D. Data · Data and Probability

Sorting by Two and Three Attributes: Venn, Carroll and Tree Diagrams

D1.1 A1About 40 minutes

Learning goals

I can sort things or people by two attributes in a Carroll diagram and a Venn diagram.
I can sort by three attributes using a tree diagram.
I can sort by three attributes using a Venn diagram with three circles.
I can read a sorting diagram to answer questions.

Words to know

attribute
A feature you can sort by, like size, pattern, shape or how someone gets to school.

Carroll diagram
A sorting table with rows and columns. Each box shows things that match one row label and one column label.

Venn diagram
A sorting diagram with circles. Things inside a circle have that attribute. Things where circles overlap have both.

overlap
The part where two or more circles cross. Things there fit every circle they are in.

tree diagram
A diagram that sorts one attribute at a time. Each branch splits into smaller branches, like a tree.

data
Facts or information we collect, like answers to a question or things we count.

Remember from Grade 2

In Grade 2 you sorted things by two attributes at once, using tables, Carroll diagrams and Venn diagrams. In Grade 3 we review those, and then sort by three attributes using a new tool called a tree diagram and a Venn diagram with three circles.

The lost-and-found bin

It is a snowy January at school. The lost-and-found bin is full of hats and mittens! The office helper wants to sort them so children can find their things fast.

Each item has three attributes we can sort by:

Kind: hat or mitten
Pattern: striped or plain
Size: small or large

We gave each item a letter so we can keep track.

A blue lost-and-found bin at school overflowing with knitted hats and mittens in many colours. — Look at all the attributes in this bin: hats or mittens, striped or plain, big or small. Sorting by attributes helps everyone find their own. AI-generated image

Item	Kind	Pattern	Size
A	hat	striped	small
B	hat	plain	large
C	mitten	striped	small
D	mitten	plain	small
E	hat	striped	large
F	mitten	striped	large
G	hat	plain	small
H	mitten	plain	small
I	hat	striped	small
J	mitten	striped	small

The lost-and-found data: 10 items, 3 attributes each.

Two attributes: a Carroll diagram

Let's sort by kind and pattern. A Carroll diagram has one attribute down the side (rows) and one across the top (columns). Each item goes in the one box that matches both of its labels.

	Striped	Plain
Hat	A, E, I	B, G
Mitten	C, F, J	D, H

Carroll diagram: lost-and-found items sorted by kind and pattern. 3 + 2 + 3 + 2 = 10 items.

Two attributes: a Venn diagram

Now let's use a Venn diagram with the circles Hats and Striped.

Striped hats have both attributes, so they go in the overlap: A, E, I.
Plain hats go in the Hats circle only: B, G.
Striped mittens go in the Striped circle only: C, F, J.
Plain mittens are not hats and not striped, so they go outside both circles, inside the box: D, H.

Venn diagram with the circles Hats and Striped. The overlap holds the striped hats. D and H are outside both circles.

Check your sorting

Every item must go somewhere, and only in one place. Count at the end: 2 + 3 + 3 + 2 = 10. That matches the 10 items in the bin, so nothing is missing.

Three attributes: a tree diagram

A tree diagram sorts one attribute at a time.

Start with all 10 items.
Split them by kind: hat or mitten.
Split each of those by pattern: striped or plain.
Split each of those by size: small or large.

Each split makes 2 branches. 2 branches, then 4, then 8 end branches. Follow a path from left to right to read a group.

Tree diagram: sort by kind, then pattern, then size. The path hat, striped, small leads to items A and I.

Worked example

Example 1: Follow a path

Use the tree diagram. Which items are mittens that are striped and small?

Start at 10 items. Follow the branch to mitten.
From mitten, follow the branch to striped.
From striped, follow the branch to small.
The end branch says C, J.

Items C and J are small striped mittens. Check with the table: C and J are both mitten, striped, small.

Worked example

Example 2: An empty branch

How many items are large plain mittens?

Follow mitten, then plain, then large.
The end branch says none.
Check the table: the plain mittens are D and H, and both are small.

0 items. A branch can be empty. That is still useful data: it tells us no large plain mittens were lost.

Three attributes: a Venn diagram with three circles

We can also use three circles: Hats, Striped and Small. The circles overlap in many places.

The very middle, where all three circles overlap, holds items that are hats and striped and small.
A part where only two circles overlap holds items with just those two attributes.
A part of one circle that does not overlap holds items with only that attribute.

Three-circle Venn diagram: Hats, Striped and Small. A and I are in the middle because they have all three attributes. Nothing is outside all three circles.

Worked example

Example 3: Where does item G go?

Item G is a plain small hat. Where does it go in the three-circle Venn diagram?

Is it a hat? Yes, so it is inside the Hats circle.
Is it striped? No, so it is not inside the Striped circle.
Is it small? Yes, so it is inside the Small circle.
So G goes where Hats and Small overlap, but outside Striped.

G goes in the part where Hats and Small overlap, outside the Striped circle.

Math talk

When you place an item, ask one question for each attribute: "Is it a hat? Is it striped? Is it small?" Saying your yes and no answers out loud helps you and your listener check your thinking.

Just for fun

What do you call a Venn diagram that sorts snacks?

A Venn-ding machine!

Hands-on activity

Sort your sock drawer (or your shoes!)

You need: 8 to 12 socks, shoes or mittens from home, 3 pieces of string or yarn, and paper.

Choose three attributes, for example: has a pattern or not, long or short, belongs to a grown-up or a child.
Tree diagram: on a big sheet of paper, draw a tree that splits by one attribute at a time. Put each item at the end of its branch.
Venn diagram: make three overlapping loops of string on the floor. Label them with your attributes. Place each item in the right part.
Count: does every item have exactly one spot? Is any part empty?

Math talk: "Which diagram made it easier to see the groups? Why?"

Your turn

Use the lost-and-found table. How many items are striped hats? Name them.

In the Venn diagram with the circles Hats and Striped, which items go outside both circles?

How many items are small?

In the tree diagram, follow hat, plain, large. What do you find?

Where does item F (a large striped mitten) go in the three-circle Venn diagram?

Why does a tree diagram that sorts by three attributes with two choices each have 8 end branches?

More practice: warm-up

Name two attributes you could use to sort mittens.

In a Venn diagram, what does it mean when something is in the overlap of two circles?

A Carroll diagram has rows Hat and Mitten, and columns Striped and Plain. How many boxes does it have?

A tree diagram splits the items into hat or mitten first. How many branches come out of the start?

In the Venn diagram with circles Hats and Striped, where does a plain large hat go?

More practice: on your own

Here is data about 8 children.

Ava: walks, has a pet, wears glasses
Ben: bus, has a pet, no glasses
Chloe: walks, no pet, no glasses
Dev: walks, has a pet, no glasses
Eli: bus, no pet, wears glasses
Fatima: bus, has a pet, wears glasses
Gus: walks, no pet, wears glasses
Hana: walks, has a pet, no glasses

Make a Carroll diagram with rows Walks and Bus, and columns Pet and No pet.

Use the same 8 children. Make a Venn diagram with the circles Walks and Wears glasses. Who goes in the overlap? Who goes outside both circles?

A tree diagram sorts the 8 children by travel, then pet, then glasses. Who is at the end of the branch bus, pet, glasses?

Make a three-circle Venn diagram with Walks, Pet and Glasses. Who goes in the middle of all three circles? Who goes in the Pet circle only?

Sort these numbers into a Venn diagram with the circles Even and Greater than 10: 4, 12, 15, 8, 20, 7.

Challenge

Look at the three-circle Venn diagram of the lost-and-found bin. Which part is empty? What kind of item would go there?

Ms. Rao says, "Everyone in the 8-children data who takes the bus has a pet." Is she right? Use a diagram to explain.

A snack table sorts 20 snacks with a tree diagram: sweet or salty, then crunchy or soft, then hot or cold. How many end branches are there? Could every end branch have exactly 3 snacks?

Check your understanding

Quiz

Venn diagram with the circles Hats and Striped: where does a striped hat go?

A tree diagram sorts by 3 attributes, each with 2 choices. How many end branches does it have?

Venn diagram with the circles Hats and Striped: where does a plain large mitten go?

Which diagram sorts one attribute at a time, with branches that split?

Use the lost-and-found Carroll diagram. Which items are striped mittens?

In a three-circle Venn diagram (Hats, Striped, Small), what goes in the middle where all three circles overlap?

For parents and teachers

What your child is learning: to sort data about people or things by two and three attributes using Carroll diagrams, Venn diagrams (including three circles) and tree diagrams (Ontario expectation D1.1). Carroll and two-circle Venn diagrams were taught in Grade 2; the tree diagram and three attributes are the Grade 3 step.

Common difficulties:

Checking only one attribute. Teach the habit of asking a yes/no question for each attribute before placing an item.
Forgetting the space outside the circles in a Venn diagram. Items with none of the attributes still belong in the diagram.
Losing their place in a tree diagram. Have your child trace the path with a finger from left to right.
Thinking an empty group is a mistake. An empty branch or region is real information.

At home: sort laundry, shoes, cutlery or toys by three attributes. Ask, "Which group has the most? Is any group empty? How do you know you didn't miss anything?" (Counting all the groups should match the starting total.)