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Investigation Questions
Use these alongside the tool above. Add number counters and drag one onto another to ‘smash’ them using +, −, × or ÷. Switch between Explore and Challenge mode to investigate.
1
In Explore mode, add a counter for 3 and a counter for 5 by typing each number and pressing Create. Drag the 3 counter onto the 5 counter. An operation popup appears with four choices: +, −, ×, ÷. Choose +. What number appears on the new counter? What colour is it? Now press ↩ to undo. This time, drag the 5 onto the 3 and choose +. Is the result the same? Does the counter land in the same place?
2
Add counters for 8 and 2. Drag 8 onto 2 and look at the merge order indicator that appears — it shows the dragged number and the target number. Choose −. What is the result? Now undo, and drag 2 onto 8 and choose −. What is the result this time? Why are the two results different? The indicator shows ‘dragged ○ target’ — explain what the order means for subtraction.
3
Add counters for 6 and 4. Before you smash them, predict the result of each of the four operations: +, −, ×, ÷. Now test each one, undoing between each. Were all your predictions correct? Which operation gives the largest result? Which gives the smallest? Which gives a result that is not a whole number?
4
Add three counters: 2, 3 and 5. Smash 2 and 3 using × to make 6. Now smash the 6 with the 5 using +. What do you get? Undo both steps and try a different approach: smash 3 and 5 using + first (making 8), then smash 2 and 8 using ×. Do you get the same final answer? Why or why not? What does this tell you about the importance of choosing which pair to combine first?
5
Look at the counter colours. Add counters for 7, −3 and 0 using the quick-tile tray. What colour is each counter? Smash 7 and −3 using +. What colour is the result? Now undo and smash them using − instead (drag 7 onto −3). What colour is the result now? Can you state the rule for when a counter is blue, red or grey?
6
Add counters for 4 and 7. Drag 4 onto 7 and choose +. Note the result. Undo, then drag 7 onto 4 and choose +. Is the result the same? Now repeat both experiments using × instead of +. What do you notice about addition and multiplication when you swap the drag order?
7
Using the same numbers 4 and 7, investigate subtraction. Drag 4 onto 7 and choose −. What is the result? Undo, then drag 7 onto 4 and choose −. What is the result now? What is the relationship between the two answers? How does the merge order indicator help you predict which way round the subtraction will go?
8
Using 12 and 3, investigate division. Drag 12 onto 3 and choose ÷. What is the result? Undo and drag 3 onto 12 and choose ÷. What is the result now? One gives a whole number and the other gives a fraction. Why does the order of division matter so much more than the order of addition?
9
Add counters for 5 and 5 — two identical numbers. Try all four operations: 5 + 5, 5 − 5, 5 × 5, 5 ÷ 5. Write down the four results. Now try the same with 3 and 3, then with 10 and 10. For which operations does the drag order not matter when both numbers are the same? Can you explain why?
10
For any number n smashed with itself, the four results are: n + n = 2n, n − n = 0, n × n = n², n ÷ n = 1. Two of these results are the same regardless of what n is. Which two? Test with n = 8, n = ½ and n = −4. Why do subtraction and division always give the same answer when both numbers are identical?
11
Using the Fractions tab in the quick-tile tray, add counters for ½ and ½. Smash them using +. What is the result? Now add counters for ⅓ and ⅓ and smash using +. What is the result? What do you expect ¼ + ¼ to give? Check using the tool. What pattern connects the denominator of the starting fraction to the result of doubling it?
12
Add counters for ½ and ⅓. Before smashing, predict: what is ½ + ⅓? Now smash using + and look at how the counter displays the fraction. Was your prediction correct? Now try ½ − ⅓. What is the result? Is it closer to 0 or to ½?
13
Add counters for ⅔ and ¾. Smash them using ×. What is the result? Does the tool simplify the fraction automatically? Now undo and try ⅔ ÷ ¾ (drag ⅔ onto ¾). What does the result look like? Can you explain why dividing by a fraction less than 1 gives a result larger than the number you started with?
14
Add a counter for ½ and a counter for 2. Smash ½ ÷ 2 (drag ½ onto 2). What is the result? Now undo and smash ½ × 2 instead. What is the result? A student says ‘dividing by 2 and multiplying by ½ give the same answer.’ Is this student confusing two different things? Test with other starting numbers to find out which claim is actually true.
15
Add counters for ½, ⅓ and ⅙. Smash ½ and ⅓ using +. What is the result? Now smash that result with ⅙ using +. What do you get? Can you explain why ½ + ⅓ + ⅙ = 1 using what you know about equivalent fractions? Use the pen tool to write the calculation with a common denominator.
16
Start with a counter for 1. Add a counter for 2 and smash 1 ÷ 2 to get ½. Add another 2 and smash ½ ÷ 2 to get ¼. Add another 2 and smash ¼ ÷ 2. Continue dividing by 2 three more times. Write down the sequence of fractions. What is the pattern? How many divisions by 2 does it take to get below 1/100?
17
Add counters for 5 and −3 from the quick-tile tray. Smash using +. What is the result? Is the counter blue or red? Now undo and try 5 + (−8) instead. What happens when the negative number has a larger absolute value than the positive number? How can you predict the sign of the result when adding a positive and a negative?
18
Add counters for −4 and −6. Before smashing, predict: what is −4 + (−6)? Will the result be more negative or less negative than either number? Smash using + and check. Now try −4 × (−6). What sign is the result? Finally try −4 − (−6), dragging −4 onto −6. Which of the four operations gives a positive result from two negatives?
19
Add counters for 3 and −3. Try all four operations. Which operation gives zero? Which gives a positive result? Which gives a negative result? Now try 3 ÷ (−3) and (−3) ÷ 3. Do they give the same result? What does this tell you about the sign rules for division?
20
A student says: ‘Multiplying two negative numbers always gives a positive result.’ Test this using −2 × (−5), then −3 × (−4), then −1 × (−7). Is the student right? Now check: is −2 × (−5) the same as 2 × 5? What about −3 × (−4) and 3 × 4? State a general rule connecting the product of two negatives to the product of their absolute values.
21
Add counters for −8 and 2. Smash −8 ÷ 2 (drag −8 onto 2). What is the result? Undo and reverse the order: 2 ÷ (−8). What is the result now? One is a whole number and one is a fraction. What sign does each have? Now try −8 ÷ (−2). What is the sign when both numbers are negative? State the complete sign rules for division.
22
Add a counter for 7 and a counter for 0. Smash using +: what is 7 + 0? Try 7 × 0. And 7 − 0. Now try 0 − 7 (drag 0 onto 7). Three of these operations leave 7 unchanged or give a predictable result. For which operation is zero ‘invisible’ (doesn’t change the other number)? For which operation does zero ‘dominate’?
23
What happens when you try 7 ÷ 0 (drag 7 onto 0)? Does the tool allow it? Now try 0 ÷ 7 (drag 0 onto 7). Is this allowed? What is the result? Why is dividing by zero impossible while dividing zero by something else is fine? Use what you know about multiplication to explain: if 7 ÷ 0 = x, then x × 0 would need to equal 7. Can it?
24
Add a counter for 8 and a counter for 1. Try all four operations: 8 + 1, 8 − 1, 8 × 1, 8 ÷ 1. Which operation leaves the number completely unchanged? This is the ‘multiplicative identity’ property. Now try the same four operations with −1 instead of 1. What does multiplying by −1 do to a number? Test with several different starting numbers including negatives and fractions.
25
Start with a counter for 2. Duplicate it (long-press) to make a second 2. Smash using × to get 4. Duplicate the 4 and smash using × to get 16. Duplicate the 16 and smash using × to get 256. Write down the sequence: 2, 4, 16, 256. What is happening? You are squaring the number each time. Now start with 3 instead. After three squarings, what number do you reach? Why does this sequence grow so quickly?
26
Switch to Challenge mode and press ‘Random 1–10’. Look at the five targets generated. They have different difficulty ratings: ★ Easy, ★★ Medium, ★★★ Hard, ★★★★ Negative, and ★★★★★ Fraction. Before attempting any, predict which will be hardest. Now work through them starting with Easy. Was your prediction correct? Which operations did you use most often for the Easy target? What about the Fraction target?
27
In Challenge mode, enter the numbers 1, 2, 3, 4 and press Generate Targets. The instruction says ‘Use all of 1, 2, 3, 4 to make each target.’ What does ‘all of’ mean? Try to reach the Easy target. Remember: you must use all four numbers in exactly three smashes, so the target is only reached when one counter remains. What happens if you create the target number after only two smashes but still have an unused counter?
28
Press ‘New Targets’ five times for the same set of numbers. Write down the Easy target each time. Do you ever see the same target twice? Are the Easy targets always small positive integers? Now compare the Hard targets. What makes them harder — is it the size of the number, or the operations required to reach it?
29
Press ‘One Big, Three Small’ to generate a set with one larger number (typically 10, 20, 50 or 100). How does having a big number change your strategy? For the Easy and Medium targets, is the big number more useful with + and −, or with × and ÷? For the Fraction target, what role does the big number play?
30
Attempt all five targets for any set of numbers. For each one, record your three smashes on paper. Now tap the target cards to reveal the tool’s solutions. Compare the tool’s approach with yours. Did you use the same operations? The same pairing order? Can the same target have more than one valid solution? Try the Negative target — how many different routes to it can you find?
31
Use the Undo Smash button to correct mistakes. But before your first smash, try a strategic approach: look at a target and think about what the final smash must be. What two numbers need to combine to make the target? Then work backwards — how can you produce those two numbers from your four starting numbers in two smashes? Is ‘working backwards’ from the target easier than ‘working forwards’ from the starting numbers? Test this strategy on the Hard and Fraction targets.
32
In Explore mode, investigate what happens when you smash n with itself for every operation. Try n = 6: you get 12, 0, 36, 1. Now try n = 10: you get 20, 0, 100, 1. The subtraction always gives 0 and the division always gives 1, regardless of n. Test with n = ½ and n = −4 to confirm. Why are 0 and 1 unavoidable here?
33
Add counters for 2, 3, 4 and 5. Smash all four using only × in all three steps: 2 × 3 = 6, then 6 × 4 = 24, then 24 × 5 = 120. Now undo everything and try a different order: 4 × 5 = 20, then 2 × 3 = 6, then 20 × 6 = 120. Try a third order of your choice. Is the final product always 120? Does the order in which you multiply numbers affect the result? Why?
34
Using the same numbers 2, 3, 4, 5 — now use only + in all three smashes. Try different pairing orders. Is the sum always 14? Now try mixing operations: use + for two smashes and × for one. How many different final results can you produce depending on which pair you multiply? List them all. Why does changing a single operation make such a big difference?
35
Start with a counter for 1. Add a counter for 2 and smash 1 ÷ 2 to get ½. Add a counter for 3 and smash ½ ÷ 3 to get ⅙. Add a counter for 4 and smash ⅙ ÷ 4 to get 1/24. You have computed 1 ÷ 2 ÷ 3 ÷ 4. The result is 1/24. Is 24 the same as 2 × 3 × 4? Now try 1 ÷ 2 ÷ 3 ÷ 5. Predict the result before checking. What is the general rule?
36
Add four counters: 8, 2, 8, 2 (duplicate the originals using long-press). Smash the first pair: 8 + 2 = 10. Smash the second pair: 8 − 2 = 6. Now smash 10 × 6 = 60. Separately, calculate 8² − 2² = 64 − 4 = 60. The same! You have just shown that (8 + 2)(8 − 2) = 8² − 2². Try with another pair: use 7, 3, 7, 3. Does it still work? This is the ‘difference of two squares’ identity — can you explain why it always holds?
37
Add a counter for any number and a counter for its reciprocal (e.g. 3 and ⅓, or 5 and ⅕). Smash using ×. What is the result? Try at least four different pairs, including ½ and 2, and ¼ and 4. Is the result always the same? What is the mathematical name for two numbers whose product is 1? Now try smashing a number and its reciprocal using +. Is there a pattern in the sums?
38
In Challenge mode, which sets of four numbers produce the widest variety of reachable targets? Try 1, 2, 3, 4 and press Generate Targets — all five targets should be reachable. Now try 1, 1, 1, 1. Are there fewer possible targets? What about 2, 2, 2, 2? Investigate: which four numbers from 1 to 10 produce the most interesting set of targets?
39
Enter the numbers 1, 2, 3, 4 in Challenge mode. Using all four in exactly three smashes, can you reach every whole number from 1 to 10? Record the three smashes for each target. Which targets were easy? Which were hard? Were any impossible? What is the largest positive integer you can reach? The most negative?
40
In Explore mode, add counters for 1, 1, 1 and 1 — four ones. Using all four in three smashes, what different results can you make? You might think only a few are possible, but remember you can use ÷ and − too. List every distinct result you can find systematically. How many are there? (Hint: think carefully about the order of operations and drag directions.)
41
The classic ’24 Game’ challenge: can you make 24 from four given numbers using +, −, ×, ÷? Try these in Challenge mode (or Explore mode): 1, 2, 3, 4 (hint: use all four operations). Then try 8, 3, 8, 3. Then try 1, 5, 5, 5. Which sets can reach 24 and which cannot? For the ones that work, is there more than one solution?
42
Investigate the Fraction target in Challenge mode. Press ‘New Targets’ ten times for the same set of numbers and write down every Fraction target you see. Do they always have small denominators? What is the largest denominator you ever see? What is the most unusual fraction that appears? Why might the tool prefer fractions with small denominators?
43
In Explore mode, start with a counter for 100. Add a counter for 3 and smash 100 ÷ 3 to get 33⅓. Now add another 3 and smash 33⅓ × 3. Do you get exactly 100 back? Try the same with 100 ÷ 7 then multiplying by 7 again. The tool uses exact fractions, so there is no rounding error. Why does exact arithmetic matter here? What would happen on a calculator that rounds to a fixed number of decimal places?
44
A student claims: ‘If you double the first number, the result always doubles.’ Test this for addition: compare 3 + 5 with 6 + 5. Is the result doubled? No — it went from 8 to 11. Now test for multiplication: compare 3 × 5 with 6 × 5. Is the result doubled? For which of the four operations is the student’s claim true, and for which is it false? Use the tool to test at least three examples for each operation.
45
Design your own Number Smash investigation. Choose between Explore and Challenge mode, decide on specific starting numbers, and write three questions that another student could explore using the tool. Consider which features to use — the drawing tools, the history panel, the undo button, or the slide boards. Test your questions yourself first, then exchange with a partner. Which of your questions led to the most interesting discoveries?