LESSON TOPIC: THE RELATIONSHIP OF ADDITION AND SUBTRACTION,

MULTIPLICATION AND DIVISION, MULTIPLICATION AND ADDITION, AND

DIVISION AND SUBTRACTIONLesson Objective: I can…

Build and clarify the relationship of addition and subtraction, multiplication and division, multiplication

and addition, and division and subtraction by evaluating identities.

LESSON 1 OPENING EXERCISE: Draw a tape diagram representing the

number 3. Add 2 more squares to your tape diagram. Write an Expression to represent how we

created a tape diagram with 5 squares. Now remove 2 squares from the tape

diagram. Adjust our original expression to represent

what we did to the tape diagram. Evaluate the expression.

CONTINUING WITH OPENING EXERCISE: Create a new diagram with 6 squares

now. Use the diagram to represent the

problem 6 + 4. Remove 4 squares from the tape

diagram. Write an expression that represents our

original problem with the removal of the squares.

Evaluate the expression.

EXERCISES:1. Predict what will happen when a tape

diagram has a large number of squares, some squares are removed, but then the same amount of squares are added back on.

2. Build a tape diagram with 10 squares. Remove 6 of them. Write an expression

to represent the tape diagram. Add 6 squares onto the tape diagram.

Alter the original expression to represent the current diagram.

Evaluate the expression.

EXERCISES CONTINUED…3. Write a number sentence, using

variables, to represent the identities we demonstrated with tape diagrams.

4. Using your knowledge of identities, fill in each of the blanks.

4 + 5 - ____ = 4 25 - ____ + 16 – 16 = 45

56 – 20 + 20 = ____ a + b - ____ = a e + ____ - f = e ____ - h + h = g

LESSON 2 OPENING EXERCISE:Draw a picture on your Ipad to represent the division and multiplication problem using a tape diagram.

1. 8 divided by 22. 3 x 2

MULTIPLICATION AND DIVISION WITH TAPE DIAGRAMS Build a tape diagram to represent 9 units. Divide the 9 units into 3 equal groups. Write an expression to represent the process

you modeled on your Ipad. Evaluate the expression. What if you multiply this by 3. Adjust your expression to show this

multiplication. Evaluate your expression. What do you notice about this expression? Write a number sentence, using variables, to

represent the identities we demonstrated with the tape diagrams.

EXPLORATORY CHALLENGE:Work with someone at your table to determine number sentences to show the relationship between multiplication and division. Use tape diagrams to support your findings.1. Create two number sentences to show

the relationship between multiplication and division. These number sentences should be identities and include variables.

LESSON 3 OPENING EXERCISE:

Write two different expressions that can be depicted by the tape diagram shown. One expression should

include addition, while the other should include multiplication.

PARTNER WORK:1. One partner build a tape diagram to

represent 2 + 2 + 2 + 2 while the other partner builds a tape diagram to represent 4 x 2.

What do you notice about the 2 tape diagrams that were created?

2. One partner builds a tape diagram to represent

3 x 4 while the other partner builds a tape diagram to represent the expression 4 + 4 + 4. Is 3 x 4 equivalent to 4 + 4 + 4? Use variables to represent the relationship

between multiplication and addition.

EXERCISES:Write an equivalent expression to demonstrate the relationship of

multiplication and addition.1. 6 + 6

2. 3 + 3 + 3 + 3 + 3 + 33. 4 + 4 + 4 + 4 + 4

4. 6 x 25. 4 X 66. 3 x 9

7. h + h + h + h + h8. 6y

EXERCISES CONTINUED… Tell whether the following number

sentences are true or false. Then explain your reasoning.

a. x + 6g – 6g = xb. 2f – 4e + 4e = 2f Write an equivalent expression to

demonstrate the relationship between addition and multiplication.

a. 6 + 6 + 6 + 6 + 4 + 4 + 4b. d + d + d + w + w + w + w + wc. a + a + b + b + b + c + c + c + c

LESSON 4 CLASSWORK: Build a tape diagram that has 20 squares. Divide the tape diagram into 4 equal sections. How many squares are in each section? Write a number sentence to demonstrate what happened. Combine your squares again to have a tape diagram with 20

squares. Now subtract 4 squares from your tape diagram. Write an

expression to demonstrate what happened. Subtract 4 more squares from your tape diagram and alter your expression to account for this change.

Subtract 4 more squares from your tape diagram and alter your expression to represent the new diagram.

Subtract 4 more squares and alter your expression to represent this new tape diagram.

Last time, subtract 4 more squares and alter your expression to represent a number sentence showing the complete transformation of the tape diagram.

DISCUSSION: What was the step by step process.

20 – 4 = 1616 – 4 = 1212 – 4 = 88 – 4 = 44 – 4 = 0

Do you see the relationship between 20 divided by 4 = 5 and 20 - 4 - 4 – 4 - 4 – 4

= 0?

20 DIVIDED BY 5 = 4 Create a number sentence when we

subtract the divisor. What is the process? What is the relationship between 20

divided by 5 = 4 and 20 – 5 – 5 – 5 – 5 = 0?

DISCUSSION: X is a number. What does 20 divided by

x = 5 mean? What must x be in this division

sentence? What is I use x and take it away creating

a subtraction expression? Write this expression. Develop two number sentences to show

the relationship between division and subtraction.

INDEPENDENT EXERCISES:Answer each of the questions using what you have learned about the relationship of division

and subtraction.1. If 12 divided by x = 3, how many times

would x have to be subtracted from 12 in order for the answer to be 0? What is the

value of x?2. 36 – f – f – f – f = 0. Write a division

sentence for this repeated subtraction sentence. What is the value of f?

3. If 24 divided by b is 12, which number is being subtracted twelve times in order for

the answer to be 0?

LESSON SUMMARY:Understanding the relationships of

operations is going to be instrumental when solving equations later in this unit.

Evaluate your learning:1 2 3 4

How will you “Sharpen your Saw?”