Operating with Decimals

# 3. Multiplying and Dividing by Powers of 10

Suggested Learning Intentions

• To understand that scientific notation can be used to describe infinitely large and infinitely small numbers
• To learn how to multiply and divide decimals using knowledge of place value in a base-10 system

Sample Success Criteria

• I can multiply and divide decimals by powers of 10 using written and mental strategies
• I can describe different kinds of numbers using scientific notations
• I can justify my thinking and solutions using manipulatives
• Place value diagram: PDF
• Modern place value diagram: PDF
• Number slider: PDF
• Calculators

Display a range of diagrams for visualising place value (e.g., by Caleb Gattegno, who is an influential mathematician and mathematics educator).

Place Value Diagram by Caleb Gattegno:

Modern Place Value Diagram:

Ask students: "What do you notice? What do you wonder"

Next, view the ClickView miniclips Multiplying by multiples of powers of 10 and Dividing by multiples of powers of 10. Sign into ClickView using your department credentials.

Facilitate a discussion about powers of ten. What are powers of ten? What happens when we multiply and divide whole numbers by powers of ten?

What connections can students make between the place value diagrams and how powers of ten are used?

The following activities provide opportunities for students to develop their number sense when multiplying and dividing decimals by powers of 10. Another important aspect of this is a continued focus on estimation and place value to ensure that students do not rely on ‘counting zeros.’

Number sliders

In this activity, students use a number slider to understand how to multiply and divide numbers by powers of ten. This activity also places emphasis on the digits moving rather than the decimal point.

As students move the digits accordingly, they will notice that zeros may need to be included (e.g. dividing by 100 makes 0.031) or the decimal point can be removed (e.g. multiplying by ten makes the number ten times bigger hence 31). Have them practice multiplying and dividing various numbers by powers of 10.

Calculator patterns – Tens Times

In this activity from NZmaths, students use place value tables to understand what happens to numbers when we divide or multiply by powers of ten. The table shows that when moving to the left the numbers grow ten times larger and when moving to the right the numbers become ten times smaller.

Ask students to explore patterns by using a constant function (multiplication) on a calculator. This will help them explore decimal places.

• multiply this by 10 (× 10) and note down the answer (30)
• press the equal sign, which shows that the answer is multiplied by 10 again (300)
• press the equal sign again (3000).

Students continue the pattern a few more times and discuss what happens to the place value of the digit 3. Students may state that the digit moves from the ones to the tens, to the hundreds etc. or that it simply moves one spot to the left each time.

Then have students do the reverse, such as divide by ten starting at 3. They will see this pattern, 0.3, 0.03, 0.003, 0.0003 etc.

Enable students by using the online tool Moving Cards to reinforce place value concepts. Choose a number between 103 and 102 and the Moving Card will show what happens when you multiply or divide that number by 0.1, by 10 or by 100.

Extend students by inviting them to view the following clip on Scientific Notation that supports them to write our very large or very small numbers in a meaningful way. Offer a range of examples such as the ones in the table below of numbers written in their standard form and scientific notation

Invite students to describe the patterns they notice. Where would you place these on a number line?

Encourage students to pay attention to the scientific form of a negative number that does not result in a negative number but rather a positive number. Make up some other examples and check on a calculator. What real life examples can you find? (Encourage students to research online).

Areas for further exploration

Moving digits is an interactive tool that can be used to construct a number in the place value chart. Multiply or divide the top line by a multiple of 10 or 100 as long as the answers remain within the place value chart.

Multiplying by 10, 100… Interactive tasks where students multiply and divide by multiples of ten. There are five levels each with a set of twelve questions. Students can check their answers regularly throughout.

Ask students to consider how powers of ten help us when multiplying and dividing decimals. For example, the answer to 12 x 15 is 180. The answer to 180 ÷ 15 = 12. Without doing any more multiplication or division, how can we use this information to estimate the answers to:

 1.2 × 15 = 18 ÷ 1.5 = 12 × 150 = 1800 ÷ 15 = 1.2 × 1.5 = 1800 ÷ 150 = 120 × 150 = 180 ÷ 1.5 =

Ask students to explain their thinking for each in two different ways. Students check their answer with a calculator.

As students to explain any errors. What do they notice? What do they wonder? Ensure students demonstrate conceptual knowledge of place value and powers of ten rather than procedural knowledge of ‘moving the decimal point over.’

Enable students by discussing the answer to 2 x 4 = 8. How can you use this information to work out 2 x 40 and 20 x 4?

Extend students by inviting students to come up with a suitable word problem to match each number sentence. E.g., 1.2 x 15 could relate to ‘What is the area of a rectangular park that is 1.2km wide and 15km long?’

Revisit the success criteria for this stage:

• I can multiply and divide decimals by powers of 10 using written and mental strategies
• I can rename decimals to assist with computations.

Monitor the degree to which students can describe the patterns when multiplying and dividing decimals by powers of 10. Do they use ideas of the base-10 system? Are they making generalisations about the patterns and connection between multiplication and division of decimals by powers of 10?

ClickView, 2020. Dividing by Multiples of Powers of 10. [Online]
Available at: https://online.clickview.com.au/libraries/videos/4488952/dividing-by-multiples-of-powers-of-10
[Accessed 15 March 2022].

ClickView, 2020. Multiplying by Multiples of Powers of 10. [Online]
Available at: https://online.clickview.com.au/libraries/videos/4488809/multiplying-by-multiples-of-powers-of-10
[Accessed 15 March 2022].

Khan Academy, n.d. Scientific Notation Examples. [Online]
[Accessed 15 March 2022].

MathsIsFun, 2020. Scientific Notation. [Online]
Available at: https://www.mathsisfun.com/numbers/scientific-notation.html
[Accessed 15 March 2022].

NZMaths, n.d. Tens Time. [Online]
Available at: https://nzmaths.co.nz/resource/tens-time
[Accessed 15 March 2022].

Siemon, D. et al., 2015. Teaching Mathematics: Foundations to Middle Years. Melbourne: Oxford University Press.

State Government of Victoria (Department of Education and Training), 2018. Moving Cards. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=c9279d6b-075f-4aa2-a8bb-c7426ff32d3a&SearchScope=All
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), 2020. Moving Digits. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=0becaf73-82b5-488c-873a-c5a2ef1fd2b9
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), 2020. Multiplying by 10, 100.... [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=8a3a9850-e1f3-4dfe-b7a8-b34020bdda8e
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Math Curriculum Companion: Calculator Patterns. [Online]
Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA216#tab1
[Accessed 15 March 2022].