Unit 2 Algebraic Reasoning, Grade 5
5E Lesson Plan Math
Grade Level: 5
Lesson Title: Unit 2 Algebraic Reasoning
Subject Area: Math
Lesson Length: 12 days
THE TEACHING PROCESS
Lesson Overview:
This unit bundles student expectations that address prime and composite numbers, grouping
symbols, simplifying expressions, and solving equations. According to the Texas Education
Agency, mathematical process standards including application, a problem-solving model, tools and
techniques, communication, representations, relationships, and justifications should be integrated
(when applicable) with content knowledge and skills so that students are prepared to use
mathematics in everyday life, society, and the workplace.
During this unit, students are introduced to the properties of prime and composite numbers. It is
encouraged to use students’ previous understandings of representing products using arrays and
area models to help support new student learning to identify prime and composite numbers by
analyzing their factors. Students examine the meaning of grouping symbols within a numeric
expression and simplify numerical expressions based on socially constructed conventions. In
addition, students represent and solve multi-step problems algebraically using an equation with a
letter representing the unknown. All operations within this unit are limited to whole numbers.
Unit Objectives:
Students will…
 Identify prime and composite numbers by analyzing their factors
 Represent products using arrays and area models to help support new learning
 Examine the meaning of grouping symbols within a numeric expression
 Simplify numerical expressions based on socially constructed conventions
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Represent and solve multi-step problems algebraically using an equation with a letter
representing the unknown
Standards addressed:
TEKS:
5.1A – Apply mathematics to problems arising in everyday life, society, and the workplace.
5.1B – Use a problem-solving model that incorporates analyzing given information, formulating a
plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving
process and the reasonableness of the solution.
5.1C –Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as appropriate,
to solve problems.
5.1D –Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
5.1E –Create and use representations to organize, record, and communicate mathematical ideas.
5.1F – Analyze mathematical relationships to connect and communicate mathematical ideas
5.1G –Display, explain, and justify mathematical ideas and arguments using precise mathematical
language in written or oral communication
5.4A – Identify prime and composite numbers.
5.4B – Represent and solve multi-step problems involving the four operations with whole numbers
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Unit 2 Algebraic Reasoning, Grade 5
using equations with a letter standing for the unknown quantity.
5.4E – Describe the meaning of parentheses and brackets in a numeric expression
5.4F – Simplify numerical expressions that do not involve exponents, including up to two levels of
grouping.
ELPS:
ELPS.c.1A—Use prior knowledge and experiences to understand meanings in English
ELPS.c.2D—Monitor understanding of spoken language during classroom instruction and
interactions and seek clarification as needed
ELPS.c.3C—Speak using a variety of grammatical structures, sentence lengths, sentence types, and
connecting words with increasing accuracy and ease as more English is aquired
ELPS.c.3D—Speak using grade-level content area vocabulary in context to internalize new English
words and build academic language proficiency
ELPS.c.4H—Read silently with increasing ease and comprehension for longer periods
ELPS.c.5B—Write using newly acquired basic vocabulary and content-based grade-level
vocabulary
ELPS.c.5F—write using a variety of grade-appropriate sentence lengths, patterns, and connecting
words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is
acquired
Misconceptions:
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Some students may simplify an expression or solve an equation from left to right rather
than using to the order of operations or grouping symbols to simplify.
Some students may simplify an expression or solve an equation by performing all like
operations first rather than using the grouping symbols to simplify.
Some students may think the equal sign means “solve this” or “the answer is” rather than
understanding that the equal sign represents a quantitative and balanced relationship.
Some students may think that the equal sign can only be placed at the end of an equation,
rather than thinking it can be placed at the beginning or end as long as the equation is
balanced. (e.g., 7 + 3 + 5 = n and n = 7 + 3 + 5)
Some students may think that the number 1 is prime rather than understanding that 1 is
neither prime nor composite.
Some students may think that all prime numbers are odd numbers and all composite
numbers are even numbers, rather than thinking of the number of factors involved.
Underdeveloped Concepts:
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Some students may not consider all the information in a problem situation before
developing an algebraic equation.
Some students may become confused when abbreviations are used in conjunction with a
letter representing an unknown in a problem situation (e.g., 6m to mean 6 meters).
Note: Please reference the 2014-2015 Implementation TAG Tool for any instructional gaps that
might be associated with the new 2012 Adopted TEKS.
Vocabulary:
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Composite number – a whole number with more than two factors
Unit 2 Algebraic Reasoning, Grade 5
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Counting (natural) numbers – the set of positive numbers that begins at one and
increases by increments of one each time {1, 2, 3, ..., n}
Dividend – the number that is being divided
Divisor– the number the dividend is being divided by
Equation – a mathematical statement composed of algebraic and/or numeric expressions
set equal to each other
Expression – a mathematical phrase, with no equal sign, that may contain a number(s), a
unknown(s), and/or an operator(s)
Factor – a number multiplied by another number to find a product
Order of operations– the rules of which calculations are performed first when simplifying
an expression
Parentheses and brackets – symbols to show a group of terms and/or expressions within
a mathematical expression
Prime number – a whole number with exactly two factors, 1 and the number itself
Product– the total when two or more factors are multiplied
Quotient – the size or measure of each group or the number of groups when the dividend
is divided by the divisor
Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
Related Vocabulary:
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Array
Grouping
Simplify
Difference
Divisible
Multiple
Multi-step
Solve
Equal
Operation
Sum
Factor list
Operator
Unknown
Factor pair
Remainder
List of Materials:
Day 1
Materials:
 Sandwich- sized plastic bags (1 per student)
 Colored tiles (at least 8 per student)
 Mini whiteboard
 Dry-erase marker and eraser
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Unit 2 Algebraic Reasoning, Grade 5
Day 2
Materials:
 Pink & green crayons (1 per student)
 Handout: Hundreds Charts (2 copies per student)
 Mini whiteboard
 Dry-erase markers and eraser
Day 3
Materials:
 Math journals
 Handout: Hundreds Charts (the same ones completed during Day 2)
 Mini whiteboard
 Dry-erase markers and eraser
Day 4
Materials:
 Mini whiteboards
 Dry-erase markers and eraser
 Handout: “Word Problems for Creating Equations”
Day 5
Materials:
 Math journals
 Handout: “Represent and Solve Problems –guided practice” (pages 1-8)
Day 6
Materials:
 Handout: “Represent and Solve Problems –guided practice” (pages 9-13) (from Day
5)
Day 7
Materials:
 Math journals
 Red pen
Day 8
Materials:
 Math journals
 Handout: “PEMDAS Rules”
 Highlighter
Day 9/ Day 10
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Unit 2 Algebraic Reasoning, Grade 5
Materials:
 Handout: “Representing Expressions”
 Handout: “Find the Missing Number”
 Handout: “Order of Operations”
Day 11
Materials:
 Performance Assessment 01
Day 12
Materials:
 Performance Assessment 02
INSTRUCTIONAL SEQUENCE
Phase 1: ENGAGE
Day 1
Materials:
 Sandwich- sized plastic bags (1 per student)
 Colored tiles (at least 8 per student)
 Mini whiteboard
 Dry-erase marker and eraser
Activity:
Summary:
Give each student a plastic bag pre-filled with at least 8 colored tiles. (If colored tiles are not
available, thick paper such as card stock cut into small squares will suffice.) Students will use the
tiles to access their prior knowledge of representing products using arrays. Students will work with
a partner using a mini whiteboard, dry erase markers, and eraser to write down the factors of the
arrays they create.
Teaching Directions:
Ask each student to take 8 tiles out of their plastic bag and place them on their desk. Have students
create an array using the 8 tiles. After they have arranged their tiles into rows and columns, have
them show their array to a partner. Each partner then writes the factor pair on the mini whiteboard
that corresponds to their array. For example, if a student arranged their tiles into an array with 2
rows with 4 columns, they would write 2 x 4 on their board. (It should only be listed once, so if
they are exactly the same, only 1 student should write it. However, if the second partner has 4 rows
with 2 columns they can write 4 x 2 on the board.)
Have students clear their tiles and try to create another, different array using the same 8 tiles.
Students should again write down the factor pairs that correspond to their second array.
After completing this activity, students should have created 4 different rectangular arrays, along
with 4 factor pairs:
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Unit 2 Algebraic Reasoning, Grade 5
Ask guiding questions to the whole group.
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Were you able to create more than one array using 8 tiles?
Yes, more than one is possible.
How many different arrays were you able to create using 8 tiles?
Four different arrays can be created.
What is a factor?
A factor is a number that is multiplied by another number to find a product.
How many factor pairs did you make?
4 different factor pairs can be created, (1 x 8) (2 x 4) (4 x 2) and
(8 x 1).
What is a composite number?
A composite number is a whole number with more than two factors.
Do you think the number 8 is a composite number?
Yes, the number 8 is composite because there are more than two different rectangular
arrays that can be made, so it has more than two factors.
Have partner 1 use the board and dry-erase marker to arrange the factors of 8 into a table:
Have partner 2 arrange the same factors of 8 into a factor “rainbow”, or list, next to the table that
partner 1 drew.
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Does organizing the factors in a table or list help you to determine whether this
number is prime or composite?
Unit 2 Algebraic Reasoning, Grade 5
Yes, you can see that 8 is composite because there are more than 2 factors: 1, 2, 4, and 8.
Next, students may erase their boards. Ask each student to remove 1 tile and place it back in the
bag so that they have 7 tiles total on their desk. Challenge each student to create an array using
only 7 tiles. Once their array is complete, have them show a partner and write the factor pair on
their board. (If 1 partner writes 1 x7, the other partner may write 7 x 1.)
After completing this activity, students should have created 2 different rectangular arrays, along
with 2 factor pairs:
Ask guiding questions to the whole group.
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Were you able to create more than one array using 7 tiles?
Yes, more than one is possible.
How many different arrays were you able to create using 7 tiles?
Only 2 different arrays are possible.
How many factor pairs did you make?
2 different factor pairs can be created, (1 x 7) and ( 7 x 1).
What is a prime number?
A prime number is a whole number with exactly two factors, 1 and the number itself.
Do you think the number 7 is a prime number?
Yes, the number 7 is prime because there are exactly 2 different rectangular arrays that
can be made, so it only has 2 factors.
How can representing a number as an array help you determine if a number is prime
or composite?
Composite numbers have MORE THAN 2 different arrays that can be made. Prime
numbers have EXACTLY 2 different rectangular arrays that can be made.
What are the advantages of being able to represent a number as an array?
Answers will vary. The tiles help you to visualize how many factors a number has, etc.
Are there any disadvantages?
Answers will vary. There may not be enough tiles for larger numbers, etc.
Have partner 1 use the board and dry-erase marker to arrange the factors of 7 into a table:
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Unit 2 Algebraic Reasoning, Grade 5
Have partner 2 arrange the factors of 7 into a factor “rainbow”, or list, next to the table that partner
1 drew.
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Does organizing the factors in a table or list help you to determine whether the
number 7 is prime or composite?
Yes, you can see that the number 7 is prime because there are exactly 2 factors, 1 and 7.
What are the advantages of being able to represent a number as a list or table of its
factors?
Answers will vary. It lists the factors in an organized way, etc.
Are there any disadvantages?
Answers will vary. It is more difficult to visualize than the tiles; some numbers might get
left out or out of order, etc.
If time allows, have students make up their own numbers (no higher than the number of tiles in the
bag) and practice creating arrays, tables, and factor “rainbows” with a partner.
What’s the teacher doing?
What are the students doing?
Raising questions and encouraging responses
Activating their previous knowledge of arrays
as they prepare to learn new information
Monitoring students as they work together
Demonstrating their understanding using the
colored tiles
Raising questions and responding to questions
Phase 2: EXPLORE
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Unit 2 Algebraic Reasoning, Grade 5
Day 2
Materials:
 Pink & green crayons (1 per student)
 Handout: Hundreds Charts (2 copies per student)
 Mini whiteboard
 Dry-erase markers and eraser
Activity:
Summary:
Students will need a pink crayon and a copy of the Hundreds Chart. Have students label the top
of the page as “Composite Numbers”. Students will explore finding all composite numbers within
the hundreds chart by first coloring all of the multiples of 2, then coloring all of the multiples of 3,
then 4, and so on, up to 10. An example of this already completed with an answer key can be
obtained from
http://learningworkroom.com/uploads/Composite_Numbers_Poster_and_Answer_Chart.pdf
Next, students will use a green crayon and a new copy of the Hundreds Chart. Have students
label the top of the page as “Prime Numbers”. Students will find all of the prime numbers on the
chart and color them green. Hint: The prime numbers will be the numbers that were not colored
pink on the composite page, except for the number 1. An example of this with an answer key can
be obtained from
http://learningworkroom.com/uploads/Prime_Numbers_Poster_and_AnswerChart.pdf
Finally, students will work with a partner to play a mini whiteboard game called “P or C?” where
they analyze the factors of a number to determine if it is prime or composite. By doing this the
student will be able to identify prime and composite numbers.
Teaching Directions:
Ask students to get a pink crayon (do not tell them about the green crayon yet, as they might be
tempted work ahead and make a mistake). Hand students a copy of the Hundreds Chart and have
them label the top of the page as “Composite Numbers”. Have students start by coloring all
multiples of 2 (NOT including 2) beginning with 2 x 2 which is 4, then 2 x3 which is 6, then 2 x4
which is 8, then 10, 12, 14, 16, 18, 20, and so on, up to 100.
When students are finished, ask:
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Look at the numbers that you have colored so far. What are all of these numbers
divisible by?
All of these numbers are evenly divisible by 2.
What else do you notice about these numbers?
They are all even.
What do you notice about the number 2?
We did not color it. It was left blank. All even numbers are colored except for the number
2.
Next, have students color all multiples of 3 (NOT including 3). Some multiples of 3 will already be
colored because they are also multiples of 2 as well. Let students know that there will be some
overlapping. In this case, they will begin with 9, since 6 will already be colored because it is a
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Unit 2 Algebraic Reasoning, Grade 5
multiple of 2. Then they will color 12, 15, 18, 21 and so on up to 99.
While students are coloring, ask:
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Think about the numbers you are coloring now. What are all of these numbers
divisible by?
All of these numbers are evenly divisible by 3.
Next, have students color all multiples of 4. As before, some multiples of 4 will overlap with
multiples of 2 and 3 and already be colored.
While students are coloring, ask:
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Think about the numbers you are coloring now. What are all of these numbers
divisible by?
All of these numbers are evenly divisible by 4.
Next have students finish coloring all multiples of 5 (NOT including 5), then 6, 7 (NOT including
7), 8, 9, and 10. While students are coloring, ask what each set of multiples are divisible by.
When students have completed coloring all the possible multiples on the page, ask:
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Look at the numbers you colored in pink. What type of numbers do you think these
are?
Composite numbers.
Why do you think that these are composite numbers?
A multiple is the product of its factors. Since these colored numbers are multiples of
numbers other than 1, then they have factors other than 1 and itself.
What is the relationship between composite numbers and the rules of divisibility?
Numbers are divided evenly by their factors. Therefore, if a number is divisible by factors
other than 1 and itself, then it is a composite number.
Have students put an “X” on the number 1 in pen/ pencil.
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Look at the numbers that you did not color. What type of numbers do you think these
are?
Prime numbers.
Why do you think that these numbers are prime?
These numbers were left blank because they are not multiples of any number other than 1
and itself. They are not divisible by any factors other than 1 and itself.
Why do you think the number 1 is crossed out?
The number 1 is neither prime nor composite.
What is the relationship between prime numbers and the rules of divisibility?
Numbers are divided evenly by their factors. Therefore, if a number is only divisible by 1
and itself, it has no other factors and it is a prime number.
Ask students to put their pink crayon aside and get a green crayon. Hand students a copy of the
Hundreds Chart and have them label the top as “Prime Numbers”. Tell students to put an “X” on
the number 1 on this page also. On this page, have students color in green all the numbers that were
left blank (except for the number 1) from the composite numbers page. On this page, the composite
numbers will be left blank. (On the previous page, the prime numbers were left blank.)
When all prime numbers have been colored green, ask:
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What do you notice about the number 0?
The number 0 is not on either of the charts.
Unit 2 Algebraic Reasoning, Grade 5
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Do you think the number 0 is a prime or composite number?
Just like the number 1, the number 0 is neither prime nor composite.
Take a look at the number 2 again, but this time on the “Prime Numbers” chart.
What do you notice?
The number 2 is the only prime number that is even.
Next, have students partner up with a mini whiteboard, dry-erase marker, and eraser. Students will
also need their composite numbers and prime numbers charts that they have colored previously.
Students will play a game called “P or C?”.
Partner 1 will start and may look at his/her chart. Partner 2 must lay their chart face down so that
they cannot see it. At the top of the mini whiteboard, Partner 1 will write P or C? Then they will
choose a number and write it in the middle of the board for Partner 2 to see. Partner 1 then asks, “P
or C?”, with “P” standing for a prime number and “C” standing for a composite number. For
example, Partner 1 may write the number 21 on the board. Partner 2 must circle either the P or the
C at the top of the board and prove their answer by writing the factors on the board. In this case,
Partner 2 would circle the “C” for composite and list the factors of 21 either in a table or “factor
rainbow” list. Next, the partners take turns and Partner 2 will choose a number to give to Partner 1
and the game continues. Each partner earns a point when they get the answer correct. If they get an
answer incorrect, they lose a point.
What’s the teacher doing?
Observing students as they work cooperatively
Asking inquiry-oriented questions to assist in
examining student thinking
What are the student’s doing?
Focused on coloring their charts correctly
Answering questions that are asked by teacher
Working collaboratively with a partner as they
participate in the “P or C” game
Phase 3: EXPLAIN
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Unit 2 Algebraic Reasoning, Grade 5
Day 3
Materials:
 Math journals
 Handout: Hundreds Charts (the same ones completed during Day 2)
 Mini whiteboard
 Dry-erase markers and eraser
Activity:
Have students glue or tape the composite numbers and prime numbers charts that they completed
on Day 2 into their math journals as a reference. (Note: these should NOT be used on a graded
assignment or during any type of assessment.)
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Do you remember what type of numbers you colored in pink on your chart?
Composite numbers
Do you remember what type of numbers you colored in green on your chart?
Prime numbers.
What do you remember about the pink numbers?
Answers will vary. Students may note that on the hundreds charts there were a lot more
pink (composite numbers) than green.
What do you remember about the green numbers?
Answers will vary. Students may recognize that there are only 25 green (prime numbers).
Students may also observe that they are all odd numbers, except for the number 2.
Have students write the formal definitions of a prime number and composite number in their
journal. Remind students that whole numbers are the set of counting (natural) numbers and zero {0,
1, 2, 3, …, n}.
Prime Number- a whole number with exactly two factors, 1 and the number itself
Composite Number- a whole number with more than two factors
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Is there anything that you would like to add to either of these definitions that would
help you remember the difference between the two types of numbers?
Answers will vary depending on what may be helpful to each student.
What can you use determine whether a number is prime or composite?
The factors of the number
What are the various ways that we can organize and analyze the factors of a number?
We can show the factors in a picture using an array, or the factors can be grouped in a
table or listed as “factor rainbow” list.
Once the factors are organized, how can you analyze them to determine whether a
number is prime or composite?
You look at how many factors there are, or how many arrays can be created. If a number
Unit 2 Algebraic Reasoning, Grade 5
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has only 2 factors- 1 and the number itself- then it is a prime number. If a number has
more than 2 factors, then it is a composite number.
What number do you think is the only common factor in ALL numbers?
Number 1
Explain to students that there is also something special about the number 0. The number 0 cannot
be expressed as a product of primes. 0 x 0 and/ or 0 x any number yields an infinite number of
factor pairs that have the product of 0.
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What other numbers are also special?
Students should explain based on exploring their charts previously that the numbers 0 and
1 are neither prime nor composite. Students should also explain that the number 2 is the
only even number that is prime.
Give students an example of four numbers to write down in their math journals.
21, 48, 61, 81
Ask students to show work in their journals of how they would analyze the factors of each number
to determine whether the numbers are prime or composite. Have students write a small “c” for
composite or “p” for prime directly above the number.
c c p c
21, 48, 61, 81
21: 1, 3 7, 21
48: 1, 2, 3, 4, 6, 8,12,16, 24, 48
61: 1,
61
81: 1, 3, 9 ,27, 81
Prime and Composite Numbers Partner Review Game:
This partner activity is a great informal assessment. The students and teacher will get instant
feedback since it is self-checking! Using the prime and composite numbers charts from the Explore
coloring activity, students will partner up with mini whiteboards, dry-erase markers, and erasers
(this can be played on notebook paper if necessary). Partner 1begins by choosing any 4 numbers
from their prime and composite numbers charts (numbers may be all prime, all composite, or any
combination of both) and writing them on the board for Partner 2 to see. Partner 2 must analyze the
factors of each of the four numbers by writing the factors of the numbers in a table or list, to
determine whether the numbers are prime or composite. Partner 2 writes a small “P” above the
numbers that he/she thinks are prime, and a small “C” above the numbers he/she thinks are
composite. Partner 1 then checks with his/her chart to determine if Player 2’s answers are correct. 1
point is awarded for each correct answer, so a player may earn up to 4 points each round. 1 point is
deducted for every incorrect answer. Partners then switch roles and continue taking turns making
up numbers from the chart to give to the other player.
Extra Credit Opportunity:
Challenge students to memorize the first 25 prime numbers from their prime numbers coloring
chart. Offer students extra credit if they can either recite (or write) them all!
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Unit 2 Algebraic Reasoning, Grade 5
What’s the teacher doing?
What are the students doing?
Providing formal definitions and explanations
Writing formal definitions in math journals
Explaining strategies on how to determine
whether a number is prime or composite
Connecting definitions to what they have
experienced
Providing examples
Monitoring students as they work examples in
their math journals
Phase 2: EXPLORE
Note: The teacher may prefer to insert the Day 11 Activity (in which the Unit 2 Performance
Assessment 1 is given) before Day 4. This is optional, but may be preferred because the Unit 2
Performance Assessment 1 covers Prime and Composite numbers, and the following days cover
Equations.
Day 4
Materials:
 Mini whiteboards
 Dry-erase markers and eraser
 Handout: “Word Problems for Creating Equations”
Activity:
During this activity, students will explore a real-life problem situation involving equations. Tell
students that the gravitational force on Earth is approximately six times greater than the
gravitational force on the moon. Because of this, they would weigh less on the moon. Tell students
that their weight on the moon is an unknown quantity, and they can use an equation to determine
their weight on the moon. Ask students to use their mini whiteboards and dry-erase marker to try
and create an equation for finding m, their weight on the moon. Students should be given time to
come up with an equation.
mx6=e
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OR e ÷ 6= m
Unit 2 Algebraic Reasoning, Grade 5
where m= weight on moon, and e= weight on earth
Challenge students use the first equation (m x 6 = e) to solve for m, their weight on the moon.
m
x 6=
e
WEIGHT ON MOON x 6 = WEIGHT ON EARTH
(unknown)
(known)
Students should be given time to discover that they need to divide their weight on Earth by 6 in
order to find their weight on the moon. For example, if a student weighs 90 lb. on Earth, they
would solve by finding 90 ÷ 6= 15. This student’s weight would be 15 lb. on the moon.
Show students that their equation could be solved this way:
m
=
m =
90 ÷ 6
15
Next, give students the “Word Problems for Creating Equations” handout. Ask students to read
the word problems and communicate and collaborate with a partner to create an equation that they
would use to solve each word problem. After writing the equation, students should use the equation
to solve.
After students have finished creating equations with their partner and solving them, have a whole
group discussion in which students explain and justify their work to the class. Use the “Word
Problems for Creating Equations- Answer Key” as a guide. Encourage students to recognize that
the equal sign can be at the beginning or end of their equation.
During the discussion, ask guiding questions:
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What is an equation?
An equation is a number sentence that has an equal sign.
Yes, an equation is a mathematical statement composed of algebraic and/or numeric
expressions set equal to each other.
How can an unknown value be represented in an equation?
Any single letter can be used to stand for the unknown quantity.
Does it matter if the equal sign is at the beginning or end of the equation?
No, the equal sign can be at the beginning or end.
Does the unknown value have to be in a certain place in the equation?
No, the unknown quantity, represented by a letter, can be in any position within the
equation.
Unit 2 Algebraic Reasoning, Grade 5
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What does “solve” mean?
To find the answer to the unknown quantity.
What’s the teacher doing?
What are the students doing?
Facilitate student thinking and guide towards
algebraic thinking
Working collaboratively with a partner to
compose equations
Encourage students to work cooperatively
Discovering how to write equations
Phase 3: EXPLAIN
Day 5
Materials:
 Math journals
 Handout: “Represent and Solve Problems –guided practice” (pages 1-8)
Activity:
Begin with a review of the vocabulary that is associated with all four operations. Have students
write the vocabulary words and definitions in their math journal.
Sum – addition
Difference – subtraction
Product –multiplication; the total when two or more factors are multiplied Quotient –
division; the size or measure of each group or the number of groups when the dividend is
divided by the divisor
Dividend – the number that is being divided
Divisor – the number that the dividend is being divided by
This lesson focuses on the handout titled “Represent and Solve Problems –guided practice”.
Have students look at pages 1-2 titled, “Represent and Solve Addition and Subtraction Problems –
guided practice”. Refer to the Teacher’s Guide and Answer Key that is provided with the handout.
Students will follow along as you guide them through the questions, allowing adequate time for
students to write an equation and solve on their handout.
During this activity, students should be leading the discussion on the explanations of their own
equations and justifying how they used their equation to solve the problem situation. Although
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Unit 2 Algebraic Reasoning, Grade 5
student equations may differ from the sample equations provided, accept all reasonable responses.
As students use their new knowledge to write equations using a letter to represent an unknown
quantity, ask guiding questions:


How are equations used to represent the relationship between quantities in problem
situations?
What is the process for writing and solving an equation from a problem situation?
Next, have students look at page 6 titled “Represent and Solve Multiplication and Division
Problems – guided practice”. Refer to the Teacher’s Guide and Answer Key that is provided with
the handout. Students will follow along as you guide them through the questions, allowing
adequate time for students to write an equation and solve on their handout.
Day 6
Materials:
 Handout: “Represent and Solve Problems –guided practice” (pages 9-13) (from Day
5)
Activity:
Today have students will need the same handout as used on Day 5 “Represent and Solve
Problems –guided practice”. Refer students to page 9 titled “Represent and Solve Multi-Step
Problems – guided practice”. Refer to the Teacher’s Guide and Answer Key that is provided with
the handout. Students will follow along as you guide them through the questions, allowing
adequate time for students to write an equation and solve on their handout.
Day 7
Materials:
 Math journals
 Red pen
Activity:
Lead a class discussion on parentheses and brackets.


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What symbols can be used when representing problem situations with an equation?
Equal sign, all four operation symbols/ operators (+, - , x, ÷), etc.
Is it helpful to be able to group numbers within an equation?
Unit 2 Algebraic Reasoning, Grade 5


Yes, some numbers need to be grouped together and solved separately.
What symbols are used to group numbers together?
Parentheses
That is correct. Parentheses ( ) are used to group numbers together. Brackets [ ]
are another symbol that may be used to group numbers together. Parentheses and
brackets are both symbols that show a group of terms and/ or expressions within a
mathematical expression.
What is an expression? How is it different from an equation?
An expression is a mathematical phrase, with NO equal sign, that may contain a
number(s), an unknown(s), and/or an operator(s.
Invite a student to come up to the white board and write a numerical expression based on the
information you give them.


5 times larger than the sum of 15 and 2
Answer: 5 x (15 + 2) or 5(15 + 2)
The quotient of the difference between a value 4 times larger than (2 + 5) and 3,
divided by 3
Answer: [4 x (2 + 5) – 3] ÷ 3
Explain to students that there are various symbols that can be used to represent multiplication,
other than the standard multiplication symbol, including: x, •, parentheses, or brackets. Have
students write down the following examples in their math journals.
Various symbols to represent multiplication
5 x 17 =
5 • 17
5(17)
(5)(17)
5[17]
[5][17]
Give students the following example to copy in their math journals.
84 ÷ 3 + 4
4x2x2
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Unit 2 Algebraic Reasoning, Grade 5
Explain to students that the fraction bar also represents a division bar. Show students that the
expression can be rewritten using a division sign with parentheses or brackets. Point out that the
top expression (numerator) and bottom expression (denominator) both stay grouped together by
parentheses/ brackets.
The fraction can be rewritten as:
(84 ÷ 3 + 4) ÷ (4 x 2 x 2)
OR
[84 ÷ 3 + 4] ÷ [4 x 2 x 2]
Next, give students another example to write in their math journals. Students will need to have a
red pen handy.
Ex: 3 × [7 + 2 - (8 + 4) ÷ 3] + 2




What do you notice about this expression?
It has both parentheses and brackets.
Since this expression has both parentheses and brackets, it has two levels of grouping.
What do you think should be solved first?
The inside parentheses (8 + 4)
Yes, the innermost grouping should be evaluated first.
Have students take their red pen and circle (8 + 4), which is inside of the parentheses. Have
students make a note out to the side that this should be evaluated first, and draw an arrow to it.
Have students solve 8 + 4 and write a 12 in red underneath.
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Unit 2 Algebraic Reasoning, Grade 5

What do you think should be solved next?
Everything that is inside of the brackets.
Have students use their regular pen/ pencil to circle everything inside of the brackets, with the 8 + 4
replaced with 12 now. Students should make a note to the side that this part of the expression
should be evaluated next, and draw an arrow to it.
Next, have students write this example in their math journal.
3 × (7 + 2 - (8 + 4) ÷ 3) + 2



What do you notice about this expression?
It is the same as the previous equation, but it has 2 sets of parentheses instead of brackets.
A double set of parentheses may be used just like the parentheses and brackets were
used in the previous expression.
Do you think this expression will be solved in the same way as the previous equation?
Yes. When both parentheses and brackets, or a double set of parentheses, is used within a
numerical expression, the inner most grouping should be evaluated first.
As in the previous example, have students take their red pen and circle (8 + 4). Have students make
a note out to the side that this should still be evaluated first, because it is inside of the innermost
parentheses, and draw an arrow to it.
Have students solve 8 + 4 and write a 12 in red underneath.

What do you think should be solved next?
Everything that is within the outer set of parentheses.
Once again, have students use their regular pen/ pencil to circle everything within the outer set of
parentheses, with the 8 + 4 replaced with 12 now. Students should make a note to the side that this
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Unit 2 Algebraic Reasoning, Grade 5
should be evaluated next, and draw an arrow to it.
Day 8
Materials:
 Math journals
 Handout: “PEMDAS Rules”
 Highlighter
Activity:
Have students write down this vocabulary word and definition in their math journal.
Parentheses and brackets – symbols to show a group of terms and/or expressions within a
mathematical expression
Order of operations – the rules of which calculations are performed first when simplifying
an equation
Explain to students that there is an acronym they can use to help them remember the order of
operations – “Please Excuse My Dear Aunt Sally”. Students should write the acronym and what
each letter stands for in their journal.
P: Parentheses/ brackets – simplify expressions inside parentheses or brackets in order from left
to right
E: Exponents *Students do not learn exponents at this grade level. Grade 6 will include whole
number exponents.
M: Multiplication/ D: Division –simplify expressions involving multiplication and/ or division in
order from left to right
A: Addition/ S: Subtraction – simplify expressions involving addition and/or subtraction in order
from left to right
Give students a laminated and hole-punched copy of the handout “PEMDAS Rules” that they can
keep in their backpack/ binder as a reference to the order of operations. (This handout was obtained
from math-aids.com at the following link:
http://www.mathaids.com/cgi/pdf_viewer_7.cgi?script_name=pemdas_rules.pl&x=124&y=26 )
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Unit 2 Algebraic Reasoning, Grade 5
Have students look again in the math journals at the last example given on Day 7:
3 × (7 + 2 - (8 + 4) ÷ 3) + 2
Students should recopy this example on a new page of their journal.
Lead students through the process of simplifying the expression in their math journal using the
order of operations.
Students should recall from yesterday that 8 + 4 should be solved first. Students should rewrite the
expression below, replacing 8 + 4 with 12.


Why do you think we evaluated (8 + 4) first?
The first step in the order of operations is “P”, which indicates that expressions inside of
parentheses and brackets are evaluated first. When there are 2 sets of parentheses, the
innermost set of parentheses is evaluated first.
What should be evaluated next?
Everything inside of the next set of (outer) parentheses should be evaluated by following
the order of operations.
Students may use a highlighter to highlight this portion of the expression, so that they remember to
concentrate on this part first.

Using the order of operations, what should be evaluated first inside of the
parentheses?
Multiplication and division come before addition and subtraction. So, 12 ÷ 3 should be
solved next.
Students should rewrite the expression below, replacing 12 ÷ 3 with 4.

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Using the order of operations, what should be evaluated next inside of the
parentheses?
Addition and subtraction should be evaluated in order from left to right, so 7 + 2 would be
evaluated first, since it is on the left.
Unit 2 Algebraic Reasoning, Grade 5
Students should rewrite the expression below, replacing 7 + 2 with 9.

Using the order of operations, what should be evaluated next?
The expression inside of the parentheses (9 – 4) should be evaluated next.
Students should rewrite the expression below, replacing 9 – 4 with 5.

Using the order of operations, what should be evaluated next?
Multiplication and division come before addition and subtraction, so 3 x 5 should be
evaluated next.
Students should rewrite the expression below, replacing 3 x 5 with 15.
Lastly, students should solve 15 + 2 to find the solution.
What’s the teacher doing?
What are the students doing?
As students use their new knowledge to
understand the meaning of parentheses/ brackets
and how to simplify expressions using the order
of operations, ask guiding questions:
 What is the meaning of parentheses
Writing definitions and notes in their math
journals
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Following along in their math journals as they
learn by example
Unit 2 Algebraic Reasoning, Grade 5




and brackets in a numeric
expression?
How can symbolic notations of
groupings be useful when simplifying
numeric expressions?
What is the process that can be used
to simplify a numeric expression?
What is the order of operations?
What is the purpose of the order of
operations?
Asking and answering questions
Phase 4: ELABORATE
Day 9/ Day 10
Materials:
 Handout: “Representing Expressions”
 Handout: “Find the Missing Number”
 Handout: “Order of Operations”
Activities:
Who Can Create the Best Word Problem Story?
Divide the class into 3 groups. Give each group an equation.
 1. 8 × 4 + n = 49
 2. (8 x 6) ÷ g = 12
 3. 17= (126 - t) ÷ 4
Challenge each group to a classroom contest in creating the best word problem that could be solved
by their equation. The word problem must be able to be solved by the equation. Each group must
also solve their equation for the unknown value. The teacher can “judge” which group came up
with the best word problem story, or invite another teacher to be the “judge”.
Representing Expressions:
In this exercise, students can practice representing 10 numerical expressions using the handout
“Representing Expressions”.
Find the Missing Number:
In this exercise, students can practice solving 16 equations for n, the unknown value, on the
handout “Find the Missing Number”. (This handout was obtained from math-aids.com at the
following link: http://www.mathaids.com/Mixed_Problems/Mixed_Problems_Missing_Numbers.html)
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Unit 2 Algebraic Reasoning, Grade 5
Simplify Expressions Using the Order of Operations:
In this exercise, students can practice simplifying 10 expressions using the order of operations on
the handout “Order of Operations”. (This handout was obtained from math-aids.com at the
following link:
http://www.mathaids.com/cgi/pdf_viewer_7.cgi?script_name=order_of_operations.pl&skill=0&type=4&language=
0&memo=&answer=1&x=173&y=27 )
Today’s Date:
Students can take turns creating an equation for the date, d, each day to write on the board. For
example, if the date is September 12, a student might write September (3 x d) = 36. The rest of
the students in the class would solve the equation to find that the date for the day, d, equals 12.
What’s the teacher doing?
What are the students doing?
Encouraging students to apply and extend the
concepts they have learned
Applying and utilizing all learned material in a
new way as they complete each elaboration
activity
Phase 5: EVALUATE
Day 11
Materials:
 Performance Assessment 01
Activity:
During days 11 and 12, students will complete Unit 2 Performance Assessments 01 and 02. The
Performance Assessments may be obtained from http://tcmpc.org/ .
Grade 5 Unit 02 PA 01
Analyze the problem situation(s) described below. Organize and record your work for
each of the following tasks. Using precise mathematical language, justify and explain each
solution process.
Mia and Thomás were discussing a famous number pattern called Fibonacci’s number
sequence. The first ten numbers in the sequence are as follows:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
1) Mia started to notice that some of the numbers were prime and some were composite.
She told Thomás that there were 6 prime numbers and 4 composite numbers in the first
ten numbers in the sequence. Thomás disagreed.
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Unit 2 Algebraic Reasoning, Grade 5
a) Design a detailed plan that Thomás can use to identify the prime and composite
numbers in the sequence using square tiles.
b) Design a detailed plan Mia can use to identify the prime and composite numbers in the
sequence using an organizational factor list.
c) Analyze and explain the relationship between the two representations to identify the
prime and composite numbers in the sequence and describe the advantages and
disadvantages of both representations.
d) Identify which numbers in the first ten numbers in Fibonacci’s sequence are prime or
composite, and explain if Thomás was correct to disagree with Mia’s statement.
Day 12
Materials:
 Performance Assessment 02
Activity:
Grade 5 Unit 02 PA 02
Analyze the problem situation(s) described below. Organize and record your work for
each of the following tasks. Using precise mathematical language, justify and explain each
solution process.
Spring Elementary School was collecting aluminum cans to recycle for money.
1) Adolfo brought in 128 pounds of cans each of the 9 months of school. Chris brought in
one big bag containing 1,474 pounds. Laura brought in 1,863 pounds, but when she got to
school, she realized that 237 pounds of the cans were actually steel cans and not
aluminum.
a) Write a numeric expression to determine the total number of pounds of aluminum cans
brought in by Adolfo, Chris, and Laura.
b) Describe the process that could be used to simplify the numeric expression
representing total number of pounds of aluminum cans brought in by the three students.
2) The teacher invited four parents to help bring all of the aluminum cans brought in by
Adolfo, Chris, and Laura to the recycling center. They divided all the aluminum cans
evenly among the four vehicles
a) Using the numeric expression representing total number of pounds of aluminum cans
brought in by the three students, write and solve an equation to determine, c, the number
of pounds of aluminum cans each vehicle needed to carry.
b) Explain the meaning of the grouping symbols used in the expression and why they are
used.
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Unit 2 Algebraic Reasoning, Grade 5
What’s the teacher doing?
What are the students doing?
Assessing student understanding
Demonstrating understanding of new concepts
learned
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