Getting Started

The purpose of this session is to explore the range of strategies already used by students to solve addition and subtraction problems. This lesson will enable you to evaluate which strategies need to be focused on in greater depth. In turn, you will identify students in your group as "expert" in particular strategies. There are two problems given as examples for exploration. You may want to use further examples of your own. Consider adapting the contexts used in these problems to further engage your students.

1. Problem 1: Sarah has $288 in the bank. She then deposits her pay for $127 from her part time job at a cafe. How much does she have now?
   Ask the students to work out the answer in their heads or by recording in some way. Give the students plenty of thinking and recording time. Ask the students to share their solutions and how they solved the problem with a peer. The following are possible responses:
   
   Place value (mentally, possibly with the support of equations on an empty number line):
   288 + 127 is just like 288 + 100 +20 +7. So that’s 388… 408… 415.
   
   Tidy numbers (mentally, possibly with the support of equations on an empty number line):
   If I tidy 288 to 300 it would be easier. To do that I need to add 12 to 288, which means I have to take 12 off the 127. So that’s 300 plus 115. 
   
   Algorithm (usually written):
   Students may visualise or record a written algorithm like this:
   
   Understanding is revealed by the language used to describe the strategy, such as, “8 plus 7 equals 15. I wrote 5 in the ones place and carried the extra ten into the tens place because 15 is made up of 5 and 10 and I can only record one digit in the ones place.”
   
   As different strategies arise ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, recording the strategies to reflect upon later in the unit on the class T chart (under the addition heading). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
    
2. Problem 2 Sarah has $466 in her bank account and spends $178 on a concert ticket. How much money does she have left in her bank account?
   Ask students to solve the problem mentally, giving them plenty of thinking and recording time. Then ask students to share their solutions and how they solved the problem with a peer. Possible responses are:
   
   Reversibility (adding on to find the difference with recording to ease memory load):
   $466 - $178 is the same as saying how much do you need to add to $178 to get $466. $178 plus $22 makes $200, plus $200 more makes $400 plus $66 makes $466. If you add up $22 plus $200 plus $66 you get $288.
   
   
   Subtracting a tidy number then compensating 
   $466 - $200 = $266. I took off $200 instead of $178 so I need to ‘pay back’ $22. $266 + $22 = $288.
   
   Equal additions:
   You round the $178 to $200 by adding $22. $466 - $200 is $266. Then you put on $22 to keep the difference the same, so it’s $288.
   
   Algorithm (usually written):
   Students may visualise or record a written algorithm like this:
   
   
   Understanding is revealed by the language used to describe the strategy, such as, “6 minus 8 doesn’t work (ignoring integers) so I changed one ten from the tens column into ones to make 16. I wrote 16 in the ones place and took one ten off in the tens place…”
   
   As different strategies arise, ask the students to explain why they chose to solve the problem in that way. Accept all the correct strategies that are elicited, and justified, recording the strategies to reflect upon later in the unit (under the ‘subtraction’ heading on the class T chart). You might also ask students to model how their strategies work with place value materials. Note that folding back to justify a strategy is often more difficult than performing the strategy.
   
   Subtraction strategies tend to be more difficult to control than addition strategies, given comparable numbers. Look out for students compensating the wrong way (taking more off) in the tidy number strategy and making errors when using the algorithm.
   
   Use your observations to plan for your subsequent teaching from the exploring section outlined below.

Exploring

Over the next two to three days, explore different strategies for addition and subtraction of whole numbers. Give the strategies a name so students can tell others which strategy they are preferencing for a given problem. Highlight when certain strategies are most efficient, for example, In the problem 357 + 189 tidy numbers would be a useful strategy because 189 is close to 200.

Follow a similar lesson structure each day to the introductory session, with students sharing their solutions to the initial questions and discussing why these questions lend themselves to the strategy being explicitly taught. Conclude each session by asking students to make statements about when the strategy would be most useful and why the certain problem is appropriate, e.g., tidy numbers when one number is close to 100 or 1000, standard place value (hundreds, tens, and ones) when no renaming is needed and reversibility when neither of the other two numbers are easy for subtraction. It is important to record examples of strategies as they will be used for reflection at the end of the unit. Some strategies may require more teaching time, greater use of materials, and more scaffolded and individualised teaching. Ensure that students who demonstrate proficiency with the strategies early on in each session have adequate opportunities for practice, extension, and supporting their peers (if appropriate).

The questions provided are intended as examples for the promotion of the identified strategies. If the students are not secure with a strategy you may need to make up some of your own questions to address student needs. Consider adapting the contexts reflected in these problems to further engage your students.

Tidy numbers then compensating

1. Room 9 are selling muesli bars at lunchtime to raise money for their camp. They had 434 at the beginning of lunchtime and sold 179, how many did they have left to sell?
   The tidy numbers strategy involves rounding a number in a question to make the question easier to solve. In the above question, 179 can be rounded to either 180 (by adding 1), or 200 (by adding 21). If 200 is subtracted (434 – 200 = 234) then the result is 21 less than the answer. 234 + 21 = 255. An empty number line shows this strategy clearly:
   
   For addition questions, one addend can be tidied by taking from the other addend. Alternatively, both addends might be tidied, and compensation used to adjust for the tidying.
2. The following questions can be used to elicit discussion about the strategy.
   • What tidy number could you use that is close to one of the numbers in the problem?
   • What do you need to do to the other number if you tidy up this number? Why?
   • Why is this strategy useful for this problem?
   • What knowledge helps you to solve a problem like this?
3. If the students do not understand the tidy numbers concept, use place value equipment (place value blocks, place value people, beaNZ) and/or an empty number line to model the problems physically. Students should be encouraged to record equations and number lines to track their thinking and reduce load on working memory.
4. Use the following questions for further practice if required:
   • Ariana has scored 739 runs for her cricket club this season. Last season she scored 294, how many did she score in total in the last two seasons?
     Shifting six between the addends gives 739 + 294 = 733 + 300 = 1033.
   • Nigel sold his guitar for $587 and his amp for $395. How much money did he make from both sales?
     Rounding 587 to 600 and 395 to 400, then compensating gives 587 + 395 = 600 + 400 – 13 – 5 = $982.
   • Farmer Samsoni has 1623 sheep, and he sells 898 sheep at the local sale. How many sheep does he have left?
     Rounding 898 to 900 gives 1623 – 898 = 1623 – 900 + 2 = 725
   • Other examples might be:
     568 + 392
     661 - 393
     1287 + 589
     1432 - 596
5. Note that the problems posed here are using a tidying up strategy rather than tidying down i.e. 103 down to 100 as in these situations place value tends to be a more useful strategy.

Place Value (written algorithm)

1. For the community hangi, 356 potatoes are peeled and there are 233 left to be peeled. How many potatoes will there be altogether?
   The place value strategy involves adding the ones, tens, hundreds, and so on. In the above problem:
   300 + 200 is added
   Then 50 + 30
   And finally 6 + 3
   As an algorithm the calculation is represented as:
   
2. The following questions can be used to elicit discussion about the strategy:
   • How can you use your knowledge of place value to solve this problem?
   • Why is this strategy useful for this problem?
3. If the students do not understand the strategy, use place value equipment to show the problems physically. Students will find it useful to record and keep track of their thinking .
4. Use the following questions for further practice if required:
   • Zac has $498 available on his eftpos card and spends $243 on a new BMX bike, how much money does he have left?
   • Other problems might be:
     3221 + 348
     4886 - 1654
     613 + 372
     784 – 473

Reversibility (adding one rather than subtracting to find the difference)

1. Faloa is helping his Mum build a path. There were 438 bricks in the pile and they used 169 of them yesterday. How many bricks have they got left to lay today?
   The reversibility strategy involves turning a subtraction problem into an addition one so the problem above becomes:
   169 + ? = 438
   Using tidy numbers to solve the problem makes calculation easier:
    
   Or
   
2. The following questions can be used to elicit discussion about the strategy:
   • How could we think of this as an addition equation?
   • What do you need to add to make it easier to solve?
   • How can you keep track of how much you have added altogether?
   • Why is this strategy useful for this problem?
   • What knowledge helps you to solve a problem like this?
3. If the students do not understand the concept, use a number line to show the problems graphically. Students will find it useful to record and keep track of their thinking using a number line.
4. Use the following questions for further practice if required:
   • At the mail sorting office there were 547 letters to be sorted, 268 of these were distributed to private boxes, how many were left to be delivered?
   • The school library has a total collection of 1034 books and 459 are issued at the moment, how many are on the shelves?
   • Other problems might be:
     628 - 342
     537 - 261
     742 - 353
     1521 - 754
     1762 - 968
     1656 - 867

Reflecting

As a conclusion to the week’s work, give the students the following five problems to solve. Pose the problems in contexts that are relevant and engaging for your students. Ask students to discuss, with a peer or in small groups, which strategy they think will be most useful for each problem and justify their view. For some problems many strategies may be equally efficient. After students have solved the problems, engage in discussion about the effectiveness of their selected strategies.

Some students may have a favourite strategy that they use, sometimes to the exclusion of all others. The best approach is to pose problems where the preferred strategy may not be the most efficient. For example, 289 + 748 is most suited to using tidy numbers and compensation.

Problems for discussion
1318 - 747
763 - 194
433 + 452
1993 + 639
4729 - 1318

You might also like to also try some problems with more than two numbers in them, such as:
721 – 373 - 89
663 - 61 - 88
63 + 422 + 49
42 + 781 + 121
84 + 343 - 89

1. Discuss the different strategies explored during the week and ask students to explain in their own words what types of problem each strategy would be useful for solving, and what types of problem each strategy would not be useful for solving. Ask the students to draw a strategy ‘from a hat’ and write questions specific to that strategy for a partner.
2. Conclude the unit by showing the students the questions asked in the initial session again. Discuss whether they would solve the problems in a different way now, and how their thinking has evolved. With a partner, students could create a short presentation demonstrating how they would solve one addition and one subtraction problem, naming the strategies used, and justifying the use of their chosen strategies.

Background Maths

A major misconception for the students is to treat decimals as two independent sets of

whole numbers separated by a decimal point.

Students who believe that 0.8 is less than 0.75 because 75 is greater than

8 show signs of this misconception. Providing operational problems where the

students must work across the decimal point helps to address the problem. Suggest to

students that the decimal point is a marker for the ones place rather than a “barrier”

that separates the wholes from the fractions.

This link between the digits on one side of the decimal point and the other is further

strengthened when students encounter decimals in the context of division where there

is a remainder.

Using Materials

Make 10 candy bars by joining 10 Unifix cubes to form each bar. Wrap each bar in a

paper sleeve to establish its “oneness”. Tell the students that this brand of candy only

comes in bars of 10 cubes.

Pose the problem: “If there are eight bars of candy and four friends, how much candy

will each person get?” (two whole bars)

Record the result as: 8 ÷ 4 = 2 wholes. Note that a whole means one.

Pose the problem: “If there were six candy bars and five friends, how much candy

would they get each?”

Let the students work out their answers with Unifix cubes.

Record the result as: 6 ÷ 5 = 1 whole and 2 tenths.

Have the students perform the calculation 6 ÷ 5 = on a calculator and compare the

result with the candy bar sharing.

Provide other examples like this to consolidate the link between the calculator

displays and the number of whole bars and tenths.

Examples might be:

“Four candy bars and five people” (4 ÷ 5 = 0.8)

“Five candy bars and two people” (5 ÷ 2 = 2.5)

“Six candy bars and four people” (6 ÷ 4 = 1.5)

Change the problems to addition and subtraction. This will require the students to link

the numbers on both sides of the decimal point.

Problem: “Henry has 3.4 candy bars, and Tania has 1.8 candy bars.

Peter thinks that they will have 4.12 bars altogether. Use the Unifix cubes and

calculator to find out if he is right.”

The students should find that the 0.4 and 0.8 combine to form another whole bar and

two-tenths, and so the answer is 5.2. The calculator will confirm this.

The students can record their ideas in this format:

With each problem, the students should model the problem with the Unifix cubes

first, predict the correct answer, and confirm it on a calculator.

Problem: “Jody has 4.3 candy bars. She thinks that if she eats 2.7 candy bars, she will

have 2.4 left. Is she right?”

Problem: “Rangi-Marie thinks that 7 x 0.4 candy bars will be 0.28. Is she right?” This

means seven lots of point four candy bars.

Problem: “Timoti thinks that 5.2 ÷ 4 = 1.3 (5.2 bars shared among four people). Is he

right?”

Using Imaging

Shielding: Set up similar problems by masking the Unifix cube model under sheets of

paper or ice-cream containers. For example: “What is 5 x 0.6?” Use five ice-cream

containers, each holding a stack of six cubes (six tenths). Label the top of each

container with 0.6.

Discuss why the answer is 3.0 and not 0.30 (Thirty tenths is the same as three ones).

Similar problems might be:

3.7 + 2.6 = (Two containers holding 3.7 and 2.6 stacks respectively)

5.1 – 2.9 = (One container holding 5.1 and another upturned into which 2.9 will be

put)

7.2 ÷ 6 = (7.2 visible to be shared among six containers)

Extend the students’ strategies into hundredths by posing problems like:

“There are five candy bars and four people. How much candy will they get each?”

The students will realise that each person will get one whole bar, but sharing the

remaining bar among four cannot be done just using tenths. Two tenths can be given

to each person, but two tenths remain.

The students may suggest that each person can then have half of a tenth. Point out that

our number system works on breaking and making into tens.

Ask, “If I break one-tenth (holding a single cube) into 10 equal pieces what will they

be called?” (hundredths) “How many hundredths will each person get in our

problem?” (five)

Record the answer as: 5 ÷ 4 = 1 whole + 2 tenths + 5 hundredths

Perform 5 ÷ 4 = 1.25 on a calculator and ask the students to explain the display. Pose

similar problems involving hundredths like:

“Three bars shared among four people” 3 ÷ 4 = (0.75)

“Seven bars shared among five people” 7 ÷ 5 = (1.4)

13 ÷ 4 = (3.25) 9.6 ÷ 8 = (1.2)

Where necessary, Unifix cubes can be used to model these problems.

Number Properties

Pose operational problems that involve working with ones, tenths, and hundredths

directly and require exchanging across the decimal point.

2.4 + 4.57 = (6.97) 6.53 – 2.7 = (3.83) 3.4 – 1.29 = (2.11)

7 ÷ 4 = (1.75) 1.25 x 6 = (7.5) 5.3 ÷ 5 = (1.06)

Game

This is another game that provides a fun context to practise addition and subtraction of decimals and opportunities for the students to explore and share additive strategies for decimals.
Note that the dice is restricted to the numbers 1, 2, and 3. This increases the likelihood of the students hitting a star, which is where the purpose of the game is fulfilled. If you cannot access a dice like this, use an ordinary one and tell the students to subtract 3 if they throw a 4, 5, or 6.
It would be a good idea to use two colours to make up the different sets of cards. This helps with the organisation of the game.
Once the game has been taught and practised, it will become a useful independent activity. You may also use it for strategy teaching by asking the students to reflect on how they did some of the star calculations.
If the students need assistance with the decimal calculations, have them use the blank decimal fraction number lines (Material Master 4-25).

Answers to Activity

Game
A game for adding and subtracting decimals

Game

This game provides a fun context to practise the addition and subtraction of decimals. It also presents opportunities for you to explore and share addition and subtraction strategies, logical thinking in selecting a winning strategy for the game play, and place value issues with decimal numbers.
The Target game board can be photocopied (see the copymaster at the end of these notes), or the students can sketch out the boxes on draft paper for a quick copy.
This game is an ideal independent activity for pairs of students to practise decimals, but it must still be taught to the group so that they understand how to play. It can also be used as a class game.
When you use the game as a teaching activity, check that the students appreciate the place value in each of the boxes. Discuss the role of the decimal point as a device that indicates the ones column to the left. When this place is identified, the students should work out the place value of the other boxes.

Strategies for adding or subtracting the decimals may also be part of the teaching activity. For example, if the dice produce the expression 2.5 + 3.6, the students may choose to add the ones first. This could be recorded on an open number line:

Other strategies may be to make 2.5 up to 3 and so on, for example, 2.5 + 0.5 + 3 + 0.1 = 6.1.
A third strategy could be 2.5 + 2.5 + 1.1 = 6.1.
These options show that the students are doing decimal addition at the advanced additive stage of thinking.
After the students have played the game a few times, have them discuss their strategies for winning. Ideas may include: “I put my first throw in the ones place unless it is a 5 or a 6 because these would be too high. I then have three throws to get a number in the other ones place that will make 5. The decimal bits should get me close to 6.” “I put the numbers under 4 in the ones place and the numbers greater than 3 in the tenths place.” Ask the students to test out the strategies to see
which ones work best for them.

Answers to Activity

Game
A game for adding and subtracting decimals

Prior knowledge.

• Use the strategy jumping the number line with whole numbers (Book 5 page 33)
• Identify the place value of the tenths hundreds and thousandths columns.
• Make combinations of tenths and hundredths that add to one

Background

In this activity students use additive strategies to solve addition and subtraction problems involving decimals.  Students need to have a good understanding of place value to make sense of the strategies with decimals.

Comments on the Exercises

Exercise 1
Asks students to solve problems by jumping a whole number and then a decimal.  For example, 6 + ? = 8.55, 6 + 2 = 8, 8 + 0.55 = 8.55 so the answer is 2.55.

Exercise 2
Asks students to solve problems by jumping to the nearest whole number, then the next whole number and then make a decimal jump. For example, 4.97 + ? = 8.12, jump 0.03 to 5, then 3 from 5 to 8, then 0.12 to 8.12

Exercises 3 and 4
Asks students to reduce the number of jumps they used in Exercise 2. For example, 3.98 + ? = 9.3, jump 0.02 to 4 then 5.3 to 9.3.

Exercise 5
Asks students to choose their own strategy to solve problems like 11.82 + ? = 38.3.

Exercises 6 and 7
Asks students to solve problems like in Exercise 5 but the numbers are larger, for example 19.88 + ? = 224.52

Written recording

Written recording of mental strategies (that is, how you thought through the problem) is important for developing sound assessment skills as it allows others to follow your reasoning and allows you to have a visual check for accidental errors. It is also something that develops over time, and needs to be discussed regularly with students. Exercises 5, 6 and 7 stress that students should solve the problems mentally, but record enough to show what they have done. Discussing or eliciting different ways of doing this is therefore an important activity, which you may choose to run either before or after setting students to work on this exercise

Share the loopy cards out among all the students in the group. The person who has “Start 1.2” ontheir loopy card begins and says the number at the top of the card and reads out loud theinstruction on that card, “Add 0.8.” The person with the answer 2.0 (1.2 + 0.8) on one of theircards reads out “2.0. Divide by 4”. This keeps going until the card with “60. Finish” comes up.

Create two arrays using animal cards to represent 6 _ 8 and 3 _ 8. Ask the students how many animals are in each array. Pose the problem of adding these sets of animals together, that is 48 + 24. Allow the students to present their strategies.

Some students may note that (6 x 8) + (3 x 8) = 9 x 8 = 72 (combining sets of eight).

Provide two similar examples, getting the students to make the arrays then solve theresulting addition problem. Suitable problems include:

(5 x 7) + (4 x 7) = 9 x 7                     (3x 6) + (6 x 6) = 9 x 6

(4 x 9) + (6 x 9) = 10 x 9                   (7 x 8) + (2 x 8) = 9 x 8

Ask the students to state what these problems have in common (both addends aremultiples of a common number).

Using Imaging

Pose problems for the students to image, beginning with the addends instead of the factors. If necessary, fold back to the materials. Possible examples include:

16 + 24 + 32 as (2 x 8) + (3 x 8) + (4 x 8) = 9 x 8

49 + 28 as (7 x 7) + (4 x 7) = 11 x 7

45 + 27 + 18 as (5 x 9) + (3 x 9) + (2 x 9) = 10 x 9

Extend the imaging to include subtraction problems:

81 – 27 as (9 x 9) – (3 x 9) = 6 x 9       64 – 48 as (8 x 8) – (6x 8) = 2 _ 8

63 – 35 as (9 x 7) – (5 x 7) = 4 x 7       90 – 36 as (10 x 9) – (4 x 9) = 6 _ 9

54 – 36 as (9 x 6) – (6 x 6) = 3 x 6 or (6 x 9) – (4 x 9) = 2 x 9

Using Number Properties

Pose addition and subtraction problems where it is helpful to identify a common factor.

48 + 56 + 24 + 32 = 20 x 8      42 + 35 + 49 + 14 = 20 x 7

72 – 27 + 45 – 36 = 6 x 9        88 – 56 – 16 + 32 = 6 x 8

120 – 54 – 48 – 18 = 0 x 6      77 – 28 + 14 – 35 = 4 x 7

Getting started

1. Begin with a true/false conjecture on board for students to discuss. Ask students:

   Do you think the statement is true or false and why?        1 – ^-4 = 5

2. Ask students to discuss their ideas with a partner. Listen for student explanations. Discuss the idea of positive and negative numbers; introduce the term integer if it does not arise in the discussion.
   An integer is a whole number that can be either greater than 0, called positive, or less than 0, called negative. Zero is neither positive nor negative
3. Ask students to explain what is happening in the equation 1 – ^-4 = 5. Ask how this equation could be demonstrated with equipment.
4. Use the counters of two different colours (e.g. black and red). Explain to students that black counters represent positive and red counters represent negative - show the number 2 as 2 black counters.
5. Ask: 
   What would happen if I added, three more black and three red?
   What would I have now? 
   Make sure students understand that the extra three black and three red cancel each other out, so the answer is still 2.
   Two integers (e.g. 3 and -3) that are the same distance from the origin (e.g. 2) in opposite directions are called opposites and when added cancel each other out making 0.
6. Use black and red pen to record the expressions you have modelled on the board: 2 +  3 = 5 and 5 + -3 = 2. Draw attention to the addition of -3.
7. Ask students, in pairs, to generate some expressions similar to the one you modelled. These expressions should: be one-step, only use the operations addition and subtraction; and demonstrate how, when added, two integers (e.g. 3 and -3) that are the same distance from the origin (e.g. 2) in opposite directions, cancel each other out making 0.  Students should use red and black counters to model their equations, before writing each expression. If time allows, students could also use digital tools, other materials, or physical movement to model their thinking. 
8. Support students as necessary. Look for students to recognise that they can use ny number of extra red and black counters, so long as these sets are equivalent.
9. Pose an equation for students to model with equipment:

           3 – 4 =

10. Discuss the answer and how they used the equipment to model it.
11. Repeat this process with the following:
    
    ^-3 + 2 =
    
    4 – ^-1 =
    
    ^- 5 + ^-2 =
12. Ask students to make up three more problems for their to swap with their partner to solve. These could also be discussed, and the solutions justified, for the class. Support small groups and individual students as necessary, making sure to use materials to model the addition and subtraction of integers.
13. Pose another true/false conjecture for discussion.

            ^-3 – 2 = ^-1

10. Conclude the session by asking students to record or discuss what they have noticed about adding and subtracting integers. Compile a summary of these ideas in a chart or modelling book.

Exploring

Over the next few days introduce the students to a selection of other models and contexts for addition and subtraction of integers, choosing from the list below. Students could, if they wished, use the double coloured counters to model some of the calculations. As they work through tasks, add any new ideas to the modelling book or chart. Pay attention to gaps in students’ knowledge, and take the opportunity of students working independently to take small-group or individualised teaching sessions that address these gaps. Ensure you model and explain how to use each of the following activities.

1. Integer learning experiences from Book 5: Teaching Addition, Subtraction and Place Value
2. Money Matters (copymaster) - Discuss the idea of cash and debts, this task could be followed up with the Figure It Out lesson Money Matters, p21, Number Year 7/8, Book 4.
3. Close to Zero game (copymaster).
4. Bonuses and Penalties game (copymaster) after playing the game get students to make up some of their own cards.
5. Integer Quick Draw game (copymaster).

Reflecting

1. Pose a variety of True/False conjectures using slightly more complex numbers, like those in the expressions below, framed in a context that is engaging and meaningful to your students (e.g. culturally or environmentally significant, relevant to current learning interests or learning from other curriculum areas). 
             3 – ^-12 = ^-15
             4 - 16 = 20
             3 + ^-17= 20
             7 - 13= ^-6
2. Revisit the summary of ideas where students have recorded what they have noticed about adding and subtracting integers. Ask students to work with a partner to develop some guidelines related to adding and subtracting integers for another classroom, for example adding a negative number is just like…, subtracting a negative number is like… This could be done on paper or in another format (e.g. video, digital presentation, acting it out, using other physical manipulatives or diagrams). 

Alternatively, you could introduce Alistair McIntosh’s thinkboard. Model completing the sections with the equation given below. Ask students to make up an equation (or provide them with one) and write it in the middle of the thinkboard. They should complete each section of the model. You could run this as a ‘carousel’ activity, in which different (random) groups of students receive a question. After giving students time, in their groups, to work out the answer to the question and fill in their thinkboards, make new groups composed of a student from each original group. Every student in the new group should share their group’s ‘thinkboard’ with their new group. This gives all students a chance to listen to each other, explain their thinking, and see different examples of integers being used in real-world contexts