Chapter 5 of 7

All 7 chapters
  1. 1Number sequences, readunderstood
  2. 2Growing shape patterns, readunderstood
  3. 3Number machines, readunderstood
  4. 4Missing numbers, readunderstood
  5. 5Odd and even rules, readunderstood
  6. 6Square and triangle numbers, readunderstood
  7. 7Jumping far ahead, readunderstood

M2.8.5 Chapter 5

Odd and even rules

Is 7 + 9 odd or even? Do I have to add them to know?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Explain why odd + odd always makes an even number
  • Test a rule with many examples before you trust it

Take 6 stones. Put them in pairs. You get 3 pairs. Nothing is left over.

Now take 7 stones. You get 3 pairs and 1 left over.

An even number makes pairs with none left over. An odd number has one left over.

Look at the last digit. A last digit of 0, 2, 4, 6 or 8 means even. Any other last digit means odd.

What happens when you add two odd numbers? Watch 3 + 5.

Draw it first

Add two odd numbers in pairs

  1. Dots in pairs. Step 1 shows 3 dots, labelled 3: one pair and one dot left over. Step 2 adds a + sign and 5 dots, labelled 5: two pairs and one dot left over. Step 3 fills in the two left over dots. Step 4 adds an = sign and 8 dots, labelled 8: four pairs, and the top pair is the filled pair.3

    1 Put 3 dots in pairs. One pair, and one dot has no partner.

  2. Dots in pairs. Step 1 shows 3 dots, labelled 3: one pair and one dot left over. Step 2 adds a + sign and 5 dots, labelled 5: two pairs and one dot left over. Step 3 fills in the two left over dots. Step 4 adds an = sign and 8 dots, labelled 8: four pairs, and the top pair is the filled pair.3+5

    2 Put 5 dots in pairs. Two pairs, and one dot has no partner.

  3. Dots in pairs. Step 1 shows 3 dots, labelled 3: one pair and one dot left over. Step 2 adds a + sign and 5 dots, labelled 5: two pairs and one dot left over. Step 3 fills in the two left over dots. Step 4 adds an = sign and 8 dots, labelled 8: four pairs, and the top pair is the filled pair.3+5

    3 Fill in the two dots with no partner. They are the left overs.

  4. Dots in pairs. Step 1 shows 3 dots, labelled 3: one pair and one dot left over. Step 2 adds a + sign and 5 dots, labelled 5: two pairs and one dot left over. Step 3 fills in the two left over dots. Step 4 adds an = sign and 8 dots, labelled 8: four pairs, and the top pair is the filled pair.3+5=8

    4 Put all the dots together. The two left overs make a pair. 8 dots, none left over.

Each odd number has one left over. The two left overs make a pair. So nothing is left over, and the total is even.

A table with the heads Add and Total is. Row 1: even + even, even. Row 2: odd + even, odd. Row 3: odd + odd, even.AddTotal iseven + evenevenodd + evenoddodd + oddeven
The three sums. Odd + odd is the one that surprises people.

Odd + odd = even is a rule. A rule tells you what happens every time.

How do you know a rule is right? Test it on many pairs. Use small numbers and big numbers.

A table with the heads Odd + odd, Total and Even. Row 1: 3 + 5, 8, yes. Row 2: 7 + 9, 16, yes. Row 3: 11 + 15, 26, yes. Row 4: 21 + 33, 54, yes.Odd + oddTotalEven3 + 58yes7 + 916yes11 + 1526yes21 + 3354yes
Four tests. Each total ends in 0, 2, 4, 6 or 8.

Many examples make you trust a rule. But you cannot test every pair of numbers. So ask why it works. The pairs picture tells you why.

One example that fails is enough to break a rule.

Worked example

Is 17 + 25 odd or even? Do not add.

  1. 17 ends in 7, so 17 is odd.
  2. 25 ends in 5, so 25 is odd.
  3. Odd + odd is even. So 17 + 25 is even.
  4. Check: 17 + 25 = 42. 42 ends in 2, so it is even.

Answer: even

See it three ways

With things

Put 3 stones in pairs, and 7 stones in pairs. Each pile has one left over. Push the piles together. The left overs make a pair.

As a picture

Dots in pairs. 3 dots, labelled 3, then a + sign, then 7 dots, labelled 7, then an = sign, then 10 dots, labelled 10. The left over dot of 3 and of 7 are filled in. In the 10, the top pair is filled in.3+7=10

In symbols

3 + 7 = 10. Odd + odd = even.

Try it

Test the rule with stones

You need: about 25 stones, beans or bottle tops, a pencil and paper.

  1. Count out 7 stones. Put them in pairs. One is left over.
  2. Count out 5 stones in a second pile. Put them in pairs. One is left over.
  3. Push the two piles together. Pair up all the stones.
  4. Write down if any stone is left over. Then try 9 and 3. Then try 11 and 1.

What you will see: The two left overs find each other every time. No stone is left over, so each total is even.

If you cannot try it, this is what happens.
Dots in pairs. 7 dots, labelled 7, then a + sign, then 5 dots, labelled 5, then an = sign, then 12 dots, labelled 12. The left over dot of 7 and of 5 are filled in. In the 12, the top pair is filled in.7+5=12

A wrong idea many people have

Three odd numbers add up to an even number.

Why it feels right: The rule worked for two odd numbers. It feels like it should work for three.

The test that breaks it: Add 1 + 3 + 5. You get 9, and 9 is odd. One example fails, so the rule is broken.

What is true: Three odd numbers make an odd total. Two left overs make a pair. The third one stays alone.

Key words

even
a number that makes pairs with none left over.
odd
a number that has one left over when you make pairs.
rule
something that is true every time.

Back to your guess

The answer is B: Even, because the two left over ones make a pair.

Odd + odd is even. Each odd number has one left over. The two left overs make a pair. So 7 + 9 is even. And 7 + 9 = 16.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which sum gives an odd number?

    Tap your answer.

    1. Not this one. 6 and 8 are both even. They make pairs with none left over. 6 + 8 = 14, which is even. Try another.

    2. Not this one. You may think two odd numbers make an odd number. Each has one left over. The two left overs make a pair. 7 + 9 = 16. Try another.

    3. Yes, that is right. 7 is odd and 8 is even. Only 7 has a left over. So 7 + 8 = 15 is odd.

    See the answer

    7 + 8

    7 is odd and 8 is even. Only 7 has a left over. So 7 + 8 = 15 is odd.

    • If you chose A: 6 and 8 are both even. They make pairs with none left over. 6 + 8 = 14, which is even.
    • If you chose B: You may think two odd numbers make an odd number. Each has one left over. The two left overs make a pair. 7 + 9 = 16.

  2. Without adding, say if 13 + 27 is odd or even.

    See the answer

    even

    13 and 27 are both odd. Odd + odd is even. Check: 13 + 27 = 40, which ends in 0.

  3. Is 9 + 6 odd or even? Draw 9 dots and 6 dots in pairs to see.

    See the answer

    odd

    9 has one left over. 6 has none. So one dot has no partner. 9 + 6 = 15, which is odd.

  4. Is 5 + 7 + 9 odd or even? Add the first two, then add 9.

    See the answer

    odd

    5 + 7 = 12 is even. 12 + 9 is even + odd, so it is odd. 12 + 9 = 21.

  5. Tariq tests "even + even = even". He tries 2 + 4, 6 + 8 and 10 + 12. It works each time. What should he do next?

    Tap your answer.

    1. Not this one. Three examples are a start, but they do not prove a rule. Test more pairs, and ask why it works. Try another.

    2. Yes, that is right. Examples help you trust a rule. One that fails breaks it. A picture of pairs shows why it works.

    3. Not this one. Many examples build trust. A picture of pairs can show why a rule works. Try another.

    See the answer

    Test more pairs, and ask why it works.

    Examples help you trust a rule. One that fails breaks it. A picture of pairs shows why it works.

    • If you chose A: Three examples are a start, but they do not prove a rule. Test more pairs, and ask why it works.
    • If you chose C: Many examples build trust. A picture of pairs can show why a rule works.

  6. From an earlier chapter

    Find the missing number: 29 + ? = 61.

    See the answer

    32

    Take 29 from 61: 61 − 29 = 32. Check: 29 + 32 = 61.

  7. Mei puts 5 dots and 3 dots in pairs. Each has one left over dot. What happens when she joins them?

    Dots in pairs. 5 dots, labelled 5, then a + sign, then 3 dots, labelled 3. The left over dot of each is filled in.53+
    Tap your answer.

    1. Not this one. Each left over now has a partner, the other left over. They make a pair. Try another.

    2. Yes, that is right. Each odd number has one left over. The two left overs make a pair. So 5 + 3 has none left over. 5 + 3 = 8, which is even.

    3. Not this one. The total needs both numbers. 3 has a left over too. The two left overs make a pair: 5 + 3 = 8. Try another.

    See the answer

    The two left overs make a pair. The total is even.

    Each odd number has one left over. The two left overs make a pair. So 5 + 3 has none left over. 5 + 3 = 8, which is even.

    • If you chose A: Each left over now has a partner, the other left over. They make a pair.
    • If you chose C: The total needs both numbers. 3 has a left over too. The two left overs make a pair: 5 + 3 = 8.

  8. Sipho adds 3 + 5 + 7. Is the total odd or even?

    Dots in pairs. 3 dots, labelled 3, then a + sign, then 5 dots, labelled 5, then a + sign, then 7 dots, labelled 7. The left over dot of each is filled in.35+7+
    Tap your answer.

    1. Yes, that is right. Each of 3, 5 and 7 has one left over. Two left overs make a pair. The third stays alone. 3 + 5 + 7 = 15, which is odd.

    2. Not this one. That rule is for two odd numbers. Here there are three. 3 + 5 = 8, and 8 + 7 = 15, which is odd. Try another.

    3. Not this one. You can tell. 3 + 5 is even, and even + odd is odd. Try another.

    See the answer

    Odd. Two left overs make a pair, but the third stays alone.

    Each of 3, 5 and 7 has one left over. Two left overs make a pair. The third stays alone. 3 + 5 + 7 = 15, which is odd.

    • If you chose B: That rule is for two odd numbers. Here there are three. 3 + 5 = 8, and 8 + 7 = 15, which is odd.
    • If you chose C: You can tell. 3 + 5 is even, and even + odd is odd.

  9. Kofi says two numbers always add to an even number. He tests 2 + 4 and 3 + 5. Both work. Is his rule true?

    Tap your answer.

    1. Not this one. 3 + 5 = 8 is even. An example that fits the rule cannot break it. Try another.

    2. Not this one. Tests that work do not prove a rule. Try 3 + 4. It makes 7, which is odd. Try another.

    3. Yes, that is right. 7 is odd. One example that fails is enough to break a rule. Tests that work only build trust.

    See the answer

    No. 3 + 4 = 7 breaks it.

    7 is odd. One example that fails is enough to break a rule. Tests that work only build trust.

    • If you chose A: 3 + 5 = 8 is even. An example that fits the rule cannot break it.
    • If you chose B: Tests that work do not prove a rule. Try 3 + 4. It makes 7, which is odd.

Keep it

Copy this into your exercise book.

Even numbers make pairs with none left over. Odd numbers have one left over. Odd + odd is even, because the two left overs make a pair. One example that fails breaks a rule.

Draw: Draw 3 dots and 5 dots in pairs. Fill in the left overs. Then draw all 8 dots in pairs.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (603 KB). It opens on any phone.