ENGAGE NY MODULE 1

L E S S O N 3 Objective: Use exponents to name place value units and explain patterns in the placement of the decimal point.

STATE THE UNIT AS A DECIMAL – CHORAL RESPONSE• 9 tenths = ______• 10 tenths = _____• 11 tenths = _____• 12 tenths = _____• 18 tenths = _____• 28 tenths = _____• 58 tenths = _____

• 9 hundredths = _____

• 10 hundredths = _____• 20 hundredths = _____• 60 hundredths = _____• 65 hundredths = _____• 87 hundredths = _____• 118 tenths = _____

MULTIPLY AND DIVIDE BY 10, 100, AND 1000

• Write two 1 thousandths disks and the number below it.• Multiply by 10. Cross out each disk and the number 2 to show that

you’re changing its value.

.0 0 2.0 2

MULTIPLY AND DIVIDE BY 10, 100, AND 1000• Using your charts, show how to multiply 0.004 x

100.• 0.004 x 100 = 0.4

• Now show how to multiply 0.004 x 1000.• 0.004 x 1000 = 4.0

• Based on the previous problems, what patterns do you see when you multiply by 10, 100, and 1000?

MULTIPLY AND DIVIDE BY 10, 100, AND 1000

• Write two 1 one and 4 thousandths disks and the number below it.• Multiply by 10. Cross out each disk and the numbers to show that

you’re changing their value.

1 . 0 0 41 0 . 0

4

• 1.004 x 10 = 10.04

MULTIPLY AND DIVIDE BY 10, 100, AND 1000• Using your charts, show how to multiply 1.024 x

100.• 1.024 x 100 = 102.4

• Now show how to multiply 1.324 x 10.• 1.324 x 10 = 13.24

• Multiply 1.324 x 1000.• 1.324 x 1000 = 1324

MULTIPLY AND DIVIDE BY 10, 100, AND 1000

• Write 4 ones disk in the table above and write the number below it.• Divide by 10. Cross out each disk and the number 4 to show you’re

changing its value.

40 .4

4 ÷ 10 = 0.4

MULTIPLY AND DIVIDE BY 10, 100, AND 1000• Using your charts, show how to divide 4.1 ÷

10.• 4.1 ÷ 10 = .41

• Now show how to divide 4.1 ÷ 100.• 4.1 ÷ 100 = .041

• Show 4.1 ÷ 1000.• 4.1 ÷ 1000 = .0041

Describe the pattern you see when dividing by 10, 100, and 1000.

APPLICATION PROBLEMJack and Kevin are creating a

mosaic by using fragments of broken tiles for art class. They want the mosaic to have 100 sections. If each section requires 31.5 tiles, how many tiles will they need to complete the mosaic? Explain your reasoning with a place value chart.

CONCEPT DEVELOPMENT – PROBLEM 1

• 10 X ____ = 10 Fill in the missing factor to complete this number sentence.

• 10 X ____ = 100 Fill in the missing factor to complete this number sentence.

• This time, using only 10 as a factor, how could you multiply to get a product of 1000? Write the multiplication sentence.

• What would the multiplication sentence be for 10000 using only 10 as a factor?

• How many factors of 10 did we have to multiply to get 1000?

• How many factors of 10 do we have to multiply to get 10000?

• Say the number sentence.

• How many zeros are in our product 10000?

• Turn and share with your partners the patterns that you see.

10

10x1

100

10x10

1000

10x10x10

10000

10x10x10x10

CONCEPT DEVELOPMENT – PROBLEM 110000 1000 100 10

10x10x10x 10

10x10x10 10x10 10x1100000

100000010x10x10x 10x10x10

10x10x10x10x10

• Using this pattern how many factors of 10 do we have to multiply to get 1 million? Work with your partner to write the multiplication sentence.

• How many factors of 10 did you use?• Why did we need 6 factors of 10?• We can use an exponent to represent how many times we use 10

as a factor. We can write 10 x 10 as 10². We say , “Ten to the second power.” The 2 is an exponent and it tells us how many times to use 10 as a factor.

• How do we express 1000 using exponents? Turn and share with your partner.• Working with your partner, complete the chart using the exponents to represent

each value in the third row.

10²10³10⁴ 10¹10⁵10⁶

CONCEPT DEVELOPMENT – PROBLEM 11000000

100000

10000 1000 100 10

10x10x10x 10x10x10

10x10x10 x10x10

10x10x10x 10

10x10x10 10x10 10x1

10⁶ 10⁵ 10⁴ 10³ 10² 10¹

• Multiply 10 one hundred times. What do you get?• Now look at the place value chart; let’s read our powers of 10 and

the equivalent values.• Since a google has 100 zeros, write it using an exponent.

CONCEPT DEVELOPMENT – PROBLEM 2• Write ten to the fifth power as a product of

tens.•10⁵ = 10x10x10x10x10

• Write ten to the seventh power as a product of tens.•10⁷ = 10x10x10x10x10x10x10

CONCEPT DEVELOPMENT – PROBLEM 310 x 100• Work with your partner to write this expression using an

exponent. Explain your reasoning.• 10³

100 x 1000• Work with your partner to write this expression using an

exponent. Explain your reasoning.• 10⁵

CONCEPT DEVELOPMENT – PROBLEM 43 x 10²• Compare this expression to the ones we’ve already

talked about. • Write 3 x 10² without using an exponent.

• 3 x 100• What’s the product?

• 300 • If you know that 3 x 100 is 300, then what is 3 x 10²?

Explain this to your partner.

CONCEPT DEVELOPMENT – PROBLEM 53.4 x 10³• Use what you learned about multiplying decimals by 10, 100,

and 1000 and your knew knowledge about exponents to solve the above problem.

• Write 3.4 x 10³ without using an exponent.• 3.4 x 1000

• What’s the product?• 3400

• If you know that 3.4 x 1000 is 3400, then what is 3.4 x 10³? Explain this to your partner.

CONCEPT DEVELOPMENT – PROBLEM 5Solve this problem.

4.021 x 10Explain how you arrived at your answer.

CONCEPT DEVELOPMENT – PROBLEM 6700 ÷ 10²• Write 700 ÷ 10² without using an exponent

and find the quotient.•700 ÷ 100 = 7

• If you know that 700 ÷ 100 is 7, then what is 700 ÷ 10²? Explain this to your partner.

CONCEPT DEVELOPMENT – PROBLEM 77.1 ÷ 10²• Write 7.1 ÷ 10² without using an exponent and

find the quotient.• 7.1 ÷ 100 = .071

• Tell your partner what you notice about the relationship between the exponents and how the values of the digits change. Discuss how you decided where to place the decimal.

CONCEPT DEVELOPMENT – PROBLEM 8Complete this pattern:

0.043 4.3 430 ______ _____ _____

• Turn and talk with your partner about the pattern on the board. How is the value of the 4 changing as we move to the next term in the sequence? Use your place value chart to explain your ideas as you complete the pattern and use an exponent to express the relationships.• 43,000 4,300,000 430,000,000

CONCEPT DEVELOPMENT – PROBLEM 9Complete this pattern:

6,300,000 ________ 630 6.3 ______

Turn and talk with your partner about the pattern on the board. How is the value of the 6 changing as we move to the next term in the sequence? Use your place value chart to explain your ideas as you complete the pattern and use an exponent to express the relationships.

• 6,300,000 63000 630 6.3 .063