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Click a tile below to add it here
Investigation Questions
Use these alongside the tool above. Add fraction tiles from the tray, snap them together for comparison, and use the Add, Subtract, Split and Duplicate buttons to explore. Long-press tray tiles to pick specific numerators.
1
Click the ½ tile in the tray to add it to the canvas. Now add a ¼ tile. Which tile is wider? Why? Add a 1 whole tile and line all three up by snapping their left edges together. Describe the relationship between the widths.
2
Add one of every unit fraction from 1 whole down to 1/10. Snap them all to the same left edge and stack them vertically. What pattern do you notice about how the widths decrease? Do they decrease by the same amount each time?
3
Add a ½ tile and a ⅓ tile side by side. Without using the Add button, can you tell from the widths alone whether ½ + ⅓ is more or less than 1? Now add a 1 whole tile above them to check.
4
Add a 1/7 tile and a 1/8 tile. They look very similar in width. Can you see the difference? Now duplicate each one several times and line up 7 copies of 1/7 against 8 copies of 1/8. What do you notice about the total widths? What does this prove?
5
Look at the tray tiles for 1/17, 1/18, 1/19 and 1/20. They are very narrow. Hover over each one to see its label. Now add all four to the canvas. Can you tell them apart by width alone? How could you check which is which without hovering?
6
Add a ½ tile to the canvas. Now add a 2/4 tile (long-press the quarters tile and choose 2). Snap them together vertically with left edges aligned. What do you notice about their widths? Try the same with 3/6, 4/8 and 5/10. What do all these fractions have in common?
7
On Canvas 1, build a stack showing all fractions equivalent to ⅓ using tiles up to twentieths. How many can you find? On Canvas 2, do the same for ¼. Which fraction has more equivalents on the wall, and why?
8
Add a 4/6 tile and a 2/3 tile. Are they the same width? Now toggle the Improper/Mixed button. Does the display change for either tile? What does this tell you about the difference between equivalent fractions and simplified fractions?
9
Can you find two fractions with different denominators that are not equivalent but are very close in width? What is the closest pair you can find? Measure by placing them on top of each other and looking at the tiny difference at the right edge.
10
A student says: “2/4 and 3/6 are different fractions because they have different numbers.” Another student says: “They’re the same fraction written differently.” Use the tiles to build an argument for one side. Which student do you agree with, and why?
11
Add three ¼ tiles to the canvas. Select all three and press Add. What does the result tile show? What does the banner show? Now place a 1 whole tile above the result. Is 3/4 more or less than 1?
12
Add a ⅓ tile and a ¼ tile. Select both and press Add. Read the banner carefully — it shows the working. What common denominator did it use? Why 12 and not some other number?
13
On a fresh canvas, investigate: what two unit fractions add up to exactly ½? How many different pairs can you find? (Hint: try ⅓ + ⅙, or ¼ + ¼.) Are there infinitely many, or does the list end?
14
Add five 1/5 tiles. Select all five and press Add. What is the result? Now try the same with six 1/6 tiles, seven 1/7 tiles, and so on. What pattern do you see? Can you explain why this always works?
15
Add a ½ tile and a ⅓ tile and a ¼ tile. Select all three and press Add. Is the result more or less than 1? Now try ½ + ⅓ + ⅙. What is special about this combination?
16
Use the tool to find three different fractions that add up to exactly 1. Can you find a set where all three fractions have different denominators? What about four fractions that add to 1?
17
Add a ¾ tile (long-press quarters, choose 3) and a ¼ tile. Select both and press Subtract. What is the result? Where does the result tile appear? Verify visually: place the result next to a ½ tile — are they the same width?
18
Add a ½ tile and a ⅓ tile. Select both and press Subtract. Read the banner — what common denominator does it use? What is the result? Now verify: add a 1/6 tile and check it matches the result’s width.
19
What happens when you subtract two fractions that are equivalent? Try subtracting ½ from 2/4. What does the tool do? Why does this make sense?
20
Start with a 1 whole tile. Subtract ⅓ from it. What is left? Now subtract another ⅓ from the result. And another. What do you end up with? What does this show about 1 ÷ 3?
21
Add a ½ tile. Select it and press Split, then choose 2. What two tiles appear? Check: are they touching? Is their combined width equal to the original ½ tile? What fraction is each piece?
22
Add a ⅓ tile and split it into 2 equal parts. What fraction does each part show? Now take one of those parts and split it into 2 again. What fraction do you have now? Keep splitting in half. Write down the sequence of fractions. What pattern do you see in the denominators?
23
Add a 1 whole tile. Split it into 3 equal parts. Now take one of the thirds and split it into 4 equal parts. What fraction is each small piece? Place 12 of these pieces in a row — do they match the width of 1 whole?
24
Split a ½ tile into 3 parts, giving you three 1/6 tiles. Now select all three and press Add. Do you get back to ½? This is the key idea: splitting and adding are inverse operations. Test this with other fractions and split numbers.
25
Add a ¾ tile (long-press quarters, choose 3). Can you split it into equal parts? Try splitting into 3. What fraction is each piece? Does 3 × (that fraction) = ¾? Now try splitting ¾ into 2. What happens? Why can’t ¾ be split into 2 equal parts using this denominator?
26
Long-press the thirds tile and add a 5/3 tile. How wide is it compared to 1 whole? Now toggle the Improper/Mixed button. What does the tile show now? Toggle back. Which representation helps you see the size more easily?
27
Add a 7/4 tile (long-press quarters, choose 7). Place a 1 whole tile above it. The 7/4 tile extends beyond 1 whole. By how much? Add a ¾ tile next to the 1 whole tile. Does 1 + ¾ match the width of 7/4?
28
Add five ⅓ tiles. Select all five and press Add. The result is 5/3. Toggle Mixed mode — it shows 1⅔. Now split the 5/3 tile into 5 parts. Do you get your original ⅓ tiles back?
29
In Mixed mode, add a 3/2 tile. It displays as 1½. Now add a 4/3 tile (displays as 1⅓). Which is larger? Can you tell from the mixed number alone? Can you tell from the tile widths? Which representation makes comparison easier?
30
Which is larger: 3/7 or 2/5? Add both tiles (long-press to create them). Stack them vertically with left edges aligned. Which tile extends further to the right? Now use Subtract to find the exact difference.
31
Put these fractions in order from smallest to largest using tiles only: ½, 3/8, 5/12, 2/5. Stack them all with left edges aligned and read the order from the right edges. Were any of them closer together than you expected?
32
Which fraction is closest to ½ without being equal to it? Try 5/11, 4/9, 7/15, 3/7, 6/13. Place each one above or below a ½ tile and look at the gap at the right edge. Which is closest?
33
Is 99/100 closer to 1 or closer to 19/20? Add both tiles and a 1 whole tile. Stack all three. Use Subtract to find the gap between each pair. Which gap is smaller?
34
Add 1/2, 1/4, 1/8, and 1/16 tiles. Select all four and press Add. What is the result? How close is it to 1? What would happen if you kept adding the next fraction in this sequence (1/32, 1/64, …)? Will the sum ever reach exactly 1?
35
Investigate unit fraction addition: 1/2 + 1/3 = 5/6. Now try 1/3 + 1/4, then 1/4 + 1/5, then 1/5 + 1/6. Write down all the results. What pattern do you notice in the numerators and denominators of the sums? Can you write a rule for 1/n + 1/(n+1)?
36
A student claims: “When you add two fractions, the result always has a bigger denominator than either fraction you started with.” Test this with at least five examples. Is the student right? Can you find a counterexample?
37
Start with 1 whole. Subtract ½. From the result, subtract ¼. From that result, subtract ⅙. Then subtract 1/8. Write down the result after each subtraction. What is the pattern? What fraction remains? If you kept going (subtract 1/16, 1/32, …), what would you approach?
38
Build a ⅓ tile and split it into 2 parts (giving 1/6 each). Take one 1/6 and split it into 3 parts (giving 1/18 each). You now have pieces of size 1/6 and 1/18. How many 1/18 pieces equal one 1/6 piece? How many 1/18 pieces equal the original 1/3? What multiplication does this demonstrate?
39
The Ancient Egyptians wrote every fraction as a sum of distinct unit fractions (no repeats). For example, 3/4 = 1/2 + 1/4. Using the tiles, can you write 5/6 as the sum of two different unit fractions? What about 7/8? Can you always do this? Find the Egyptian fraction representation for 2/7 (hint: you may need three unit fractions).
40
On Canvas 1, build a stack of fractions all equivalent to ½. On Canvas 2, build a stack for ⅓. On Canvas 3, build one for ¼. Which denominators from 2–20 have a fraction equivalent to ½? Which have one equivalent to ⅓? What determines whether a denominator can produce a fraction equivalent to a given unit fraction?
41
Investigate: for which values of n does 1/n + 1/(n+1) simplify to something with a smaller denominator? Test n = 1 through n = 10. When does the sum simplify, and when is it already in simplest form? Can you spot the pattern?
42
Challenge: using only unit fraction tiles and the Add button, build a tile that equals exactly 1 whole using the fewest possible tiles. Now try it with the most tiles you can (all different denominators). What is the maximum number of distinct unit fractions that sum to exactly 1?
43
A fraction is called a Farey neighbour of another if no simpler fraction exists between them. For example, 1/3 and 1/2 are Farey neighbours because no fraction with a denominator ≤ 3 sits between them. Using tiles, find all Farey neighbour pairs between 0 and 1 using denominators up to 5. Line each pair up and look at the gap between their right edges.
44
Design your own Fraction Tiles investigation. Choose a mathematical focus (equivalence, addition, subtraction, comparing, or splitting), decide which tool features to use, and write three questions that another student could explore. Test them yourself across different canvases first, then exchange with a partner. Which of your questions led to the most interesting discoveries?