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L1-04 · Count within a condition

When a question says "of the squares" or "among the kids with glasses", it picks out a smaller group. That group becomes the whole. The bottom number of your fraction is the size of that group, not the size of everything.

Level 1 icon Level 1 · Grades 4–6 · about 40 minutes
Big idea. When a question says “of the squares” or “among the kids with glasses”, it picks out a smaller group. That group becomes the whole. The bottom number of your fraction is the size of that group, not the size of everything.
Before this lesson. L1-01, L1-02
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: Which group are we counting?

Maya and Leo cut out 24 paper shapes for an art project. Some are squares and some are circles. Some are red and some are blue. They spread them all out on the table.

Maya counts the red squares. There are 6. “So 6 out of 24 shapes are red squares,” she says.

Then Leo asks a new question: “Of the squares, what fraction is red?” Now only the squares matter. Maya pushes the circles to one side. There are 8 squares, and 6 of them are red. The answer is 6 out of 8. The top number stayed 6, but the bottom number changed from 24 to 8. So if Leo picks one square without looking, the probability that it is red is 6 out of 8.

“Now try this,” says Maya. “Of the red shapes, what fraction are squares?” This time they keep only the red shapes. There are 10 red shapes, and 6 of them are squares. That is 6 out of 10.

Leo is surprised. “Both questions are about red squares, but the answers are different!” Maya nods. “The question tells you which group is the whole. Different group, different bottom number.”

Keeping only one group changes the bottom number of the fraction.

What the animation shows. A grid shows 24 shapes. The 6 red squares light up: 6 out of 24. Then every shape that is not a square fades away, and the fraction becomes 6 out of 8. Next, only the red shapes stay, and the fraction becomes 6 out of 10, before a two-way table shows all the counts.

Hands-on: Shape sort and two-way table

About 20 minutes.

You need:

Steps:

  1. Make 20 small paper cards. On each card, draw a square or a circle in one of two colors. You choose how many of each.

  2. Mix up the cards on the table. Count how many there are of each kind, like red squares and blue circles.

  3. Draw a two-way table. Put the colors in the rows and the shapes in the columns. Add a total row and a total column.

  4. Write your counts in the table. Check that the totals add up to 20.

  5. Answer: of the squares, what fraction is red? Use the square total as the bottom number.

  6. Answer: of the red shapes, what fraction are squares? Use the red total as the bottom number.

  7. Compare your two answers. Circle the bottom number of each fraction in your table.

Practice together

In a class of 20 kids, 8 wear glasses. Of the kids with glasses, 3 have a pet dog. What fraction of the kids with glasses have a dog?

Check your answer

The question says “of the kids with glasses”, so the whole is the 8 kids with glasses. 3 of them have a dog. The answer is 3 out of 8, not 3 out of 20.

Try it yourself

1. A box has 5 red squares, 5 blue squares, 2 red circles, and 8 blue circles. Of the circles, what fraction is red?

Check your answer

There are 2 + 8 = 10 circles, and 2 of them are red. So the answer is 2 out of 10, or one fifth.

2. Use the same box. Of the red shapes, what fraction are circles?

Check your answer

There are 5 + 2 = 7 red shapes, and 2 of them are circles. The answer is 2 out of 7. It is not the same as 2 out of 10, because the group is different.

Talk about it

Check yourself

1. A bag has 6 red squares, 2 blue squares, 4 red circles, and 12 blue circles. Of the squares, what fraction is red?

Show the answer

C. 6 out of 8

The question is about the squares only. There are 8 squares, and 6 of them are red, so the answer is 6 out of 8.

2. In a club, 12 kids play soccer. Of the soccer players, 4 also play piano. Which total goes on the bottom of the fraction?

Show the answer

B. The 12 kids who play soccer

The words “of the soccer players” make the soccer players the whole. So the bottom number is 12, and the answer is 4 out of 12.

3. Remember Lesson 2. A fair spinner has 4 equal parts, and 1 part is yellow. What is the probability of not landing on yellow? (Review of L1-02)

Show the answer

B. 3 out of 4

The probability of yellow is 1 out of 4. So not yellow is 1 take away one fourth, which is 3 out of 4.

Words to know

condition
A rule that picks out a smaller group, like “only the squares”.
two-way table
A table that sorts things two ways at once, like by color in the rows and by shape in the columns.
probability
A number from 0 to 1 that says how likely something is, according to a model.
For grown-ups: about this lesson

Why it matters. This lesson is an informal start on conditional probability. OpenStax Introductory Statistics defines the probability of A given B as the probability of A and B divided by the probability of B, which with counts means using only the cases in B as the whole Barbara Illowsky and Susan Dean, 2023. Children practice reading the condition from the question and choosing the matching total from a two-way table. Comparing “red given square” with “square given red” shows early that the two directions are not the same.

Watch for these mix-ups.

  • The bottom number is always the total of everything. (When a question says “of the ...”, the bottom number is the size of that group.)

  • “Of the squares, how many are red?” and “of the red shapes, how many are squares?” have the same answer. (They share a top number but use different groups as the whole.)

  • A difference between groups shows that one thing causes the other. (A two-way table shows a pattern in the counts, not a cause; Lesson 6 looks at common causes.)

To make it easier. Use 12 shapes and only one question at a time. Physically move the shapes that are not in the group off the table before counting.

To make it harder. Ask the child to make a set of shapes where “of the squares, what fraction is red?” equals “of the red shapes, what fraction are squares?”. (This happens when there are as many squares as red shapes.)

Limits. The fractions here are counts within one set of shapes. They describe that set only. The lesson does not yet use the words “given that” or the formal notation for conditional probability.

On the computer. Grown-ups can see where this idea leads later in Lab: Bayes’ rule, written for older learners.

Sources for this guide. Barbara Illowsky and Susan Dean, 2023

Next: L1-05 · Assign a fair comparison

References
  1. Barbara Illowsky and Susan Dean. (2023). 3.1 Terminology. OpenStax, Introductory Statistics 2e. https://openstax.org/books/introductory-statistics-2e/pages/3-1-terminology