Double-Sided Counters – Interactive Negative Numbers Tool

1

Switch to Explore mode. Drag three positive (red) counters into the workspace. Toggle the Value button on. What value do you see? Now add two negative (blue) counters. What is the value now? Explain in your own words why the value changed.

2

Starting with an empty workspace in Explore mode, add five negative counters. What is the value? Now add positive counters one at a time, toggling Value on after each one. Describe what happens to the value each time. At what point does the value become zero? What happens if you keep adding positives?

3

Turn the Value toggle off. Build a collection of counters where there are more negatives than positives. Ask a partner to look at the counters and predict whether the value is positive, negative, or zero. How can you tell just by looking, without counting exactly?

4

Turn Count on and build several different collections that all have the same value. For example, can you make a value of 2 using four counters? Six counters? Ten counters? What do all your collections have in common?

5

Can you build a collection with a value of zero using exactly six counters? What about eight counters? Can you do it with five counters? What must be true about the total number of counters for a zero value to be possible?

6

Add one positive and one negative counter to the workspace. Drag one onto the other to form a zero pair. Watch the animation. What value did the pair contribute before they were removed? Why do we call it a ‘zero pair’?

7

Build a collection with four positives and four negatives. Before pressing Find Zero Pairs, predict: how many zero pairs will form? What value will remain? Now press the button and check. Were you right?

8

Build a collection with five positives and three negatives. Press Find Zero Pairs. How many pairs formed? How many counters remain, and what is their sign? Can you state a general rule: if you have P positives and N negatives, how many zero pairs form, and what remains?

9

Turn the Value toggle off. Add seven positive counters and some negative counters (don’t tell your partner how many). Press Find Zero Pairs. Your partner must look at what remains and work out: (a) the value, and (b) how many negative counters you originally added. How do they figure out part (b)?

10

Start with an empty workspace. Press the ±0 Zero Pair button five times. What is the value? Will it ever be anything other than zero, no matter how many times you press it? Explain why. Now remove just one counter by tapping it. What is the value? Why?

11

Switch to Addition mode. In column A, build the number 3 (three positives). In column B, build the number 2 (two positives). Toggle Number A, Number B, and Result on. What does the expression show? Does this match what you expected?

12

In Addition mode, build 4 in column A and (−2) in column B using two negatives. Reveal All. What is the result? Now press the swap button ⇄. What does the expression show now? Is the result the same or different? What does this tell you about addition?

13

Set up 5 + (−5) by building five positives in column A and five negatives in column B. Before revealing, predict the result. Now use Find Zero Pairs in each column. What happens? Reveal the result. What is special about adding a number and its opposite?

14

Keep all reveals hidden. Build a mystery addition where column A has some positives and column B has some negatives. Show your partner only the counters. Can they work out the result before you reveal it? Now try one where both columns have a mix of positives and negatives. Is this harder?

15

Investigate: when you add a positive number to a negative number, is the result always negative? Build at least four different examples using the tool. Can you state a rule for when the result is positive, when it is negative, and when it is zero?

16

Set up the addition (−3) + (−4) by building three negatives in column A and four negatives in column B. Reveal the result. Now try (−1) + (−6). And (−5) + (−2). What do you notice about adding two negative numbers? Is the result always negative? Is it always ‘bigger’ than either number? Be careful with your language here.

17

Switch to Subtraction mode. Build 5 in column A and 2 in column B. Reveal All. What is the result? Now swap columns using ⇄. What expression do you see now? Is the result the same? What does this tell you about subtraction compared to addition?

18

Set up 3 − (−2). Build three positives in column A and two negatives in column B. Reveal the result. Many students expect the answer to be 1. What is it actually? Using the counters, explain why subtracting a negative makes the result larger.

19

Investigate subtraction of two negative numbers: set up (−2) − (−5). Build two negatives in column A and five negatives in column B. Predict the result, then reveal. Is it positive or negative? Try (−5) − (−2) as well. How do the two results compare?

20

Set up 4 − 4 and reveal the result. Now set up 4 − (−4) and reveal. Finally, set up (−4) − 4 and reveal. You have three calculations that all use the number 4 twice. Why are all three results different? What role does the sign play?

21

Using the swap button ⇄, investigate this claim: ‘A − B always gives the opposite sign to B − A.’ Test it with at least four different pairs of numbers (try positives, negatives, and mixtures). Does the claim hold? Can you explain why using zero pairs?

22

In Addition mode, set up any calculation and note the result. Press ⇄ to swap. Does the result change? Try at least five different additions, including ones with negative numbers. Write a conjecture about addition based on your findings.

23

Now do the same in Subtraction mode. Set up any subtraction and note the result. Press ⇄. Does the result change? Try five different subtractions. Write a conjecture about subtraction.

24

In Subtraction mode, set up 7 − 3 and note the result (4). Press ⇄ to get 3 − 7 and note the new result (−4). What is the relationship between 4 and −4? Try three more examples. Can you write a formula connecting (A − B) and (B − A)?

25

Find a subtraction where swapping does not change the result. What must be true about A and B? How many solutions are there?

26

In Addition mode, build counters in both columns but keep all reveals off. Reveal only the Result. Can your partner work backwards to find a possible pair of numbers A and B? Is there more than one answer? How many possibilities can they find?

27

Set up a subtraction, then reveal only Number A and the Result, keeping Number B hidden. Challenge a partner to figure out what Number B must be. What strategy do they use? Now try revealing only Number B and the Result instead. Is this easier or harder?

28

In Explore mode, build a collection of counters with the Value hidden. Turn Count on so your partner can see how many positives and negatives there are. Can they calculate the value before you reveal it? What calculation are they doing?

29

Set up an addition where Number A is hidden and equals (−3), Number B is hidden and equals 5, and the Result is revealed as 2. A student says ‘the answer is 2, so both numbers must be positive.’ Reveal the numbers. Why is the student’s reasoning wrong? What does this teach us about making assumptions?

30

In Addition mode, set up 1 + (−1), then 2 + (−2), then 3 + (−3). What is the result each time? Use Find Zero Pairs to see why. Now try 10 + (−10) without building it. What will the result be? Write a general rule.

31

Investigate: start with 5 + 1, then 5 + 0, then 5 + (−1), then 5 + (−2). What happens to the result as the second number decreases by one each time? Continue the pattern. What would 5 + (−10) be? Can you explain why this pattern must continue?

32

Set up (−3) − (−1), then (−3) − (−2), then (−3) − (−3). What happens to the result as you subtract increasingly negative numbers? Now try (−3) − (−4) and (−3) − (−5). Describe the pattern and explain it using counters.

33

A student claims: ‘Subtracting a negative number is the same as adding the corresponding positive.’ Test this by comparing 4 − (−3) and 4 + 3 in the tool. Try at least four more pairs. Does the claim always hold? Can you explain why using zero pairs?

34

Investigate which pairs of integers A and B satisfy A + B = A − B. Use the tool to test different values. What must B equal? Can you prove it?

35

Using the tool, find all the different ways to make a result of 5 from an addition of two integers. How many are there if each number must be between −10 and 10? What if there is no restriction? Organise your results systematically.

36

A ‘sign swap’ means replacing every positive counter with a negative and vice versa. Build any collection in Explore mode and note the value. Now clear and build the sign-swapped version. What is the relationship between the two values? Does this always hold? What happens if you sign-swap both columns in Addition mode?

37

Challenge: find two numbers A and B where the value of A + B is equal to the value of A − B. Use the tool to test your ideas. Now find two numbers where A + B = B − A. Is this possible?

38

Design your own double-sided counters investigation. Choose a mode, decide which reveals to hide, and write a question that another student could explore. Test it yourself first, then exchange with a partner.