Chapter 5 of 7

All 7 chapters
  1. 1Tally and frequency tables, readunderstood
  2. 2Pictograms with a key, readunderstood
  3. 3Bar charts, readunderstood
  4. 4Tables with rows and columns, readunderstood
  5. 5Sorting diagrams, readunderstood
  6. 6Line graphs, readunderstood
  7. 7Your own survey, readunderstood

M2.7.5 Chapter 5

Sorting diagrams

A number fits neither circle. Where do I put it?

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

  • Sort numbers in a Venn diagram with two circles
  • Sort numbers into a 2 by 2 table using two rules

Mei has ten cards. They show the numbers 1 to 10.

She sorts them with two rules. Rule one: the number is even. Rule two: the number is over 6.

Over 6 means 7 and up. The number 6 is not over 6.

Some cards fit one rule. Some fit both. Some fit neither. A Venn diagram sorts them. It is two circles that cross.

Draw it first

Sort the cards with two circles

  1. Two circles that cross. The left circle has the rule even. The right circle has the rule over 6. Step 1 draws the empty circles. Step 2 puts 2, 4 and 6 in the even circle only, and 7 and 9 in the over 6 circle only. Step 3 puts 8 and 10 where the circles cross. Step 4 draws a box round both circles and puts 1, 3 and 5 outside the circles.evenover 6

    1 Draw two circles that cross. Write one rule on each. Write even on one and over 6 on the other.

  2. Two circles that cross. The left circle has the rule even. The right circle has the rule over 6. Step 1 draws the empty circles. Step 2 puts 2, 4 and 6 in the even circle only, and 7 and 9 in the over 6 circle only. Step 3 puts 8 and 10 where the circles cross. Step 4 draws a box round both circles and puts 1, 3 and 5 outside the circles.evenover 624679

    2 A card that fits one rule goes in that circle only. 2, 4 and 6 are even. 7 and 9 are over 6.

  3. Two circles that cross. The left circle has the rule even. The right circle has the rule over 6. Step 1 draws the empty circles. Step 2 puts 2, 4 and 6 in the even circle only, and 7 and 9 in the over 6 circle only. Step 3 puts 8 and 10 where the circles cross. Step 4 draws a box round both circles and puts 1, 3 and 5 outside the circles.evenover 624681079

    3 A card that fits both rules goes where the circles cross. 8 and 10 are even and over 6.

  4. Two circles that cross. The left circle has the rule even. The right circle has the rule over 6. Step 1 draws the empty circles. Step 2 puts 2, 4 and 6 in the even circle only, and 7 and 9 in the over 6 circle only. Step 3 puts 8 and 10 where the circles cross. Step 4 draws a box round both circles and puts 1, 3 and 5 outside the circles.evenover 624681079135

    4 Draw a box round the circles. 1, 3 and 5 fit neither rule. They go outside the circles, inside the box.

The part where the circles cross is the overlap. A card in the overlap fits both rules.

A card that fits neither rule still has a place. It goes outside both circles, inside the box. Do not leave it out.

Worked example

Where does the card 6 go in Mei's diagram?

  1. Is 6 even? Yes.
  2. Is 6 over 6? No. Over 6 means 7 and up.
  3. It fits one rule only. It goes in the even circle, outside the overlap.

Answer: In the even circle only.

You can sort the same cards in a 2 by 2 table. It has two rows and two columns. Each card gets one box.

A 2 by 2 table. The corner says Numbers. The columns are even and odd. The rows are over 6 and 6 or less. Over 6 and even holds 8, 10. Over 6 and odd holds 7, 9. 6 or less and even holds 2, 4, 6. 6 or less and odd holds 1, 3, 5.Numbersevenoddover 68, 107, 96 or less2, 4, 61, 3, 5
The shaded box matches the overlap. These cards are even and over 6.

A number that is not even is odd. Look at 1, 3 and 5. They fit neither rule. In the table they share one box. It is the box for odd and 6 or less. In the Venn diagram they sit outside.

Worked example

Where does the card 7 go in the 2 by 2 table?

  1. Is 7 even? No. It is odd. Use the odd column.
  2. Is 7 over 6? Yes. Use the over 6 row.
  3. Read the box where the odd column meets the over 6 row.

Answer: In the box for odd and over 6.

See it three ways

With things

Write 1 to 6 on six scraps. Lay two loops of string that cross. Name the loops even and under 4. Place each scrap.

As a picture

Two circles that cross inside a box. The left circle is even and holds 4 and 6. The right circle is under 4 and holds 1 and 3. The overlap holds 2. The number 5 is outside both circles.evenunder 4462135

In symbols

2 is even and under 4, so it is in the overlap. 5 is odd and not under 4, so it is outside. Count the places: 2 + 1 + 2 + 1 = 6. All six cards are there.

Try it

Sort number cards in two loops

You need: 10 scraps of paper, a pencil, string, or a stick to draw loops in the dirt.

  1. Write the numbers 1 to 10 on the scraps. Put one number on each.
  2. Make two loops that cross. Use string, or draw them in the dirt.
  3. Name the loops. One loop is odd. The other loop is under 5.
  4. Put each card in its place. Cards that fit both rules go where the loops cross.
  5. Put the cards that fit neither rule outside both loops.

What you will see: All 10 cards have a place. 1 and 3 are in the overlap. 6, 8 and 10 are outside both loops.

If you cannot try it, this is what happens.
Two circles that cross inside a box. The left circle is odd and holds 5, 7 and 9. The right circle is under 5 and holds 2 and 4. The overlap holds 1 and 3. The numbers 6, 8 and 10 are outside both circles.oddunder 557913246810

A wrong idea many people have

A number that fits neither rule can be left out.

Why it feels right: The circles are where the rules are. A number with no rule seems to have no place.

The test that breaks it: Sort all 10 cards. Count the cards inside the circles. Count the cards still in your hand. You must place 10 cards. The box holds the rest.

What is true: Every number has a place. One that fits neither rule goes outside both circles, inside the box.

Key words

Venn diagram
two circles that cross. It sorts things by two rules.
overlap
the part where the two circles cross. It holds what fits both rules.
2 by 2 table
a table with two rows and two columns. It sorts things by two rules.

Back to your guess

The answer is B: Outside both circles, but still on the page.

Every number goes somewhere. One that fits neither rule goes outside both circles, inside the box.

Check yourself

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

  1. Mei has the card 8. The rules are even and over 6. Where does 8 go?

    Two circles that cross. The left circle is labelled even. The right circle is labelled over 6. Both are empty.evenover 6
    Tap your answer.

    1. Not this one. You checked one rule. Is 8 over 6 too? Yes, so it fits both rules. Try another.

    2. Yes, that is right. 8 is even. 8 is over 6. It fits both rules, so it goes where the circles cross.

    3. Not this one. 8 is even, so it fits a rule. Outside is for cards that fit neither rule. Try another.

    See the answer

    In the overlap.

    8 is even. 8 is over 6. It fits both rules, so it goes where the circles cross.

    • If you chose A: You checked one rule. Is 8 over 6 too? Yes, so it fits both rules.
    • If you chose C: 8 is even, so it fits a rule. Outside is for cards that fit neither rule.

  2. Mei has the card 3. The rules are even and over 6. Where does 3 go?

    Tap your answer.

    1. Not this one. The overlap is for cards that fit both rules. 3 is odd and not over 6. Try another.

    2. Yes, that is right. 3 is odd, so it is not even. 3 is not over 6. It fits neither rule.

    3. Not this one. Every card has a place. Such a card goes outside both circles, inside the box. Try another.

    See the answer

    Outside both circles, inside the box.

    3 is odd, so it is not even. 3 is not over 6. It fits neither rule.

    • If you chose A: The overlap is for cards that fit both rules. 3 is odd and not over 6.
    • If you chose C: Every card has a place. Such a card goes outside both circles, inside the box.

  3. The rules are odd and over 8. The numbers are 5, 8, 9, 12 and 15. Which numbers go in the overlap?

    See the answer

    9 and 15

    5 is odd, but not over 8. 8 and 12 are even. 9 and 15 are odd and over 8.

  4. Sort 2, 3, 4, 6, 7, 8 and 9 into this table. Which numbers go in the shaded box?

    A 2 by 2 table with empty boxes. The corner says Numbers. The columns are even and odd. The rows are under 5 and 5 or more. The box for even and 5 or more is shaded.Numbersevenoddunder 55 or more
    See the answer

    6 and 8

    The shaded box is for even numbers that are 5 or more. 6 and 8 are even and 5 or more.

  5. Joseph sorts the cards 1 to 12. The rules are even and over 8. How many cards go outside both circles?

    See the answer

    4

    A card goes outside if it is odd and 8 or less. Those cards are 1, 3, 5 and 7.

  6. From an earlier chapter

    Amina picks mangoes and limes. Use the table. How many mangoes did she pick in all three days?

    A table with the heads fruit, Mon, Tue and Wed. The mango row reads 4, 6 and 5. The lime row reads 3, 2 and 7.fruitMonTueWedmango465lime327
    See the answer

    15

    Read along the mango row. 4 + 6 + 5 = 15.

  7. Look at the diagram. How many numbers are inside the under 6 circle?

    Two circles that cross inside a box. The left circle is even and holds 6 and 8. The right circle is under 6 and holds 1, 3, 5. The overlap holds 2 and 4. The numbers 7 and 9 are outside both circles.evenunder 6682413579
    Tap your answer.

    1. Not this one. You left out the overlap. 2 and 4 are inside the under 6 circle too. Try another.

    2. Not this one. You counted the overlap twice. It is already inside the circle. Try another.

    3. Yes, that is right. The under 6 circle holds 1, 3 and 5. The overlap holds 2 and 4. 3 + 2 = 5.

    See the answer

    5

    The under 6 circle holds 1, 3 and 5. The overlap holds 2 and 4. 3 + 2 = 5.

    • If you chose A: You left out the overlap. 2 and 4 are inside the under 6 circle too.
    • If you chose B: You counted the overlap twice. It is already inside the circle.

  8. Where does the number 5 go in this table?

    A 2 by 2 table with empty boxes. The corner says Numbers. The columns are even and odd. The rows are under 5 and 5 or more.Numbersevenoddunder 55 or more
    Tap your answer.

    1. Yes, that is right. 5 is odd, so use the odd column. 5 is not under 5, so use the 5 or more row.

    2. Not this one. Under 5 means less than 5. The number 5 is not less than 5. Try another.

    3. Not this one. An even number shares into pairs with none left. 5 leaves one over, so it is odd. Try another.

    See the answer

    In the box for odd and 5 or more.

    5 is odd, so use the odd column. 5 is not under 5, so use the 5 or more row.

    • If you chose B: Under 5 means less than 5. The number 5 is not less than 5.
    • If you chose C: An even number shares into pairs with none left. 5 leaves one over, so it is odd.

  9. The rules are even and under 7. The cards are 1 to 10. One card is in the wrong place. Which one?

    Two circles that cross inside a box. The left circle is even and holds 6, 8, 10. The right circle is under 7 and holds 1, 3, 5. The overlap holds 2 and 4. The numbers 7 and 9 are outside both circles.evenunder 768102413579
    Tap your answer.

    1. Not this one. Under 7 means 6 and less, so 7 is not under 7. It is odd too, so it is outside. Try another.

    2. Yes, that is right. 6 is even and 6 is under 7. It fits both rules, so it belongs in the overlap.

    3. Not this one. 8 is even, but it is not under 7. So the even circle only is the right place. Try another.

    See the answer

    6

    6 is even and 6 is under 7. It fits both rules, so it belongs in the overlap.

    • If you chose A: Under 7 means 6 and less, so 7 is not under 7. It is odd too, so it is outside.
    • If you chose C: 8 is even, but it is not under 7. So the even circle only is the right place.

Keep it

Copy this into your exercise book.

A Venn diagram sorts by two rules. A number that fits both goes in the overlap. A number that fits neither goes outside both circles, inside the box. A 2 by 2 table sorts the same way.

Draw: Two crossing circles in a box. Write even and over 6 on them. Put the numbers 1 to 10 in.

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 (828 KB). It opens on any phone.