Mathematics
Grade 4 – Unit 1 (SAMPLE)
Unit 1: Multiples and Factors
Possible time frame:
10 days
In this unit students develop understanding of multiples and factors, applying their understanding of multiplication from the previous year. This understanding lays a strong
foundation for students to develop, discuss, and use efficient, and accurate, computational strategies involving multi-digit numbers. These concepts and the terms “prime” and
“composite” are new to Grade 4, so they are introduced early in the year to give students ample time to develop and apply understanding.
Supporting Cluster Standards
Gain familiarity with factors and multiples.
4.OA.B.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors.
Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole
number in the range 1-100 is prime or composite.
Standards Clarification
For assessment purposes, one
or both factors should be
greater than 5.
Additional Cluster Standards
Standards Clarification
Generate and analyze patterns.
4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the
rule itself. For example, given the rule, “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the
terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
While working on 4.AO.C.5,
students use manipulatives to
determine whether a number
is prime or composite.
Although there are shape
patterns in arrays, the focus of
this unit is number patterns.
4.OA.C.5 is repeated in Unit 13,
where the focus will be on
identifying shape patterns.
Focus Standards for Mathematical Practice
MP.3 Construct viable arguments and
critique the reasoning of others.
MP.7 Look for and make use of
structure.
The focus of this unit is to use student’s understanding of the concept and language to discuss the structure of multiples and factors.
(MP.3, MP.7)
Review the Grade 4 sample year-long scope and sequence associated with this unit plan.
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
What will students know and be able to do by the end of this unit?
Students will demonstrate an understanding of the unit focus and meet the expectations of the Common Core State Standards on the unit assessments.
Standards
Unit Assessment
Objectives and
Formative Tasks
The standards for this unit
include:
Students will demonstrate
mastery of the content
through assessment items
and tasks requiring:
Objectives and tasks
aligned to the CCSS prepare
students to meet the
expectations of the unit
assessments.
4.OA.B.4 Find all factor pairs
for a whole number in the
range 1-100. Recognize that a
whole number is a multiple of
each of its factors. Determine
whether a given whole
number in the range 1-100 is a
multiple of a given one-digit
number. Determine whether a
given whole number in the
range 1-100 is prime or
composite.
•
•
•
•
Conceptual
Understanding
Procedural Skill and
Fluency
Application
Math Practices
Concepts and Skills
Each objective is broken
down into the key concepts
and skills students should
learn in order to master
objectives.
4.OA.C.5 Generate a number
or shape pattern that follows a
given rule. Identify apparent
features of the pattern that
were not explicit in the rule
itself.
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
Sample End-of-unit Assessment Items:
1)
What factor of 12 is missing in the list of numbers?
1, 2, 3, 4, _____, 12
2)
3)
On the chart, circle all the numbers that have 4 as a factor.
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Which of these numbers is a prime number?
A.
4)
17
6
B.
27
C.
67
D.
81
The pattern below is found by adding 5 to the first term to get the second term. Then subtract 2
from the second term to get the third term. The pattern of adding 5, then subtracting 2 to get
the next two terms continues. Write the next two numbers in the number pattern.
1
6
4
9
7
12
10
_____ _____
Explain why after the first term of 1, the pattern alternates with 2 evens then 2 odds.
5)
The table below shows some number pairs. The following rule was used to find each number in
column B.
Rule: Multiply the number in Column A by itself and then add 3. Find the missing number, using
the same rule.
A
2
3
5
8
6)
B
7
12
28
?
Which number is both a factor of 100 and a multiple of 5?
A.
4
B.
40
C.
50
D. 80
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
7)
Peter made the statement below.
“The number 32 is a multiple of 8. That means all of the factors of 8 are also factors of 32.”
Is Peter’s statement correct? Why?
8)
Draw a model to show the factor pairs of 18. Is 18 a prime number or a composite number?
How do you know?
9)
Draw a model to show the factor pairs of 5. Is 5 a prime number or a composite number? How
do you know?
10)
Randi bought some tickets to win a bicycle. The tickets are listed below. The winning ticket
number was described as being a multiple of 2 and having the most factors. What was the
winning ticket number? How many factors are there? Explain your reasoning.
18
48
17
64
4
Mathematics
Grade 4 – Unit 1 (SAMPLE)
Sample End of Unit Assessment Task:
I.
The two-eyed space creatures, three-eyed space creatures, and four-eyed space creatures are
having a contest to create a group with 24 total eyes.
a.
How many two-eyed creatures are needed to make a group with 24 total eyes? How
many three-eyed creatures needed to make a group of 24 total eyes? How many foureyed creatures are needed to make a group with 24 total eyes? Complete the chart
below.
Creature Groups
Two-eyed
Three-eyed
Four-eyed
Groups Made
b.
The creatures decided to have a contest to create a group with 40 total eyes. Only
groups that can form a group of with 40 eyes can participate. Can all three groups listed
in part a participate in this contest? Tell how you know.
c.
If other creature groups decided to join in the contest with the 40 total eyes, which of
the following creature groups could participate: one-eyed creatures, five-eyed
creatures, seven-eyed creatures, or eight-eyed creatures? Explain your thinking.
d.
The three-eyed creatures tell the six-eyed creatures that they cannot participate in the
contest in part c. Is this true? Why, or why not? Please explain your thinking.
Adapted from: “The Contest” Item #43081-43804
http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
II.
The table below shows a list of numbers. For every number listed in the table, multiply it by 2
then add 1.
a.
Record the results of your calculations in the right column of the table.
Number
0
1
2
3
4
5
10
Multiply by 2 then add one
b.
What do you notice about the numbers you entered into the table?
c.
Sherri decided to apply the rule, multiply by 2, and add 1, to the numbers 6—9. Below is
the table she created. Sherri noticed that all the numbers she entered are odd. Explain
why all of the numbers are odd.
Number
6
7
8
9
Double the number plus one
13
15
17
21
Adapted from: http://www.illustrativemathematics.org/illustrations/487
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
Sample End-of-unit Assessment Item Responses:
1)
What factor of 12 is missing in the list of numbers? (4.OA.C.5)
Solution: 6
1, 2, 3, 4, _6_, 12
2)
On the chart, circle all the numbers that have 4 as a factor. (4.OA.B.4)
3)
4)
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Which of these numbers is a prime number? (4.OA.B.4)
Solution: C, 67
A. 6
B. 27
C. 67
D.
81
The pattern below is found by adding 5 to the first term to get the second term. Then subtract 2
from the second term to get the third term. The pattern of adding 5, then subtracting 2 to get
the next two terms continues. Write the next two numbers in the number pattern. (4.OA.C.5)
1
6
4
9
7
12
10
__15__
__13___
Explain why after the first term of 1, the pattern alternates with 2 evens then 2 odds.
Solution: The pattern alternates with two evens and two odds because subtracting two even
numbers will always be even, subtracting odd and even numbers will always be odd, adding two
odds will always be even, and adding an even and an odd number will always be odd. Therefore,
adding 5 to 1, an odd number, results in an even number. Then subtracting 2 will result in
another even number. Next adding 5 to the even number will result in an odd number. Since
subtracting 2 from an odd number will result in another odd number, the next number in the
pattern is odd. This pattern of two evens followed by two odd numbers will continue after the
number 1.
5)
The table below shows some number pairs. The following rule was used to find each number in
column B. (4.OA.C.5)
Rule: Multiply the number in Column A by itself and then add 3. Find the missing number, using
the same rule.
Solution: 67
A
B
2
7
3
12
5
28
8
67
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
6)
7)
Which number is both a factor of 100 and a multiple of 5? (4.OA.B.4)
Solution: C, 50
A. 4
B. 40
C. 50
D. 500
Peter made the statement below. (4.OA.B.4)
“The number 32 is a multiple of 8. That means all of the factors of 8 are also factors of 32.”
Is Peter’s statement correct? Why or Why not?
Solution: Yes, his statement is correct. The factors of 8 are 1, 2, 4, and 8. 32 = 8 x 4 and 8 = 2 x 4
that means 32 = (2 x 4) x 4. Also, 1 is a factor of all numbers. So, all factors of 8 are also factors
of 32.
Students might also say “The factors of 8 are 1, 2, 4, and 8. 1, 2, 4, and 8 all divide 32 without a
remainder, so they are all factors of 32.” This is also an acceptable response.
8)
Draw a model to show the factor pairs of 18. Is 18 a prime number or a composite number?
How do you know? (4.OA.B.4)
Solution: Students should include models to show the following factor pairs (1x18, 2x9, and 3x6)
18 is a composite number because 18 has more than one factor pair. Students do not need to
provide all six models below—one model for each factor pair should be included.
9)
Draw a model to show the factor pairs of 5. Is 5 a prime number or a composite number? How
do you know? (4.OA.B.4)
Solution: Students should only show the model of 5x1. 5 is a prime number because 5 has
exactly two factors, or one factor pair, one and itself. Students only need to show one of the
models below.
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
10)
Randi bought some tickets to win a bicycle. The tickets are listed below. The winning ticket
number was described as being a multiple of 2 and having the most factors. What was the
winning ticket number? How many factors are there? Explain your reasoning. (4.OA.B.4)
Solution: Even numbers are multiples of two. Therefore, the ticket numbers that are multiples of
2 are 18, 48, and 64.
Ticket number 18 has factors as follows: 1, 2, 3, 6, 9, and 18. Ticket number 48 has factors as
follows: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. Ticket number 64 has factors as follows: 1, 2, 4, 8,
16, 32 and 64.
The winning ticket number is 48. Ticket 48 is a multiple of 2 and ticket number 48 has the most
factors which is 10 factors.
18
48
17
64
9
Mathematics
Grade 4 – Unit 1 (SAMPLE)
Sample End of Unit Assessment Task Response: (4.OA.B.4)
I.
The two-eyed space creatures, three-eyed space creatures, and four-eyed space creatures are
having a contest to create a group with 24 total eyes.
a.
How many two-eyed creatures are needed to make a group with 24 total eyes? How
many three-eyed creatures needed to make a group of 24 total eyes? How many foureyed creatures are needed to make a group with 24 total eyes? Complete the chart
below.
Creature Groups
Two-eyed
Three-eyed
Four-eyed
Groups Made
12
8
6
b.
The creatures decided to have a contest to create a group with 40 total eyes. Only
groups that can form a group of with 40 eyes can participate. Can all three groups listed
in part a participate in this contest? Tell how you know.
Solution: No, the three-eyed creatures could not have a group to make a group with 40
total eyes. Three-eyed groups would only be multiples of 3 (for example, 3, 6, 9, 12, 15,
18, 21, 24, 27, 30, 33, 36, 39, 42, 45, etc…). They could have a group with a total of 39 or
42.
c.
If other creature groups decided to join in the contest with the 40 total eyes, which of
the following creature groups could participate: one-eyed creatures, five-eyed
creatures, seven-eyed creatures, or eight-eyed creatures? Explain your thinking.
Solution: The one-eyed, five-eyed and eight-eyed can participate because they are
factors of 40. The seven-eyed could not participate because 7 is not a factor of 40.
d.
The three-eyed creatures tell the six-eyed creatures that they cannot participate in the
contest in part c. Is this true? Why, or why not? Please explain your thinking.
Solution: Yes, it is true. The six-eyed groups would only be multiples of 6 (for example,
6, 12, 18, 24, 30, 36, 42, etc…). They could have a group with a total of 36 or 42.
10
Mathematics
Grade 4 – Unit 1 (SAMPLE)
II.
The table below shows a list of numbers. For every number listed in the table, multiply it by 2
then add 1. (4.OA.B.4)
a.
Record the results of your calculations in the right column of the table.
Number
0
1
2
3
4
5
10
Multiply by 2 then add one
1
3
5
7
9
11
21
b.
What do you notice about the numbers you entered into the table?
Solution: There are several patterns that students might see. For example, they might
notice that the result increases by two when the number increases by one. Hopefully,
some students will notice that all the numbers they entered are odd.
c.
Sherri decided to apply the rule, multiply by 2, and add 1, to the numbers 6—9. Below is
the table she created. Sherri noticed that all the numbers she entered are odd. Explain
why all of the numbers are odd.
Solution: The values in the right column are all odd because multiplying an even or an
odd number by 2 always results in an even number. An even number plus 1 will always
result in an add number.
Number
6
7
8
9
Double the number plus one
13
15
17
21
11
Mathematics
Grade 4 – Unit 1 (SAMPLE)
Possible Pacing and Sequence of Standards
Content and Practice Standards
Gain familiarity with factors and multiples.
4.OA.B.4 Find all factor pairs for a whole number in
the range 1-100. Recognize that a whole number is
a multiple of each of its factors. Determine whether
a given whole number in the range 1-100 is a
multiple of a given one-digit number. Determine
whether a given whole number in the range 1-100 is
prime or composite.
Generate and analyze patterns.
4.OA.C.5 Generate a number or shape pattern that
follows a given rule. Identify apparent features of
the pattern that were not explicit in the rule itself.
For example, given the rule, “Add 3” and the starting
number 1, generate terms in the resulting sequence
and observe that the terms appear to alternate
between odd and even numbers. Explain informally
why the numbers will continue to alternate in this
way.
Possible Connections to Standards for
Mathematical Practices
MP.1 Make sense of problems and persevere in
solving them.
Students will need to make sense of the application
task and persevere to determine how many hot dogs
and hot dog buns to buy to feed a given number of
Possible Pacing and Sequence
Days 1-5
Objectives:
Students will list all of the factor pairs for any whole number in the range 1—100.
Students will determine multiples of a given whole number (1—100).
Students will investigate to determine if a number is prime or composite.
Concepts and Skills:
• Understand the terms “factors” and “multiples.”
• Find factors of a given whole number using different strategies (making rectangles with colored
tiles or grid paper, and number charts).
• Use different strategies to show and explain why a number is prime or composite
• Understand that 1 is neither prime nor composite.
• Use skip counting by factors to determine multiples.
• Explain whether a given whole number is a multiple of a given one-digit number.
Sample Tasks:
1) Use the list of numbers below to answer the questions.
8, 9, 19, 27, 31, 33, 56
a. Which numbers are multiples of 2? Show how you know.
b. Which numbers are multiples of 3? Show how you know.
c. Which numbers are multiples of 5? Show how you know.
2) Use models to show all factor pairs for 47. Is this number prime or composite? How do you know?
12
Mathematics
Grade 4 – Unit 1 (SAMPLE)
people. Students continually evaluate their
reasoning by asking, “Does this make sense?”
MP.2 Reason abstractly and quantitatively.
In order for students to determine whether answers
are reasonable, students have to think about their
answer in the given context which requires
quantitative reasoning.
Day 6
Objectives:
Students apply the understandings of factors and multiples to solve a real-world problem.
Application Task Description:
This task uses student understandings of multiples to determine how much will need to be purchased for
a class picnic. While some of the problems may seem like students are asked to find the least common
multiple (LCM) or greatest common factor (GCF), these skills are not required and should not be taught
at this time. Those concepts are reserved for grade 6.
Days 7-8
Objectives:
Students will generate a pattern that follows a given rule.
MP.3 Construct viable arguments and critique the
reasoning of others.
Students will make logical arguments and respond to
the mathematical thinking of others when discussing
their work. Students may use objects or drawings to Students will identify and explain additional patterns or special behaviors in a pattern that go beyond the
construct arguments while critiquing the reasoning
given rule.
of others.
Concepts and Skills:
MP.4 Model with mathematics.
• Create and extend number patterns that repeat or grow.
Students will represent factor pairs in various ways
• Explain why other patterns that are not stated in the given rule (i.e., why all resulting numbers
including building rectangular arrays.
are even, all numbers are odd, numbers alternate even/odd, etc…) occur.
MP.6 Attend to precision.
Students will attend to precision in the vocabulary
they use in their explanations. They will also attend
to precision when finding all factor pairs for a given
number.
MP.7 Look for and make use of structure.
Students will look closely to discern patterns or
structures. Students will make sense of structure
when finding multiples and factors of a given
number.
Sample Tasks:
1) There are 3 candies in the jar. Each day 2 candies are added. How many candies are in the jar on the
5th day? After completing the table, look at the numbers in the Candies column. Find at least one pattern
you see that is not a part of the rule? Explain the pattern in terms of how the number of candies changes.
Day
Calculation
Candies
0
2x0+3
3
1
2x1+3
5
2
2x2+3
7
3
2x3+3
9
4
5
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Mathematics
Grade 4 – Unit 1 (SAMPLE)
MP.8 Look for and express regularity in repeated
reasoning.
Students will look for and express regularity in
repeated reasoning when finding the next number
when given a rule. Students may notice if
calculations are repeated and look for both general
methods and shortcuts.
2) Use the rule multiply by 3, subtract 1 to write the numbers in the pattern. What do you observe about
the numbers in the pattern? Explain your observations.
2, _____, ______, ______, _____
Days : End of Unit Assessment 9-10
14
Mathematics
Grade 4 – Unit 1 (SAMPLE)
Application Task:
Teacher Note: The following problem is often used to find the least common multiple or greatest
common factor. Understanding least common multiple or greatest common factor is not a requirement
of 4th grade students as the skill/concept is left for Grade 6. Students can complete these problems by
applying their understanding of multiples without understanding the concept of LCM or GCF.
Isaac is planning a picnic for his classmates. Everyone coming to the picnic wants to eat hot dogs. He
knows that hot dogs at the local store are sold 10 in a package. The hot dogs buns are sold 8 in a
package.
1. What is the least number of packages of hot dogs and hot dog buns Isaac can buy to have
exactly the same amount of hot dogs and buns? Show how you determined your answer.
2. Isaac discovers that 30 people are coming to the picnic and everyone wants to eat 2 hot dogs!
What is the least number of packages of hot dogs Isaac will have to buy to feed all 30 people?
What is the least number of packages of buns Isaac will have to buy to feed all 30 people? Show
how you know.
Total number of hot
dogs and buns needed
Number in package
Total packages needed
Hot dogs
Hot dog buns
3. Isaac noticed he would have more hot dog buns than hot dogs. He decided not to waste food so
he wanted to feed the teachers in the other grades.
a. What is the least number of packages of hot dogs and hot dog buns will Isaac need to buy to
have exactly the same number of hot dogs and buns to feed all 30 people two hot dogs
each? Tell how you know.
b. How many extra hot dogs will there be to feed the teachers in the other grades? Explain
how you know.
c. If each teacher in the other grades eats two hot hogs, how many teachers can eat? Explain
your reasoning.
4. Isaac asked some of the students in other classes to bring some other items for the picnic. Help
Isaac’s friends figure out what they need to buy.
a. Tanya will bring the ketchup. She already has 3 bottles and each bottle has enough for 20
hot dogs. How many more bottles of ketchup will Tanya need to buy for there to be enough
ketchup for all of the hot dogs? Show how you found your answer.
15
Mathematics
Grade 4 – Unit 1 (SAMPLE)
b. Jordan found boxes of mustard packets. One box has 10 mustard packets. How many packs
of mustard must Jordan buy to have enough mustard for all of the hot dogs, if one packet is
used for each hot dog? Show how you found your answer.
16
Mathematics
Grade 4 – Unit 1 (SAMPLE)
Application Task:
Isaac is planning a picnic for his classmates. Everyone coming to the picnic wants to eat hot dogs. He
knows that hot dogs at the local store are sold 10 in a package. The hot dogs buns are sold 8 in a
package.
1. What is the least number of packages of hot dogs and hot dog buns Isaac can buy to have
exactly the same amount of hot dogs and buns? Show how you determined your answer.
Solution:
hot dog packages
hot dog bun packages
Isaac would need to buy 4 packages of hot dogs and 5 packages
of hot dog buns to have the exactly the same number of hot dogs
and hot dog buns. Count by multiples of 10 and multiples of 8.
2. Isaac discovers that 30 people are coming to the picnic and everyone wants to eat 2 hot dogs!
What is the least number of packages of hot dogs Isaac will have to buy to feed all 30 people?
What is the least number of packages of buns Isaac will have to buy to feed all 30 people? Show
how you know.
Hot dogs
Hot dog buns
Total number of hot
dogs and buns needed
30 people x 2 hot dogs =
60 hot dogs
30 people x 2 hot dogs =
60 hot dog buns
Number in package
10 in package
8 in package
Total packages needed
6 packages of hot dogs x 10 = 60
hot dogs
8 packages of hot dog buns x 8 =
64 hot dog buns
7 packages would only be 56 buns
(7 x 8) which is not enough.
3. Isaac noticed he would have more hot dog buns than hot dogs. He decided not to waste food so
he wanted to feed the teachers in the other grades.
a. What is the least number of packages of hot dogs and hot dog buns will Isaac need to buy to
have exactly the same number of hot dogs and buns and still be able to feed all 30 people
two hot dogs each? Tell how you know.
Solution: 30 x 2 = 60 hot dogs will be needed to feed all 30 people.
hot dog packages
hot dog bun packages
Since Isaac will need 60 hot dogs, he will need to buy 8 packages of hot dogs and 10
packages of hot dog buns to have exactly the same number of hot dogs and hot dog
buns.
17
Mathematics
Grade 4 – Unit 1 (SAMPLE)
b. How many extra hot dogs will there be to feed the teachers in the other grades? Explain
how you know.
Solution: 80 total hot dogs will be made. 60 of them will be used to feed the other
people. So, there will be 20 hot dogs left to feed the teachers in the other grades.
c. If each teacher in the other grades eats two hot hogs, how many teachers can eat? Explain
your reasoning.
Solution: There will be 20 hot dogs left over. If each teacher eats two hot dogs, then 10
teachers can eat.
4. Isaac asked some of the students in other classes to bring some other items for the picnic. Help
Isaac’s friends figure out what they need to buy.
a. Tanya will bring the ketchup. She already has 3 bottles and each bottle has enough for 20
hot dogs. How many more bottles of ketchup will Tanya need to buy for there to be enough
ketchup for all of the hot dogs? Show how you found your answer.
Solution: 3 bottles x 20 servings = 60 servings total
80 servings needed – 60 servings = 20 more servings needed
1 bottle has 20 servings so Tanya needs to buy 1 more bottle of ketchup.
b. Jordan found boxes of mustard packets. One box has 10 mustard packets. How many packs
of mustard must Jordan buy to have enough mustard for all of the hot dogs, if one packet is
used for each hot dog? Show how you found your answer.
Solution: 80 packets are needed. 16 boxes x 5 packets = 80 packets. So Jordan needs to
buy 16 boxes of mustard packets.
18