Math Detours · Grade 5 · Around the corner · 1

Odd plus odd is always even

Not because you checked a hundred examples. Draw odd numbers as pairs with one left over and you can see it must be true.

Name ____________________Date ____________

Tonight's idea

3 + 5 = 8. 7 + 9 = 16. Odd plus odd keeps coming out even. But the odd numbers never end. Check a million sums and you still have not checked them all.

So do not check. Look at what odd and even mean.

An even number of dots can be lined up in pairs with none left over. An odd number of dots makes pairs with exactly one spare dot.

+ = 5: one spare 7: one spare 12: the spares pair up
5 + 7. Each odd number brings one spare dot. Two spares make one more pair.

Now add two odd numbers. Each one brings its pairs and one spare. The two spares pair up with each other. Everything is in pairs, so the total is even.

The picture never asked how many pairs there were. It works for 5 + 7 and for 1001 + 4999 in the same way. An explanation that covers every case at once is called a proof.

Watch one: odd plus even

Is 7 + 4 odd or even? Do not add yet. Think in pairs.

7 is three pairs and one spare. 4 is two pairs and no spare. Put them together: five pairs, and one spare with nothing to pair with. So the total is odd. Check: 7 + 4 = 11.

Nothing here depended on 7 and 4. Odd plus even is always odd.

Your turn

  1. Do not add. Just say odd or even. (a) 37 + 45   (b) 128 + 56   (c) 99 + 250   (d) 4001 + 2999
  2. Draw 3 + 9 as dots in pairs, like the picture above. Ring the two spare dots that join up.
  3. (a) Add three odd numbers, such as 3 + 5 + 7. Is the total odd or even? (b) What about four odd numbers? (c) What about nine odd numbers? Use spare dots to explain.
  4. Pick three numbers from this list that add up to 30. You may use a number twice.
    1, 3, 5, 7, 9, 11, 13, 15
    Either find them, or explain why nobody ever will.

Stretch: odd times odd

3 × 5 means 5 + 5 + 5. Use what you found in exercise 3 to explain why odd × odd is always odd. Then explain why even × any whole number is always even.

Answers

  1. (a) Even: odd + odd. (b) Even: even + even has no spares at all. (c) Odd: odd + even. (d) Even: odd + odd. The totals are 82, 184, 349 and 7000.
  2. 3 is one pair and a spare. 9 is four pairs and a spare. The two spares join to make a sixth pair: 12.
  3. (a) Odd. Three spares: two pair up and one is left. \(3 + 5 + 7 = 15\). (b) Even. Four spares make two pairs. (c) Odd. Nine spares make four pairs with one left. The rule: an odd number of odd numbers gives an odd total. An even number of odd numbers gives an even total.
  4. Nobody will. Every number in the list is odd, and three odd numbers always add up to an odd number. 30 is even. You did not have to try all the choices to know.

Stretch. Odd × odd is an odd number of odd numbers added together, like 5 + 5 + 5. By exercise 3 the total is odd. Even × anything: write it as an even number of equal rows. The rows pair up, two by two, with none left over. So the total is even, whatever is in each row. For example 4 × 7 is 7 + 7 and 7 + 7.

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