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Stage 5 of 20

Complements of Ten

Master complements of 10 as connected addition and subtraction bonds, then use them to bridge through ten, reason about balance, and check sums.

Start the interactive drill 18 checkable exercises

What You'll Learn

  • Recall every ordered pair of whole numbers that totals 10
  • Explain complements as two parts of a fixed whole
  • Apply a complement to bridge efficiently through ten
  • Extend the same relationship to missing-addend and subtraction questions

Why This Matters

Complements reduce many calculations to one recognized relationship. They support addition over ten, subtraction from ten, regrouping, change-making, and later work with multiples of ten and one hundred.

The deeper lesson is conservation: parts may shift while a chosen whole remains constant. This is a central habit in flexible mental arithmetic.

How Complements of Ten Works

A complement of ten is the amount a number needs to make a complete ten. The core pairs are 0 and 10, 1 and 9, 2 and 8, 3 and 7, 4 and 6, and 5 and 5. Each pair can be read in both orders, so the complement of 8 is 2 and the complement of 2 is 8.

These pairs form a balanced pattern. As the first part rises by one, the second falls by one, while the whole stays fixed at ten. This invariant is more useful than a disconnected list: if 3 + 7 is known, then shifting one from 7 to 3 gives 4 + 6, still ten.

Complements make crossing ten economical. For 8 + 5, identify 2 as the complement of 8. Decompose 5 into 2 and 3, complete 10, then attach the remaining 3. The equality works because 5 has only been regrouped: 8 + (2 + 3) = (8 + 2) + 3.

Subtraction and missing-addend questions expose the same bond. Ten minus 6 is 4 because 6 and 4 compose ten. In 7 + ? = 10, the question mark is not a guess; it names the absent part of a known whole. Stage 3 inverse reasoning and Stage 4 count-up reasoning both support this conclusion.

Ten is a strategic landmark because our place-value system groups ones in tens. Once these bonds are secure, the pattern scales: 30 needs 70 to make 100, while 63 needs 37. That extension is not the main calculation target here, but it reveals why complement thinking later helps with money and estimation.

Mental Walkthroughs

Read the strategy, then prove you can use it: every example ends with a matching problem on new numbers that you answer and check right here. Answer it and press New numbers for as many fresh problems as you like.

Name the missing complement

7 + ? = 10

  1. 1 I treat 10 as the whole and 7 as one known part.
  2. 2 I count from 7 to 10: 8, 9, 10, which is three steps.

7 + 3 = 10

Your turn — try it on new numbers

3 + ? = 10

Bridge through ten

8 + 6

  1. 1 The complement of 8 is 2.
  2. 2 I split 6 into 2 and 4, then make 8 + 2 = 10.
  3. 3 I add the remaining 4 to reach 14.

8 + 6 = 14

Your turn — try it on new numbers

10 − 2 = ?

Read the inverse bond

10 − 4

  1. 1 I ask which part joins 4 to form the whole 10.
  2. 2 The known complement pair is 4 and 6.

10 − 4 = 6

Your turn — try it on new numbers

10 − 8 = ?

Use balance to recover a pair

What complements 6?

  1. 1 I recall 5 + 5 = 10.
  2. 2 If one part rises from 5 to 6, the other falls from 5 to 4.

The complement of 6 is 4.

Your turn — try it on new numbers

10 − 1 = ?

Watch for These Missteps

Giving the total instead of the complement

When asked for the complement of 7, 10 is the target whole, not the missing part. Solve 7 + ? = 10 to obtain 3.

Using the complement twice

In 8 + 5, use 2 from the 5 to reach ten, then add only the leftover 3. Adding 5 again duplicates part of the original addend.

Omitting zero and ten

The full whole-number pattern includes 0 + 10 and 10 + 0. Zero contributes no quantity, so its partner must supply all ten.

Complements of Ten in Everyday Life

A ten-slot egg carton with six eggs has four empty spaces. The filled and empty spaces are complementary parts of the fixed capacity: 6 + 4 = 10. If three more eggs are added, one space remains because 9 and 1 complete the carton.

With ten counters used as a game target, a player holding 7 needs 3 more. The question is a missing-addend problem, and the complement gives the answer without recounting the whole collection.

Pro Tips

  • • Practice pairs in both directions: answer 3 for 7 and answer 7 for 3.
  • • Picture filled and empty spaces in one ten-frame rather than two separate groups.
  • • Use 5 + 5 as the midpoint and adjust opposite parts equally.
  • • Before crossing ten, name the complement and the leftover amount separately.

Practice Exercises

Work each problem mentally, then type the answer and check it. Nothing is marked correct until you enter a value that actually equals the result, and every problem keeps its mental-steps note in reach if you get stuck.

  1. 1. 6 + ? = 10
  2. 2. 3 + ? = 10
  3. 3. 9 + ? = 10
  4. 4. 10 − 7 = ?
  5. 5. 10 − 2 = ?
  6. 6. 5 + ? = 10
  7. 7. 0 + ? = 10
  8. 8. 8 + ? = 10
  9. 9. 7 + 5 = ?
  10. 10. 9 + 6 = ?
  11. 11. 8 + 4 = ?
  12. 12. 6 + 7 = ?
  13. 13. 10 − 9 = ?
  14. 14. 10 − 5 = ?
  15. 15. 4 + ? = 10
  16. 16. ? + 2 = 10
  17. 17. 8 + 5 = ?
  18. 18. 10 − 0 = ?

Drill Stage 5 with unlimited problems

Type your answer and press Enter. Every problem is generated on the spot and uses complements of ten, asked in both directions. Get it wrong and the steps below show how that exact problem is solved mentally.

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Problem 1

6 + ? = 10

Stage 5 Questions

Is a complement the same as an opposite number?

No. Here a complement is the amount needed to reach the chosen whole, 10. Opposite numbers concern equal distances from zero.

Why include 5 + 5 only once?

Reversing this pair produces the same ordered appearance because its two parts are equal; it is the midpoint of the complement pattern.

Do complements only work with ten?

Complements can be defined for any target whole. Ten is emphasized because it is the basic grouping unit in decimal place value.