Taken away
0
Drag counters here to take them away
+1 positive
·
−1 negative
|
Tap a counter to flip it · Drag onto its opposite to make zero · Drag to the bank (or long-press) to remove
+1
Positive
−1
Negative
Investigation Questions
Use these alongside the tool above. Build numbers from positive and negative counters, tap a counter to flip it, form zero pairs, and switch between Explore, Add and Subtract. In Subtract, take counters away into the tray — adding zero pairs first when you need to.
1
In Explore mode, drag three positive (red) counters onto the mat. Turn the Value toggle 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
Put one positive counter on the mat. Now tap it. What happens? Tap it again. A counter is “double-sided” — one side is +1 and the other is −1. How does flipping a counter change the value of the mat?
3
Starting from an empty mat, add five negative counters. What is the value? Now flip them to positive one at a time, turning Value on after each flip. Describe what happens to the value each time. At what point does the value become zero?
4
Turn the Value toggle off. Build a collection where there are more negatives than positives. Ask a partner to predict whether the value is positive, negative or zero. How can they tell just by looking, without counting exactly?
5
Turn Count on. Build several different collections that all have a value of 2 — first using four counters, then six, then ten. What do all your collections have in common? (Look at the difference between the number of positives and negatives.)
6
Can you build a value of zero using exactly six counters? Eight counters? Can you do it with five? What must be true about the total number of counters for a value of zero to be possible?
7
Add one positive and one negative counter to the mat. Drag one onto the other to form a zero pair. Watch the animation. What value did the pair contribute before they vanished? Why do we call it a “zero pair”?
8
Press the ±0 Zero Pair button five times. What is the value of the mat? Will it ever be anything other than zero, no matter how many times you press it? Explain why. This is the key idea behind subtraction later: adding zero pairs never changes the value.
9
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? Press the button and check. Were you right?
10
Build 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: with P positives and N negatives, how many zero pairs form, and what is left?
11
Make a value of +3 in as many different ways as you can — using zero pairs to “hide” extra counters. (For example, four positives and one negative.) How many representations can you find? Why are there infinitely many?
12
Switch to Add. Build 3 on the mat (three positives), then press “Add the second number”. Now add 2 more positives. Turn First, Second and Result on. What does the number sentence show? Does it match what you expected?
13
Set up 4 + (−2): build four positives, press “Add the second number”, then add two negatives. Use Find Zero Pairs to simplify. Reveal the Result. Why is the answer 2 and not 6? Which counters cancelled, and which were left?
14
Set up 5 + (−5): five positives, then five negatives. Predict the result before revealing. Press Find Zero Pairs. What happens to every counter? What is special about adding a number and its opposite?
15
A student writes (−3) + (−4) = 7, saying “two negatives make a positive”. Build it: three negatives, add four more negatives. What is the real answer? Use the counters to explain why the “two negatives make a positive” rule does not apply to addition.
16
Keep all reveals hidden. Build a mystery addition where the first number is positive and the second is negative. Show a partner only the counters. Can they work out the result before you reveal it? What strategy do they use?
17
Investigate: when you add a positive and a negative number, is the result always negative? Build at least four examples. State a rule for when the result is positive, when it is negative, and when it is zero.
18
Switch to Subtract. Build 5 (five positives) and press “Take away the second number”. Now drag 2 positive counters into the tray. Reveal the Result. What calculation have you just modelled, and what is the answer?
19
Set up 3 − 5: build three positives and press “Take away the second number”. You need to take away five positives, but there are only three. Add zero pairs (±0 Zero Pair) until you have five positives, then take five away. What is left? Why is the answer negative?
20
The big one: set up 3 − (−2). Build three positives, press “Take away the second number”. You need to take away two negatives — but there are none on the mat. Add two zero pairs (the value stays 3), then take the two negatives into the tray. Many students predict 1. What is the answer really? Explain why taking away a negative makes the result larger.
21
Set up (−2) − (−5): two negatives, then take away five negatives (add zero pairs as needed). Reveal the result. Is it positive or negative? Now set up (−5) − (−2). How do the two results compare?
22
Set up 4 − (−3). Solve it with take-away and note the result. Now turn on the “add the opposite” toggle in the number sentence. What addition does it show? Set up 4 + 3 separately and check it gives the same answer. Test three more pairs. Can you explain, using zero pairs, why subtracting a negative is the same as adding a positive?
23
Model 4 − 4, then 4 − (−4), then (−4) − 4. All three use the number 4 twice, yet all three results are different. Solve each with the tool. Why are they different? What role does the sign play?
24
Before you build anything, predict how many zero pairs you will need to add for: 6 − (−1), 6 − (−4), 6 − (−7). Then model each one. What is the connection between the number you are subtracting and the number of zero pairs you must add?
25
In Add, build a calculation but keep all reveals off. Reveal only the Result. Can a partner work backwards to find a possible first and second number? How many different answers can they find?
26
Set up a subtraction, then reveal only the First number and the Result, keeping the Second hidden. Challenge a partner to work out what was taken away. Now reveal only the Second and Result instead. Which is easier?
27
In Explore, build a collection with the Value hidden but Count on, so a partner can see how many positives and negatives there are. Can they calculate the value before you reveal it? What calculation are they doing?
28
Build an addition where the first number is (−3), the second is 5, and the result is 2 — but keep the first two hidden, showing only Result = 2. A student says “the answer is 2, so both numbers must be positive.” Reveal the numbers. Why is the student wrong? What does this teach us about assuming?
29
In Add, model 1 + (−1), then 2 + (−2), then 3 + (−3). What is the result each time? Use Find Zero Pairs to see why. Predict 10 + (−10) without building it. Write a general rule.
30
Model 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? Why must this pattern continue?
31
In Subtract, model (−3) − (−1), then (−3) − (−2), then (−3) − (−3), then (−3) − (−4), then (−3) − (−5). What happens to the result as you subtract increasingly negative numbers? Describe and explain the pattern using counters.
32
Model 7 − 3 and note the result. Now model 3 − 7. What is the relationship between the two answers? Try three more pairs. Can you write a connection between (A − B) and (B − A)?
33
Investigate which pairs of integers A and B make A + B equal to A − B. Use the tool to test values. What must B be? Can you prove it using zero pairs?
34
Find all the different ways to make a result of 5 from an addition of two integers between −10 and 10. How many are there? Organise your results systematically. What if there is no restriction on the numbers?
35
A “sign swap” means flipping every counter to its opposite. Build any collection in Explore and note the value, then flip every counter. What is the relationship between the two values? Does this always hold? Explain using the idea of the opposite of a number.
36
Find two numbers A and B where A − B equals A + B. Then try to find two numbers where A + B equals B − A. Is that possible? Use the tool to test your ideas and justify your conclusions.
37
Design your own counters investigation. Choose Explore, Add or Subtract, decide which reveals to hide, and write a question for another student to explore. Test it yourself first, then swap with a partner.