Inverse operations

What it looks like when your child can do it

• — Asked what undoes adding 9, your child says subtracting 9 without a pause, and can say it the other way around too.
• — Given 4 x 6 = 24, your child can say the two division facts that run back the other way: 24 ÷ 4 = 6 and 24 ÷ 6 = 4.
• — When your child finds a missing number, they put it back in and check that it works, without being asked.

How to teach it at the kitchen table

1. 01

   Start with a real undo

   Put 5 crackers on a plate, add 3, then take 3 away. Ask what happened. Taking 3 away put the plate back where it started. That is all an inverse operation is: a second move that cancels the first.

2. 02

   Name the two pairs out loud

   Adding and subtracting undo each other. Multiplying and dividing undo each other. Write the two pairs on an index card and keep it on the table while you work.

3. 03

   Play the mystery number game

   Say: I think of a number, multiply it by 6, and get 42. What was my number? Your child divides 42 by 6 to get 7. Then trade places and let your child make up the puzzles for you.

4. 04

   Ask what was done, then what undoes it

   Before any arithmetic, ask two questions. What was done to the mystery number? What undoes that? These are the same two questions the workbook asks on its equations page.

5. 05

   Check every answer by putting it back

   Once your child has an answer, put it back into the first sentence. If 7 x 6 really makes 42, the answer holds. If it does not, the undo was the wrong one, and that is the thing to look at.

You will need: Paper and a pencil · A handful of counters, like beans or coins · An index card for the two undo pairs

Try it together

1. Try 1

   Say: I think of a number and add 48. I get 125. What was my number?

   Listen for  77, because 125 - 48 = 77. Check: 77 + 48 = 125. Watch that your child subtracts the 48 and does not add it.

2. Try 2

   Say: I think of a number and multiply it by 5. I get 115. What was my number?

   Listen for  23, because 115 ÷ 5 = 23. Check: 5 x 23 = 115. If your child says 110, they subtracted the 5. Ask whether the 5 was added or multiplied.

3. Try 3

   Say: I think of a number and divide it by 3. I get 12. What was my number?

   Listen for  36, because multiplying undoes dividing: 12 x 3 = 36. Check: 36 ÷ 3 = 12. This one runs in the other direction, so it is worth a slow second look.

4. Try 4

   Say: I think of a number and subtract 2.5. I get 10. What was my number?

   Listen for  12.5, because adding undoes subtracting: 10 + 2.5 = 12.5. Check: 12.5 - 2.5 = 10. Decimals follow the same rule as whole numbers.

5. Try 5

   A fuel bill of 276 dollars is split evenly among 4 families. Before any arithmetic, ask your child to name the undo. Then find each family's share.

   Listen for  69 dollars. Four equal shares make 276, so divide by 4: 276 ÷ 4 = 69. Check: 4 x 69 = 276. A child who says divide by 4 before computing has the idea.

Watch for

• — Undoing a multiplication by subtracting. In 5n = 115 the 5 looks like something to take away, so a child writes 110. The check catches it: 5 x 110 is 550, nowhere near 115.

If it is not there yet

• — Go back to fact families. Write 3, 5 and 8 on a card and find all four facts: 3 + 5 = 8, 5 + 3 = 8, 8 - 5 = 3 and 8 - 3 = 5. Then do the same with 4, 6 and 24 for multiplying and dividing.
• — Use real objects for the undo. Add counters to a pile and take them away, or deal cards into equal piles and gather them back up, until the pairs feel familiar.

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 22

  Solving One-Step Equations

  Needs it because solving is undoing, and a child has to know which operation undoes which before the moves on this sheet make sense.