Suggested 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

Display a range of supermarket dockets. Invite students to mentally estimate how much money Joe needs in his bank account to confidently ‘Tap and Go’.

Joe has the following items on his shopping list…

- 2kg apples at $3.70 per kg
- A can of baby formula for $14.95
- Dishwashing tablets for $12.99
- 3 packets of Tim Tams at $3.65 each
- 20 litres of petrol at $1.36 per litre.

Ask students to share what strategies they are using to work out the answer e.g., addition, multiplication, rounding to the nearest whole number/dollar.

Post the following guiding questions and use the Think, Pair, Share routine for students to consider their ideas:

- How do we use our understanding of place value when adding and subtracting decimals?
- What can help us to add and subtract decimals?

**Number talks**

Follow up with Number Talks that focus on estimation and mental strategies when adding and subtracting decimals. Number talks are a great way to encourage fluency using student reasoning. Number talks are a very student-focused activity, and this positions students as independent maths learners, as they do most of the heavy lifting.

To introduce the number talks, write a problem on the board. (Several suggestions are provided below.)

**0.45 + 2.36 **

or

**9.754 - 0.2**

or

**73.46 + 6.2 + 0.582 **

or

**"If I remove 1.54m of tape from a 5.5m roll, how much will be left?"**

Leave the problem written on the board and ask students to individually attempt to solve it. This can take some time - it is important to allow students this time to think through their responses.

When students are ready to provide a response, they should use some form of subtle visual indication. The visual cue should be subtle so that other students are not so aware (for example, a hand in the air is highly visible, an indication of fingers held to the chest facing forward is less so). The visual cue can also indicate a level of confidence in the response (one finger for not confident, three fingers for very confident and so forth).

Once all students are ready with a response, hand a white board marker over to one of the students indicating confidence and ask them to record their response on the board. The student then provides their solution, explaining to the class how they got their answer. Ask if any students had another way to complete the problem and ask students indicating an alternative method to share their response.

For each student response, it is important that you do not identify correct or incorrect answers or ask others to do so. At this point, it is all about the strategies that students are using, not whether the answer is correct or not.

Once there are several strategies recorded on the board, lead the class through a discussion of these strategies by selecting some of the following questions:

- What is similar about these strategies? What is different?
- Which of these strategies do we think is more efficient and why?
- In what situations would these strategies be easier or harder to use?
- What knowledge are we using when we use these strategies?
- How can we check to see if we have the answer correct?
- Which strategy will you use next time?

Reasoning is used in this activity through students justifying their responses through explaining their strategies as well as through students being challenged to consider which of the class strategies are most efficient.

Fluency is being developed through students identifying the most efficient strategies to complete the calculation.

The following activities provide opportunities for students to develop their estimation and fluency when adding and subtracting with decimals. It is important to maintain a focus on students’ conceptual understanding by applying their knowledge of place value with the decimals.

**1. Closest to 10**

Using the fishbowl strategy, demonstrate how to play Closest to 10 (boards are available in Materials and texts). Each player has a playing board. Using three 6- or 10-sided dice, students decide prior to rolling the dice at their turn, to choose only two dice and the third must be a zero (This will be helpful near the end of 7 rolls). Students then roll the dice and decide which column to place each digit. For example, on my first roll, I rolled 1, 4 and 0. On my second roll, I rolled a 6, 0 and 2. The winner is the person whose final total is closest to 10.

Facilitate a discussion about the strategies students used and what they needed to think about to get ‘Closest to 10’. How did students work out who was closest to 10 if one student was a little bit over or a little bit under, e.g., 10.32 or 9.85?

**Enable** students by offering a game board to tenths. Encourage students to use colour in a set of ten decimats as they go to support their thinking. Also focus on what aspect of decimals students are experiencing difficulties with. For example, Closest to 10 is designed to support students build their understanding of decimal place value by making connections to see that 12.8 is the same as 12.80. As students are estimating, also note the way they consider the value of the decimals. For example, do students understand that 0.25 is between 0.2 and 0.3?

**Extend** students by offering a gameboard with thousandths. Again, focus on the language students are using.

**2. Metre ruler**

This activity helps students investigate tenths, hundredths and thousandths using a metre ruler. Here they will work with:

- decimetres = 10cm, to represent one tenth (0.1)
- centimetres = 1cm, to represent one hundredth (0.01) and
- millimetres = 1mm, to represent one-thousandth (0.001).

Ask students to repeatedly roll three ten-sided dice to generate numbers to add and subtract. For example, a student might first roll a 3, a 5, and another 5. Have this student 'start', by finding 0.355m on their metre ruler. The student then rolls to create another 3-digit decimal number to add. Say this student next rolled a 4, a 3 and an 8. This student now needs to add 0.355 and 0.438 and find that point (0.793) on their metre ruler. Now have the student roll again, and this time subtract the number. For example, 0.793 - 0.691.

Encourage students to estimate their solutions each time they add or subtract. As they add and subtract various numbers, ask students to use the ruler to show their 'jumps' (i.e., the distance that represents the number they are adding or subtracting).

Concentrate on the language being used. For example, renaming 12.80 + 14.99 as 12 and 80 hundredths plus 14 and 99 hundredths will help with mental computation. Observe students who intuitively use the compensation strategy: we know that 0.99 is one hundredth away from one, so if we round this number to 15 and add 12.80, we get 27.80. The next step is to subtract one hundredth from the answer which gives 27.79. Note that teaching specific strategies is not recommended. Instead, notice what strategies students are using and assist them in moving towards more efficient methods through class discussions.

Pose challenging questions such as: 0.671 - 0.31 + 0.424, where students must use the full metre ruler.

**Extend **students by having them represent this on an empty number line to think about the scale and place value of numbers. This will also assist with mental computation.

**3. Open decimal problems**

The following task has been reproduced with permission from Sullivan & Lilburn (2006, p. 30).

In these calculations, some numbers are missing. What might the missing numbers be?

3._ _ +_.7_ = 6._3

3._1 + _.47 + 0._ _ = 8.68

Note how students solve this. Who does it by trial and error? Who develops a system such as deciding on one of the numbers and then subtracting it from 8.68 to see what is left to work with? How many different correct solutions can they find?

**Areas for further exploration**

The following open problems may be useful for students to consolidate their understanding of adding and subtracting with decimals:

Display the list of the top 10 fastest 100m sprint runners in the world (World Athletics, n.d.).

- Usain Bolt 9.58
- Tyson Gay 9.69
- Yohan Blake 9.69
- Asafa Powell 9.72
- Justin Gatlin 9.74
- Christian Coleman 9.76
- Nesta Carter 9.78
- Maurice Greene 9.79
- Steve Mullings 9.80
- Richard Thomson 9.82

Commence with a number talk about decimals in context such as "9.58 seconds - what does this mean?"

Pose a range of questions, focusing on students' ability to estimate and calculate:

- What is the smallest and biggest difference in times?
- How much faster was athlete x than athlete y?
- How could you mentally estimate the total times?
- What strategies could you use to calculate the actual total of times?

Monitor:

- How are students focusing on the meanings of the numbers and the decimal point?
- What strategies are they developing for adding/subtracting decimals? Students will likely intuitively move toward a formal approach of lining up the decimal, but it is important that they can explain why/how it works.

Revisit the success criteria for this stage:

- I can round appropriately based on the context
- I can explain the reasonableness of my estimations when working with decimals
- I can apply my knowledge of addition and subtraction strategies when working with decimals
- I can rename decimals to assist with computation.

Students reflect on their learning using a Thinkboard (see the Materials and texts section above) to demonstrate their understanding. Students can roll dice to create an addition and subtraction decimal problem to annotate in their Thinkboard.

While students may use a formal algorithm to check their answers, it is important that students are using a range of flexible strategies to estimate and compute decimal problems to demonstrate a sound understanding of place value knowledge.

Kaplinsky, R., 2020. *Open Middle. *[Online]

Available at: https://www.openmiddle.com/

[Accessed 15 March 2022].

Smith, J., 2014. *Beyond Traditional Math - The Fishbowl: A Peer Modelling Strategy. *[Online]

Available at: https://beyondtraditionalmath.wordpress.com/2014/02/16/the-fishbowl-a-peer-modeling-strategy/

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), 2020. *Classroom talk techniques: Think, pair, share. *[Online]

Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/english/literacy/speakinglistening/Pages/exampleclasstalk.aspx#:~:text=Think%2C%20pair%2C%20share,to%20'turn%20and%20talk'.&text=During%20an%20inquiry%2C%20students%20may,studen

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. *Maths Curriculum Companion: Metre Ruler. *[Online]

Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA214

[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. *Maths Curriculum Companion: Number Talks. *[Online]

Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA214

[Accessed 15 March 2022].

Sullivan, P. & Lilburn., P., 2004. *Open-Ended Maths Activities: Using Good Questions to Enhance Learning in Mathematics. *Melbourne: Oxford.

World Athletics, 2019. *100 Metres Men. *[Online]

Available at: https://www.worldathletics.org/records/all-time-toplists/sprints/100-metres/outdoor/men/senior

[Accessed 15 March 2022].

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

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

4. Multiplying and Dividing Decimals

EXPLORESuggested 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