Why We Do The Things We Do: Teaching Students to Think Algebraically Using The Properties of Real
Numbers
Martha Barker and Ellen Forbes
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Day 1
Law and Order
Students will be introduced to the new topic of properties of real numbers, be able to
recognize and manipulate terms using the commutative property of both addition and
multiplication
This lesson begins with an overview of how properties play a role in manipulation of
terms in expressions and equations. The students will identify the commutative property
with addition and multiplication as well as gain an understanding in how to apply the
concept in real world problem solving.
Day 2
ZERO, our hero. He’s the ONE!
Students will recognize and manipulate terms using the multiplicative property of zero
and the identity property of both addition and multiplication.
The students will identify the multiplicative property of zero and the identity property of
both addition and multiplication as well as gain an understanding in how to apply the
concept in real world problem solving.
Day 3
To do or to undo…that is the e-question….
Students will recognize and manipulate terms using the Equality Properties of Addition,
Subtraction, Multiplication and Division, and Inverse Property of Multiplication and
Addition
The students will identify the Equality Properties of Addition, Subtraction, Multiplication
and Division, and Inverse Property of Multiplication and Addition as well as gain an
understanding in how to apply the concept in real world problem solving. Quiz on Day 1
and Day 2 material.
Day 4
Make new friends, but keep the old….
Students will recognize and manipulate terms using the Associative Property of Addition
and Multiplication
The students will identify the Associative Property of Addition and Multiplication as well
as gain an understanding in how to apply the concept in real world problem solving.
Quiz on Day 1-3 material.
Day 5
One for you and one for me, two for you and two for me….
Students will recognize and manipulate terms using the Distributive Property of
Multiplication over Addition and Subtraction.
The students will identify the Distributive Property of Multiplication over Addition and
Subtraction as well as gain an understanding in how to apply the concept in real world
problem solving. Quiz on Days 1 – 4 material.
Day 6
If One for All and All for One, then Equality for One, Equality for All!
REGULAR EDUCATION CLASSES: REVIEW FOR UNIT TEST and quiz on day's 1
– 5 material.
ADVANCED CLASSES: Students will recognize and manipulate terms using Reflexive,
Symmetric, and Transitive Properties of Equality
The students will identify Reflexive, Symmetric, and Transitive Properties of Equality as
well as gain an understanding in how to apply the concept in real world problem solving.
Quiz on Days 1 – 5 material.
Day 7
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Practice what you preach
Unit Assessment or review for unit assessment.
REGULAR EDUCATION CLASSES: Unit Assessment
ADVANCED CLASSES: Quiz 1-6 material and review
Day 8
Practice makes perfect
Practice with practical applications or unit assessment
REGULAR EDUCATION CLASSES: computer lab to apply property concepts to on
line with virtual manipulative website
ADVANCED CLASSES: Unit Assessment
Why We Do The Things We Do: Teaching Students to Think Algebraically Using The Properties of Real
Numbers
Martha Barker and Ellen Forbes
We will introduce the Properties of Real numbers using manipulatives and other activities for students to
identify and differentiate between the properties. We will demonstrate the steps in developing property
recognition and applying the properties in real world applications. Activities will be modeled to aid teachers in
incorporating these strategies within their lessons to ensure successful student comprehension.
Activities:
Tile Drawing Sheet
This is an activity modified from the Hands on Stanards, Deluxe Edition, The First Source for Introducing Math
Manipulatives (Grades 5-6), ETA Cuisenaire, pp 100-105. Students use square tiles of different colors (or any
two to three different types of medium as variables) to represent the properties of real numbers and color the
individual squares on centimeter grid paper as a pictorial representation of each property. There are two
different sheets utilized, grid paper and algebraic format.
Property Cards
These cards can be used in several different ways: as an introduction to the properties, property identification
with descriptions, sorting, identifying property names, matching descriptions and/or examples with the titles,
etc. They can be used as flashcards to study for a test.
Puzzle Pages
There are several different types of puzzle activities reinforcing vocabulary and property identification. These
are in the form of crosswords and word searches with examples (more advanced) and with terms (intro study).
Title of Lesson/Activity: LAW AND ORDER
Objective: The student will discover the commutative property using concrete manipulatives demonstrating
movement of values in an equation. The student will build from concrete to pictorial methods of demonstration
and further to abstract concepts with practical applications of the commutative property of addition and of
multiplication.
SOL:
a)
b)
c)
d)
e)
7.3 “The student will identify and apply the following properties of operations with real numbers:
the commutative and associative properties for addition and multiplication;
the distributive property;
the additive and multiplicative identity properties;
the additive and multiplicative inverse properties; and
the multiplicative property of zero.”
Materials: paper, pencil, worksheet (optional), dice, color tiles, or some manipulatives of different colors,
handout of properties chart.
Differentiation: Worksheet with the operations already in a chart to fill out with words listed in a word bank.
Vocabulary: property, elements, variable, operations, commutative, sum, product, factors, addends,
Anticipatory Set: Imagine you and your friends are trying to add up the total of a dinner bill. You have
declared a race and the first person who adds up the bill does not have to pay for his or her dinner! You want to
win, but the list of dinner prices are $11.95, 9.95, and 7.95. How can you add these numbers together without a
calculator? First round to the nearest dollar, $12, 10, and 8. Now you have $22 plus $8. Would it be ok to add
the 12 and 8 to get 20, then add 10? Is that ok, coming up with a different total or the same?
Process:
1. Warm up activity on the overhead or board for the students to complete upon entering the class room. 2.
Lesson.
Direct Instruction: There’s a concept in math called a PROPERTY, which is a characteristic rather than an
action. We refer to this as a rule that governs addition and multiplication. An operation is an action (multiply,
divide, add, and subtract). Properties are accepted to be true and are supported with axioms.
Let’s review what we know. Equality What does equality mean? How does an expression differ from an
equation? Early in elementary school, we learned how to add two numbers together. We learned that 4 + 3 = 7
and 3 + 4 = 7. What if we put these two expressions together and wrote it like this: 4 + 3 = 3 + 4. Would this
be a true statement? What if we used 4 and 3 as factors, would you get the same result with 4 ∙ 3 = 3 ∙ 4? With
addition or multiplication, the order does not affect the sum (for addition) or the product (for multiplication).
This is referred to as the Commutative Property of Addition or the Commutative Property of Multiplication.
There are several properties that we will discuss and use over the course of this unit. You will be introduced to
a property, be asked to identify the property, practice how it is used, and explain how to solve an equation using
the properties. We will begin with the Commutative Property, as it is the most familiar. Later you will work
with the identity and inverse properties, the equality properties, zero and distributive properties, and the
associative property.
Go back to the introduction example. How can you quickly and easily add those three numbers faster than your
friends?
Guided Practice: Distribute the tiles to pairs or groups of students. Each group/pair should have different
colored tiles in multiples. Show students with tiles on the board, overhead, or smartboard the left side of the
equation for the commutative property. Ask each student to create the other side as equal to the given tiles. Ex.
green blue blue = blue green blue. Let the students create their own and challenge others in the group. Show
students how this will work with multiplication, example 3 ∙ 2 = 2 ∙ 3 using three groups of two green tiles or
two groups of three green tiles and each side of the equation has 6 tiles. Walk the room to check for
understanding as the students produce more examples. Have the students DRAW examples of the tiles in their
notes.
Direct Instruction: The commutative property for addition states that changing the order of the addends does
not change the sum. For example, 5 + 4 = 4 + 5. When we see this property in algebraic terms, it is usually
written like this: a + b = b + a. The commutative property for multiplication states that changing the order of
the factors does not change the product. For example, 5 · 4 = 4 · 5. Again, in algebraic terms, this property
reads: a · b = b·a. Subtraction and division are not commutative. Order does matter for these operations as
seen in these examples, 5 – 4  4 – 5, and 10 / 5  5 / 10.

Students will take notes for definitions and Commutative Property examples in binder.
factora term that is multiplied
operations something that causes a change, mathematically addition, subtraction, multiplication, and division
sumthe solution to an addition problem
productthe solution to a multiplication problem
addenda term that is added
equation a mathematical sentence that states that two expressions are equal.
expression? a mathematical phrase that contains numbers, variables, and operations without an equal or
inequality sign. Is 2a + 7 = 43 an expression? No, it’s an equation. What about 2a + 7? Yes.
Independent Practice: Take a look at some more examples, copy these into your notes.
4+5∙2=5∙2+4
4+2∙5=4+5∙2
4∙2∙5=4+2+5
4∙2+5=4∙5+2
Which of these represents the commutative property and which operation is applied with this property? For any
equation that you choose that is NOT an example of the commutative property, explain why it is not.
Answers: 4 + 5 ∙ 2 = 5 ∙ 2 + 4
Commutative of Addition
4+2∙5=4+5∙2
Commutative of Multiplication
4∙2∙5=4+2+5
NOT equal and not a property
4∙2+5=4∙5+2
NOT equal. OOO states that multiplication must be completed PRIOR to
any addition, so changing the addends of 2 and 5 would have the four multiplied by a 2 on the left and a 5 on
the right.
Guided Practice: Word problems on the board/overhead/smartboard of how commutative property can be
used in real life.
Independent Practice: Complete two word problems using the commutative property.
Direct Instruction: Since this is one of many properties we will study, direct the students to the hand out chart
to fill in the property, description, example. This chart will be used to summarize each property as we go over
them.
Homework: worksheet with examples, determining if the example supports this property and of creating the
other side of the commutative property equation.
Closure: We will be discussing several properties of real numbers. Today we specifically discussed the
commutative property. Write down an example and a non-example of the commutative property and describe
in complete sentences what it means.
Assessment: Use the exit cards to determine if students understand what the commutative property is and is
not. Review of homework will also assist in assessing student understanding.
Teacher reflection:
Created by Martha Barker and Ellen Forbes
REFLEXIVE, SYMMETRIC, and TRANSITIVE PROPERTIES
Algebra Properties
Objective:The student will demonstrate Reflexive, Symmetric, and Transitive Properties using pictorial
representations to apply knowledge of the properties from drawing through the abstract level.
Materials: Paper, Pencil, Property Summary Sheet, Tiles, 5 different colored pencils, 1-centimeter graph
paper, overhead, and markers of various colors
Vocabulary: reflect; symmetric; transitive; “if… then” statement;
Bellringers: Identify properties from previous lessons.
A) 29 + 56 = 56 + 29
B) df = fd
C) (5m) 2x (3a) = (3a) 2x (5m)
D) 2 + (3 + 1) = (2 + 3) + 1
Anticipatory Set: If Susie Q was born in Mathews, and Mathews in
Virginia, then Susie Q was born in Virginia. These last few
properties deal with conditional type statements. Where
the statement on the left is equal to what is on the right.
Instruction: Starting with the most simple situation . . .
In case A: For any number, a, a is equal to a.
This makes me think of a person looking into a mirror, what
word is associated with a picture and a mirror. (Reflection)
Reflection indicates that what is on one side Is the same that is on the other side. So when
analyzing this type of
property it would be a = a or 4 = 4. This property is
called the Reflection Property. With tiles we would show
this with an example similar to 3 = 3 where we would insert
an equal sign between……. (the same number on each side of equal).
In case B: For any numbers, a, b and c, “If a + b = c; then c equals a + b”. For most four-year
olds the lunch time special is a sandwich; “If peanut butter and jelly is a sandwich, then a
sandwich equals peanut butter and jelly.”
So lets turn that into mathematics. If 4 + 6 equals 10, then
10 equals 6 + 4. This is known as Symmetric Property.
It is the same on each side. Be careful not to confuse this
commutative property. The one additional clue in this would
be the additional words “if, then”.
In case C: For any numbers, a, b, c, d, and e, “If a + b = c
and c = d + e, then a + b = d + e” This seems very confusing with this many different letters, but
when replaced with actual numbers or real world situations, it becomes much simpler. If 4 + 1 =
5 and 5 = 2 + 3, then 4 + 1= 2+3.
The clue to this is that we can eliminate the 5 and 5 and then each side is equal but in different
forms. Once you understand the concept you will see there is an extra “piece”
In the middle. One student called this the “oreo” effect.
Guided Practice:
Students should take out their Tile Drawing page.
On the drawing sheet we can illustrate these three new properties and explain how they work
and their clues.
Independent Practice:
Identify the following properties
A) If 2 + 6 = 8 then 8 = 6 + 2
B) km = km
C) If 3 • 4 = 12 and 12 = 2 • 6, then 3 • 4 = 2 • 6
D) If 5 + 9 = 14 and 14 = 2 • 7, then 5 + 9 = 2 • 7
E) If 9 + 2 = 11, then 11 = 2 + 9
Using the GREEN card set; Identify the properties that were
discussed today. Allow 15 – 20 minutes and allow students
to change partners.
Homework: Property Identification Worksheet (Advanced) Cumulative
List.
Closure: Exit Slip explaining in student words how Reflexive, Symmetric,
and Transitive Properties are alike and different.
Assessment: Quiz for cumulative properties given the next day.
Created by Martha Barker and Ellen Forbes
Created by Martha Barker and Ellen Forbes