Lesson 3 Patterns in Multiplication

- Let’s look at patterns in multiplication of a fraction by a whole number.

Warm-up Choral Count: $\frac{1}{4}$ and $\frac{1}{8}$

Count by $\frac{1}{4}$ , starting at 0
Count by $\frac{1}{8}$ , starting at 0

Activity 1 Describe the Pattern

Here are two tables with expressions. Find the value of each expression. Use a diagram if you find it helpful.
Leave the last two rows of each table blank for now.
Set A
expression
value
$1 \times \frac{1}{8}$
$2 \times \frac{1}{8}$
$3 \times \frac{1}{8}$
$4 \times \frac{1}{8}$
$5 \times \frac{1}{8}$
$6 \times \frac{1}{8}$
Set B
expression
value
$2 \times \frac{1}{3}$
$2 \times \frac{1}{4}$
$2 \times \frac{1}{5}$
$2 \times \frac{1}{6}$
$2 \times \frac{1}{7}$
$2 \times \frac{1}{8}$
Study your completed tables. What patterns do you see in how the expressions and values are related?
In the last two rows of the table of Set A, write $\frac{11}{8}$ and $\frac{13}{8}$ in the “value” column. Write the expressions with that value.
In the last two rows of the table of Set B, write $\frac{2}{12}$ and $\frac{2}{15}$ in the “value” column. Write the expressions with that value.

Activity 2 What’s Missing?

Problem 1

Use the patterns you observed earlier to complete each equation so that it’s true.

$5 \times \frac{1}{10} =$
$8 \times \frac{1}{6} =$
$4 \times$ $= \frac{4}{5}$
$6 \times$ $= \frac{6}{10}$
$\times \frac{1}{4} = \frac{3}{4}$
$\times \frac{1}{12} = \frac{7}{12}$

Problem 2

Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right.

Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12.
Step 2: Write the fraction you received as a product of a whole number and a unit fraction.
Step 3: Draw a diagram to represent the expression you just received.
Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.

Practice Problem

Problem 1

Kiran’s cat eats $\frac{1}{2}$ cup of food each day.

How much food does Kiran’s cat eat in a week?
Draw a diagram to represent the situation.

expression	value
$1 \times \frac{1}{8}$
$2 \times \frac{1}{8}$
$3 \times \frac{1}{8}$
$4 \times \frac{1}{8}$
$5 \times \frac{1}{8}$
$6 \times \frac{1}{8}$

expression	value
$2 \times \frac{1}{3}$
$2 \times \frac{1}{4}$
$2 \times \frac{1}{5}$
$2 \times \frac{1}{6}$
$2 \times \frac{1}{7}$
$2 \times \frac{1}{8}$

Warm-up Choral Count: 14 and 18

Activity 1 Describe the Pattern

Activity 2 What’s Missing?

Problem 1

Problem 2

Practice Problem

Problem 1

Warm-up Choral Count: $\frac{1}{4}$ and $\frac{1}{8}$