Start
0
Op
Number
1
Addition
Subtraction
Print Worksheet
Investigation Questions
Use these alongside the tool above. Build calculations, animate jumps, compare with Line B, toggle Hide Labels, and switch between Explore and Practice modes to investigate.
1
In Explore mode, set Start to 3, Operation to +, and Number to 5. Press Go to lock in the equation. What appears in the equation bar? Now press Go again to start the animation. Describe what you see. Where does the dot start? Where does it end? What does the arc represent?
2
Set up 8 + 4 on the 0 to 20 range. Before pressing Go, predict: will the answer fit on the number line? Now try it. What happens? Change the number to 3 instead. Does it fit now? What is the largest number you can add to 8 and still stay on this number line?
3
Set up 12 − 5. Press Go to see the equation, then Go again to animate. Which direction does the arc go? What colour is it? Now set up 12 + 5. How is the arc different? What do the direction and colour tell you about the operation?
4
Set up 10 + 5 and animate it. Now press Reset, set up 5 + 10, and animate that. Compare the two arcs. Do they end at the same place? Do they look the same? What property of addition does this show?
5
Switch to the −10 to 10 range. Set Start to 0, Operation to +, Number to 1. Press Go twice to animate. Now use the ▲ arrow on Number to increase it to 2, then animate again. Keep increasing and animating. What happens to the arc as the number gets larger? Describe how the height of the arc changes.
6
Switch to −10 to 10. Set Start to 3 and Number to 5, with Operation set to −. Before animating, predict: will the answer be positive or negative? Animate and check. Now try Start 3, subtract 3. What is special about this result? Where does the dot land?
7
Set up −4 + 7 (Start = −4, Operation = +, Number = 7). Watch the animation carefully. The arc splits into two parts. Describe what happens at zero. What is the dark circle with ‘0’ that appears? Why does the tool break the jump into two pieces here?
8
Set up 6 − 9. Animate step by step. At which step does the dot cross zero? Now set up −3 + 8. Does this also cross zero? Compare the two bridge animations. What do all calculations that cross zero have in common?
9
Investigate: starting from a positive number, what is the smallest subtraction that gives a negative result? Try Start = 1, subtract 2. Now try Start = 5, subtract 6. Can you state a general rule: if Start is positive, what must be true about the number you subtract to get a negative answer?
10
Set up −7 − 3 and animate it. The result is −10. Now set up −7 + 3 and animate. The result is −4. Both start at −7, but one goes further left and the other goes right. Explain in your own words what adding does versus subtracting when you start at a negative number.
11
Switch to Practice mode and select only the type (+) − (−). Generate a question. Look at the equation carefully — it shows something like 3 − (−5). Before pressing Go, predict: will the answer be larger or smaller than the starting number? Animate and check. Try at least five questions. What do you notice?
12
In Explore mode, set up 4 − (−3). Which direction does the arc go — left or right? Many students expect subtraction to always go left. Why does subtracting a negative go right? Now compare with 4 + 3. What do you notice about the two results?
13
A student says: ‘Subtracting a negative is the same as adding the positive.’ Test this claim by comparing these pairs: 5 − (−2) and 5 + 2; then −3 − (−4) and −3 + 4; then 0 − (−7) and 0 + 7. Use the tool to animate each one. Does the claim always hold? Can you explain why using the direction of the arcs?
14
Set up −2 − (−8). Predict: will the answer be positive or negative? Now animate. Were you right? Try −6 − (−4). Is the pattern the same? Can you state a rule for when subtracting a negative from a negative gives a positive result?
15
In Explore mode, toggle Line B on. Click the Line A equation bar to select it and set up 3 + 5. Now click Line B and set up 5 + 3. Animate both. Compare the arcs. Do they start and end at the same places? What is the same and what is different about the two animations?
16
With Line B on, set up 8 − 3 on Line A and 8 + (−3) on Line B. Animate both. Compare the results, the arc directions, and the positions of the dots. Are these two calculations doing the same thing? Use this to explain the relationship between subtraction and adding a negative.
17
Set up 6 − 4 on Line A and 4 − 6 on Line B. Animate both. What is the relationship between the two answers? Try three more pairs where you swap the start and the number. Can you write a general rule connecting A − B and B − A?
18
Set up −3 + 7 on Line A and −3 + 3, then −3 + 4 on Line B (adjust between animations). You are exploring what happens when you add different amounts to the same starting number. How does the ending position change as the number increases by 1 each time?
19
Challenge: using Line A and Line B, find two different calculations that give the same result. For example, can you find an addition and a subtraction that both equal 4? How many different pairs can you find?
20
Set the animation to Step by Step. Set up 5 + 3 and press Go to lock in the equation. Now press Go repeatedly. How many steps does it take to complete the animation? What happens at each step? Why is the last step labelled Show Answer instead of Next Step?
21
Using Step by Step on the −10 to 10 range, set up −4 + 9. This crosses zero, so the jump splits into a bridge. How many steps does this take compared to a jump that doesn’t cross zero? What extra visual appears at zero?
22
Now switch to Play All and animate the same calculation (−4 + 9). How is the experience different? In a classroom, when might a teacher choose Step by Step, and when might they choose Play All? Discuss with a partner.
23
Toggle Hide Labels on. The number line now shows only a bare line with arrows. Set up 0 + 6 in Explore mode and animate step by step. When the starting dot appears, its label is revealed beneath it. The arc animates and the end dot appears — but without a label. Press Reveal to show it. Before you pressed Reveal, how could you have worked out where the dot landed?
24
With Hide Labels on, set up 3 + 4 on the 0 to 20 range. After the arc animates, the end dot sits on the bare line with no label. A partner must count the position from the start label to predict the answer. Reveal to check. Take turns setting up calculations for each other. Who gets the most right?
25
Turn Hide Labels on with the −10 to 10 range. Set up −2 + 7. This crosses zero. After the bridge arc, the zero badge appears — this is the only landmark on the bare line. Use it as a reference point to predict where the final dot lands. Is it easier or harder to predict when zero is visible?
26
With Hide Labels on, switch to Practice mode. Select only (+) + (+) on the 0 to 20 range. Generate a question and animate step by step. You see the equation (with ▢ for the answer), the start label, the arc label showing the jump size, but no end label until you press Reveal. Describe the reasoning you use to find the answer.
27
Now try Hide Labels in Practice mode with (−) + (+) on −10 to 10. The start label appears as a negative number. The arc crosses zero. Can you still predict the landing position? What makes this harder than a calculation that stays on one side of zero?
28
In Practice mode, select only (+) + (+) on the 0 to 20 range. Generate ten questions using New Question. What do all the starting numbers have in common? What do all the second numbers have in common? Can the answer ever be negative with this type?
29
Now select only (+) − (+) → (+). Generate several questions. What must be true about the starting number and the number being subtracted for the answer to be positive? Now switch to (+) − (+) → (−). What changes?
30
Switch to the −10 to 10 range and select all types using Select All. Generate 20 questions, keeping a tally of how many give positive answers, negative answers, and zero. Is there a roughly equal split? Are any results more common than others?
31
Select only (−) + (−). Generate several questions. Is the result always negative? Is the result always further from zero than either starting number? Can you explain why adding two negatives always moves further left?
32
Select only (−) − (−). Generate several questions. Sometimes the answer is positive and sometimes it is negative. Can you work out the rule? When does subtracting a negative from a negative give a positive result?
33
In Explore mode, set up 10 + 1, then animate. Reset, then try 10 + 2, then 10 + 3, and so on. Watch where the end dot lands each time. Now try 10 − 1, 10 − 2, 10 − 3. What pattern do you see in the landing positions? What would 10 − 11 give?
34
Start with 0 + 5 and note the result. Now try 1 + 4, then 2 + 3, then 3 + 2, then 4 + 1, then 5 + 0. All these pairs add up to 5. Using Line B, animate two of these pairs side by side. What is the same and what is different about the arcs?
35
On −10 to 10, animate 5 − 8 and note the result (−3). Now animate −3 + 8. What do you get? Try another pair: 3 − 7 = −4, then −4 + 7. What is happening? Can you explain why A − B followed by adding B always gets you back to A?
36
A student claims: ‘Adding a number and then subtracting the same number always gets you back to where you started.’ Test this using the tool. Try it with positive numbers, negative numbers, and zero. Does it always work? Can you find a case where it fails?
37
On −10 to 10, set up 0 + 1, then 0 + 2, then 0 + 3, going up to 0 + 10. Now do 0 − 1, 0 − 2, down to 0 − 10. Describe the symmetry you see. What does this tell you about the relationship between positive and negative numbers?
38
Investigate: for the calculation A + B on the −10 to 10 range, what is the largest possible result? What values of A and B give this maximum? What is the smallest (most negative) possible result? What about the largest possible result of A − B?
39
On −10 to 10, use Step by Step to animate 3 − 5. The jump crosses zero. Step through slowly. How many arcs appear? Where does the first arc end? What is the zero badge telling you? Why is it useful to break the journey at zero?
40
Animate −4 + 10 step by step. The bridge goes from −4 to 0 (a jump of +4), then from 0 to 6 (a jump of +6). Check: does 4 + 6 = 10? Will the two bridge parts always add up to the total jump? Test with two more examples that cross zero.
41
A teacher says: ‘When you cross zero, think of it as two separate problems.’ Set up −3 + 8. The bridge shows −3 to 0 (which is +3) and 0 to 5 (which is +5). So the problem becomes: ‘How far from −3 to 0?’ and then ‘How far from 0 to 5?’ Try this thinking with −7 + 9 and 4 − 6. Does the ‘two problems’ approach help?
42
Find all the calculations on the −10 to 10 range where the bridge arc from the start to zero and the arc from zero to the end are exactly the same length. What must be true about the start number and the result for this to happen?
43
How many different calculations on the 0 to 20 range give a result of exactly 10? Remember, you can use any start number and either addition or subtraction. List them systematically. Is there a pattern?
44
On −10 to 10, find a calculation where the arc is as tall as possible (covers the most distance). Now find one where the arc is as short as possible. What determines the height of the arc?
45
Print a worksheet using With questions with only (+) + (+) selected on 0 to 20. Without using the tool, solve all six questions by drawing your own arcs on the printed number lines. Now generate a second worksheet with (−) + (+) on −10 to 10. Which worksheet was harder? Why?
46
Using Practice mode with all types selected, generate questions and sort them into categories: ‘I can solve this in my head’, ‘I need the number line to help’, and ‘I find this really challenging.’ What makes a question fall into each category? Compare your sorting with a partner’s.
47
Investigate: is there a calculation where the starting number, the second number, and the result are all the same? What about where all three are different signs? Use the tool to test your ideas.
48
Design your own number line investigation. Choose a mode, a range, and specific question types. Write a question that another student could explore using the tool. Test it yourself first to make sure it leads to interesting discoveries, then exchange with a partner.