- Pick a denominator, then tap a piece of the long bar: tapping the third quarter gives ΒΎ. Use Wholes + first for a mixed number such as 1ΒΎ. Or type a fraction (2/3, 1 3/4, 0.75) and press Add bar. β¨ opens a keypad.
- π² New question makes one for this mode at the chosen level: 1 same denominator, 2 one is a multiple of the other, 3 the denominators share a factor, 4 co-prime denominators, 5 mixed numbers.
- Every bar is drawn to the same scale: one outlined box is 1 whole.
- Tap a bar and choose a new denominator underneath: each third cut into 4 makes twelfths. The thick lines show the old pieces. β marks a denominator that works for every bar. You can join pieces too (4/8 as 1/2).
- π‘ Common denominator cuts every bar into the smallest one for you.
- When every bar has pieces the same size, count them and tap the ? (or press Space). In Add the answer bar shows the pieces end to end; in Subtract the pieces taken away are crossed out. Thick lines on the answer show how it simplifies.
- οΌ Add: two to four fractions. β Subtract: the second from the first (β Swap if the second is bigger). β Compare: which is bigger? Each mode keeps its own work.
Type an answer and press Check. It is drawn in purple under the real one, so a wrong answer such as 1/3 + 1/4 = 2/7 can be seen to be too small. In Compare, tap <, = or >. π on (the usual) keeps the answer hidden until you tap ?; off, it shows as soon as the pieces match.
Shortcuts only work after you’ve clicked on the tool, so they never get in the way of the rest of the page.
Press π Questions (or Q) to pick a question, even in full screen. Set it up loads it (β© brings back what was there). π¨ prints this board, or 8 questions like it with answers on a second page. π copies a link that opens the tool as it is now. Nothing is sent anywhere, and nothing is stored on this device.
Investigation questions
64 questions. Work through these with the tool above. Set it up puts the fractions on the board for you (β© undo brings back what was there), and Show on board puts the question on screen, full size.
First experiments
The board shows 1/5 + 2/5. Each outlined box is 1 whole. Count the shaded fifths and predict the answer, then tap the ? to check.
Teacher notes
3/5. The answer bar shows the 1 and the 2 fifths end to end. The denominator stays 5 because the pieces are still fifths; only the number of pieces changes. Watch for 3/10.
Make 3/8 yourself: tap 8 on the denominator tabs, then tap the third piece of the long bar. Make 4/8 the same way. Before you tap ?, how many eighths are shaded altogether?
Teacher notes
7/8. Tapping the third piece shades three of the eighths, which is what the 3 in 3/8 means.
Switch to β Subtract. The board shows 4/5 β 1/5. On the answer bar the piece taken away is crossed out. How many fifths are left?
Teacher notes
3/5. Subtraction here is “take away”: start with 4 fifths, cross out 1, count what is left.
In β Compare, which is bigger, 2/3 or 3/5? Decide by looking at the lengths, tap <, = or > next to β My answer, then tap the ?.
Teacher notes
2/3 > 3/5: 10/15 against 9/15, a difference of only 1/15. Ask how sure students were from the picture alone: a common denominator turns a judgement into a count.
Press π² New question five times on level 1 and say each answer before you tap ?. What do the answers have in common with their questions?
Teacher notes
Before any simplifying, every answer has the same denominator as the question: the numerators add and the size of the pieces stays the same. Some answers then simplify, and some go over 1.
Same denominators
Here is 2/6 + 4/6. Predict what the answer bar will look like. What happens when the shaded pieces fill a whole?
Teacher notes
6/6 = 1: the answer bar fills exactly one whole. 6/6, 1 and 12/12 are all names for one whole.
Do you add the numerators, the denominators, or both? For 3/7 + 2/7 a student wrote 5/14. Their answer is on the board in purple. Is it right? How can the bars show it isn’t?
Teacher notes
3/7 + 2/7 = 5/7. 5/14 is smaller than 3/7 on its own (6/14), so it can’t be the total. Adding the denominators would make the pieces smaller, but adding never changes the size of the pieces.
Try 1/7 + 1/7, then 2/7 + 2/7, then 3/7 + 3/7. Before you set it up, predict 4/7 + 4/7. What will be different about this one?
Teacher notes
8/7 = 1 1/7. It is the first answer that goes over 1, so the answer bar needs a second whole.
Predict 5/8 + 7/8, then reveal it. The answer can be written three ways. What do the thick lines on the answer bar show?
Teacher notes
12/8 = 3/2 = 1 1/2. The thick lines join the eighths in fours, making halves: 12 eighths are 3 halves, which is one whole and a half.
The board shows 7/8 β 3/8. Tap the 3/8 bar, then tap 8 on the denominator tabs and the fifth piece, to replace it with 5/8. What happens to the crossed-out pieces? What would 7/8 β 7/8 look like?
Teacher notes
7/8 β 3/8 = 4/8 = 1/2, and 7/8 β 5/8 = 2/8 = 1/4. More pieces crossed out, fewer left. 7/8 β 7/8 would cross out every shaded piece: 0.
Equivalent fractions
Tap the 1/2 bar and cut its pieces into quarters, then sixths, then eighths. What happens to the number of shaded pieces? What never changes?
Teacher notes
2/4, 3/6, 4/8. The number of pieces goes up by the same factor as the denominator, so the shaded length never changes: multiply the numerator and the denominator by the same number.
Cut 2/3 into twelfths. How many twelfths is it? Into how many pieces has each third been cut? (The thick lines show the old thirds.)
Teacher notes
8/12. Each third is cut into 4, so there are 4 times as many pieces and 4 times as many shaded: 2 Γ 4 = 8 and 3 Γ 4 = 12.
Tap the 1/4 bar and read its list of new denominators. Then tap the 1/3 bar. Why are the lists different? What do the β denominators have in common?
Teacher notes
Quarters can be cut into multiples of 4 (8, 12, 16 β¦) and thirds into multiples of 3 (6, 9, 12 β¦). The β ones (12, 24, 36, 48) are common multiples of 3 and 4, so they work for both bars.
A student says “to find an equivalent fraction you double the top and the bottom”. Tap the 1/5 bar. Is doubling the only choice? What is the rule behind every choice in the list?
Teacher notes
Doubling (2/10) is one of many: Γ3 gives 3/15, Γ4 gives 4/20 and so on up to Γ12, 12/60. Any whole number works, as long as the numerator and the denominator are multiplied by the same one.
Tap the 3/5 bar. Can you cut it into eighths? Why not?
Teacher notes
No: 8 isn’t in the list. Fifths can only be cut into equal pieces that make a multiple of 5 (10, 15, 20 β¦). In eighths 3/5 would be 4.8 eighths, not a whole number of pieces.
Tap the 4/8 bar and choose 1/2 (Γ· 4). What happened to the pieces? Now tap the 6/8 bar. Can it be joined the same way?
Teacher notes
4/8 joins in fours to make 1/2. 6/8 can only be joined in pairs, to make 3/4: 6 eighths don’t make a whole number of groups of four. Simplifying is cutting in reverse.
Are 2/6 and 3/9 the same? Tap ? to find out, then tap each bar and join its pieces to make 1/3.
Teacher notes
They are equal: both are 1/3 (6/18 and 6/18). 2/6 joins in pairs and 3/9 in threes, and both become 1/3.
A student says “add the same number to the top and the bottom to get an equivalent fraction, so 1/2 = 2/3”. Use Compare to show them otherwise.
Teacher notes
2/3 is bigger than 1/2, by 1/6. Adding the same number to the top and the bottom changes the size; multiplying both by the same number doesn’t (2/4, 3/6 β¦).
Adding with different denominators
Set up 1/2 + 1/4. The board says the pieces are different sizes. Which bar needs cutting? Tap it and make the pieces match, then count.
Teacher notes
1/2 = 2/4, so 2/4 + 1/4 = 3/4. Only the half needs cutting, because 4 is a multiple of 2.
Try 1/3 + 1/6. Which fraction needs converting, and why only one? Does the answer simplify?
Teacher notes
1/3 = 2/6, so 2/6 + 1/6 = 3/6 = 1/2. Sixths are already a multiple of thirds; the answer then simplifies, and the thick lines on the answer bar group the sixths in threes.
Set up 1/3 + 1/4. This time both bars need cutting. Before tapping anything, think of a denominator that works for both. Then tap a bar and look for the β .
Teacher notes
4/12 + 3/12 = 7/12. 12 is the smallest common multiple of 3 and 4. Students who try to cut the thirds into quarters find there is no such button.
Add 2/3 + 3/5 using fifteenths. Then tap each bar and change both to thirtieths. Do you get the same answer? Which was easier?
Teacher notes
10/15 + 9/15 = 19/15 = 1 4/15. With thirtieths: 20/30 + 18/30 = 38/30, which simplifies to the same 19/15. The smallest common denominator saves the simplifying.
This board uses twelfths for 1/2 + 1/3. Reveal the answer and simplify it. Now change both bars to sixths. Which route was shorter?
Teacher notes
6/12 + 4/12 = 10/12 = 5/6. In sixths, 3/6 + 2/6 = 5/6 straight away. Any common denominator works; the smallest one means less simplifying.
Set up 1/2 + 1/3 + 1/6. Can you find one denominator that works for all three bars? Predict the total first.
Teacher notes
3/6 + 2/6 + 1/6 = 6/6 = 1. 6 is a multiple of 2, 3 and 6. This is the classic “one whole from three unit fractions”.
For 3/4 + 5/6, a student multiplies the denominators to get 24. Does that work? Is it the smallest common denominator?
Teacher notes
24 works (18/24 + 20/24 = 38/24) but 12 is smaller: 9/12 + 10/12 = 19/12 = 1 7/12. When the denominators share a factor (here 2), the product is never the smallest.
Before converting, will 2/5 + 1/3 be more or less than 1/2? Then work it out.
Teacher notes
More: 6/15 + 5/15 = 11/15, and a half is 7.5/15. Estimating against 1/2 is a quick check on any answer.
Find a common denominator for 3/8 + 5/12 in your head before pressing π‘ Common denominator. Did you get the same as the tool?
Teacher notes
24: 9/24 + 10/24 = 19/24. 8 Γ 12 = 96 also works, but 24 is the smallest number in both times tables.
Subtracting
Set up 3/4 β 1/3. Make the pieces the same size and find the answer. Which pieces are crossed out?
Teacher notes
9/12 β 4/12 = 5/12. The last 4 of the 9 shaded twelfths are crossed out; 5 are left.
The board shows 1/2 β 2/3, and it says the answer would be negative. Press β Swap. Why does order matter in subtraction but not in addition?
Teacher notes
2/3 β 1/2 = 4/6 β 3/6 = 1/6. The other way round it is β1/6. Swapping the fractions in an addition gives the same total; in a subtraction it changes the sign.
Estimate 5/6 β 1/3: more or less than 1/2? Then work it out.
Teacher notes
Neither: 5/6 β 2/6 = 3/6 = 1/2 exactly. A nice surprise for students who were sure it was one or the other.
Work out 3/5 β 1/2. Look at the two bars: is the answer a big or a small fraction?
Teacher notes
6/10 β 5/10 = 1/10. The bars are close in length, so the difference is small: one tenth.
In 7/8 β 3/4, only one bar needs cutting. Which one, and why?
Teacher notes
The 3/4: quarters can be cut into eighths (3/4 = 6/8), but eighths can’t be joined into quarters here because 7 is odd. 7/8 β 6/8 = 1/8.
4/5 β 3/4: the two bars look almost the same length. How big is the difference? How many pieces do you need to show it?
Teacher notes
16/20 β 15/20 = 1/20. You need twentieths: 1 piece out of 20 is the gap you can just see at the ends of the bars.
Set up 1 β 3/8. How many eighths make 1 whole? Tap the 1 bar and cut it.
Teacher notes
1 = 8/8, so 8/8 β 3/8 = 5/8. Writing 1 as a fraction with the right denominator is the step students most often skip.
Mixed numbers
Set up 3/4 + 1/2 and reveal the answer. It is shown as 5/4 = 1 1/4. What does the 1 mean on the answer bar?
Teacher notes
3/4 + 2/4 = 5/4 = 1 1/4. The answer bar fills one whole box and a quarter of the next: the 1 is the full box.
Make 1 1/2 with the builder: press Wholes + once, tap 2, then tap the first piece. Then add 2/3 and work out the total.
Teacher notes
1 1/2 + 2/3 = 1 3/6 + 4/6 = 13/6 = 2 1/6. The fraction parts add to more than 1, so a whole carries over.
Work out 1 3/4 + 2 2/3. Is it quicker to add the wholes and the fractions separately? Check both ways give the same answer.
Teacher notes
53/12 = 4 5/12. Separately: 1 + 2 = 3, and 3/4 + 2/3 = 17/12 = 1 5/12, so 4 5/12. Both give the same, but the fractions can carry a whole.
2 1/4 β 1 2/3. A student says “2 β 1 = 1 and 2/3 β 1/4 = 5/12, so 1 5/12”. Their answer is in purple. What went wrong?
Teacher notes
7/12 (27/12 β 20/12). The student subtracted the smaller fraction from the bigger one. 1/4 is less than 2/3, so a whole has to be swapped for twelfths: 2 3/12 = 1 15/12, then 1 15/12 β 1 8/12 = 7/12.
Work out 3 β 1 1/3. How many thirds are in 3?
Teacher notes
3 = 9/3 and 1 1/3 = 4/3, so 5/3 = 1 2/3.
Improper fractions work too. Work out 5/3 + 7/4 and write the answer as a mixed number.
Teacher notes
20/12 + 21/12 = 41/12 = 3 5/12.
Which is bigger, 1 2/3 or 5/3? Predict, then check.
Teacher notes
They are equal: 1 2/3 is 3 thirds and 2 more, which is 5/3. The bars are the same length.
In β Subtract, press π² New question on level 5 a few times. Look for questions where the fraction part of the first number is smaller than the fraction part of the second. How do you deal with those?
Teacher notes
Those need a whole exchanged for pieces before taking away (or both numbers written as improper fractions), as in 2 1/4 β 1 2/3.
Comparing
Which is bigger, 3/4 or 4/5? Each is one piece short of 1 whole. Does that help?
Teacher notes
3/4 < 4/5 (15/20 < 16/20). Both are one piece short of 1, but a fifth is a smaller piece than a quarter, so 4/5 is closer to 1.
Many students think 1/7 is bigger than 1/5 “because 7 is bigger than 5”. Compare them. How would you convince that student?
Teacher notes
1/7 < 1/5, by 2/35. Cutting a whole into more pieces makes each piece smaller: the bigger the denominator, the smaller the unit fraction.
Compare 5/8 and 3/5 by eye first. Then cut both bars into the same pieces and count.
Teacher notes
5/8 > 3/5: 25/40 against 24/40, a difference of 1/40. Too close to be sure by eye: the common denominator settles it.
Are 2/3 and 4/6 the same size? Predict, then check.
Teacher notes
Equal: 4/6 = 2/3. The bars are the same length; the 4/6 bar just has its pieces cut in half.
3/7 and 2/5 are both less than 1/2. Which is bigger? Predict, then use a common denominator.
Teacher notes
3/7 > 2/5: 15/35 against 14/35, by 1/35. Both are just under a half, so the benchmark of 1/2 doesn’t decide it.
Compare 5/12 and 3/8. What is the smallest denominator that works for both?
Teacher notes
24: 10/24 > 9/24, so 5/12 > 3/8. 24 is the smallest common multiple of 12 and 8.
4/9 and 5/11 are both just under 1/2. Which is closer to a half? Which is bigger?
Teacher notes
4/9 < 5/11 (44/99 < 45/99). 4/9 is 1/18 below a half and 5/11 is 1/22 below, so 5/11 is closer, and bigger.
A student says 3/8 is bigger than 3/5 because 8 is bigger than 5. Compare them. Do you need a common denominator this time?
Teacher notes
3/8 < 3/5. No conversion is needed: both have 3 pieces, and fifths are bigger pieces than eighths. With equal numerators, the smaller denominator wins.
Misconceptions
A student adds the tops and adds the bottoms: 1/3 + 1/4 = 2/7. Their answer is in purple under the real one. How do the bars show it can’t be right?
Teacher notes
The answer is 7/12. 2/7 is less than 1/3 on its own, so it can’t be the total of 1/3 and something more. Adding the denominators makes the pieces smaller.
For 3/4 β 1/2, a student subtracts the tops and the bottoms and gets 2/2. Check their answer.
Teacher notes
The answer is 1/4. 2/2 is 1 whole, which is more than the 3/4 they started with: you can’t take something away and end up with more.
1/2 + 1/2 = 2/4? Check the student’s answer.
Teacher notes
1/2 + 1/2 = 1. 2/4 is only a half, the same as one of the two parts. Two halves make a whole, which most students know already, so this one surprises them.
For 2/3 + 1/6, a student changes 2/3 into 2/6 and gets 3/6. Their answer is in purple. Tap the 2/3 bar: what is 2/3 in sixths really?
Teacher notes
2/3 = 4/6, so the answer is 5/6. Changing only the denominator cuts the pieces smaller without making more of them: 2/6 is half of 2/3.
1 1/2 + 1 1/2: a student writes 2 2/4. Check it.
Teacher notes
The answer is 3. The wholes were right (2), but 1/2 + 1/2 was added as 2/4 instead of 1, which loses a whole.
Practice with the generator
Choose level 2 (One is a multiple) and press π² New question. Type your answer next to β My answer and press Check before you tap ?. Do five. What is the extra step compared with level 1?
Teacher notes
Only one bar needs cutting each time: the one with the smaller denominator, because the other denominator is already a multiple of it.
On level 3 (Share a factor), spot the common factor of the two denominators before tapping anything. Is the smallest common denominator smaller than the two denominators multiplied together?
Teacher notes
Always smaller: the denominators share a factor (2 or 3), so the product counts it twice. For example sixths and ninths need eighteenths, not fifty-fourths.
On level 4 (Co-prime) the denominators share no factor. Is the smallest common denominator always their product? Try five questions.
Teacher notes
Yes. When two numbers share no factor other than 1, their lowest common multiple is their product: thirds and fifths need fifteenths.
Open investigations
Is 1/2 + 1/3 the same as 1/3 + 1/2? Does the order matter in subtraction? Use β Swap in Subtract to test your claim.
Teacher notes
Addition: 5/6 either way. Subtraction: 1/2 β 1/3 = 1/6, and 1/3 β 1/2 is negative (β1/6), so the order matters.
Find two fractions with different denominators that add to exactly 1. How many pairs can you find? Then find three fractions that do it.
Teacher notes
The second fraction is always 1 minus the first, written with any equivalent denominator: 1/3 + 4/6, 3/4 + 2/8, 2/5 + 6/10 β¦ Three: 1/2 + 1/3 + 1/6 is the classic.
What is the largest total you can make by adding two proper fractions with denominators up to 10?
Teacher notes
With two different fractions, 9/10 + 8/9 = 161/90 = 1 71/90. If the same fraction can be used twice, 9/10 + 9/10 = 1 4/5. The largest proper fractions are the ones closest to 1.
Use Subtract to find fractions with different denominators whose difference is exactly 1/6. Then find some with a difference of 1/12. Do you notice a pattern?
Teacher notes
1/2 β 1/3, 2/3 β 1/2 and 1/3 β 1/6 all make 1/6; 1/3 β 1/4, 1/4 β 1/6 and 3/4 β 2/3 all make 1/12. One pattern: 1/n β 1/(n + 1) = 1/(n(n + 1)).
Find two unit fractions (numerator 1) that add to 1/2. Is there more than one pair?
Teacher notes
Only two pairs: 1/4 + 1/4 and 1/3 + 1/6. With different denominators, 1/3 + 1/6 is the only one.
Design three addition or subtraction questions: one easy, one medium and one hard. Swap with a partner and solve them with the tool. What made the hard one hard?
Teacher notes
Typical answers: different denominators that share no factor, mixed numbers that need a whole exchanged, or an answer that needs simplifying. The levels of π² New question follow the same order.
No questions match that filter.