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Rational Number Project

Fraction Operations and
Initial Decimal Ideas
Lesson 22: Overview

• Patty paper
• Two color pencils or
• Student sheets A and B
• Transparencies 1 and 2
Students use patty paper (an area model) to multiply
fractions. Students develop the algorithm for multiplying
fractions by noticing patterns related to the patty paper
Teaching Actions

Warm Up

Determine the fraction of the square that is
colored gray.

Show A and B together then discuss. Show B and
C together and discuss


Fraction: Fraction:
Fraction: Fraction:

The students should be able to do
the warm up problems mentally.
Please give them a minute to think
of the fractional amounts and have
them write the fractions on a piece
of scratch paper.

The purpose of squares A and B is
to focus the students’ attention on
getting equal sized pieces. Students
may suggest that you draw in lines
on the transparency to show equal
sized pieces when you discuss the

The emphasis for Square C is on
multiplication. Try to get students
to explain how they calculated the
fraction colored gray. Some
students may notice that the total
number of equal sized pieces
(denominator) can be found by
multiplying 7 by 4 and the number
of colored in pieces can be found
by multiplying 4 by 2 (numerator).

The emphasis on Square D is a
combination of strategies for the
previous squares. Multiplication
can be used to count both the
number of shaded rectangles as
well as the number of total
rectangles when determining the
fraction of the square that is
shaded. This is an important part
of explaining why the algorithm
for multiplication of fractions


Teaching Actions

Large Group Introduction

1. Explain to the students that we are going to develop a
rule for multiplying fractions today. To begin this
exploration, ask them to look at the picture on
transparency 1. Explain that the grayed in part
represents the amount of cake Kathy has left from

Kathy’s Cake

2. Ask the students to think about how much cake Kathy
has left. Record the fractional amount below the cake.

3. Explain that Kathy wants to eat two-fifths of the
remaining cake. Ask the students to think how Kathy
could find two-fifths of the remaining cake.

4. Have a volunteer come up and shade in two-fifths of
the three-sevenths of a cake with a dark marker. It
may look something like what is shown below.

There is four-sevenths of a cake

The student may or may not draw a
picture as shown to the left. The
sevenths and the fifths were chosen
for this example so the students will
draw horizontal and vertical lines
to show the total number of pieces.
It is easier to explain why the
multiplication algorithm works if
students draw one fraction
vertically and the fraction of that
fraction horizontally.


Teaching Actions

5. Ask the students to explain what fraction of the whole
cake will Kathy eat ( ). Ask a few students how they
determined this fraction. [Students most likely will
draw in the horizontal lines across the whole square
to explain why the total number of parts is 35].

6. Ask: What multiplication sentence matches the
actions we made on the picture of the patty paper?

Small Group/Partner Work

7. Assign Student Pages A & B. You may want to go
over the first question with the students so they know
how to complete the class work.

Wrap Up

8. Review select problems from the Student Pages. To
help students construct the rule for multiplying
fractions, guide the discussion with these questions:

• How did you show of ?

• How did the picture change when you
did that?

• If you extend the horizontal lines
across the square, how many total
parts is the square partitioned into?

• Where is a 3 by 5 rectangle in your

Teaching Actions

o Where is a 2 by 1 rectangle in your

9. Ask the students to picture a piece of patty paper with
colored. Show transparency 2.


10. Write a 3 in the denominator of the first fraction in the
multiplication sentence.

Ask the students to explain why multiplying 9 by 3
will give you the denominator of the product.

Ask: What number will we write in the denominator
of the result fraction? (27) Where is the 3 by 9
rectangle in the picture?

Ask: What does the 27 represent” (The number of
equal-sized pieces that the unit is partitioned.) You
can write three horizontal lines on the transparency to
show the 27 equal sized pieces.

11. Write a 2 in the numerator of the multiplication

Ask: What number should we write in the numerator
of the product? (10)

Students have trouble explaining
why the multiplication algorithm
works. The goal of the student
pages and the wrap up is to help
them verbalize why the algorithm
works using the patty paper

Students should be able to explain
that each of the 9ths will be cut
into 3rds. Multiplying 9 by 3
counts the number of total pieces
the unit is cut into. One-third of
one-ninth is one- twenty-seventh.


Teaching Actions

Ask: What does the 10 represent? (The 10 pieces
represent the number of pieces that are darkly
shaded.) You may want to show two-thirds of five-ninths
on the transparency to show how
multiplication helps you show the product. Where is
the 2 by 5 rectangle in the picture?

12. Explain that the “algorithm” for multiplying fractions
is to multiply the numerators and the dominators.
The product of the numerators will give you the
number of pieces in the answer and the product of
the denominators will give you the number of pieces
in the unit.

Write the following number sentence with letters:

multiplication algorithm

• Picture to symbols
• Symbols to picture to symbols
• Pictures to symbols to verbal