Grade by Grade
D. Data · Data and Probability

Sorting with Venn Diagrams

D1.1About 35 minutes
Learning goals
  • I can sort things into a Venn diagram with two circles.
  • I can explain why some things go in the middle where the circles overlap.
  • I can explain why some things go outside both circles.
  • I can show the same sorting in a Venn diagram and a Carroll diagram.

Words to know

Venn diagram
A sorting picture with circles inside a box. Each circle has a label. Things that fit both labels go where the circles overlap.
overlap
The middle part where two circles cross. Things here fit BOTH labels.
outside
The space inside the box but not in any circle. Things here fit NEITHER label.
both
Fits the first label AND the second label.
neither
Fits NOT the first label and NOT the second label.

Let's get started!

Ms. Kaur's class asked: "Do you have a dog? Do you have a cat?"

Ava has a dog. Ben has a cat. Then Chloe said, "I have a dog and a cat!" And Dev said, "I have no pets."

Where should Chloe go? Where should Dev go? A Venn diagram can help!

Parts of a Venn diagram

A Venn diagram has a box with two circles inside. The circles overlap in the middle, like two hula hoops on the floor.

  • The left circle is for things that fit the first label.
  • The right circle is for things that fit the second label.
  • The middle (overlap) is for things that fit both labels.
  • The space outside the circles (but inside the box) is for things that fit neither label.
Has a dogHas a catAvaEliChloeGusBenFatimaDevHana
A Venn diagram of pets. Ava and Eli have only a dog. Ben and Fatima have only a cat. Chloe and Gus have a dog and a cat. Dev and Hana have neither: Dev has no pets and Hana has only a fish.
Worked example

Reading the pet Venn diagram

Use the Venn diagram above. How many children have a dog?

  1. Find the Has a dog circle. That is the whole left circle, including the middle.
  2. In the left part only: Ava and Eli. That is 2.
  3. In the middle: Chloe and Gus. They have a dog too! That is 2 more.
  4. Add: 2 + 2 = 4.

4 children have a dog: Ava, Eli, Chloe and Gus.

Don't forget the middle!

The middle belongs to both circles. When you count everyone in a circle, count the middle part too.

Let's ask more questions about the pet Venn diagram:

  • Who has a dog and a cat? Chloe and Gus (the middle).
  • Who has a cat but not a dog? Ben and Fatima.
  • Who has neither a dog nor a cat? Dev and Hana (outside). Hana has a fish, but a fish is not a dog or a cat!
  • How many children were asked? 2 + 2 + 2 + 2 = 8.
Just for fun

Silly Venn diagram: label one circle Yellow and the other Things you can eat. What goes in the middle? A banana! 🍌 And a school bus? Yellow circle only, please. Don't eat the bus! 🚌

Worked example

Sorting shapes in a Venn diagram

Sort these shapes. The labels are Red and Has 4 sides.

red square, red triangle, blue square, red circle, yellow triangle, green rectangle

  1. Take one shape at a time. Ask: Is it red? Does it have 4 sides?
  2. Red square: red? Yes. 4 sides? Yes. Both, so it goes in the middle.
  3. Red triangle: red? Yes. 4 sides? No, 3 sides. It goes in the Red circle only.
  4. Red circle: red, no sides. Red circle only.
  5. Blue square and green rectangle: not red, but 4 sides. 4 sides circle only.
  6. Yellow triangle: not red, 3 sides. Neither! It goes outside the circles.
  7. Check: 2 + 1 + 2 + 1 = 6 shapes.

Red only: red triangle, red circle. Both: red square. 4 sides only: blue square, green rectangle. Outside: yellow triangle.

Red4 sidesred △red ○red □blue □green ▭yellow △
The shapes sorted in a Venn diagram. △ triangle, ○ circle, □ square, ▭ rectangle.

Venn diagram or Carroll diagram?

They show the same sorting in different ways! Here is the shape sorting as a Carroll diagram:

Has 4 sidesNot 4 sides
Redred squarered triangle, red circle
Not redblue square, green rectangleyellow triangle
The same shapes in a Carroll diagram.
Venn diagram partCarroll diagram box
Middle (both)Red and Has 4 sides
Left circle onlyRed and Not 4 sides
Right circle onlyNot red and Has 4 sides
OutsideNot red and Not 4 sides
Each part of a Venn diagram matches one box of a Carroll diagram.

When the middle is empty

Sometimes nothing fits both labels. Then the middle stays empty, and that's okay! For example, with the labels Has 3 sides and Has 4 sides, no shape can be in both circles.

Hands-on activity

Hula hoop Venn diagram

You need: two loops of string, yarn or skipping ropes (or two hula hoops), sticky notes, and about 10 small toys or kitchen things.

  1. Make two big circles on the floor that overlap in the middle.
  2. Write a label for each circle, for example Plastic and Blue.
  3. Sort each thing. Things that are plastic and blue go in the middle. Things that are neither go outside the circles.
  4. Count each part. How many things are in the Plastic circle altogether (remember the middle)?
  5. Change the labels and sort again: Used in the kitchen and Made of metal.

Math talk: "Why did you put this in the middle?"

Your turn

1

In a Venn diagram, what goes in the middle where the circles overlap?

2

In a Venn diagram, what goes outside the circles?

3

Look at the pet Venn diagram. How many children have a cat?

4

Labels: Red and Has 4 sides. Where does a red rectangle go?

5

Labels: Red and Has 4 sides. Where does a blue circle go?

6

Labels: Can swim and Has legs. Where does a fish go?

7

Same labels. Where does a frog go?

More practice: warm-up

1

How many circles does our Venn diagram have?

2

Labels: Fruit and Red. Where does a strawberry go?

3

Labels: Fruit and Red. Where does a banana go?

4

Labels: Fruit and Red. Where does a red fire truck go?

5

Labels: Fruit and Red. Where does a carrot go?

More practice: on your own

1

A Venn diagram has the labels Likes skating and Likes skiing. 3 kids are in skating only, 2 are in the middle, 4 are in skiing only, and 1 is outside. How many kids like skating?

2

Same Venn diagram. How many kids like skiing?

3

Same Venn diagram. How many kids were asked in all?

4

Same Venn diagram. How many kids like only one of the two?

5

Labels: Has 3 sides and Has 4 sides. Why is the middle empty?

Challenge

1

Draw (or describe) a Carroll diagram that shows the pet Venn diagram. Who goes in the Has a dog, Not a cat box?

2

Labels: Numbers greater than 10 and Numbers less than 20. Where do 5, 15, 25 go?

3

Leo says, "6 kids are in the Likes hockey circle and 5 kids are in the Likes soccer circle, so 11 kids were asked." Two kids are in the middle and nobody is outside. Is Leo right?

Check your understanding

Quiz

1.

In a Venn diagram, things that fit both labels go...

2.

Labels: Red and Fruit. Where does a red apple go?

3.

Labels: Red and Fruit. Where does a blue mitten go?

4.

A Venn diagram: Dog only 3, middle 1, Cat only 2, outside 4. How many have a dog?

5.

Same Venn diagram. How many were asked in all?

6.

Which part of a Carroll diagram matches the outside of a Venn diagram?

For parents and teachers

What your child is learning: to sort by two attributes using a Venn diagram (two overlapping circles) and to see that it shows the same information as a Carroll diagram (Ontario expectation D1.1).

Common difficulties:

  • Forgetting that the middle belongs to both circles. When asked "How many have a dog?", children often count only the left part. Trace the whole circle with a finger, middle included.
  • Leaving out items that fit neither label. Remind your child that the box around the circles is part of the diagram, and outside items still belong.
  • Adding the two circle totals to get the number of people. This counts the middle twice (see the Challenge question).

At home: use two loops of string or two belts on the floor. Sort toys, clothes or groceries with labels like Soft / Blue or Food / Comes in a can. Let your child choose the labels and explain each placement.