Number 1 Arithmetic
Mathematical Ideas:
Number 1 provides a revision and extension of arithmetic techniques from the primary school, and an introduction to some of the basic concepts of number theory.
Because students/classes vary in their background, the initial content may have to be varied by the teacher, and teachers should use their own judgement about how much
addition/subtraction/multiplication facts need to be covered. They should be mindful though, that many students leave primary school proficient in arithmetic, and not all
students need to recover the basics.
The scheme emphasises multiplication as an appropriate starting point, because of 1) the difficulty that many students face with multiplication techniques and 2) its
applications in high school algebra (expanding brackets etc). Emphasis is also placed on building up mental pictures rather than on the use of a particular algorithm.
Students should have at hand a variety of techniques for solving arithmetic problems, and be encouraged ALWAYS to check the reasonableness of their answers.
The Number Theory covered (factors, multiples and primes) relies on students both understanding concepts AND learning vocabulary. For less able students, it provides
many opportunities to practice the basic number links (eg multiplication/division) needed to perform arithmetic.
Many students arrive in High School with a collection of misconceptions concerning place value. Some believe that the longer the decimal, the bigger it is, some the
contrary (Stacey, K. & Steinle, V. (1998)). The initial lessons on decimals use the conflicts between these various misconceptions to allow students to build a more accurate
visualization of decimal numbers, which is enhanced by the use of (virtual) manipulatives. Decimal arithmetic (particularly addition) is used to build this mental picture.
As with decimals, it is important that students have a clear mental picture of directed numbers. Two models can be suggested; one. The one used in this scheme uses the
number line and builds on students’ familiar experiences of ordering directed numbers on a line (eg a thermometer). The second model relies on the idea of cancelling +/quantities (the “IOU” model using money), and can be introduced if students have problems with the first model. Multiplication is initially seen as repeated addition; most
students seem to accept multiplying two negatives as a way of completing the pattern in the times-table.
Ideas covered:
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Arithmetic. Teachers may need to supplement the basic course which emphasises multiplication
Number Theory (factors/multiples/primes)
Order of operations (BEDMAS)
Powers and roots
Decimal Numbers and Place Value
Decimal arithmetic
Directed Numbers (including arithmetic with directed numbers)
Curriculum Coverage:
NC
Arithmetic
Number Theory
(factors/multiples/primes)
2
Demonstrate the ability to
use the multiplication facts;
write and solve story
problems which involve
whole numbers, using
addition, subtraction,
multiplication, or division;
write and solve story
problems which require a
choice of any combination
of the four arithmetic
operations.
Recall the basic addition
and subtraction facts;
mentally perform
calculations involving
addition and subtraction;
demonstrate the ability to
use the multiplication facts;
write and solve story
problems which involve
whole numbers, using
addition, subtraction,
multiplication, or division;
write and solve story
problems which require a
choice of any combination
of the four arithmetic
operations.
[Suggested Learning
Experience: exploring number
patterns showing multiples.]
3
Order of operations
(BEDMAS)
Powers and roots
Decimal Numbers and
Place Value. Decimal
Arithmetic.
Read any 3-digit whole
number; explain the
meaning of the digits in 2or 3-digit whole numbers;
order any set of three or
more whole numbers (up to
99).
Demonstrate knowledge
of the conventions for
order of operations.
Explain the meaning of the
digits in any whole number;
explain the meaning of the
digits in decimal numbers
with up to 3 decimal places;
order decimals with up to 3
decimal places.
Directed Numbers
4
5
6
Make sensible estimates
and check the
reasonableness of answers;
explain satisfactory
algorithms for addition,
subtraction, and
multiplication
[Suggested Learning
Experience: exploring factors
of numbers by investigating
rectangular (composite)
numbers and
line (prime) numbers]
explain the meaning and
evaluate powers of whole
numbers;
[Suggested Learning
Experience: Students
investigate the representation
of any integer as a product of
primes, and show how to use
this property in determining
greatest common divisors, and
lowest common multiples].
express the values of
square roots in
approximate and exact
forms.
[Suggested Learning
Experience: Investigate
surds and other irrational
numbers]
Explain the meaning of
negative numbers;.
negative numbers, using
practical activities or
models if needed.
Round numbers sensibly.
Solve problems involving
positive and
Pedagogy:
Because of the place of this topic at the start of Year 9, it is especially important for the teacher to build a learning community in their classroom, giving opportunities for
pair and group work. Groups should be structured by the teacher for various purposes, rather than relying on friendship groupings. The department website gives some
suggestions for grouping. The first lesson introduces the students to each other, and gives the teacher a chance to get to know them, and something of their mathematical
experience.
Alongside of the lesson activities are both an enhancement and an extension course. Both should be used by class teachers. It is suggested that teachers put aside one
lesson a week to work on enhancement activities, that free students from the demands of assessment, allow them to “take risks”, and introduce them to a variety of
“maths rich” environments.
Focus is put on developing a “can do” attitude (Managing Self), and students are given strategies for dealing with situations when they are stuck. A variety of problem
situations are introduced, and students are encouraged to believe that when they are stuck they can, maybe with scaffolding, at least make progress, even if they can’t go
onto solve the problem. The scheme provides two such activities (“Sports Hall” and “Towers”).
The department are beginning to integrate the use of virtual manipulatives into their teaching, as far as the available technology allows. These “objects” allow students to
build up concrete mental pictures of some of the basic objects and concepts of mathematics, before they begin to abstract them into algebra later in the year.
Reading:
The incidence of misconceptions of decimal notation amongst students in Grades 5 to 10
Stacey, K. & Steinle, V. (1998)
NUMBER 1 LESSONS
Lesson 1
Support
Main
Lesson2
Lesson 3-4
1.
2.
Go through Course Booklet, equipment, HW expectations etc.
What is Maths? MindMap. Whole Class. Extract: Number, Statistics, Geometry, Algebra,
Measurement.
3. The answer is 10. Put the title “The answer is 10” in the centre of the page. Divide the page
(roughly) into 5 sectors (Number, Statistics etc), and in each one give examples of questions
with answer of 10.
Think (silence->Pair (random)->Share (2 pairs). Produce poster as a group, taking a response in turn
from each person in the group. Each person may pass.
LO:
Expectations for the year
Learning names
Participation
Modelling group work for future use
Assessing prior knowledge
For Support students, this should be seen as times
table practice, or multiplication of powers of ten.
LO: Estimate the answer to a multiplication
problem
2.
3.
SC:
Every person in the room contributed an idea in class, pair and group situations.
SC: Can estimate 23x47 to 20x50=1000
1. Two digit by two digit multiplication.
Introduce concept of estimating by rough
(informal) rounding, eg 13x47 is approx.
10x50=500.
Insist that approx. is used over coming lessons.
Ask for a range of methods of multiplying (ad hoc
and algorithm). See attachment. Spend 2 lessons
on algorithms – emphasise “box method”.
“Closest Product Game”: approx. and checking
skills (“last digit” check).
LO:
Perform multiplication using a range of different
algorithms
Estimate the answer to a multiplication problem
SC: Can estimate, and calculate, without a
calculator 23x47
As for Main, but extend with Near Misses 1, Near
Misses 2 and Near Misses 3.
Extension
Effective teachers provide students with opportunities to work both independently and collaboratively
to make sense of ideas.
Support
Main
Extension
Lesson 5-6
Lesson 7
Lesson 8-9
“Sports Hall” worksheet
“I’m Stuck” Cards and poster
Factor trees. Introduce “prime” as the end points.
Express a number as a product of prime factors.
Factor trees continued.
LO: 1) uses multiplication to solve a problem
2) uses “resilience” in solving a problem (managing
self)
Pairs (random)
Hand out sheet. Read (individually). Brainstorm
“what to do when stuck”. Hand out cards.
Separate into two piles – useful in this problem.
Not useful in this problem. Order the useful ones.
LO:
Understands “factor”/”factor
pair”/”prime”/”product of primes factors”.
LO: Understands “factor”/”factor pair”. Can draw a
factor tree.
Using factor trees to derive highest common
factors. “Highest Common Factor” worksheet.
Definition: A prime is a number with exactly two
factors.
LO: Can use a factor tree to derive the highest
common factor of two (or more) numbers.
Lesson ender: “Fizz-Buzz” Game
Lesson ender: “Fizz-Buzz” Game
Start problem as whole class.
As for main, but product of prime factors in index
form.
Managing Self
Support
Main
Lesson 10
Lesson 11
Lesson 12
Primes. Understand the concept of a prime
number. Recognise primes up to 100. Primes are
learnt by experience of “playing” with primes.
Discuss primes as the “ends” of factor trees.
Primes are numbers with two factors. Few
examples and then sieve. Then Pirate Maze (using
sieve).
Factors Multiples and Primes.
Factors Multiples and Primes.
Review vocab.: factor, factor pair, multiple,
prime, prime factor, common factor, HCF,
common multiple, LCM. Introduce square
number.
Revise vocab.: factor, factor pair, multiple, prime,
prime factor, common factor, HCF, common
multiple, LCM, square number.
LO: Understand “prime number”
Eratosthenes’s Sieve (needs 100 grid).
Pirate Prime Maze from 10Ticks.
SC: Can explain why 17 (for example) is prime.
LO: Learns the meaning of, and distinguishes
between “factor”, “multiple” and “prime”
“Factors and Multiples game’. Talk about setting
targets; if we can’t solve the problem, at least we
can improve our score. “Starting from Scratch” is
another possible problem solving Strategy.
LO: Learns the meaning of, and distinguishes
between “factor”, “multiple” and “prime”
Complete a poster showing the vocab, with
examples.
Run “prime factorization” animation.
Worksheet: “poster”
Extension
Lesson ender: “Fizz-Buzz” Game
As above, plus:
Is 221 prime? Explain.
What is the highest prime you can find?
(use prime tester.lgo to test).
SC: Can name 3 factors of 12, and 3 multiples of
12. Can identify primes up to at least 20.
Extension: Effective teachers are able to facilitate
classroom dialogue that is focused on
mathematical argumentation.
Managing Self. Effective teachers shape
mathematical language by modelling appropriate
terms and communicating their meaning in ways
that students understand.
SC: can give an example of each of the words in the
above list.
Support
Main
Lesson 13-14
Lesson 15
Lesson 16
“Towers” problem
Powers and Roots
BEDMAS
LO: Understands “powers of 2”, “powers of 3 ” etc
and associated notation. Understands “power”,
“exponent” and “base”. Can calculate a power on
a calculator.
LO: Can calculate any number to a power, using a
calculator.
LO: Can use BEDMAS
Alpha text book Chapter 5, Powers and roots.
Learn Alberta BEDMAS manipulative.
Alpha text book chapter 6
Four Fours Problem
(Also Decimals Diagnostic as a pretest for next
lesson)
SC: Can calculate 25 and (-3)4 on a calculator.
Extension group can calculate √(5041) on a
calculator, and (√2)2 without a calculator.
SC: Can calculate (without a calculator) 2+3x5 using
the BEDMAS convention.
“The Towers” worksheet.
Use “what to do when stuck” cards.
Summarize with vocabulary and the use of a
calculator.
SC: can read 24, and calculate it on a calculator.
Understands that it means 2x2x2x2
Roots (section 5.8) is extension only.
Extension
Managing Self
Effective teachers carefully select tools and
representations to provide support for
students' thinking.
Support
Main
Lesson 17-18
Lesson 19
Lesson 20-21
Ordering Decimal Numbers.
Decimals/Place Value
Adding and subtracting decimals.
LO: Can order ANY group of decimal numbers.
LO: Can order ANY group of decimal numbers.
LO: Can add or subtract two decimal numbers.
Group the class by misconception. Attempt to
have a “Longer-Larger”, “Shorter-Larger” and
“Task-Expert” in each group, if possible. Allow
each group 5 minutes to discuss and then mark
each of the homeworks, and then discuss as a
class.
Use decimals grid to discuss the values of a
decimal number. (unit square, tenths, hundredths
etc).
Use decimals grid and Decimals Blocks from the
National Library of Virtual Manipulatives.
10Ticks L4P4 p.29-33 Ordering decimal numbers.
Repeat diagnostic
Make sure that students have a clear picture before
allowing the algorithm.
SC: Each group can nominate a person who can
explain why eg 0.56 is bigger than 0.496 and
0.437 is smaller than 0.6
SC: Whole class “task expert”
Then some text book exercises on adding and
subtracting decimals.
SC: class can add eg 1.45 and 1.36
Multiplying and dividing decimals (emphasise the
use of estimation to check reasonableness).
Extension
Effective teachers are able to facilitate classroom
dialogue that is focused on mathematical
argumentation.
Effective teachers carefully select tools and
representations to provide support for students'
thinking.
Support
Main
Lesson 22-24
Lesson 25
Lesson 26-27
Directed Numbers
Arithmetic with directed numbers
Revision
LO: Can order a list of directed numbers
Can add and subtract negative numbers
LO: Can multiply and divide using directed
numbers
Can use a calculator to perform arithmetic
with directed numbers
LO: Can develop a plan to revise for an assessment.
Introduction of negative numbers, using a
temperature/thermometer model (half a lesson).
Learn Alberta has a short introductory video.
‘Thermometer” is also useful.
“Walk the Line”. The physical/concrete aspect of
this lesson is enormously important, and should be
referred to often.
Use the “Multiplication table”, and complete the
three quadrants (+/+, +/- and -/+) using repeated
addition. Ask students to “guess’ the -/- quadrant,
or (ultimate proof!) try it on their calculator.
Discuss division also.
Discuss (using Year 9 Revision section on website)
how to revise, and sources of help and questions.
Very few students will have revised for anything at
primary school. Emphasise the need to do lots of
questions, and allow access to lots of questions.
SC: By the end of lesson 26, ALL students can name
the topic/topics they intend to revise in lesson 27,
and how they plan to revise.
p.58-60 in Alpha text book
When students are fairly confident with “walking
the number line”, then text book type exercises
can be introduced. Eg 10Ticks L5P4 p3-6
SC: class can calculate (both mentally and on
calculator) eg 3x-4, -4x-5, 50/-5.
SC: class can calculate eg 4-7, 4+-6, -3+-4, 2--8
Extension
Students who have a good grasp of the concepts
of directed numbers should still complete some
text book exercises, but can then try the more
“puzzle like” activities in 10Ticks L5P4 p7-10
Effective teachers carefully select tools and
representations to provide support for
students' thinking.
Managing Self