The Walking Method for Multiplication
1
Stand at 0 on the number line.
2
Look at the second number — if it’s positive, face right →. If negative, face left ←. Its size is your step size.
3
Look at the first number — if it’s positive, walk forward. If negative, walk backward. Its size is the number of steps.
4
Where you land is the answer!
Why does (−) × (−) = (+)? If you face left and walk backward, you move right — into positive territory!
First Number (A)
2
×
Second Number (B)
3
Multiplication
Print Worksheet
Investigation Questions
Use these alongside the tool above. Build calculations using the walking method, animate step by step or all at once, compare with Line B, toggle Hide Labels, and switch between Explore and Practice modes to investigate.
1
In Explore mode on the −12 to 12 range, set First Number (A) to 2 and Second Number (B) to 3. Press Go. Watch the animation carefully. What appears below the number line at zero? Which direction does the arrow point? What do the arcs show? Where does the final dot land? Describe the full journey in your own words.
2
Open the How it Works panel by pressing ? in the toolbar. Read the four steps of the walking method. Now set up 3 × 4 and animate it. As you watch, identify each step: where did you start? Which way did you face? How many steps did you take? How far was each step? Close the panel and explain the method to a partner without looking.
3
Set up 1 × 5 and press Go. Now try 2 × 5, then 3 × 5, then 4 × 5. Watch the arcs each time. What stays the same across all four animations? What changes? What does the first number control?
4
Now keep the first number fixed at 3 and change the second number: try 3 × 1, then 3 × 2, then 3 × 3, then 3 × 4. What stays the same? What changes? What does the second number control?
5
Set up 4 × 3 and animate it. Note where the dot lands. Now set up 3 × 4 and animate. Do they give the same answer? Do the journeys look the same? Describe the difference between the two animations even though the results are equal.
6
Set up 2 × 3 on the −12 to 12 range. Before pressing Go, look at the equation bar. The 2 and the 3 are shown in different colours. Which colour is the first number? Which colour is the second? Now press Go and watch the animation. Which colour are the arcs? Which colour are the landing dots? What is the connection?
7
Switch to the −12 to 12 range. Set up 2 × (−3). Look at the equation bar. The second number has changed colour compared to question 6. What colour is it now? Press Go. What colour are the arcs and the direction arrow? What does the colour change tell you about the sign of the second number?
8
Now set up (−2) × 3. Look at the first number in the equation. What colour is it? How is this different from when the first number was positive? Press Go. What colour are the landing dots? Can you now state the full colour rule: what determines the colour of each element on the number line?
9
Set up (−3) × (−4) and look at the equation bar before pressing Go. How many different colours can you see in the equation? Press Go and watch. The arcs are one colour and the dots are another. A student watching says ‘I can tell which number controls the arcs just by matching colours.’ Explain what they mean.
10
Switch to the −12 to 12 range. Set up 2 × (−3). Before pressing Go, use the walking method to predict: which direction will you face? Will you walk forward or backward? Where will you land? Now animate and check. Were you right?
11
Set up 3 × 4 and animate it. Now set up 3 × (−4) and animate. Both have the same first number and the same step size. What is different about the two journeys? What is the relationship between the two answers?
12
Set up 2 × 5 and note the result. Now try 2 × (−5). What do you notice about the two results? Repeat with 3 × 4 and 3 × (−4). Can you state a rule about what happens to the result when you change the sign of the second number?
13
Set up (−2) × 3. The first number is negative, so you walk backward. But you face right (because 3 is positive). Predict which direction you actually move. Animate and check. Now compare with 2 × 3. What is the relationship between the two results?
14
Investigate: set up 4 × 2, then (−4) × 2, then 4 × (−2), then (−4) × (−2). Record all four results. Two of these are positive and two are negative. Which combinations of signs give a positive result? Which give a negative result?
15
Set up (−2) × (−3). Before pressing Go, work through the walking method carefully: the second number is −3, so which direction do you face? The first number is −2, so do you walk forward or backward? If you face left and walk backward, which direction do you actually move? Predict the answer, then animate to check.
16
Try these four calculations and record the results: 3 × 2, (−3) × 2, 3 × (−2), (−3) × (−2). Arrange your results. How many are positive? How many are negative? A student says ‘two negatives make a positive.’ Is this a good description? When exactly do two negatives make a positive in multiplication?
17
Set up (−1) × (−1) and animate. Now try (−2) × (−2), then (−3) × (−3). What do you notice about the results? Are they always positive? Now try (−1) × (−5) and (−5) × (−1). Same result? Same journey?
18
Open the How it Works panel and read the insight box at the bottom. It explains why negative × negative = positive using the walking method. Now close the panel and explain this in your own words to a partner, using a specific example. Can you convince them without using the tool?
19
Toggle Line B on. Click the Line A equation bar and set up 3 × 4. Click Line B and set up 4 × 3. Animate both. Both give 12, but describe how the two journeys are different. What does this tell you about multiplication?
20
With Line B on, set up 2 × 6 on Line A and 6 × 2 on Line B. Animate both. On which line are the arcs taller? On which line are there more arcs? Which journey looks ‘faster’ and which looks more ‘detailed’? Despite these differences, do both reach the same answer?
21
Set up 3 × (−4) on Line A and (−3) × 4 on Line B. Before animating, predict: will the results be the same? Will the journeys look the same? Now animate both. Were you right? What is different about the two paths?
22
Use Line A and Line B to find two different multiplications that land on the same answer. For example, can you find two different ways to reach 12? How about −12? What about 0? How many different multiplications can you find that give the same result?
23
Set up 5 × 2 on Line A and 5 × (−2) on Line B. Animate both. The two journeys start the same way (same facing direction and step count) but end in different places. Why? What does this comparison show about the effect of changing the sign of the second number?
24
Set the animation to Step by Step. Set up 3 × 4 and press Go. The first click shows the direction arrow and the facing indicator. Press Next Step repeatedly. How many clicks does it take to complete the full animation? What appears at each click? Why does the last button say Show Answer instead of Next Step?
25
Using Step by Step, animate 5 × 3, then 3 × 5. Count the number of clicks for each. Which takes more clicks? Why? Does this match what you would expect from the walking method?
26
Now switch to Play All and animate 4 × 3. How is the experience different from Step by Step? In a classroom, when might a teacher choose Step by Step, and when might they choose Play All? Discuss with a partner.
27
Toggle Hide Labels on. The number line now shows only a bare line with zero marked. Set up 2 × 3 and animate. The landing dots appear with their values shown. Before the answer is revealed, can you work out the answer from the dot labels alone? What strategies do you use?
28
With Hide Labels on, switch to the 0 to 48 range (where labels show every 2). Set up 3 × 5 and animate. The landing dots still show their values even though the tick labels are hidden. Why is this especially helpful on the larger ranges?
29
With Hide Labels on, set up 2 × (−4) on the −12 to 12 range. Animate step by step. At each step, a dot appears with its label. A partner must predict where the next dot will land before you press Next Step. Take turns. Who predicts correctly more often?
30
Switch to Practice mode. Select only (+) × (+) on the 0 to 24 range. Press New Question ten times, noting each equation. What do you observe about the numbers chosen? Is the result always positive? Can the result ever be zero? Can it ever be negative?
31
Select only (+) × (−) on the −12 to 12 range. Generate several questions. Is the result always negative? Now select only (−) × (+) and generate several more. Are these results also always negative? How do the journeys differ between the two types, even though the sign of the result is the same?
32
Select only (−) × (−) on the −12 to 12 range. Generate several questions. Is the result always positive? A student says ‘this doesn’t make sense — I’m multiplying two negatives and getting a positive.’ Use the walking method animation to explain to them why this is correct.
33
Switch to the −24 to 24 range and select Select All to enable all four question types. Generate 20 questions, keeping a tally of positive results, negative results, and zero. Is there an equal split? Are any outcomes more common? Can you explain why?
34
In Explore mode, set up 1 × 3, then 2 × 3, then 3 × 3, then 4 × 3. Watch where each dot lands. What is happening to the result each time the first number increases by 1? Without animating, predict where 8 × 3 would land. Check your answer.
35
Now investigate what happens as the second number changes: try 4 × 1, then 4 × 2, then 4 × 3. How does the step size change each time? How does the final position change? What would 4 × 0 give? The tool skips zero, but can you predict the answer from the pattern?
36
Set up 3 × 2, then 3 × 1, then mentally continue to 3 × 0, then use the tool for 3 × (−1), then 3 × (−2). Write down the sequence of results. What pattern do you see? Does this pattern help explain why 3 × (−1) = −3?
37
A student claims: ‘Multiplying by a bigger number always gives a bigger answer.’ Test this by comparing 3 × 5 and 3 × (−5). Is −5 bigger or smaller than 5? Is the result bigger or smaller? Now try (−3) × 5 and (−3) × (−5). What is happening? When is the student’s claim true and when is it false?
38
Investigate: when does A × B give the same result as B × A? Use the tool to test at least six pairs, including pairs with negatives. Does this always hold? Use Line B to compare the journeys. The answers are always equal, but are the journeys ever identical?
39
Set up 2 × (−6) and note the result (−12). Now set up (−2) × 6 and note its result (−12). These give the same answer. Can you find other pairs where a positive times a negative equals a negative times a positive? Is there a general pattern?
40
Start on the 0 to 24 range. Set up 4 × 6 = 24 — the dot lands right at the end. What is the largest product you can make on this range? What factor pairs give it? Now switch to 0 to 48. What is the largest product now?
41
On the −12 to 12 range, what is the most negative result you can produce? What values of A and B give this minimum? Now try to make the most positive result. What is it? Are the two extremes symmetric?
42
Switch between the 0 to 24 and 0 to 48 ranges. On 0 to 24, labels appear at every integer. On 0 to 48, labels appear at every 2. Look at the unlabelled tick marks between the labels. How can a student use these smaller ticks to find odd-numbered positions? Set up 3 × 5 = 15 on the 0 to 48 range and check that the dot lands on an unlabelled tick.
43
Set up 3 × 4 and animate. The result is 12. Now think: what subtraction or addition would produce the same journey — starting at 0 and making three jumps of 4 to the right? How is repeated addition related to multiplication? Use the number line visual to explain.
44
Set up 4 × (−3) and animate. You see 4 jumps of 3 going left. Now think about this as repeated subtraction: 0 − 3 − 3 − 3 − 3. Does this give the same result? What does this tell you about multiplying by a negative number?
45
A student says: ‘I know that 12 ÷ 3 = 4 because 4 × 3 = 12.’ Use the tool to animate 4 × 3. The dot lands at 12 after four jumps of 3. Now imagine ‘undoing’ those jumps. How many jumps of 3 does it take to get from 12 back to 0? What does this tell you about the relationship between multiplication and division?
46
Press Print and select With questions with only (+) × (+) selected on the 0 to 24 range. 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 the −12 to 12 range. Which worksheet was harder? Why?
47
Print a Blank number lines worksheet on the −12 to 12 range. Write your own multiplication questions for a partner to solve by drawing arcs. Include at least one from each sign type: (+) × (+), (+) × (−), (−) × (+), and (−) × (−). Exchange worksheets and mark each other’s work using the tool to check.
48
How many different multiplications on the −12 to 12 range give a result of exactly 12? List them systematically, including negative factors. How many give exactly −12? Is the number the same? Why?
49
Find a multiplication where the first number, the second number, and the result are all the same. Is this possible with a positive number? A negative number? How many solutions are there?
50
Using the tool, investigate this claim: ‘If you double the first number, the result doubles.’ Test it with at least four examples, including negatives. Does it always hold? Now test: ‘If you double both numbers, the result doubles.’ Is this true? What actually happens to the result?
51
Challenge: find two different multiplications where the journeys look as different as possible but give the same result. Think about using different signs, different numbers of arcs, and different step sizes. Use Line B to display them side by side. Describe what makes them look so different despite having the same answer.
52
Design your own multiplication number line investigation. Choose a mode, a range, and specific settings. 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.