategy used, e.g., concrete or visual
representations, repeated division by prime numbers,
or factor trees.
• Provide an example of a prime number and explain
why it is a prime number.
• Provide an example of a composite number and
explain why it is a composite number.
• Solve a given problem involving factors or multiples.
Preparation and Planning
Materials
• Optional: counters
Masters
• Chapter Review—Frequently Asked
Questions p. 83
• Chapter 3 Test pp. 84–86
• Optional: Number Lines, Masters Booklet p. 33
Extra Practice
Workbook p. 26
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Chapter Review—
Frequently Asked
Questions p. 83
Chapter 3 Test pp. 84–86
N7. Demonstrate an understanding of integers,
concretely, pictorially, and symbolically.
[C, CN, R, V]
Achievement Indicators
• Place given integers on a number line and explain
how integers are ordered.
• Compare two integers; represent their relationship
using the symbols<,>, and ⴝ, and verify using a
number line.
• Order given integers in ascending or descending
order.
Optional: Number Lines,
Masters Booklet p. 33
N9. Explain and apply the order of operations, excluding
exponents, with and without technology (limited to
whole numbers).
[CN, ME, PS, T]
Achievement Indicator
• Apply the order of operations to solve multi-step
problems with or without technology, e.g., computer,
calculator.
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Using the Chapter Review
Use these pages to consolidate and assess students’
understanding of the concepts developed in the chapter. The
Practice questions can be used for assessment of learning.
Refer to the assessment chart for the details of each question.
Alternatively, use the Practice questions as a practice test,
and then administer Chapter 3 Test pp. 84–86. The scoring
guides and rubrics provided for the Practice questions can
also be used for the test questions: each question on the test
corresponds to the Practice question of the same number.
Frequently Asked Questions
(Individual/Groups)
Have students read the Frequently Asked Questions (FAQs)
on Student Book page 102 and create a new example for each
question in their own notes. Then have students summarize
Copyright © 2010 Nelson Education Ltd.
the answers to the FAQs in their own words, as a way of
reflecting on the concepts.
Alternatively, have students complete Chapter Review—
Frequently Asked Questions p. 83 with their Student
Books closed. Discuss students’ answers, and then compare
these answers with those in the Student Book. Students can
refer to the answers to the FAQs as they work through the
Practice questions.
Practice (Individual)
Most students will be able to complete Questions 1 to 14 in
class. Assign any uncompleted questions for homework.
Some students may want to use materials and/or masters that
were used in this chapter’s lessons. Provide students with
counters, number lines, or 100 charts, as needed, to complete
the questions.
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Assessment of Learning—What to look for in student work
Specific Outcome and Process Focus: N3 [R, V]
Question 1, written answer
• How do these arrays show the factors of 16?
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• often draws insightful and logical
conclusions using knowledge of
factors
• in many situations, draws logical
conclusions using knowledge of
factors
• sometimes draws simple, logical
conclusions using knowledge of
factors
• rarely draws conclusions from a
mathematical situation using
knowledge of factors
Question 2, written answer
• Which number from 10 to 20 has an odd number of factors? Explain how you identified the factors.
Specific Outcome and Process Focus: N3 [R]
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• often draws insightful and logical
conclusions using knowledge of
factors
• in many situations, draws logical
conclusions using knowledge of
factors
• sometimes draws simple, logical
conclusions using knowledge of
factors
• rarely draws conclusions from a
mathematical situation using
knowledge of factors
Question 3, short answer
• Maddy listed these factors of 48: 1, 2, 4, 5, 8, 16, and 48.
a) Which number listed is not a factor of 48?
b) Which factors are missing?
Specific Outcome and Process Focus: N3 [R]
(Score 1 point for each correct answer for a total out of 5.)
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Assessment of Learning—What to look for in student work
Specific Outcome and Process Focus: N3 [R]
Question 4, short answer
• List the first five multiples of each number.
a) 7
b) 6
c) 9
d) 40
(Score 1 point for each correct answer for a total out of 4.)
Specific Outcome and Process Focus: N3 [PS]
Question 5, written answer
• Every five years, Statistics Canada conducts a census to collect data about Canadians.
A census was conducted in 2006. Will a census be conducted in 2036? Explain your thinking.
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• demonstrates an insightful
understanding of the problem
• demonstrates a complete
understanding of the problem
• demonstrates a basic understanding
of the problem
• demonstrates a limited
understanding of the problem
• chooses an efficient and effective
strategy
• chooses an appropriate and
workable strategy
• chooses a simplistic and/or routine
strategy
• chooses an inappropriate or
unworkable strategy
Specific Outcome and Process Focus: N3 [R]
Question 6, short answer, written answer
a) Write two prime numbers. How do you know that these are prime numbers?
b) Write two composite numbers. How do you know that these are composite numbers?
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• provides a precise explanation of
prime and composite numbers
• provides a clear and logical
explanation of prime and
composite numbers
• provides a partially clear explanation
of prime and composite numbers
• provides a vague and/or inaccurate
explanation of prime and composite
numbers
Specific Outcome and Process Focus: N3 [R]
Question 7, written answer
• Is there any multiple of 6 that is a prime number? Explain your thinking.
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• draws insightful and logical
conclusions when determining
whether any multiple of 6 is a
prime number
• draws logical conclusions when
determining whether any multiple
of 6 is a prime number
• draws simple, logical conclusions when
determining whether any multiple of a
number is a prime number
• does not draw conclusions when
determining whether any multiple of
a number is a prime number
• makes an insightful generalization
when determining whether any
multiple of a number is a prime
number
• makes a logical generalization
when determining whether any
multiple of a number is a prime
number
• makes a simple generalization when
determining whether any multiple of
a number is a prime number
• is unable to make a generalization
when determining whether any
multiple of a number is a prime
number
Question 8, short answer
• Jennifer divided a number by the prime number 3. Then she divided her result by 3.
Her final answer is 3. What number did she divide by 3?
Specific Outcome and Process Focus: N3 [R]
(Score 1 point for each correct answer for a total out of 1.)
Question 9, short answer
• A number between 40 and 80 is a multiple of 7. Another factor of the number is 9. What is the number?
Specific Outcome and Process Focus: N3 [PS, R]
(Score 1 point for each correct answer for a total out of 1.)
(Continued on next page)
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Assessment of Learning—What to look for in student work
Specific Outcome and Process Focus: N7 [C]
Question 10, written answer
• Holly has a goal to learn 10 new French words each week. She uses integers to show whether she has learned
more or fewer words than her goal. What do you think the integers –3, 0, and 3 represent? Explain.
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• provides a precise and insightful
explanation of the meaning of
positive and negative integers
• provides a clear and logical
explanation of the meaning of
positive and negative integers
• provides a partially clear explanation
of the meaning of positive and
negative integers
• provides a vague and/or inaccurate
explanation of the meaning of
positive and negative integers
Specific Outcome and Process Focus: N7 [V]
Question 11, written answer, model
• How do you know that 5 3? Use a number line.
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• uses insightful visual
representations that verify
whether 5 3
• uses meaningful visual
representations that verify
whether 5 3
• uses simple visual representations
that verify whether 5 3
• uses unclear visual representations
that verify whether 5 3
Specific Outcome and Process Focus: N7 [C, V]
Question 12, short answer, model
• Order these temperatures from coldest to warmest. Show your work.
(Score 1 point for the order of the temperatures for a total out of 1.)
Specific Outcome and Process Focus: N9 [ME]
Question 13, short answer
• Calculate. Use the rules for order of operations.
a) 12 7 4 2
b) (100 50 2 1) 76
(Score 1 point for each correct answer for a total out of 4.)
c) (4 7) 2 12 2
Question 14, short answer, written answer
a) Calculate (2 1) (4 3).
b) How can you use the numbers from 1 to 4 and any operations with brackets to make an expression that equals 2?
d) 6 5 4 2 1
Specific Outcome and Process Focus: N9 [PS]
Work meets standard
of excellence
Work meets standard
of proficiency
Work meets
acceptable standard
Work does not yet meet
acceptable standard
• shows flexibility and insight with
operations and brackets when
solving the problem, adapting if
necessary
• shows thoughtfulness with
operations and brackets when
solving the problem
• shows understanding with operations
and brackets when solving the
problem
• attempts to solve problem
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Chapter Task
A Block Dropping Game
STUDENT BOOK PAGE 105
SPECIFIC OUTCOME
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
Achievement Indicator
• Solve a given problem involving factors or multiples.
Preparation and Planning
Pacing
10–15 min Introduction
30–45 min Using the Task
Materials
• Optional: counters
Masters
• Chapter 3 Task pp. 87–88
• Optional: 1 cm Grid Paper, Masters Booklet
p. 22
Mathematical
Process Focus
PS (Problem Solving), R (Reasoning), and
V (Visualization)
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
Chapter 3 Task pp. 87–88
Optional: 1 cm Grid Paper,
Masters Booklet p. 22
Using the Chapter Task
Use this task as an opportunity to assess students’
understanding of the concepts developed in the chapter
and their ability to apply them in a rich problem-solving
situation. Refer to the assessment chart on page 71 for the
details of each part of the task.
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Chapter Task: A Block Dropping Game
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Introduction (Whole Class) ± 10–15 min
In this video game, the player uses the cursor to grab
rectangular blocks that the computer drops from the top of
the screen in order to form a square on the screen. Students
should be comfortable with factors and multiples before
playing. Ask students how they could determine whether a
number is a factor of another number. You may wish to
activate existing knowledge by having students identify the
factors of a number, such as 36.
Using the Task (Individual) ± 30–45 min
Together, read all the information on Student Book page 105,
including the central question. For Prompts A, C, and D,
encourage students to use words such as factor and multiple in
their explanations. Students may find it helpful to create a
factor rainbow for 12 before working through the prompts.
You may want to provide counters to assist students in
identifying the factors of 12.
Students should work through the task independently.
Remind students to use the Task Checklist as a way to help
them produce an excellent solution. Some students may be
able to work through the task as it is described on the student
page; however, most will benefit from using Chapter 3 Task
pp. 87–88 to plan and record work. As students work
through the task, observe and/or interview individuals to see
how they are interpreting and carrying out the task.
Possible Solutions to Chapter Task
A. For example, I can use six copies of the 2-by-3 block to
form the top row of the 12-by-12 square because 2 is a
factor of 12. Then I can make three more rows like the
first row to have a total of four rows, because 3 is also a
factor of 12.
2
12
The 2-by-5 block cannot be used to make the square
because 5 is not a factor of 12. So the computer cannot
make rows or columns of the 2-by-5 block to fit the
12-by-12 square.
The 1-by-2 block can be used to make the square
because 1 and 2 are both factors of 12. So the computer
can make rows and columns of copies of the 1-by-2
block to fit the 12-by-12 square.
C. 1-by-1, 2-by-2, 3-by-3, 4-by-4, 6-by-6, and 12-by-12;
for example, each side length is a factor of 12. So copies
of the square blocks can be used by the computer to
make the square.
D. For example, my game has rectangular and square blocks
dropping from the screen and two squares: a 15-by-15
square and a 12-by-12 square. The computer can make
either square if both the length and the width of the
blocks are factors of 15 or 12. So a player would choose a
block depending on whether both lengths of the sides are
factors of 15 or factors of 12. For example, a player
would choose 3-by-5 for the 15-by-15 square, and 2-by4 for the 12-by-12 square. A block that is 4-by-5,
however, would not be chosen because both dimensions
are not factors of 15 or 12.
Adapting the Task
3
You can adapt the task in the Student Book to suit the needs
of your students. For example:
• Use Chapter 3 Task pp. 87–88.
• Have students work in pairs or small groups.
• Challenge students to identify blocks that can be used to
fill a rectangular game board, such as a 16-by-24 board,
rather than the square game board.
12
B. For example, the 3-by-4 block can be used to make the
square because 3 and 4 are both factors of 12. So the
computer can make rows and columns of copies of the
3-by-4 block to fit the 12-by-12 square.
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Assessment of Learning—What to look for in student work
Outcome
N3. Demonstrate an
understanding of factors
and multiples by
• determining multiples
and factors of numbers
less than 100
• identifying prime and
composite numbers
• solving problems
involving multiples.
[PS, R, V]
Work meets
standard of
excellence
Work meets
standard of
proficiency
Work meets
acceptable
standard
Work does not yet
meet acceptable
standard
• often draws insightful and
logical conclusions and
recognizes inappropriately
drawn conclusions without
prompting
• comprehensively analyzes
situations and makes
insightful generalizations
• chooses efficient and
effective strategies when
applying knowledge of
multiples and factors
• in many situations, draws
logical conclusions and
recognizes inappropriately
drawn conclusions when
prompted
• completely analyzes
situations and makes logical
generalizations
• chooses workable and
reasonable strategies when
applying knowledge of
multiples and factors
• sometimes draws simple,
logical conclusions and
sometimes recognizes
inappropriately drawn
conclusions when prompted
• superficially analyzes
situations and makes simple
generalizations
• chooses partially appropriate
and workable strategies
when applying knowledge of
multiples and factors
• rarely draws conclusions
from a mathematical
situation and usually does
not recognize inappropriately
drawn conclusions
• is unable to analyze
situations and make
generalizations
• chooses inappropriate and/or
unworkable strategies when
applying knowledge of
multiples and factors
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Chapter Task: A Block Dropping Game
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STUDENT BOOK PAGES
106–107
Chapters 1–3
Cumulative Review
SPECIFIC OUTCOMES
N1. Demonstrate an understanding of place value for
numbers
• greater than one million
• less than one thousandth.
[C, CN, R, T]
Preparation and Planning
Materials
•
•
•
•
Masters
• Table of Values, Chapter 1 p. 63
• Balance Scales, Chapter 1 p. 68
• Place Value Chart to Hundred Millions,
Masters Booklet p.42
• Decimal Place Values Chart to Millionths,
Masters Booklet p.45
Nelson Website
Visit www.nelson.com/mathfocus and follow
the links to Nelson Math Focus 6, Chapter 3.
N2. Solve problems involving large numbers, using
technology.
[ME, PS, T]
N3. Demonstrate an understanding of factors and
multiples by
• determining multiples and factors of numbers less
than 100
• identifying prime and composite numbers
• solving problems involving multiples.
[PS, R, V]
counters
calculator
grid paper
chart paper
Tables of Values, Chapter 1
p. 63
Balance Scales, Chapter 1
p. 68
Place Value Chart to
Hundred Mullions,
Masters Booklet p. 42
Decimal Place Value Chart
to Millionths, Masters
Booklet p. 45
N7. Demonstrate an understanding of integers,
concretely, pictorially and symbolically.
[C, CN, R, V]
N9. Explain and apply the order of operations, excluding
exponents, with and without technology (limited to
whole numbers).
[CN, ME, PS, T]
PR1. Demonstrate an understanding of the relationships
within tables of values to solve problems.
[C, CN, PS, R]
PR3. Represent generalizations arising from number
relationships using equations with letter variables.
[C, CN, PS, R, V]
PR4. Demonstrate and explain the meaning of
preservation of equality concretely, pictorially and
symbolically.
[C, CN, PS, R, V]
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Using the Cumulative Review
The questions on Student Book pages 106–107 provide
practice with multiple-choice questions while reviewing the
concepts developed in Chapters 1 to 3.
Question
Answer
Grade 6 Outcome
Chapter
1
D
PR1
1
2
A
PR3
1
3
C
PR4
1
4
B
N1
2
5
D
N2
2
6
A
N1
2
7
C
N1
2
8
D
N1
2
9
B
N3
3
10
D
N3
3
11
A
N3
3
12
B
N3
3
13
D
N7
3
14
A
N9
3
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Chapter 3
Family Letter
Dear Parent/Caregiver:
Over the next three weeks, your child will be learning about identifying factors and
multiples of numbers and how to determine whether a number is prime or composite.
Your child will also learn how to represent, compare, and order integers, and will
perform a series of calculations using the rules for order of operations. Your child will
have many opportunities to apply knowledge of factors, multiples, and integers in
solving realistic problems.
To reinforce the concepts your child is learning at school, you and your child can
work on some at-home activities such as these:
• Have your child model factors of numbers less than 100 by putting numbers of
items in equal groups. Your child can also calculate multiples of smaller numbers
they encounter, such as the number of snack packages in three or four boxes.
• Your child can measure and record the daily high and low temperatures during the
week and then place the temperatures on a number line. Have your child order the
temperatures from coldest to warmest or warmest to coldest. Your child can also
compare temperatures from different cities.
• Have your child solve any skill-testing questions found on cereal boxes or other
contest entry forms, and have your child explain how he/she applied the rules for
order of operations to arrive at the correct answer.
You may want to visit the Nelson website at www.nelson.com/mathfocus for more
suggestions to help your child learn mathematics and develop a positive attitude
toward learning mathematics. As well, you can check the Nelson website for links to
other websites that provide online tutorials, math problems, brainteasers, and
challenges.
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Name:
Scaffolding for Getting Started
Date:
Page 1
STUDENT BOOK PAGES 68–69
Banner Designs
Daniel is making a banner for Heritage Day. It has 30 squares.
He coloured every second square red to represent one of the four
colours on an Aboriginal medicine wheel.
He drew a symbol to represent an eagle in every third square.
? How can you predict how many coloured squares will
have a symbol on them?
A. Continue Daniel’s banner to 18 squares.
Colour every second square red.
Sketch an eagle symbol in every third square.
B. Circle the red squares that have an eagle.
Why does the pattern 6, 12, 18, … represent the
red squares with an eagle?
How can you use a number pattern to predict the next
red square with an eagle?
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Scaffolding for Getting Started
Page 2
STUDENT BOOK PAGES 68–69
C. Daniel’s banner has 30 squares. Suppose you want to figure out
how many red squares have an eagle. How could you skip count to
figure out the number of red squares with an eagle?
How many squares are there altogether?
Use your answers above to write a multiplication equation you could
use to figure out how many red squares have an eagle.
How many red squares on Daniel’s banner have an eagle symbol?
Explain what you did.
D. Design a banner with a different number of squares on grid paper. Use
one of the symbols below and another colour from the medicine wheel.
bear
drum
horse
fish
How can you predict the number of coloured squares that have a
symbol on them?
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Scaffolding for Lesson 2, Question 3
STUDENT BOOK PAGE 76
3. What is the same about a list of multiples of 3 and 9?
What is different?
• List the multiples of 3 up to 30:
This is the same
as skip counting
by 3s from 3
to 30.
• List the multiples of 9 less than 30:
• How are the two lists of multiples the same?
• How are the two lists of multiples different?
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String Ar t
Curious Math: String Ar t
STUDENT BOOK PAGE 77
4
5
6
2
3
48
47
46
45
44
7
43
8
42
9
41
40
10
39
11
12
38
13
37
14
36
35
15
34
16
33
17
18
32
19
31
20
21
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Chapter 3: Number Relationships
30
22
23
24
25
26
27
28
29
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Mid-Chapter Review⎯Frequently Asked Questions
STUDENT BOOK PAGES 86–87
Q: What are some ways to identify factors?
A:
Q: What are some ways to identify multiples?
A:
Q: How are prime and composite numbers different?
A:
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Four in a Row Game Board
Math Game: Four in a Row
STUDENT BOOK PAGE 101
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
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Calculation Cards
Math Game: Four in a Row
2×8–2×2
2×1×5
(3 + 4 × 3) ÷ 3
1+2×3+8÷2
3+2×2+2
1+2×3
2×3+4–4
4×4–4×2
12÷ 2– 2 × 2
10 – 3 × 3
2 × (1 + 2) – 3
16 – 2 × 6
STUDENT BOOK PAGE 101
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Calculation Cards
Math Game: Four in a Row Page 2
STUDENT BOOK PAGE 101
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Chapter Review⎯Frequently Asked Questions
STUDENT BOOK PAGE 102
Q: How can you represent and compare integers?
A:
Q: What are the rules for order of operations?
A:
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Chapter 3 Test
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Page 1
1. How do these arrays show the factors of 18?
2. Which number from 20 to 30 has exactly three factors?
Explain how you identified the factors.
3. Natalie listed these factors of 72: 1, 2, 6, 9, 10, 12, 18, and 36.
a) Which number listed is not a factor of 72?
b) Which factors are missing?
4. List the first five multiples of each number.
a) 8
b) 4
c) 12
d) 15
5. The Winter Olympics were held in 2006. If the Winter Olympics are
held every four years, will they be held in 2044? Explain your thinking.
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6. a) Write one prime number. How do you know that this is a
prime number?
b) Write three composite numbers. How do you know that these
are composite numbers?
7. Andrew says that if a number is even, it is not a prime number.
Is Andrew correct? Explain your thinking.
8. Garret divided the number 30 by a prime number.
Then he divided his answer by another prime number.
His answer is 3. What prime numbers did he divide by?
30
3
9. A number between 40 and 50 is a multiple of 8.
12 is also a factor of that number.
What is the number?
10. Sam used integers to compare three math marks to his first
math mark in October. What do you think the integers –5, 0, and
5 represent? Explain.
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Chapter 3 Test
Page 3
11. How do you know that –2 > –6? Use a number line.
12. Elements melt and freeze at different temperatures. Order these
temperatures from coldest to warmest. Show your work.
Melting Points of Elements
Melting Point
of Chlorine
Melting Point
of Helium
Melting Point
of Salt
Melting Point
of Silver
Melting Point
of Mercury
–101 ⬚C
⫺272 ⬚C
⫹98 ⬚C
⫹961 ⬚C
–38.72 ⬚C
13. Calculate. Use the rules for order of operations.
a) 20 ⫺ 3 ⫻ 5 ⫹ 6
c) (15 ⫺ 6) ⫼ 3 ⫹ 4 ⫻ 2
b) 45 ⫼ (7 ⫹ 8) ⫻ 2
d) 24 ⫼ 6 ⫹ 5 ⫻ 3 ⫹ 1
14. a) Calculate 2 ⫼ 2 ⫻ (2 ⫹ 2).
b) How can you use four 2s and any operations plus
brackets to make an expression that equals 5?
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Chapter 3 Task
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A Block Dropping Game
STUDENT BOOK PAGE 105
In a video game, blocks shaped like rectangles drop from
the top of the screen. You grab blocks that you think could
form a square. The computer copies the blocks you grab
and tries to make the square.
Task Checklist
K Did you use factors
or multiples to help
solve the problem?
K Did you check your
calculations?
K Did you include
diagrams?
K Did you explain your
thinking clearly?
? Which blocks should you grab to make the square?
Read the Task Checklist above before you begin.
A. How do you know copies of the 2-by-3 block can be used
to make the 12-by-12 square? Use a diagram to explain.
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B. Which of these blocks can be used to make
the square?
• 3-by-4
• 2-by-5
• 1-by-2
C. Suppose that square blocks drop from the top of
the screen. Which blocks would you grab? Explain.
D. Design a similar video game with rectangular blocks and
square blocks dropping from the top of the screen. How
can a player decide which block to grab?
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Answers for Chapter 3 Masters
Scaffolding for Getting Star ted pp. 75–76
A.
E
E
E
E
E
E
E
E
E
E
E
E
B.
For example, I saw the pattern 6, 12, 18,… Every 6th square is red with an
eagle, so the next square that should be red with an eagle is the 24th square.
C. 6, 12, 18, 24, 30
30 squares altogether
⫻ 6 ⫽ 30
Five red squares have an eagle.
For example, there are 30 squares and every 6th square is red with an eagle.
I divided to find the number of red squares with an eagle: 30 ⫼ 6 ⫽ 5.
D. For example, I’ll create a banner with 100 squares. I’ll colour every second square yellow.
Every fifth square will have the symbol for a horse. I’ll figure out how many yellow squares
will have a horse.
horse
horse
horse
Every 10th square is yellow with a horse symbol. There are 100 squares, so I divided
100 by 10: 100 ⫼ 10 ⫽ 10, so there will be 10 squares that are yellow with a horse symbol.
Scaffolding for Lesson 2, Question 3, p. 77
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
9, 18, 27
The numbers in both lists are multiples of 3.
Sum of the numbers in both lists are multiples of 9, but sum in the list of multiples of 3
are not multiple of 9
Chapter 3 Test pp. 84–86
1. Each array has 18 circles, and each number of rows and columns represents a factor.
The arrays show 1 ⫻ 18, 2 ⫻ 9, 3 ⫻ 6.
2. 25. For example, I knew 23 and 29 had only two factors because they are prime.
So I used mental math to identify the factors of 20, 21, 22, 24, 25, 26, 27, 28, and 30.
20 has factors 1, 2, 4, 5, 10, and 20: 6 factors
21 has factors 1, 3, 7, and 21: 4 factors
22 has factors 1, 2, 11 and 22: 4 factors
24 has factors 1, 2, 3, 4, 6, 8, 12, and 24: 8 factors
25 has factors 1, 5, and 25: 3 factors
26 has factors 1, 2, 13, and 26: 4 factors
27 has factors 1, 3, 9, and 27: 4 factors
28 has factors 1, 2, 4, 7, 14, and 28: 6 factors.
30 has factors 1, 2, 3, 5, 6, 10, 15, and 30: 8 factors
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3. a) 10
b) 3, 4
4. a) 8, 16, 24, 32, 40
b) 4, 8, 12, 16, 20
c) 12, 24, 36, 48, 60
d) 15, 30, 45, 60, 75
5. No; The Winter Olympics were held in 2006, so they will also be held
in 2010, 2014, 2018, 2022, 2026, 2030, 2034, 2038, and 2042.
They will not be held in 2044.
6. a) For example, 11 is prime because it has only 2 different factors, 1 and itself.
b) For example, 10, 12, and 14 because each has more than 2 different factors.
7. Andrew is not correct. 2 is even and is a prime number.
8. 2 and 5; for example, if he started with 30 and ended with 3, he must have
5 30
divided by prime numbers that multiply to 10. The only prime numbers that
multiply to 10 are 2 and 5.
2
6
9. 48; for example, I wrote a list of the multiples of 8 to determine the multiples
3
between 40 and 50:
8, 16, 24, 32, 40, 48
The only number between 40 and 50 that is also a multiple of 12 is 48.
10. For example, –5 means the test score is 5 less than his first mark; 0 means
the test score is the same as his first mark; 15 means the test score is 5 more
than his first mark.
11. ⫺2 is to the right of ⫺6 on the number line, so ⫺2 > 26.
–6
–2
–10
0
12. For example, I recorded the temperatures on a number line.
–38.72 °C
–272 °C
Helium
–101 °C
0 °C +98°
Chlorine
Salt
Mercury
+961 °C
Silver
From coldest to warmest, the temperatures are ⫺272 ⬚C, ⫺101 ⬚C, ⫺38.72 ⬚C, ⫹98 ⬚C, ⫹961 ⬚C.
13. a) 20 ⫺ 3 ⫻ 5 ⫹ 6
⫽ 20 ⫺ 15 ⫹ 6
⫽5⫹6
⫽ 11
b) 45 ⫼ (7 ⫹ 8) ⫻ 2
⫽ 45 ⫼ 15 ⫻ 2
⫽3⫻2
⫽6
c) (15 ⫺ 6) ⫼ 3 ⫹ 4 ⫻ 2
⫽9⫼3⫹4⫻2
⫽3⫹4⫻2
⫽3⫹8
⫽ 11
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