Tiles are drawn to scale: tap one, then the tray, and the new tile joins on to its end.
- Tap a tile in the tray to add that unit fraction. It goes on the next free row, starting at the left edge, so tiles line up ready to compare.
- Tap a tile on the board first and the new tile joins on to its end. Keep tapping the tray to build a row: ¼, ¼, ¼ …
- Long-press or right-click a tray tile (or turn on 🔢 Any numerator) to choose the numerator: one ¾ tile, or three ¼ tiles joined end to end. Or type a fraction such as 3/4, 7/4, 1 2/3, 0.4 or 35% and press Make tile. ⌨ opens a keypad.
- Every tile is drawn to scale: its length is its value times the length of 1 whole. Use − and + beside “1 whole” to change that length. The tiles all change together.
- Drag a tile to move it. Edges snap to other tiles’ edges and to 0 on the number line, with a blue guide line. Drag across empty board to select several. Drop a tile on the 🗑 corner to delete it. Long-press a tile to duplicate it.
- Tap a tile for its buttons. With two or more selected, the buttons change.
- ✂ Split cuts a tile into 2 to 12 equal parts, or into unit fractions (¾ into three ¼s).
- = Rename gives a tile an equivalent name, ½ as 3/6, without changing its length.
- × Copies joins copies end to end: five copies of ⅕ make 1 whole.
- ⊞ Add puts the tiles end to end and the answer underneath, so you can see they match. The working appears under the board: ⅓ + ¼ = 4/12 + 3/12 = 7/12.
- ⊟ Subtract (two tiles) puts the shorter under the longer and fills the gap with the difference.
- ↕ Line up stacks tiles from the same left edge, shortest at the top, and writes them in order with < and =.
- Show as switches every label between fractions, mixed numbers, decimals and percentages. Recurring decimals have dots over the repeating digits.
- 📏 Number line puts a line marked 0, 1, 2 under the tiles, with marks for the denominator of the selected tile. ▥ Parts shows the unit parts inside each tile. ◻ Plain makes every tile the same colour. 🏷 Hide labels hides every name so the class has to judge by length.
- Each denominator has its own colour, the same in the tray and on the board.
🙈 hides the answer to every Add, Subtract, Split, Rename and Line up behind a “?” until you tap it (or press Space), so the class can predict first.
Shortcuts only work after you’ve clicked on the tool, so they never get in the way of the rest of the page.
Ten boards, each with its own undo. Press 📋 Questions (or Q) to pick a question, even in full screen. Set it up puts the tiles on this board if it’s empty, or on the next empty one, so your work is never lost. 🔗 copies a link that opens the tool with this board. 🖨 prints a sheet of tile questions with answers.
Nothing is sent anywhere, and nothing is stored on this device.
Investigation questions
65 questions. Work through these with the tool above. Set it up puts the right tiles on the board for you (on an empty board, so nothing you have made is lost), and Show on board puts the question on screen, full size.
First experiments
The board shows 1 whole, ½ and ¼, all starting at the same left edge. Predict: how many ¼ tiles fit along the ½ tile? Tap the ¼ tile, then tap ¼ in the tray. The new tile joins on to its end. Were you right?
Teacher notes
Two: two quarters make the same length as one half. Tapping a tile first and then the tray builds a row. Watch for students who expect ¼ to be longer because 4 is bigger than 2.
Here are the unit fractions from 1 whole down to 1/10. Do the tiles get shorter by the same amount each time? Select the ½ and the ⅓ and press ⊟ Subtract, then do the same with ⅓ and ¼.
Teacher notes
No. ½ − ⅓ = 1/6 but ⅓ − ¼ = 1/12, and the gaps keep shrinking: 1/9 − 1/10 is only 1/90. The bigger the denominator, the smaller the piece, and the less difference one more piece makes.
Make ¾ three different ways: long-press (or right-click) the ¼ in the tray and choose 3; tap the ¼ on the board, then tap ¼ in the tray twice more; type 3/4 in the box and press Make tile. Line them up. What is the same and what is different?
Teacher notes
All three are the same length. One is a single tile with its quarters marked by the ▥ Parts lines; the other is three separate quarter tiles. ¾ means three lots of ¼.
Press − next to “1 whole” a few times, then +. What changes? Is ½ always half the length of the 1 whole tile?
Teacher notes
Every tile shrinks or grows together, so ½ is always exactly half of the whole. A fraction is a fraction of a whole: “½” has no fixed size until you know what the whole is.
The labels are hidden. Put these four tiles in order from shortest to longest just by looking, then tap each one to reveal it. Which pair was hardest to tell apart?
Teacher notes
Shortest to longest: 1/20, 1/19, 1/18, 1/17. Each is only about 5–6% longer than the one before, so a stack with left edges lined up is the only reliable way to compare them by eye.
Equivalence
Predict which of these tiles are exactly the same length as ½. Select them all and press ↕ Line up to check.
Teacher notes
2/4, 3/6, 4/8 and 5/10 are all equal to ½; 3/5 is longer (it is 1/10 more). In each equal one, the numerator is half the denominator. The ▥ Parts lines show ½ cut into 2, 3, 4 and 5 pieces.
Tap the bottom ½ tile and press = Rename, then choose 3/6. What changed, and what stayed the same? Rename it again as 4/8.
Teacher notes
The name, the colour and the parts lines change; the length doesn’t. Renaming ½ as 3/6 multiplies the numerator and denominator by 3: the same length, cut into three times as many pieces.
Tap the ⅓ and press = Rename. Using denominators up to 20, how many other names does ⅓ have? Now try the ¼. Which has more, and why?
Teacher notes
⅓ has 5 others with denominators up to 20 (2/6, 3/9, 4/12, 5/15, 6/18); ¼ has 4 (2/8, 3/12, 4/16, 5/20). There are more multiples of 3 than multiples of 4 up to 20.
Is 4/6 the same as ⅔? Select both and press ↕ Line up. Then rename the 4/6 in its simplest form. How do the parts lines show why?
Teacher notes
Yes: they line up exactly, and Rename gives ⅔. Each third of the ⅔ tile is two sixths of the 4/6 tile. Dividing the numerator and denominator by 2 joins the pieces in pairs.
A student says: “To make an equivalent fraction, add the same number to the top and the bottom, so ½ = 2/3 = 3/4.” Use the tiles to convince them otherwise.
Teacher notes
½, ⅔ and ¾ get longer each time, so they are not equal. Multiplying works instead: 2/4 lines up with ½. Adding the same number to the top and the bottom changes the size; multiplying both by the same number doesn’t.
One student says 2/4 and 3/6 are different fractions because the numbers are different. Another says they are the same number written two ways. Select both and press ⊟ Subtract. Who is right?
Teacher notes
The second student. The difference is 0: they are exactly the same length, both equal to ½. Different names, one number.
Comparing and ordering
Which is larger, 3/7 or ⅖? Predict, then select both and press ↕ Line up. Read the working under the board.
Teacher notes
3/7 is larger: with a common denominator, 3/7 = 15/35 and ⅖ = 14/35, so the difference is only 1/35. Lining up from the same left edge makes a tiny difference visible.
Put ½, ⅜, 5/12 and ⅖ in order from smallest to largest. Write your order down first, then press ↕ Line up.
Teacher notes
⅜ < ⅖ < 5/12 < ½ (0.375, 0.4, 0.417, 0.5). They are all within ⅛ of each other, which is why comparing by eye from a loose pile is unreliable.
These three fractions all have the same numerator. Predict the order, then line them up. What rule do you notice?
Teacher notes
⅜ < ⅗ < ¾. With the same numerator, the bigger the denominator, the smaller the fraction, because each piece is smaller. Students who compare denominators like whole numbers get this backwards.
Which is larger, 5/6 or ⅞? Look at the gap each one leaves before the end of the 1 whole tile.
Teacher notes
⅞ is larger. Each is one piece short of 1 whole, and the missing ⅛ is smaller than the missing ⅙. Comparing how far from 1 each one is avoids finding a common denominator.
Which of 5/11, 4/9, 7/15, 3/7 and 6/13 is closest to ½? All of them are a little less than ½. Line them up with the ½ tile and look at the gaps, then check with ⊟ Subtract.
Teacher notes
7/15 is closest: it is 1/30 less than ½. The gaps are 1/22, 1/18, 1/30, 1/14 and 1/26. Each is half a piece short of ½, so the smallest pieces (fifteenths) give the smallest gap.
Is 99/100 closer to 1 or to 19/20? Use ⊟ Subtract on each pair.
Teacher notes
Closer to 1: 1 − 99/100 = 1/100, while 99/100 − 19/20 = 4/100 = 1/25. The gap to 1 is hard to see at all, so the subtraction settles it.
Without lining them up, say whether each tile is more or less than ½. Then reveal the labels by tapping, and check against the ½ tile.
Teacher notes
5/9 and 7/12 are more than ½; 4/9 and 5/12 are less. Half of 9 is 4½ and half of 12 is 6, so compare each numerator with half its denominator.
Adding fractions
Predict what three quarters add up to. Select the three ¼ tiles and press ⊞ Add. Where does the answer go, and does it match the row?
Teacher notes
¾. The pieces line up end to end with the answer underneath, the same length. A common wrong answer is 3/12: adding the denominators as well as the numerators.
Select the ⅓ and the ¼ and press ⊞ Add. Read the working under the board. Why does it use twelfths?
Teacher notes
⅓ + ¼ = 4/12 + 3/12 = 7/12. 12 is the lowest common multiple of 3 and 4: the smallest pieces that both a third and a quarter can be cut into exactly. 24ths would work too (8/24 + 6/24 = 14/24 = 7/12).
A student says ½ + ⅓ = ⅖, adding the tops and adding the bottoms. Predict whether the ⅖ tile is right, then select ½ and ⅓ and press ⊞ Add.
Teacher notes
½ + ⅓ = 5/6. The ⅖ tile is shorter than the ½ tile on its own, so it can’t be the total. Adding the denominators is the most common fraction addition error.
Which pairs of unit fractions add up to exactly ½? Try ⅓ + ⅙ and ¼ + ¼, then look for others. Does the list go on for ever?
Teacher notes
Only two pairs: ⅓ + ⅙ and ¼ + ¼. If one tile is shorter than ¼ the other must be longer than ¼ but less than ½, which leaves only ⅓, and ½ − ⅓ = ⅙.
Tap the ⅕ tile and press × Copies, then choose 5. What do five fifths make? Predict how many sevenths make 1, then check.
Teacher notes
5 × ⅕ = 5/5 = 1, and 7 sevenths make 1. Any fraction with the same numerator and denominator is one whole.
Is ½ + ⅓ + ¼ more or less than 1? Predict, then press ⊞ Add. Now try ½ + ⅓ + ⅙.
Teacher notes
½ + ⅓ + ¼ = 13/12 = 1 1/12, a little more than 1. ½ + ⅓ + ⅙ = 1 exactly (6/12 + 4/12 + 2/12 = 12/12).
Find three fractions with different denominators that add up to exactly 1. Then find four.
Teacher notes
Three: ½ + ⅓ + ⅙. Four: ½ + ¼ + ⅙ + 1/12, or ½ + ⅓ + 1/9 + 1/18. There are many more; check each with ⊞ Add.
Add ½ + ¼ + ⅛ + 1/16. How far short of 1 is the answer? What happens to the gap if you keep going?
Teacher notes
15/16, which is 1/16 short of 1. Each new tile fills half the gap that is left, so the total gets closer and closer to 1 but never reaches it.
Subtracting fractions
Predict ¾ − ¼. Select both and press ⊟ Subtract. Where does the answer appear?
Teacher notes
¾ − ¼ = 2/4 = ½. The ¼ goes under the ¾ and the answer fills the gap to the end of the ¾: subtraction as “how much longer”.
Select ½ and ⅓ and press ⊟ Subtract. What common denominator does the working use? Make a ⅙ tile and check it fits the gap.
Teacher notes
½ − ⅓ = 3/6 − 2/6 = ⅙. The new ⅙ tile is exactly the length of the gap.
A student says ¾ − ½ = 2/2 = 1, subtracting the tops and the bottoms. Predict the answer, then press ⊟ Subtract.
Teacher notes
¾ − ½ = ¼. Subtracting the denominators as well gives 2/2, which is longer than either tile, so it can’t be the difference.
How much more do you need to add to ⅝ to make 1? Predict, then subtract.
Teacher notes
⅜: 1 − ⅝ = 8/8 − 5/8 = 3/8. Writing 1 as 8/8 is the key step.
Subtract ⅓ from 1 whole. Now subtract ⅓ from the answer, and then from that answer. What do you end up with?
Teacher notes
1 − ⅓ = ⅔, then ⅔ − ⅓ = ⅓, then ⅓ − ⅓ = 0. One whole is three thirds, so you can take a third away exactly three times.
Start with 1 whole. Subtract ½, then subtract ¼ from the answer, then ⅛, then 1/16. Write down each answer. What pattern do you notice?
Teacher notes
The answers are ½, ¼, ⅛ and 1/16: each time, what is left equals the tile you just took away. The amount left halves every time and gets closer and closer to 0.
Splitting and renaming
Predict what you get if you split ½ into 2 equal parts. Tap the ½, press ✂ Split and choose 2. Are the pieces the same length as the ¼ below?
Teacher notes
Two quarters, end to end, each the same length as the ¼ tile. Half of a half is a quarter: ½ ÷ 2 = ¼.
Split the ⅓ into 2 equal parts. Then split one of the pieces into 2, and keep going. Write down the fractions you get. What happens to the denominator each time?
Teacher notes
⅙, 1/12, 1/24, 1/48: the denominator doubles each time you halve the piece. Halving a fraction doubles its denominator, not halves it.
Split the 1 whole into 3 equal parts. Then split one of the thirds into 4. What is each small piece? How many of them would make 1 whole?
Teacher notes
Each small piece is 1/12, and 12 of them make 1 whole: ⅓ ÷ 4 = 1/12 because 3 × 4 = 12 pieces fill the whole.
Split the ½ into 3 equal parts. Select the three pieces and press ⊞ Add. Do you get back to ½?
Teacher notes
Yes: ⅙ + ⅙ + ⅙ = 3/6 = ½. Splitting into equal parts and adding them back are inverses.
Before you press anything, predict what each piece will be if you split ¾ into 3 equal parts. Then predict for 2 equal parts. Use ✂ Split to check each one (↩ undo in between).
Teacher notes
¾ ÷ 3 = ¼ and ¾ ÷ 2 = ⅜. Half of three quarters is three eighths: cut each quarter in half and take three of the halves. With 🙈 on, the answer stays hidden until you tap it.
Split ⅔ into 4 equal parts. The working says each part is ⅙. Rename one piece as 2/12. Why does the tool show ⅙ first?
Teacher notes
⅔ ÷ 4 = 2/12, which simplifies to ⅙: the tool gives the simplest form. Both names are right and the tile is the same length.
Split the ⅓ into 2, then split one ⅙ into 3. How many 1/18 pieces make ⅙? How many make the whole ⅓?
Teacher notes
3 pieces of 1/18 make ⅙, and 6 make ⅓. ⅓ ÷ 2 ÷ 3 = 1/18, so dividing by 2 then by 3 is the same as dividing by 6.
Improper fractions and mixed numbers
How does 5/3 compare with 1 whole? Switch Show as to 1¾ Mixed. What does the tile show now? Which way of writing it helps you see its size?
Teacher notes
5/3 = 1⅔: it is one whole and two thirds more. The improper fraction counts the thirds; the mixed number says how many wholes and what is left.
Is 7/4 the same length as 1 + ¾? Predict, then select both tiles in the bottom row and press ⊞ Add.
Teacher notes
Yes: 1 + ¾ = 4/4 + 3/4 = 7/4 = 1¾. Four of the seven quarters make the whole.
Add the five ⅓ tiles. Then tap the answer, press ✂ Split and choose Unit fractions. Do you get your five thirds back?
Teacher notes
5 × ⅓ = 5/3 = 1⅔, and splitting 5/3 into unit fractions gives five ⅓ tiles again.
In mixed numbers, which is larger: 1½ or 1⅓? Can you tell from the mixed numbers, or do you need the tiles?
Teacher notes
1½ is larger: both have 1 whole, and ½ > ⅓. With mixed numbers, compare the wholes first, then the fractions; with 3/2 and 4/3 you need a common denominator (9/6 and 8/6).
Is 11/4 bigger or smaller than 2¾? Predict, then select the 2 and the ¾ and press ⊞ Add.
Teacher notes
They are equal: 2¾ = 8/4 + 3/4 = 11/4. The 1 whole here is shorter so that 2¾ fits on the board.
Fractions, decimals and percentages
Switch Show as to 0.75. Which of these fractions have decimals that stop?
Teacher notes
½ = 0.5, ¼ = 0.25, ⅕ = 0.2, ⅛ = 0.125, 1/10 = 0.1. ⅓ = 0.333…, shown as 0.3 with a dot over the 3: it never stops.
Look at the decimals for thirds, sixths, ninths and sevenths. Which denominators give recurring decimals? Where do the dots go for 1/7?
Teacher notes
⅓ = 0.333…, ⅙ = 0.1666…, 1/9 = 0.111…, 1/7 = 0.142857142857… and 1/12 = 0.08333… For 1/7 the dots go over the 1 and the 7. A fraction in its simplest form stops only if the denominator has no prime factors other than 2 and 5. The dots go over the first and last digits of the repeating block.
Switch to 75%. Predict the percentage for ⅛, then for ⅜. Were you right?
Teacher notes
⅛ = 12.5%, ⅜ = 37.5%, ⅝ = 62.5%, and 1 = 100%. Each eighth is 12.5%, so count up in 12.5s.
Type 0.4 in the box and press Make tile. What fraction does the tile become? Is it longer or shorter than ⅜?
Teacher notes
0.4 becomes ⅖, which is longer than ⅜ (0.4 > 0.375). Converting to decimals is a quick way to compare.
In percentages, ⅓ shows as 33.3% with a dot over the last 3, because it recurs. What do three thirds add up to? Select them and press ⊞ Add.
Teacher notes
100%: 3 × ⅓ = 1 exactly. Rounding ⅓ to 33.3% and adding three of them gives 99.9%, which is why exact fractions matter.
On a number line
Turn on 📏 Number line and tap the ¾ tile. Where does the tile end on the line? Where would ¾ of the way to 2 be?
Teacher notes
The tile ends at the third quarter mark between 0 and 1. ¾ is a number, a point on the line, as well as a piece of a whole. ¾ of the way to 2 is 1½.
Between which two whole numbers does 7/3 lie? Predict, then check on the number line.
Teacher notes
Between 2 and 3: 7/3 = 2⅓. Six thirds make 2, and there is one third more.
Is ⅔ closer to ½ or to 1? Use the number line, then check with ⊟ Subtract.
Teacher notes
Closer to ½: ⅔ − ½ = ⅙, while 1 − ⅔ = ⅓.
Switch Show as to decimals with the number line on. Tap the ⅕ tile. What are the marks labelled now?
Teacher notes
The fifths are labelled 0.2, 0.4, 0.6 and 0.8. The same points have fraction names and decimal names.
Predictions and misconceptions
Predict: which is bigger, ⅓ or ⅕? Lots of students choose ⅕ because 5 is bigger than 3. Tap each tile to reveal it.
Teacher notes
⅓ is bigger. Dividing the whole into more pieces makes each piece smaller.
A student says ⅓ + ⅓ = 2/6. Predict whether the 2/6 tile is the right length, then add the two thirds.
Teacher notes
⅓ + ⅓ = ⅔, which is twice as long as 2/6 (= ⅓). The pieces stay thirds when you add them; only how many you have changes.
With 🙈 on, select ⅖ and ⅓ and press ⊞ Add. Predict the answer before you tap the “?”.
Teacher notes
⅖ + ⅓ = 6/15 + 5/15 = 11/15. With 🙈 on, the answer and the working wait until you tap or press Space.
Split the 2 tile into 4 equal parts and the 1 tile into 2 equal parts. A student says “a quarter is always smaller than a half”. Compare a piece from each.
Teacher notes
Each piece is ½ of a whole: a quarter of 2 is the same length as half of 1. A fraction depends on the whole it is a fraction of.
Two of these hidden tiles are the same length. Which ones? Tap to reveal and check.
Teacher notes
⅓ and 2/6 are the same length. With the labels hidden, students have to judge by length rather than by the numbers.
Turn on ◻ Plain so every tile is the same colour, then hide the labels. Can the class still put the tiles in order? Try it with a partner: one builds, one orders.
Teacher notes
7/12 < ⅗ < ⅝ < ⅔ (about 0.583, 0.6, 0.625, 0.667). Without colour or labels, length is the only clue, which is the point of the tiles.
Open investigations
The Ancient Egyptians wrote fractions as sums of different unit fractions, like ¾ = ½ + ¼. Write ⅚ and ⅞ this way. Can 2/7 be written with just two unit fractions?
Teacher notes
⅚ = ½ + ⅓ and ⅞ = ½ + ¼ + ⅛ (⅞ needs three: with two, the bigger would have to be ½, leaving ⅜). 2/7 = ¼ + 1/28: type 1/28 in the box to check. Every fraction less than 1 can be written this way.
Using unit fractions with different denominators from ½ to 1/20, what is the largest number of tiles you can find that add up to exactly 1?
Teacher notes
8, and there are only two ways: ¼ + ⅕ + ⅙ + 1/9 + 1/10 + 1/15 + 1/18 + 1/20, and ⅓ + ⅕ + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20. Small denominators use up the whole too quickly, so the answers lean on 15ths, 18ths and 20ths.
For which denominators from 2 to 20 is there a fraction equal to ½? To ⅓? To ¼? What decides it?
Teacher notes
Equal to ½: every even denominator (10 of them). Equal to ⅓: the multiples of 3 (6 of them). Equal to ¼: the multiples of 4 (5). A denominator works when it is a multiple of the unit fraction’s denominator.
Add 1/n + 1/(n + 1) for n = 2 to 7: ½ + ⅓, ⅓ + ¼ and so on. What do you notice about the numerators and denominators? Does the answer ever simplify?
Teacher notes
The answers are 5/6, 7/12, 9/20, 11/30, 13/42 and 15/56: the numerator is the sum of the two denominators and the denominator is their product. It never simplifies, because 2n + 1 shares no factor with n or n + 1.
Line up ⅓ and ½. Type the fraction you get by adding the numerators and adding the denominators, (1 + 1)/(3 + 2) = 2/5, and press Make tile. Where does it sit? Try it with ⅔ and ¾.
Teacher notes
⅖ sits between ⅓ and ½, and 5/7 sits between ⅔ and ¾. This “mediant” always lands between the two fractions. It is not their sum, which is a useful contrast with the add-the-tops-and-bottoms mistake.
Design your own Fraction Tiles investigation. Choose a focus (equivalence, comparing, adding, subtracting, splitting), set up a board, and write three questions for a partner. Use 🔗 to send them the board.
Teacher notes
Good questions start with a prediction and have an answer the tiles can check. The 🔗 link opens the tool with exactly the tiles on the board.
No questions match that filter.