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Investigation Questions
Use these alongside the tool above. Create number tiles, split into factor pairs (✂), decompose into primes (⚛), pull out factors (÷), generate multiples (×), and drag tiles onto each other to multiply.
1
Create a tile for the number 7. What colour is it? Now create tiles for 2, 3, 5 and 11. Each prime has its own colour. Can you work out the colour code? Now create 4. Why is it white? What about 1 — why is it grey?
2
Create tiles for every number from 2 to 12. Sort them by dragging: primes on the left, composites on the right. How many of each are there? How can you tell at a glance which are prime just from the colour?
3
Create a tile for 30 and look at its tooltip (hover or check Number Facts). Now create tiles for 2, 3 and 5. Compare their colours with the colour of the 30 tile. What do you notice? Repeat with 12. Can you explain why composite tiles are white while their prime building blocks have colours?
4
Create tiles for 4, 8, 16, 32. They are all powers of 2. Tap each one — what does the prime feedback toast say? Now try 9, 27, 81. What do all the primes-of-2 tiles and primes-of-3 tiles have in common? What is different?
5
Without creating them, predict: will 51 be coloured or white? What about 91? Create both tiles and tap them to check. Were you surprised? What makes these numbers deceptive?
6
Create a tile for 12. Tap it to select it, then open Number Facts (ℹ). How many factors does 12 have? How many factor pairs? Now close the panel and press the ✂ button on the 12 tile. Count the tiles that appear. How does the number of tiles relate to the number of factor pairs?
7
Create a tile for 24 and press ✂ to split it into factor pairs. Look at the pairs that appear. Which pair has the two numbers closest together? Which pair has the biggest difference between the two numbers? Use the pen to circle the pair you think is most ‘useful’ and explain your choice.
8
Press ✂ on a tile for 36. Now do the same for 48 on a new board. Which number has more factor pairs? Before checking, predict: does 60 have more or fewer factor pairs than 48? Create 60 and check. What is it about these numbers that gives them so many factors?
9
Create tiles for 2, 3, 5, 7 and 11. Try pressing ✂ on each one. What happens? Why don’t prime numbers have a factor pairs button? Now try 4 — the smallest composite. How many factor pairs does it produce?
10
Investigate: create tiles for 1, 4, 9, 16, 25 and 36 (the square numbers). Press ✂ on each composite. What do you notice about one of the factor pairs every time? How is a square number’s middle factor pair different from those of non-square numbers?
11
Create a tile for 64 and press ✂. Now do the same for 72 on a new board. Both numbers are close in value, but one has significantly more factor pairs. Which one, and why? What role does the prime factorisation play in determining how many factors a number has?
12
Create a tile for 12 and press the ⚛ button. What tiles appear? Write down the prime factorisation shown in the toast. Now create a fresh 12 and press ✂ instead. Compare the two results. How are they different? Which one breaks the number down further?
13
Create tiles for 30, 60 and 120. Press ⚛ on each one (use a different board for each). Write down the prime factorisation of all three. What stays the same across all three? What changes? How is the factorisation of 120 related to that of 60?
14
Create a tile for 100 and press ⚛. How many prime factor tiles appear? Now try 128. How many appear this time? Why does 128 produce so many more tiles than 100, despite being only slightly larger?
15
Investigate using ⚛: create tiles for 8, 27, 125 and 343. Decompose each one. What do all four prime factorisations have in common? These are all powers of primes — what is special about the factorisation of a number that is a power of a single prime?
16
Create a tile for 360 (use the Favourites row). Press ⚛ to decompose it. Count the prime tiles: how many 2s, how many 3s, how many 5s? Now select the original 360 tile before decomposing and check Number Facts. Does the prime factorisation shown in the panel match the tiles on the canvas?
17
A student claims: ‘If two numbers have the same prime factorisation, they must be the same number.’ Test this by decomposing several different numbers. Can you find two different numbers that produce the same set of prime tiles? What does this tell you about the Fundamental Theorem of Arithmetic?
18
Create a tile for 24. Press the ÷ button and look at the popup grid. How many factors can you pull out? Pull out a 3. What two tiles appear? Check: does 3 × 8 = 24? Now take the 8 tile and pull out a 2. What do you have now? Keep pulling factors until every tile is prime. What have you done?
19
Create a tile for 60. Pull out the factor 5 using ÷. You should see a 5 tile and a 12 tile. Now drag the 5 tile onto the 12 tile. What happens? What is the product? Dragging one tile onto another multiplies them — you have just reversed the pull operation. Try this with other factor-and-quotient pairs.
20
Create two tiles: 6 and 8. Drag one onto the other to merge them. What is the result? Now create 4 and 12 and merge those. Same result? Use ✂ on 48 to see all the different pairs that multiply to make 48. How many pairs from the grid could you have used to build 48 by merging?
21
Create a tile for 100. Pull out the factor 10 using ÷. You get 10 and 10. Now pull a 2 out of one of the 10 tiles, giving you 2 and 5. Pull a 2 out of the other 10 as well. You should now have 2, 5, 2, 5. Merge them back together in a different order: merge the two 2s first (making 4), then the two 5s (making 25), then merge 4 and 25. Do you get 100 again? What does this show about multiplication?
22
Create a tile for 7 and a tile for 5. Merge them by dragging one onto the other. What do you get? Now press ✂ on the 35 tile. What factor pairs appear? One pair should be 5 × 7 — the two numbers you started with. Does every pair of primes you merge produce a number with exactly two factor pairs? Why?
23
Investigate: create a tile for 2 and duplicate it five times (long-press each tile). You have six 2 tiles. Merge two of them to make 4. Merge another pair to make 4. Merge the last pair to make 4. Now merge your three 4 tiles step by step. What final number do you get? Is this the same as 2⁶? Check using Number Facts.
24
Create a tile for 5 and press the × button. What four tiles appear? Write down all five numbers (including the original). What do you notice about the gaps between consecutive multiples? Now try the same with a tile for 8. Is the pattern the same?
25
Create a tile for 1 and press ×. What multiples appear? Now try 2, then 3. The multiples of 1 are just the counting numbers. The multiples of 2 are all even. What can you say about the multiples of 3? Check using Number Facts on each tile — are they all composite?
26
Create a tile for 7 and press ×. Look at the four new tiles: 14, 21, 28, 35. Tap each one. Are any of them prime? Now try the multiples of 11. Are any of those prime? Investigate: can a multiple of a number (other than ×1) ever be prime? Explain why or why not.
27
Create a tile for 250 and press ×. The tool generates multiples up to 999 and skips any that would exceed this. How many multiples appear? What is the largest? Now try 300, then 500. At what point does the tool run out of space for all four multiples?
28
Create a tile for 12 and press × to see its first four multiples (24, 36, 48, 60). Now select the 24 tile and check Number Facts. How many factors does 24 have? Check 36, 48 and 60 too. Is there a pattern in how many factors the multiples have? Do multiples always have more factors than the original number?
29
Create a tile for 6 and press ×. Now create a tile for 8 and press ×. Look at both sets of multiples. Do any numbers appear in both sets? These are common multiples. What is the smallest common multiple you can see? Is this the LCM of 6 and 8? How does the LCM relate to the factor pairs of each number?
30
Create a tile for 1 and tap it to select it. Open Number Facts. What type badge does it show? Is 1 prime, composite, or neither? Why do mathematicians put 1 in its own category? What other facts can you read from the panel?
31
Create a tile for 12 and open its Number Facts. It shows 6 factors and 3 factor pairs. Now create 24 and check. Does doubling a number double the number of factors? Try going from 24 to 48. What actually happens to the factor count when you double a number?
32
Use Number Facts to find a number between 1 and 100 that has exactly 2 factors. Now find one with exactly 3 factors, then 4, then 6, then 12. What type of number always has exactly 2 factors? What must be true about a number for it to have an odd number of factors?
33
Open Number Facts for 36. It shows badges for ‘Composite’, ‘Even’, and ‘Square’. Can you find a number that is Composite, Odd, and Square? What about Prime and Even? How about a number that has all three of: Composite, Even, and Cube?
34
Use Number Facts to compare 24 and 36. Which has more factors? Which has more factor pairs? Do they share any factors in common? What is the highest factor they share? This number has a special name — what is it, and how can you find it from the two lists of factors?
35
Create a tile for 100 and open Number Facts. Look at the first 5 multiples. Now create a tile for 10 and check its first 5 multiples. One of 10’s multiples should be 100. And one of 100’s multiples should be a multiple of 10 too. What is the relationship between multiples and factors? If A is a factor of B, is B always a multiple of A?
36
Create tiles for 6, 12, 18, 24 and 30. Open Number Facts for each and write down their factor counts: how many factors does each have? These are all multiples of 6, but do they all have the same number of factors? What determines how many factors a multiple of 6 has?
37
Use ✂ on 12, 24 and 36 (each on a separate board). For each number, find the factor pair where the two numbers are closest together. For 12 it is 3 × 4 (difference 1). For 24? For 36? When does a number have a factor pair with difference 1? What is special about these numbers?
38
Create a tile for 60 and decompose it (⚛). Now create a tile for 90 and decompose it. Look at the prime tiles from each. What primes do they share? Merge the shared primes together. What number do you get? This is the HCF (highest common factor) of 60 and 90. Can you verify using Number Facts?
39
Investigate the ‘prime deserts’: create tiles for every number from 24 to 30. Tap each tile to see if it is prime or composite. How many consecutive composite numbers did you find? Now try 90 to 96. Can you find a run of 5 or more consecutive composites? What is the longest run you can find below 100?
40
A student says: ‘The more factors a number has, the larger it must be.’ Test this claim. Using Number Facts, find the smallest number with exactly 6 factors. Now find the smallest number with 8 factors. Is 12 (which has 6 factors) larger or smaller than 128 (which has 8 factors)? Is the student right?
41
Create tiles for 2, 4, 8, 16 and 32 (powers of 2). Use Number Facts to check the factor count of each: 2, 3, 4, 5, 6. Now try powers of 3: 3, 9, 27, 81. What are their factor counts? Can you spot the formula? If a number is pⁿ (a prime raised to a power), how many factors does it have?
42
Create a tile for 12 and press × to generate multiples. Now take the 24 tile and press × to generate its multiples. Some of these will be multiples of 12 too. Which ones? Is every multiple of 24 also a multiple of 12? Is every multiple of 12 also a multiple of 24? What rule connects these two ideas?
43
Using the 🎲 random composite button, generate five 2-digit composites. For each one, use Number Facts to record the number of factors. Which of your numbers is the ‘most composite’ (has the most factors)? Now switch to 3-digit composites and generate five more. Do larger numbers tend to have more factors?
44
A number is called ‘abundant’ if the sum of its proper factors (every factor except itself) is greater than the number. For example, the proper factors of 12 are 1, 2, 3, 4, 6, which sum to 16 > 12, so 12 is abundant. Use Number Facts to test the first ten composite numbers. Which are abundant? Can you find the smallest abundant number?
45
Investigate highly composite numbers — numbers that have more factors than any smaller number. Start with 1 (1 factor). The next is 2 (2 factors). Then 4 (3 factors). Can you find the complete sequence up to 100? Use Number Facts to check factor counts. What do you notice about the prime factorisations of these special numbers?
46
On a fresh board, create tiles for 2, 3 and 5. Using only these prime tiles and the merge operation (drag-to-multiply), can you build every number from 2 to 30? Which numbers can you build and which are impossible? What additional primes would you need to tile to reach every number up to 30?
47
Explore twin primes using the tool: create tiles for numbers that differ by 2 (like 11 and 13, or 29 and 31) and tap each to confirm they are both prime. How many twin prime pairs can you find below 100? Do you think twin primes ever stop? Use Number Facts on the number between each twin pair — what do you notice about it?
48
Challenge: start with a tile for 360. Using only the ÷ (pull factor) and merge (drag-to-multiply) operations — no creating new tiles — can you rearrange 360 into exactly three tiles that are all the same number? What would that number need to be? Is it possible? What about rearranging into two equal tiles? What must be true about a number for this to be possible?
49
Design your own Factor & Multiple Tiles investigation. Choose a mathematical focus (prime factorisation, HCF, LCM, factor counting, or something else), decide which tool features to use, and write three questions that another student could explore. Test them yourself across different boards first, then exchange with a partner. Which of your questions led to the most interesting discoveries?