Equivalent fractions

What it looks like when your child can do it

• — Your child folds a strip in half, then in half again, and says 1/2 and 2/4 cover the same amount.
• — Asked for a fraction equal to 3/5, your child multiplies the top and bottom by 2 and writes 6/10.
• — Your child simplifies 6/8 to 3/4 by dividing both numbers by 2, and knows the amount did not change.

How to teach it at the kitchen table

1. 01

   Show it with paper strips

   Fold one strip in half and shade one half. Fold a second strip into fourths and shade two. Lay them side by side. The shaded parts match, so 1/2 and 2/4 are equal.

2. 02

   Cut every piece the same way

   Cutting each half into 2 pieces doubles the number of pieces and doubles the number shaded. That is why you multiply the top and the bottom by the same number.

3. 03

   Multiply both numbers, never add

   2/3 with the top and bottom times 2 is 4/6. Adding 1 to the top and bottom gives 3/4, a different amount. Doubling a recipe doubles every ingredient; it does not add one cup of each.

4. 04

   Divide both numbers to simplify

   Going the other way gives smaller numbers for the same amount. 6/8 with the top and bottom divided by 2 is 3/4. Stop when no number bigger than 1 divides both.

5. 05

   Check by cross-multiplying

   To see whether 3/4 and 9/12 are equal, multiply across: 3 x 12 = 36 and 4 x 9 = 36. Matching products mean the fractions are equal.

You will need: Paper strips · A pencil

Try it together

1. Try 1

   Fill in the blank: 2/3 = ?/15.

   Listen for  10. The bottom went from 3 to 15, which is times 5, so the top goes from 2 to 10. Watch for 14, which comes from adding 12 to the top because 12 was added to the bottom.

2. Try 2

   Simplify 12/18 as far as it goes.

   Listen for  2/3. Dividing by 6 gets there in one step. If your child divides by 2 and stops at 6/9, ask whether 6 and 9 still share a number. They share 3.

3. Try 3

   Are 4/6 and 6/9 equal?

   Listen for  Yes. Both simplify to 2/3, and cross-multiplying gives 4 x 9 = 36 and 6 x 6 = 36. Either check is fine.

4. Try 4

   Is 3/5 equal to 5/7? Each number in 5/7 is 2 more than the matching number in 3/5.

   Listen for  No. Cross-multiplying gives 3 x 7 = 21 and 5 x 5 = 25, which do not match. Adding the same number to the top and bottom changes the amount.

5. Try 5

   Name three fractions equal to 3/4.

   Listen for  Many answers work, such as 6/8, 9/12 and 12/16. Each one multiplies the top and bottom of 3/4 by the same number. Check one by cross-multiplying.

Watch for

• — Adding the same number to the top and bottom. It feels fair and it changes the amount: 1/2 plus 1 on each becomes 2/3.

If it is not there yet

• — Go back to paper strips folded into halves, fourths and eighths, and line them up to find matching amounts before writing any numbers.
• — Practice the multiplication facts that come up most, the 2s through the 6s, since every equivalent fraction is a multiplication.

A few days of this is normal, and it is time well spent. The lesson will still be there, and it will go faster.

Where this skill comes up

• Grade 6 Math Practice Workbook · Lesson 6

  Ratio Tables and Double Number Lines

  Needs it because equivalent ratios are made the same way, multiplying both parts by the same number.