Fraction Visualizer — Why Flip and Multiply?

Type two fractions and an operation, and the picture works the problem out step by step: bars are cut into equal pieces to find a common denominator, a square is shaded both ways for multiplication, and division is shown three ways, including why you flip the second fraction and multiply. Each step has one sentence to read aloud and the calculation beside it changes at the same time, so a parent can explain homework within a minute.

How to use

1. Pick "+ Add", "− Subtract", "× Multiply" or "÷ Divide" and type the "First fraction" and "Second fraction", for example 3/4, 1 1/2 or 2.
2. Press "Next step" and read the sentence aloud while the picture changes. "Previous step" goes back, "Play" runs through all steps, and "Speed" sets how fast.
3. Under "Picture" choose "Bars", "Circles" or "Number line" for adding and subtracting; under "Answer as" choose "Mixed number" or "Improper fraction".
4. For division, choose a "Way to divide": "How many fit?", "Common denominator" or "Why flip and multiply".
5. Press "New problem" for another one. With "Practice: answer first" ticked, the child types an answer and presses "Check" before the steps open.
6. "Full screen" puts the picture on a classroom screen; the arrow keys, the space bar and F also work.

How it works

• All values are exact: numerators and denominators are whole numbers, results are simplified with the greatest common divisor, and decimals are never used. Denominators go from 1 to 24 and each number up to 20 wholes.
• Mixed numbers are first written as improper fractions: whole part × denominator + numerator. A fraction such as 2/4 is drawn as typed and simplified at the end.
• "How many fit?" switches to the common denominator view when more than 40 copies would have to be laid out. Subtraction needs the first fraction to be at least as large as the second.
• The last problem and the chosen pictures are remembered in this browser only.

Why do you flip and multiply when dividing fractions?

Write the division as one big fraction: 3/4 on top, 1/8 at the bottom. Multiplying the top and the bottom of a fraction by the same number does not change its value. Choose the number that turns the bottom into 1, which is the reciprocal 8/1, because 1/8 × 8 = 1. The top becomes 3/4 × 8 = 6, and anything divided by 1 is itself, so 3/4 ÷ 1/8 = 3/4 × 8 = 6.

In the "Why flip and multiply" view both bars stretch by the same factor: the bar for 1/8 grows to exactly 1, and the bar for 3/4 grows by the same amount to 6. The "How many fit?" view checks the answer another way: six pieces of 1/8 fit exactly into 3/4.

The common denominator method gives the same result without flipping: 3/4 = 6/8, and 6 eighths divided by 1 eighth is 6 ÷ 1 = 6. When the pieces have the same size, dividing fractions is just dividing the numbers of pieces.

Adding fractions with different denominators, shown with pictures

Pieces can only be counted together when they are the same size. For 3/4 + 5/6, quarters and sixths differ, so both are cut into twelfths, because 12 is the smallest number that 4 and 6 both divide into. Each quarter is cut into 3, so 3/4 = 9/12, and each sixth into 2, so 5/6 = 10/12. Then 9 + 10 = 19 twelfths, which is 19/12 or 1 7/12.

A frequent mistake is adding the tops and the bottoms: 3/4 + 5/6 is not 8/10. The picture shows why: 8/10 is less than 1, but 3/4 alone is already most of a whole and 5/6 adds almost another whole.

At the end the pieces are joined into bigger pieces where possible, which is simplifying. 1/6 + 1/3 = 3/6, and joining the sixths two at a time gives 1/2.

Multiplying fractions with the area model

2/3 × 3/4 can be read as 3/4 of 2/3. The square is cut into 3 columns and 2 are shaded, which is 2/3. Then it is cut into 4 rows and 3 are hatched. The part shaded both ways has 2 × 3 = 6 small pieces, and the whole square has 3 × 4 = 12, so the product is 6/12 = 1/2. This is why you multiply the numerators and the denominators.

With a whole number or a mixed number the model uses several squares: 1 1/2 × 2/3 spans two squares across, and the shaded overlap is 3 × 2 = 6 pieces of size 1/6, which is 1.

Questions and answers

Why do we flip the second fraction when dividing?

Dividing by a fraction is the same as multiplying by its reciprocal, because multiplying the divisor by its reciprocal makes it 1 and the dividend must be multiplied by the same number. For example 3/4 ÷ 1/8 = 3/4 × 8 = 6.

How do you add fractions with different denominators?

Cut both into pieces of the same size using the least common multiple of the denominators, then add the numerators. 3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 1 7/12.

What does 3/4 ÷ 1/8 mean?

It asks how many pieces of 1/8 fit into 3/4. Six fit exactly, so the answer is 6.

Why is a fraction times a fraction smaller?

Multiplying by a fraction less than 1 takes only part of a number. 3/4 of 2/3 is 1/2, which is less than 2/3.

How do you divide a whole number by a fraction?

Write the whole number over 1 and multiply by the reciprocal: 3 ÷ 1/2 = 3 × 2 = 6, because six halves fit into 3.

How do you change an improper fraction to a mixed number?

Divide the numerator by the denominator: the quotient is the whole part and the remainder is the new numerator. 19/12 = 1 7/12.