Unit 2: Algebraic Reasoning
5th Grade
5E Lesson Plan Math
Grade Level: 5
Lesson Title: First six weeks, Unit 2: Algebraic Reasoning
Subject Area: Math
Lesson Length: 12 days
(There are 2 parts: Part 1= Prime and Composite Numbers.
Part 2= Algebra)
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…
-be introduced to the properties of prime and composite numbers.
-access prior knowledge of area models and arrays to represent products
-scaffold understanding of prime and composite numbers through prior knowledge and analyze their factors.
-examine the meaning of grouping symbols within a numeric expression and simplify numerical expressions based on socially
constructed conventions.
-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 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,
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Unit 2: Algebraic Reasoning
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including mental math, estimation, and number sense as appropriate, to solve problems.
5.1D Communicate mathematical ideas, reasoning, and their implications using multiple representatives, 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 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 , ELPS.c.2D , ELPS.c.3C , ELPS.c.3D , ELPS.c.4H , ELPS.c.5B , ELPS.c.5F, , ELPS.c.5G
Misconceptions:
-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:
-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).
Vocabulary:
composite number- a whole number with more than two factors
counting (natural) numbers - the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ...,
n}
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dividend- 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 number(s), unknown(s), and/or 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 facors 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:
array, difference, equal, factor list, factor pair, grouping, multi-step, operation, operator, remainder, simplify, solve, sum, unknown
List of Materials:
Day 1
Math journal
Handout: Divisibility Rules
Handout: Am I a Factor?
Handout: Find the Factor Practice
Handout: Common Factor Practice Part 1
Handout: Common Factor Practice Part 2
Day 2
Color tiles (30 per pair of students)
Math journal
Handout: Factor Pair Area Model Practice
Day 3
Color tiles
YouTube video: http://www.youtube.com/watch?v=V08g_lkKj6Q
Handout: Sieve of Eratosthenes
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Day 4
Math journal
Prime and composite number game: http://www.sheppardsoftware.com/mathgames/numbers/fruit_shoot_prime.htm
Poster board or bulletin board paper
Markers
Day 5
Performance Assessment
Day 6
Handout: What’s My Rep? Cards (Prepare prior to instruction for every 2 students- copy on cardstock, laminate, and cut apart)
Handout: What’s My Rep? Recording Sheet
Day 7
3 index cards (per student)
Scissors (1 pair per student)
Teacher resource: Stamp Collection Problem
Math journal
Handout: Diagram and Equation Practice (Addition and Subtraction)
Teacher resource: Basketball Problem
Whiteboards (1 per student)
Dry erase markers (1 per student)
Handout: Diagram and Equation Practice (Multiplication and Division)
Day 8
Whiteboards (1 per student)
Dry erase markers (1 per student)
Teacher resource: Trading Cards Problem
Handout: Combined Equations Practice (save to discuss on Day 9)
Day 9
Handout: Combined Equations Practice (from Day 8)
Handout: PEMDAS Practice
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Day 10
Order of Operations Game: http://www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html
Handout: PEMDAS Practice 2
Day 11
Teacher resource: Express Line
Handout: Express Line
Teacher resource: Processing Tables
Handout: Processing Tables
Handout: Real-world Expressions and Equations
Day 12
Performance Assessment
INSTRUCTIONAL SEQUENCE:
Part 1: Prime and Composite Numbers (Days 1-5),
Part 2: Algebra (Days 6-12).
Part 1: Prime and Composite Numbers
Day 1
Phases: ENGAGE, EXPLORE, EXPLAIN
Day 1
Materials:
Math journal
Handout: Divisibility Rules
Handout: Am I a Factor?
Handout: Find the Factor Practice
Handout: Common Factor Practice Part 1
Handout: Common Factor Practice Part 2
Activity:
Phase: ENGAGE-Students use logic and reasoning skills to review and identify factors.
1.
Display the following numbers for the class to see: 1, 2, 3, 4, 6, 12. Place students in pairs and have them talk with their partner
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Unit 2: Algebraic Reasoning
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about what these numbers have in common. They may write ideas down in their math journals. Allow time for student
discussion/writing. Whole group, discuss the student pairs' ideas.
What’s the teacher doing?
Walking the room to listen to students' prior knowledge/ideas,
assisting as needed.
As a whole group, ask the following questions:






What are the students doing?
Discussing the numbers with their partners.
Recording ideas in their journals.
Answering questions that the teacher poses to the class.
What do the numbers have in common? Answers may
vary.
What is a factor? (A number multiplied by another
number to find a particular product.)
How can you prove that each of the numbers is a
factor of 12? Answers may vary. 12 is divisible evenly by
each of the numbers; when I multiply certain pairs of the
numbers, I get 12; etc.
What does it mean to find the factors of a number?
Answers may vary. A factor is a number multiplied by
another number to find a particular product. To find the
factors of a number means to find numbers that can be
multiplied together to get that product; etc.
What numbers from the list can be multiplied together
to make the product 12? (1 x 12, 2 x 6, 3 x 4). These are
called factor pairs.)
What is the difference between listing the factors and
factor pairs? Answers may vary. When you list the
factors, they are all the numbers in pairs but written
together (1, 2, 3, 4, 6, 12). When you are listing the factor
pairs, they are listed together with a multiplication symbol
between the pair (1 x 12, 2 x 6, 3 x 4); etc.
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2.
Display all the pairs of factors for 12 for the class to see.
1 x 12
2x6
3x4
Explain to students that 2 x 6 and 6 x 2 are not both listed in
this situation because they represent the same two factors
and that each pair of numbers is called a factor pair.
Allow time for students to determine if any other whole
numbers are missing. Monitor and check for understanding.
Discuss the strategies used to determine all the factors.
Ask:

How do you know that there are not more pairs of
whole numbers whose product is 12? Answers may
vary. Whole numbers between 1 and 12 were checked
until factors began to repeat: 1 x 12, 2 x 6, 3 x 4, 4 x 3, 6 x
2  factors are now repeating; etc.
Review with the class the various ways to find the factors of a
number.
(Organizational Factor Lists):
1
2
3
6
7
14
21
42
Go on to discuss other ways to show the factors of a number.
Factor Table:
42
1
2
3
6
42
21
14
7
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Factor T-Chart:
42
1 42
2 21
3 14
6 7
Array:
7 by 6 = 7 x 6
Topics:
 Factor pairs
You can use the following handouts for students that are
struggling and need a reteach/review:
Divisibility Rules, Am I a Factor?, Find the Factor
Practice and Common Factor Practice Part 1 and 2.
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Unit 2: Algebraic Reasoning
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Day 2
Phases: ENGAGE, EXPLORE, EXPLAIN
Day 2
Materials:
Color tiles (30 per pair of students)
Math journal
Handout: Factor Pair Area Model Practice
Activity:
Phases: EXPLORE, EXPLAIN- Students investigate and discuss a real-life problem situation. Students identify factor pairs in order to
solve the problem. Students are introduced to the concept of prime and composite numbers.
What’s the teacher doing?
EXPLORE and EXPLAIN
What are the students doing?
Students investigate and discuss a real-life problem situation.
Students identify factor pairs in order to solve the problem.
Identifying factors.
1. Prior to instruction, create a Bag of Color Tiles for every 2
students by placing 30 color tiles in a plastic zip bag.
2. Place students in pairs and distribute a Bag of Color Tiles to
each pair.
Display the following question for the class to see:
A fifth grade class is preparing for a class
party. Twenty-four cupcakes are to be served at the
party. The cupcakes can be arranged in a single
row or in equal rows on the buffet table. How many
different possible arrangements can the 24
cupcakes be arranged?
Solving the problem using their tiles.
3.
Ask:
 How can you use the color tiles to represent the
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


solutions to this problem? Answers may vary. I can
create all the possible arrangements by building the rows
that 24 cupcakes can be arranged in; etc.
Can the cupcakes in this problem be arranged in one
row? (yes)
What would that look like? (1 row with 24 cupcakes
(tiles) in a row)
Is it possible for the cupcakes to be arranged in five
even, or equal rows? Why or why not? (no) Answers
may vary. The rows have to all have the same number of
students and 24 cupcakes cannot be split evenly into 5
rows; etc.
4. Instruct student pairs to use their color tiles to demonstrate all
the possible ways the cupcakes could be arranged and record
a model and description of each rectangular model built in
their math journals. Allow time for students to find and record
all the possible arrangements. Monitor and assess student
pairs to check for understanding. Have a class discussion
about the different arrangements for the problem situation.
Ask:
 How many different area models did you form with the
24 tiles, representing the rows for 24 cupcakes? (8
area models)
 What are they? (1 x 24, 2 x 12, 3 x 8, 4 x 6, 24 x 1, 12 x 2,
8 x 3, and 6 x 4)
 How many ways can the cupcakes be arranged on the
party table? Explain. (8 ways, because there are 8
different area models or arrangements.)
 What are some things the students might consider
when deciding which arrangement to use? Answers
may vary. How big the table is; etc.
 What are the factors of 24? 1, 2, 3, 4, 6, 8, 12, 24
 How can you be sure that you have listed all of the
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
factors of 24? Answer may vary. None of the other
numbers less than 24 will work evenly; every number in
our list has its partner; etc.
How many factor pairs can be made from the factors
of 24? Explain. (4 pairs (1 x 24, 2 x 12, 3 x 8, and 4 x 6),
because those are the pairs of numbers that can be
multiplied to make 24. Even though the representation for
this problem situation is different, 24 x 1, 12 x 2, 8 x 3, and
6 x 4 do not need to be listed because they have the same
2 factors as a pair already listed.)
5. Distribute handout: Factor Pair Area Model Practice to each
student. Instruct students to complete the handout. Remind
students that they may use their Bag of Color Tiles, if needed.
Allow time for students to complete the activity. Monitor and
assess students to check for understanding.
Complete "Factor Pair Area Model Practice".
Students create arrays for prime numbers (they don't know what
prime means yet) and then discuss the difference between the
arrays for prime numbers and composite numbers with teacher.
Write the following numbers on the board: 5, 17, 23.
In pairs, have students create an array for the following
numbers: 5, 17, 23.
Ask :
*What is different about these numbers and their arrays?
They only have one factor pair, and only one possible array
(vertical/horizontal).
Explain that numbers with only one factor pair are called
Prime numbers.
Tell them that numbers that have more than one factor pair
are called Composite. Tell students that they will learn more
about these numbers tomorrow.
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Unit 2: Algebraic Reasoning
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Part 1: Prime and Composite Numbers
Day 3
Phases: EXPLORE, EXPLAIN
Day 3
Materials:
Color tiles
YouTube video: http://www.youtube.com/watch?v=V08g_lkKj6Q
Handout: Sieve of Eratosthenes
Activity:
Phases: EXPLORE, EXPLAIN- Introduce the concept of prime and composite numbers.
Remind students about the arrays they made yesterday for 5, 17, What are the students doing?
and 23, and ask :
*What are these special types of number called that only
have one factor pair? Prime numbers.
Provide students with color tiles and have them create arrays for
the following numbers in pairs:
Students are working with their tiles to create arrays and sketching
the results.
4, 7, 13, 15, 18, 25, 29
Monitor students to check for understanding.
Once students have completed their arrays, tell them to separate
their numbers' arrays in two groups: 1) those who have more than
one possible array, and 2) those who have only one possible
array (Vertical/horizontal format does not matter. i.e: 1 X 7 is the
same as 7 X 1).
Have students sketch their arrays (separated into 2 groups) by
Students come up with their own definitions for Prime and
those numbers that are Prime, and those that are Composite. Ask Composite numbers and record them in their math journals.
students to turn and talk to their partner about creating a good
definition for Prime numbers and Composite numbers. Give them
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Unit 2: Algebraic Reasoning
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a few moments to discuss with their partners and then they can
share with the class.
Tell students that there is a cool way to find out if a number is
Prime or Composite.
Display the video (linked below) to your students about the Sieve
of Eratosthenes:
http://www.youtube.com/watch?v=V08g_lkKj6Q
Have students complete the handout: Sieve of Eratosthenes.
Students complete the "Sieve of Eratosthenes".
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Part 1: Prime and Composite Numbers
Day 4
Phases: EXPLORE, EXPLAIN, ELABORATE
Day 4
Materials:
Math journal
Prime and composite number game: http://www.sheppardsoftware.com/mathgames/numbers/fruit_shoot_prime.htm
Poster board or bulletin board paper
Markers
Activity:
Phases: EXPLORE, EXPLAIN, ELABORATE- Find factor pairs by using factor trees, t-charts, factor tables, and rainbow factoring.
What is the teacher doing?
What are the students doing?
Draw an array for the number 5.
Ask: What does it mean if I can only create an array for a
number that has only 1 row? It means it only has 1 factor pair,
1 and itself.
If a number has only 1 factor pair it is called a prime number.
Show some other examples of prime number arrays. Have
students take a few moments to come up with their own
examples in their journals, and then share with the class.
Students come up with examples of prime number arrays in their
journals.
Use the number 42, demonstrate finding its factors, but this time
with a Factor Tree:
Students review their own definitions for prime and composite in
their math journals.
Explain that a factor tree is special, because it allows us to see a
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Unit 2: Algebraic Reasoning
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number's prime factors. Take a moment to define and discuss
prime and composite.
Prime denotes a number that has exactly 2 factors, 1 and itself.
Composite denotes a number that has 3 or more factors.
The bottom row of a factor tree (if done correctly) should contain
only prime factors of a number. When multiplied together, the
prime factors should equal the number being factored.
Discuss some other "special numbers":
- 2 is the only even prime number
- 1 is the only common factor in all of the factor pairs of prime
numbers. 1 is neither prime nor composite.
- 0 is neither prime nor composite
Have students complete a journal entry in their math journals to
compare and contrast prime and composite numbers. Tell them
that they may illustrate and provide examples with their writing.
Provide students with time to share their writing with their table
groups.
Students write a journal entry comparing and contrasting prime and
composite numbers, then share with their table group.
If there is extra time, students can go to the website:
http://www.sheppardsoftware.com/mathgames/numbers/fruit
_shoot_prime.htm
Students play the online game.
and play the prime and composite number game.
Review the various ways to find factors of a number. Explain to
students that in each way (Organizational Factor Lists) you can
determine if a number is prime or composite:
In a T-Chart, a number will only have one pair of factors:
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Unit 2: Algebraic Reasoning
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11
1
11
A factor table is the same way:
11
1
11
In Rainbow Factoring:
5
1
5
Students can complete the assignment: Create a poster to
demonstrate how to prove whether a number is prime or
composite (see details at right).
Have students create a poster demonstrating ways to prove
whether a number is prime or composite. They may use any of the
Organizational Factor Lists, Factor Tree, Arrays, drawings, etc.
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Unit 2: Algebraic Reasoning
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Part 1: Prime and Composite Numbers
Day 5
Phases: EVALUATE
Day 5
Activity:
Phase: EVALUATE- Students complete the Performance Assessment from the IFD.
Analyze the problem situation 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 Thomas 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 Thomas that there were 6 prime
numbers and 4 composite numbers in the first ten numbers in the sequence. Thomas disagreed.
a) Design a detailed plan that Thomas can used to identify the prime and composite numbers in the sequence using square tiles.
b) Design a detailed plan that 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 numbers sequence are prime or composite, and explain if Thomas was
correct to disagree with Mia's statement.
What is the teacher doing?
What are the students doing?
Monitor students as they work on the Performance Assessment Completing the Performance Assessment.
to determine if any re-teaching is necessary prior to the unit
assessment(s).
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Unit 2: Algebraic Reasoning
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Part 2: Algebra
Day 6
Phases: ENGAGE, EXPLORE, EXPLAIN
Day 6
Materials:
Handout: What’s My Rep? Cards (Prepare prior to instruction for every 2 students- copy on cardstock, laminate, and cut apart)
Handout: What’s My Rep? Recording Sheet
Activity:
Phases: ENGAGE, EXPLORE, EXPLAIN- Students are introduced to algebraic expressions and equations by reminding them that
there are often multiple ways to say the same thing. Students investigate and discuss that there are many ways to represent the same
mathematical situation.
What is the teacher doing?
1. Display a series of icons/symbols/pictures for the class to
see (on projector/document camera/smartboard). Some
examples might include a stop sign or other traffic signs,
traffic light, text messaging emoticons, etc.
What are the students doing?
;-p
Ask students to identify what each symbol/picture means.
Ask:
 What's the point of saying the same thing more than
one way? Answers may vary.
 Why would you need to know there are so many
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Unit 2: Algebraic Reasoning
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words or phrases that mean the same thing?
Answers may vary. Not all people use the same words to
express themselves; etc.
1. Prior to instruction, create a class resource: What’s My
"What's My Rep?"
Rep? Cards for every 2 students by copying on cardstock,
cutting apart, laminating, and placing in a plastic zip bag.
2. Remind students that there are many ways to say the same
thing. Explain to students that this is true for most spoken
languages as well as for the language of mathematics. When
it comes to equations, there are many ways to represent the
same mathematical situation.
3. Place students in pairs and distribute a class resource:
What’s My Rep? Cards to each pair and handout: What’s
My Rep? Recording Sheet to each student.
4. Instruct student pairs to match the problem situation cards
from class resource: What’s My Rep? Cards to the
appropriate equations and diagrams and then record their
matches on their handout: What’s My Rep? Recording
Sheet. Allow time for students to determine how to match
the cards. Remind students that they may turn over all the
cards at once and begin matching, or they may separate the
cards into piles and start with the problem card and then try
to match; however, regardless of the way the students begin
matching, all groups must have 1 person read each problem
situation aloud to their partner before trying to match the
representations to it. Monitor and assess student pairs to
check for understanding. Facilitate a class discussion about
the activity.
Ask:
 How did you know this equation/diagram went with
this problem situation? Answers may vary. I looked for
the cards that had the same numbers; I looked for
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Unit 2: Algebraic Reasoning
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

pictures that showed splitting, adding on; etc.
How did you know you were correct when you
selected a particular equation/diagram to go with this
problem? Answers may vary. My partner and I agreed
on the cards; I solved the problem and all 4 cards
showed the same answer; etc.
What does this matching activity tell you about word
problems? Answers may vary. Word problems can have
more than 1 representation; a model for a word problem
can help us find the appropriate equation because it
shows what is happening in the story, and helps us solve
the problem more accurately and easily; etc.
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Unit 2: Algebraic Reasoning
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Part 2: Algebra
Day 7
Phases: EXPLORE, EXPLAIN
Day 7
Materials:
3 index cards (per student)
Scissors (1 pair per student)
Teacher resource: Stamp Collection Problem
Math journal
Handout: Diagram and Equation Practice (Addition and Subtraction)
Teacher resource: Basketball Problem
Whiteboards (1 per student)
Dry erase markers (1 per student)
Handout: Diagram and Equation Practice (Multiplication and Division)
Activity:
Phases: EXPLORE, EXPLAIN- Students explore and diagram mathematical equations algebraically using words, letter, and/or labels.
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Unit 2: Algebraic Reasoning
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What's the teacher doing?
What are the students doing?
1. Distribute 3 index cards and a pair of scissors to each student. Instruct students to cut 2 of
their index cards in half and label the remaining uncut index card with the word “whole” at the
top of the card. Instruct students to label 3 of the cut index cards with the following: "=,"
"part," and "part," as shown below. Explain to students that the second half of the second
card will be left blank for students to determine and record the operation needed to solve the
problem.
Whole
Part
Part
=
2. Remind students that a Part-Part-Whole Model can be used to represent the numbers in an
equation or word problem and shows the relationships the numbers have with each other.
Instruct students to place the word labels on the upper part of each card to leave room for
the numbers in the problem.
Part
Part
Whole
3. Display teacher resource: Stamp Collection Problem. Invite a student volunteer to read the
problem aloud.
4. Instruct students to examine the information from the problem on the displayed teacher
resource: Stamp Collection Problem. Facilitate a class discussion about how to fill in the
Part-Part-Whole Model.
Ask:
 What information do you already know from the problem? Where would you record
that information? (I know that Marco now has 122 stamps, and that is the whole. I also
know that he added 15 stamps to his collection and that is 1 part.)
 What are trying to find? How would you record that? (I am trying to find out how
many stamps Marco had in the beginning. That is our missing information. I could write a
question mark to show that we don't know the answer yet.)
5. Instruct students to use the information from the displayed teacher resource: Stamp
Collection Problem to record the known information and a question mark on the second
"Stamp Collection Problem"
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Unit 2: Algebraic Reasoning
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“part” card.
Part
Part
15
stamps
?
Whole
122 stamps
6. Instruct students to rearrange their cards to form an equation for the problem, recording the
appropriate operation symbol on the blank index card. Facilitate a class discussion to debrief
student solutions.
Whole
Part
__
122 stamps
Part
Part
Part
15
stamps
?
?
Whole
Part
122 stamps
15
stamps
+
+
Whole
122 stamps
15
stamps
Part
15
stamps
=
Part
?
=
Part
=
Whole
122 stamps
Whole
122 stamps
Part
__
?
?
Part
=
15
stamps
Ask:
 How can both an addition equation and a subtraction equation be used to
represent this problem? Answers may vary. You can either take away the 15 that he
added to his collection or think about the problem like he had "some," then added 15
more; etc.
 What does the question mark represent in this equation? (The number of stamps
Marco had in his collection in the beginning, the same thing that it represented in the
diagram.)
 Is there any other way to show the missing information other than with a question
mark? Explain. (yes) Answers may vary. I could use an empty box or a letter to
represent the information we are looking for; etc.
7. Instruct students to record their diagrams with a letter, instead of a question mark, in their
math journal (e.g., 122 – 15 = y, 122 - y = 15, 15 + y = 122, or y + 15 = 122). Explain to
students that letters (any letters) in equations represent numbers or values. Allow time for
students to solve the equation and complete their diagram. Monitor and assess students to
23
Unit 2: Algebraic Reasoning
5th Grade
check for understanding. Facilitate a class discussion about the diagrams and equations.
Ask:
 What is the value of y in the stamp collection problem? How do you know? (107,
because 122 - 15 = 107 or 15 + 107 = 122.)
 How does a diagram help you decide which operation to use to solve a problem?
Answers may vary. The diagram helps me organize the information and decide what the
problem is asking. It helps me decide what the “whole” is and what is happening to the
whole, or how to get to the whole. A model helps me set up an equation I can use to find
the solution properly; etc.
8. Instruct students to turn their index cards over to start a new problem.
9. Present the following scenario to the class: "Marco has 107 stamps in his collection and
added 15 more stamps. How many stamps does he have now?" Instruct students to create a
diagram and equation to match the new problem.
Part
Part
107
stamps
15
stamps
Whole
Part
107
stamps
+
Part
15
stamps
=
Whole
122 stamps
122 stamps
Allow time for students to fill in the given information and rearrange their index cards. Monitor
and assess students to check for understanding. Facilitate a class discussion about the
differences between this problem scenario and the first problem scenario presented.
Ask:
 What is the known information in this problem? (We know the 2 parts of the problem
and need to find the whole.)
 What could you put in place of the whole if you do not know the solution? Answers
may vary. A box, a question mark, or a letter; etc.
10. Instruct students to record a letter and a label on the “whole” index card such as: “y stamps,”
then record their diagram and equation in their math journals.
Ask:
 Is the part-part-whole diagram the only way to represent an addition or subtraction
equation? How do you know? (no) Answers may vary. Other diagrams could include
number lines, charts, tables; etc.
11. Distribute handout: Diagram and Equation Practice (Addition and Subtraction) to each
student as independent practice or homework.
“Diagram and Equation Practice"
24
Unit 2: Algebraic Reasoning
5th Grade
------------------------------------------------------------------------------------1. Distribute a whiteboard and dry erase marker to each student.
2. Display teacher resource: Basketball Problem. Invite a student volunteer to read the
problem aloud.
3. Instruct students to use their whiteboards and dry erase markers to create a diagram that
could be used to solve this problem, using the letter “t” to represent the value of the unknown
in this problem. Remind students to label their diagrams to facilitate their understanding of
the problem. Allow time for students to complete their diagram. Monitor and assess students
to check for understanding.
4. Place students in pairs. Instruct students to share their diagrams with their partner. Facilitate
a class discussion about the diagrams drawn.
5. Invite a student who created a part-part-whole diagram to model the problem to share their
diagram with the class. Facilitate a class discussion about the part-part-whole diagram.
Ask:
 How could you use a part-part-whole diagram to model this problem? Answers may
vary. Separating the whole (54) into 6 equal parts; etc.
t
t
t
t
t
"Basketball Problem"
t
54 players

Why is the whole separated into 6 equal parts? (because there are 6 teams with an
equal number of players on each team)
 What does the “t” represent in this diagram? (the number of players on each team)
 Is the part-part-whole diagram the only way to represent this problem? Explain (no)
Answers may vary. This is a division problem, so I could also use an area model or a
number line; etc.
6. Facilitate a class discussion about why students selected a particular model.
Ask:
 Which model(s) seemed to work best for problems involving small numbers? Why?
Answers may vary.
 Which model(s) seemed to work best for problems involving larger numbers?
Why? Answers may vary.
25
Unit 2: Algebraic Reasoning
5th Grade

Which model(s) did not seem to work best for problems involving larger numbers?
Why? Answers may vary.
 What types of information is needed to be able to use each model? Answers may
vary.
7. Instruct students to use their diagram to write an equation on their whiteboards that could be
used to solve the basketball problem and record their diagram and equation in their math
journals. Allow time for students to complete their equations. Monitor and assess students to
check for understanding. Facilitate a class discussion to debrief student solutions.
Ask:
 What are possible equations that could be used to solve this problem? (54 ÷ 6 = t,
54 ÷ t = 6, 6 x t = 54, t x 6 = 54)
 What does t represent in your equation? (the number of players on each team)
 How many players are on each team? How do you know? (9, because 54 ÷ 6 = 9 and
9 x 6 = 54)
8. Remind students that they have used index cards to change their subtraction problems into
addition problems.
Ask:
 How could you reword the basketball problem to make it a multiplication problem,
where you are trying to determine the total number of players? Answers may vary.
There are 6 teams of basketball players with 9 players on each team. How many
basketball players are there altogether?
Record this problem for the class to see, preferably underneath or near the displayed
teacher resource: Basketball Problem.
 Could you create a diagram and equation to represent a multiplication situation for
the new basketball problem? Explain. (yes) Answers may vary. I could use a part-partwhole diagram or an area model; etc.
 If you used a part-part-whole model, what would “t” represent in the problem?
Explain. (It would represent the whole or total number of players.)
 How many equal parts would make up the whole? How do you know? (6, because
there are 6 teams)
9. Instruct students to use the information from the new basketball problem to create a partpart-whole diagram on their whiteboards and then discuss their diagrams with their partner.
Allow time for students to complete their diagrams. Monitor and assess students to check for
26
Unit 2: Algebraic Reasoning
5th Grade
understanding. Facilitate a class discussion about the diagrams created.
Ask:
 How do your diagrams differ for the multiplication problem than for the division
problem? Answers may vary, but could include how the diagrams now show a known
value for the equal groupings, whereas for the division diagram, the value of each
grouping was unknown; etc.
9
9
9
t
9
9
9
total players

What equation is represented by this diagram? How do you know? (6 x 9 = t or 9 x 6
= t) Answers may vary, but could include because I have 6 groups with 9 in each group
and that equals the total number of players; etc.
10. Instruct students to record their multiplication situation, part-part-whole diagram, and
corresponding equation in their math journals.
11. Instruct student pairs to generate a different diagram that could be used to represent this
problem. Allow time for students to complete their diagrams. Monitor and assess students to
check for understanding. Facilitate a class discussion about the diagrams created and why
students selected a particular model.
Ask:
 Which model(s) seemed to work best for problems involving small numbers? Why?
Answers may vary.
 Which model(s) seemed to work best for problems involving larger numbers?
Why? Answers may vary.
 Which model(s) did not seem to work best for problems involving larger numbers?
Why? Answers may vary.
 What types of information is needed to be able to use each model? Answers may
vary.
“Diagram and Equation Practice
12. Distribute handout: Diagram and Equation Practice (Multiplication and Division) to each (Multiplication and Division)”
student as independent practice or homework.
27
Unit 2: Algebraic Reasoning
5th Grade
Discuss the difference between equation and expression as a class.
Equation
Something = Something Else.
An expression can be contained in an equation.
An expression combines numbers and symbols, but does not contain an equal sign (there
is no balance). An equation contains expressions that are separated by an equal sign.
Part 2: Algebra
Day 8
Phases: EXPLORE, EXPLAIN
Day 8
Materials:
Whiteboards (1 per student)
Dry erase markers (1 per student)
Teacher resource: Trading Cards Problem
Handout: Combined Equations Practice (save to discuss on Day 9)
Activity:
Phases: EXPLORE, EXPLAIN- Students investigate the order of operations, its purpose, and the importance of parentheses and
brackets.
28
Unit 2: Algebraic Reasoning
5th Grade
What is the teacher doing?
Students:
1. Distribute a whiteboard and dry erase marker to each student.
2. Record the following pair of equations for the class to see:
16 – 4 + 2 = y
16 – (4 + 2) = y
Ask:
 How are these equations different? (one has parentheses; one does not)
 What does it mean when an expression or equation has parentheses? (The
operations in parentheses must be completed first.)
3. Instruct students to use their whiteboard and markers to write and find the value of
each equation. Allow time for students to solve the equations. Monitor and assess
students to check for understanding. Facilitate a class discussion to debrief student
solutions.
Ask:
 What is the value of y in the first equation? (14)
 What is the value of y in the second equation? (10)
 Does inserting the parentheses around the 4 + 2 change the value of the
equation? How do you know? (yes) Answers may vary. In the first equation,
only 4 is being removed from 16, but in the second equation where you have to
add 4 and 2 first, 6 is being removed from 16; etc.
4. Place students in pairs. Record each of the following practice problems for the class
to see. Instruct students to use their whiteboards and dry eraser marker to find the
solution to each equation and share their solution with their partner. Allow time for
students to solve each equation. Monitor and assess students to check for
understanding. Facilitate a class discussion to debrief student solutions.
Discussion.
7 x 10 + 2 = a
24 – 2 + 5 = a
5 + 14 – 3 = a
7 x (10 + 2) = a
24 – (2 + 5) = a
5 + (14 – 3) = a
29
Unit 2: Algebraic Reasoning
5th Grade
What is the teacher doing?
What are the students
doing?
7 x 10 + 2 = a
24 – 2 + 5 = a
5 + 14 – 3 = a
7 x (10 + 2) = a
24 – (2 + 5) = a
5 + (14 – 3) = a
Ask:
 Was the solution the same for each pair of problems? If not, which pairs had the same
solution and which pairs had different solutions? (No, the first 2 pairs had different
solutions, and the last pair had the same solution of 16.)
 Why do you think the solution to the last pair was the same? Answers may vary. The total
being removed from the problem did not change, and when adding, the sum does not depend
on the order of addends (e.g., 2 + 4 = 6 and 4 + 2 = 6); etc.
5. Display teacher resource: Trading Cards Problem. Instruct students to use their
whiteboards and dry erase markers to model the problem using the letter “C” to represent the
value of the unknown. Remind students to label their diagrams to facilitate their understanding
of the problem.
Trading cards he
started with
4
Pack 1
8
Pack 2
8
"Trading Cards Problem"
Pack 3
8
C: Trading cards altogether
30
Unit 2: Algebraic Reasoning
5th Grade
Trading cards he
started with
4
Pack 1
8
Pack 2
8
Pack 3
8
C: Trading cards altogether
Allow time for students to complete their diagram. Monitor and assess students to check for
understanding. Facilitate a class discussion about the diagrams created,
6. Instruct students to use their diagram to write an equation on their whiteboards that could be used
to solve the problem situation and record their diagram and equation in their math journals.
Ask:
 What does “C “represent in this problem? (The number of trading cards altogether.)
 What equation did you write to represent this problem situation? Answers may vary. 4 +
(3 x 8) = C, (3 x 8) + 4 = C; etc.
 What is the value of C in this equation? (28)
7. Invite a student who created a table to model the problem to share their diagram and explain
how the table represents the problem situation. If no student used this model, introduce it as an
alternate model to the class. Facilitate a class discussion about this alternate model.
Ask:
 Why did you start with “0” card packs? Answers may vary. Even though he had 4 cards to
begin with, he started out with “0” packs of cards. So, he had no packs, but 4 extra cards
needed to be recorded”; etc.
 How did you know the number of cards was 12 when he had 1 card pack? Answers may
vary. There were 8 cards in each pack and I added 8 to the original 4 cards he already had;
etc.
8. Distribute handout: Combined Equations Practice to each student. Instruct students to
complete the handout. Allow time for students to complete the activity. Monitor and assess
students to check for understanding. Facilitate a class discussion to debrief student solutions, as
needed.
"Combined Equations
Practice"
31
Unit 2: Algebraic Reasoning
5th Grade
TEACHER NOTE
Alternate model:
Number of
Number of
Card Packs
Cards
0
4
1
12
2
20
3
28
Part 2: Algebra
Day 9
Phases: EXPLORE, EXPLAIN
Day 9
Materials:
Handout: Combined Equations Practice (from Day 8)
Handout: PEMDAS Practice
Activity:
Phases: EXPLORE, EXPLAIN- Students investigate the order of operations, its purpose, and the importance of parentheses and
brackets.
32
Unit 2: Algebraic Reasoning
5th Grade
Teacher:
Review "Combined Equations Practice" and discuss answers.
Students:
Review Order of Operations
Discuss the significance of brackets. Explain that brackets are used to group and define order as
well, with innermost parentheses being doing first in order of operations:
F = 2 [ 5 + (9÷3)]
In this equation above, the innermost parentheses is done first (9÷3). Then everything left in the
brackets is done [5 + 3]. Then multiply 2 times that answer: 2 x 8= 16.
F= 16
Teach students about PEMDAS:
Parentheses
Exponents (explain they will learn about this later)
Multiplication
Division
Addition
Subtraction
They can remember the acronym with the phrase: "Please excuse my dear Aunt Sally".
When we solve an equation, we are "simplifying" its numeric expression(s).
Ask students:
Why is the order of operations important? It helps us know in what order to solve an equation.
What is the purpose for having specific rules for order of operations? Because in math there is
a right and wrong answer. We want the "right" answer to be right for everyone, not just subjective to
each person.
On the white board, document camera, etc, write the following for the class to see:
G= 5 x (2 + 17)
G= (2 + 17) x 5
33
Unit 2: Algebraic Reasoning
5th Grade
G= 5(17 + 2)
5 x (17 + 2) = G
Have students solve for G in each equation.
Ask them:
What do you notice about all the answers? They are all the same answer. G = 95
Why are the answers all the same? Because the order of operations, or PEMDAS tells what order
to work out the problem, not left to right.
Students complete PEMDAS Practice.
Complete "PEMDAS Practice".
Part 2: Algebra
Day 10
Phases: ENGAGE, EXPLORE, EXPLAIN
Day 10
Materials:
Order of Operations Game: http://www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html
Handout: PEMDAS Practice 2
Activity:
Phases: EXPLORE, EXPLAIN, ELABORATE- Students continue to explore Order of Operations.
What's the teacher doing?
Review the Order of Operations with students. Students can link to the "Order of Operations
Millionaire Game Below":
What are the students
doing?
Play the Order of Operations
Millionaire Game.
http://www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html
You can divide the class into 2 teams, have them play individually, or play in pairs.
Students complete PEMDAS Practice 2.
Complete "PEMDAS Practice
2".
34
Unit 2: Algebraic Reasoning
5th Grade
Part 2: Algebra
Day 11
Phases: EXPLORE, EXPLAIN, ELABORATE
Day 11
Materials:
Teacher resource: Express Line
Handout: Express Line
Teacher resource: Processing Tables
Handout: Processing Tables
Handout: Real-world Expressions and Equations
Activity:
Phases: EXPLORE, EXPLAIN, ELABORATE- Students use numbers and operational symbols to represent a verbal phrase
symbolically. Students use given input and output values to write a numerical process and the expression that symbolically represents
the relationship. Students write an expression or equation to represent the relationships in a real-life problem situation.
1. Display the teacher resource: Express Line.
2. Place students in groups of 4. Distribute handout: Express Line to each student. Instruct student
groups to use numbers and one or more operational symbol to represent each verbal phrase. Allow
students 10 minutes to complete the activity. Monitor and assess student groups to check for
understanding. Facilitate a class discussion about the expressions generated for each problem.
Ask:
What operation is indicated by the word sum? (addition)
What operation is indicated by three more than? (addition)
What operation is indicated by three less than? (subtraction)
What process is indicated by the word product? (multiplication)
What process is indicated by the word double? (multiplication)
3. Display teacher resource: Processing Tables.
4. Distribute handout: Processing Tables to each student. Instruct student groups to use the given
input and output value to complete the table to represent the relationship between the values and
record an expression to symbolically represent the relationship. Allow time for students to complete
Students:
Complete "Express Line" and
"Processing Tables".
35
Unit 2: Algebraic Reasoning
5th Grade
the activity.
Monitor and assess student groups to check for understanding. Facilitate a class discussion to
debrief student solutions.
Ask: What part of the numerical process represents the input? (The numbers from the first
column
of the table.)
What part of the numerical process represents the output? (The numbers from the last column
of the table.)
How can you use the information in the table to represent the symbolic form for this
problem situation? (Look for patterns in the table and translate the patterns to a numerical
representation.)
How can you use the information in the table to represent the verbal form for this problem
situation? (Translate the symbolic representation to a verbal representation by relating the symbols
in the expression to the problem situation.)
--------------------------------------------------------------------------------------------------------------------------
"Real World Expressions and
Equations"
Students write an expression or equation to represent the relationships in a real-life problem
situation.
Instructional Procedures:
1. Distribute handout: Real-world Expressions and Equations to each student. Instruct students to
record the expression or equation that best represents the real-life problem situation. Allow students
to complete the handout as independent practice or homework.
36
Unit 2: Algebraic Reasoning
5th Grade
Part 2: Algebra
Day 12
Phases: EVALUATE
Day 12
Activity:
Phases: EVALUATE- Students complete the Performance Assessment from the IFD.
Performance Assessment: 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 as 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 Lara 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.
37
Unit 2: Algebraic Reasoning
5th Grade
38
Unit 2: Algebraic Reasoning
5th Grade
39