Suggested Learning Intentions

- To recognise the appropriate operation when solving decimal problems in context
- To use my number sense when multiplying and dividing decimal problems

Sample Success Criteria

- I can select the operation and use effective strategies when solving decimal problems
- I can explain my thinking using pictures, diagrams and number sentences when solving decimal problems
- I can explain and justify my solutions using manipulatives or other representations of the mathematics

A strong understanding of multiplication and division with whole numbers and a well-developed understanding of place value are necessary when operating with decimals. It is helpful to introduce experiences that involve using materials to demonstrate what is happening when multiplying and dividing with decimals and simultaneously focus on conceptual strategies such as benchmarking when estimating to give the problem meaning.

Pose and discuss the following examples with students, using the decimal thinkboard for multiplication and division (available in the Materials and texts section above).

Use these prompts to help:

- How are they the same?
- How are they different?
- What could help us estimate the answer?
- What could help us work out the answer?
- What do you notice happens to the value of the number when we multiply and divide decimals?
- What other word problems can students think of?

Access additional information on using grids with decimal multiplication from the Mathematics Curriculum Companion.

This task has been borrowed from ‘Decimal Word Problems’ (Sullivan, 2018, pp. 90-91) and is reproduced with permission.

Students need practice at reading and decoding word problems. These questions use similar numbers so the focus is on choosing the type of calculation. The method for performing the calculation is not obvious. It uses the idea of rate, which is used in all such consumption statements.

To indicate the intended learning, you might write: ‘Even if there are only small differences between word problems, the way to solve them can differ. It’s important to read word problems carefully.’

To introduce the problem, make sets of cards from ‘Decimal match-up’ (available in the Materials and texts section above). Either laminate them to use again later or print them and let students paste them into their books. Students can sort them out in small groups and discuss the placement of the cards. Invite students to explain how they sorted the cards. This task assumes that students have had some experience with decimal calculations, especially with estimating the answers.

**Learning Task**

My car has a fuel consumption of 9 L/100 km.

- How far can I travel if I use 90 L of fuel?
- How much fuel do I use in travelling 1000 km?
- How far can I travel on 100 L of fuel?
- How much fuel do I use in travelling 9 km?

Work out the answers to these problems in two different ways. How are these problems similar and how are they different? (Focus on the context and number sentences to decide which operation to use).

Explain that 9 L/100 km means 9 L of fuel for each 100 km travelled. This lesson is best conducted in small groups. Encourage students to explain their reasoning and justify solutions. Students can use a calculator to find a solution. The calculator won’t necessarily help with solving the problem if students are unable to choose the appropriate operation for solving it.

**Enable** students by telling them that your car has fuel consumption of 9 L/100 km. Ask: How far can I travel if I use 90 L of fuel? How much fuel do I use in travelling 1000 km?

**Extend** students by asking them to create another set of problems like those in the learning task.

Possible solutions for the learning task:

- 1000 km
- 90 L
- 1111.11 km
- 0.82 L

**Areas for further exploration**

**1. Fuel consumption**

Ask students to use the information below to write a question to match each of the three calculations. Essentially the students are asking themselves: *What question would this calculation solve?*

A car uses 2.5 kg of carbon dioxide for each litre of fuel used. My car has a fuel efficiency of 9 L/100 km. I travel 20000 km in a year.

- 2.5 x 9 ÷ 100
- 20 000 ÷ 100 x 9
- 2.5 x 9 ÷ 100 x 20000

**2. Electricity consumption**

Ask students to assume that electricity causes 1.21 kg of carbon dioxide per kwh (kilowatt hours) of emissions, and that each house of 4 people uses around 20 kwh per day. Ask: *What are the emissions of carbon dioxide for a house of 4 people for one year?*

**3. Red meat consumption**

Ask students to assume that there is 4.44 kg of carbon dioxide per dollar of red meat, and that a family of 4 spends $50 per week on red meat. Ask: *What are the emissions of carbon dioxide from that family’s meat consumption for one year?*

**4. Car emissions**

Ask students to estimate how far their family drives their car in a year. Assume that there is 2.50 kg of carbon dioxide per litre of fuel. Ask: *If a car uses 7 L/100 km, what are the emissions for a year if the car is driven the same distance as your family’s car?*

Multiplying and dividing decimals exposes students to a range of mathematical skills and concepts which you may wish to review and consolidate with students. Consider the progress made by students when determining which skills and concepts you select to review.

Three suggested hints and concepts for review are presented below.

**1. Multiplication and division do not always increase and decrease (respectively) the value of numbers.**

Materials: Maze Board (available in Materials and Texts), Calculator, Counters

Instructions

- The aim of the game is to have the smallest number on your calculator when the game ends.
- Both players enter 100 on their calculator and the counter is placed on the START.
- Player A chooses to move the counter along a single line segment and performs the operation on his or her calculator.
- Player B moves the counter from its new position along a single line segment (but not back along the line segment their opponent has just used) and performs this operation on their calculator.
- The game continues with each player in turn moving the counter and performing the operation that is on the line segment they have just passed.
- The calculator is not cleared between turns.
- The game ends when one player reaches the Finish and the winner is the player with the smallest number on their calculator at that point in the game. Alternatively, you could choose to end the game after a certain time has elapsed and the player with the smaller number at that point is the winner.

Adapt the game by:

- Changing the aim to the largest number wins, or nearest to a particular number e.g., 127.6
- Playing with three players
- Letting each player have their own counter
- Not allowing any segments to be retraced. The game ends when the finish is reached or there are no further moves possible
- Removing the word ‘finish’ and asking students to keep moving around the game board until the time is up.

Source: DET (n.d., p. 52)

**2. It is helpful to estimate the product when multiplying by decimals larger (and smaller) than one. **

- The aim of this game is to be the player to reach the target of 100 or as close as possible to it
- The first player enters a number on the calculator (some ‘ugly’ number less than 100 is usual but any number could be displayed)
- The second player must multiply this number by another number so that the result will be as close as possible to 100
- The first player then multiplies this new number, trying to get closer still to 100
- The players take it in turns until one player hits the target by getting 100 on the calculator (digits after the decimal point do not count, so 100.xxx is fine)
- The game can be recorded on the task sheet or blank paper.

Here is an example game:

Source: DET (n.d., p. 58)

**3. It is important to read word problems carefully to recognise the appropriate operation and consider ways of solving it.**

You have bought 28.4 kg of lollies for the school fete. If you give 4 kg to one classroom so that they can make small show bags, how many classrooms will receive lollies? Check that students conceptually understand the problem. For example, they may say, “this means how many lots of 4 kg are in 28.4 kg?

Can the students give a reasonable estimate before solving it accurately with pen and paper or a calculator?

In each small bag, the teacher places 0.1 kg of lollies because other goodies will be placed inside the bags as well. How many bags can be made per classroom? How many bags altogether? If each bag sold for $5.89, what are the profits?

Source: DET (n.d., b)

**Assessment considerations include:**

- Do students recognise which operation to use for decimal problems?
- Do students rely on repeated addition for multiplication of decimals?
- Do students demonstrate well developed number sense when multiplying/dividing a decimal by a whole number? For example:
- Do students make reasonable estimations?
- Do students' explanations and justifications demonstrate conceptual understanding? This could include renaming to whole numbers and using understanding of place value to locate the decimal point. For example, 3.2 x 4 is the same as 32 tenths x 4 = 128 tenths

State Government of Victoria (Department of Education and Training), n.d. *Fractions and Decimals Online Interview Classroom Activities. *[Online]

Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=71215a8e-f63d-4fd5-8625-2dde17151d06&SearchScope=All

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. *Maths Curriculum Companion: Multiply decimals by whole numbers and divide by non-zero whole numbers. *[Online]

Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA215

[Accessed 15 March 2022].

Sullivan, P., 2018. *Challenging Mathematical Tasks. *South Melbourne: Oxford University Press.

Other stages

1. Decimal Number Sense

EXPLORESuggested Learning Intentions

- To understand decimal place value to the thousandths
- To understand the relative size of decimals

Sample Success Criteria

- I can describe decimal place value to thousandths
- I can explain and justify my solutions using a variety of manipulatives such as place value blocks, counters or decimats
- I can use my knowledge of place value to compare, order and round decimals

2. Adding and Subtracting Decimals

EXPLORESuggested Learning Intentions

- To solve decimal problems using rounding and estimation
- To use my understanding of place value to solve decimal problems

Sample Success Criteria

- I can round appropriately based on the context
- I can explain the reasonableness of my estimations when working with decimals
- I can rename decimals to assist with computation
- I can model my solution using manipulatives such as place value blocks or decimats
- I can apply my knowledge of addition and subtraction strategies when working with decimals

3. Multiplying and Dividing by Powers of 10

EXPLORESuggested 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