Expressions and Equations

The Relationships of the Operations

Common Core: Engage New York6.EE.1, 6.EE.2, 6.EE.3 and 6.EE.4

Lesson 4: The Relationship of Division & Subtraction

Focus Standard: 6.EE.A.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+ ) 𝑥 to produce the equivalent expression 6+3𝑥; apply the distributive property to the expression 24 +18 𝑥 𝑦 to produce the equivalent expression 6(4 +3 )𝑥 𝑦 ; apply properties of operations to + + 𝑦 𝑦 𝑦 to produce the equivalent expression 3𝑦.

What does 6.EE.A.1 cover?

Write and evaluate numeric expressions involving whole-number exponents.

What does 6.EE.A.2 cover?• Write, read, and evaluate expressions in which letters stand for numbers

a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract 𝑦from 5” as 5−𝑦.

b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8+7) as a product of two factors; view (8+7) as both a single entity and a sum of two terms.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas = 3 𝑉 𝑠 and =6 2 𝐴 𝑠to find the volume and surface area of a cube with sides of length =1/2𝑠 .

What does 6.EE.A.3 cover?

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+ ) 𝑥 to produce the equivalent expression 6+3𝑥; apply the distributive property to the expression 24 +18 𝑥 𝑦to produce the equivalent expression 6(4 +3 )𝑥 𝑦 ; apply properties of operations to + + 𝑦 𝑦 𝑦 to produce the equivalent expression 3𝑦.

What does 6.EE.A.4 cover?

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions + + 𝑦 𝑦 𝑦 and 3 𝑦 are equivalent because they name the same number regardless of which number 𝑦stands for.

Table of ContentsDate Title Page

2/6/14 NEW Focus 8- Algebraic Expression Scale and Chart Fresh Left

2/6/14 F8 Engage NY Lesson 1 –Add and subtract Relationship

Fresh Left

2/7/14 F8 Engage NY Lesson 2 – Multiplication and Division Relationship

Fresh Left

2/10/14 F8 Engage NY Lesson 3-Multiplication and Addition Relationship

Fresh Left

2/11/14 F8 Engage NY Lesson 4-Division and Subtraction Relationship

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Focus 8 Algebraic Expression Learning Goal

I am able to work with numerical expressions and use letters to represent unknowns in problem solving situations, investigate and apply properties of operation in numerical contexts, such as the associative, distributive, and commutative properties. I can build on my understanding of inverse operations to solve algebraic expressions.

Today, my learning target is to…

•Build and clarify the relationship of division and subtraction by determining that

12÷ =4 means 𝑥12− − − − =0𝑥 𝑥 𝑥 𝑥

MY PROGRESS

CHART

Before we start the Learning Target

Lesson, think about the Learning Target

for today….

How much prior knowledge do you

have regarding that goal?

Chart your prior knowledge using your pre-target score icon.

Lesson Preparation NEEDED!

Lesson Notes • Students will continue to use the squares from Lessons 1

and 3 to create tape diagrams. Each pair of students will

need 30 squares to complete the activities.

Actual Size- print and cut 2 sets per pair of students

Activity & Discussion

Activity & Discussion

Activity & Discussion- Discuss the process step by step to determine that the number of times the divisor was subtracted from the dividend is the same number as the quotient.

• Do you recognize a relationship between and ? If so, what is it?

• If you subtract the divisor from the dividend 5 times (the quotient) there will be no remaining squares.

Activity & Discussion- Discuss the process step by step to determine that the number of times the divisor was subtracted from the dividend is the same number as the quotient.• Let’s take a look at a similar problem, , to see if the quotient is the number

of times the divisor is subtracted from the dividend.• Let’s create a number sentence when we subtract the divisor.

Discuss the process to determine that the number of times the divisor is subtracted from the dividend is the same number as the quotient.

Determine the relationship between and is the same as

Is this relationship ALWAYS true? Let’s try to prove that it IS!

Proving the Relationship- Model the following set of tape diagrams with leading questions for discussion.

• is a number. What does mean?• Exactly five ’s can be subtracted from twenty.

• What must be in this division sentence?• Four

• Let’s keep taking away until we reach zero.

Model taking each away and creating subtraction expressions to record.• Build a subtraction expression.

• Is • YES!

• Develop two number sentences using numbers and letters to show the relationship of division and subtraction.• Possible answers:

Model taking each away and creating subtraction expressions to record.• Or, means that 5 can be subtracted

exactly number of times from Is it true when ? Subtract five from twenty four times.

• Let’s keep taking away until we reach zero.

• Build a subtraction expression.

• What two operations are we relating in the problems we completed?• Division and subtraction

Student Practice-Record your answer in your math workbook.

Answers

Student Practice- Record your answers in your math workbook.

Answers

Student Practice- Record your answers in your math workbook.

Answers

Student Problem Set- Record your answer in your math workbook

Student Problem Set Answers

Closing (4 min)Display the graphic organizer provided at the end of the lesson. Copies of the organizer can be made for students to follow along and record. • In each of the circles, we can place an operation to satisfy

the organizer. In the last four lessons, we have discovered that each operation has a relationship with other operations, whether they are inverse operations or they are repeats of another.

Place the addition symbol in the upper left hand circle. • Let’s start with addition. What is the inverse operation of

addition? • Subtraction.

Closing (4 min)Place the subtraction symbol in the upper right hand circle. • After our discussion today, repeated subtraction can be represented by

which operation? • Division.

Place the division symbol in the lower right hand circle. • Which operation is the inverse of division? • Multiplication.

Place the multiplication symbol in the lower left hand circle. • Let’s see if this is correct. Is multiplication the repeat operation of

addition? • Yes.

Understanding the relationships of operations is going to be instrumental when solving equations later in this unit.

Student Exit Ticket for Lesson 4The Relationship of Division & Subtraction

Exit Ticket - Lesson 4 Solutions

Today, I achieved my learning target by…

•Building and clarifying the relationship of division and subtraction by determining that

12÷ =4 means 12− − − − =0𝑥 𝑥 𝑥 𝑥 𝑥

MY PROGRESS

CHART

Before we start the Learning Target

Lesson, think about the Learning Target

for today….

How much prior knowledge do you

have regarding that goal?

Chart your prior knowledge using your pre-target score icon.

The End of Lesson 4

The Relationship of Division & Subtraction