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Rational Number Project

Additional Notes to the Teacher
Lesson 22

The goal for this lesson is for students to develop the rule or algorithm for multiplying two

The students will use pictures of patty paper to partition fractions of fractions . We found that
students seem to be able to see multiplication ideas better if they draw one fraction using
vertical lines and then finding the fraction of this fraction using horizontal lines. In the Kathy’s
cake example, the students are told that Kathy has three-sevenths of a cake left over as shown in
the picture below.

Kathy’s Cake

Kathy is then going to eat two -fifths of this amount of cake. Although you could draw in
lines to find two-fifths of the three-sevenths it would be difficult to see. Please model
and encourage your students to draw in horizontal lines when they partition the three-sevenths
into fifths. To explain the amount of the darkly shaded part of the cake, students will need to
extend the horizontal lines across the whole patty paper. If students folded the patty paper first
into sevenths and then into fifths, the paper would have creases showing the horizontal

two-fifths of three-sevenths darkly shaded

By the end of the lesson you want students to be able to explain that the product of the
show the size of the pieces. When you take fifths of sevenths you will get pieces
that are thirty-fifths of the unit. The product of the numerators is more difficult to explain. The
students should be able to explain that the total number of darkly shaded pieces can be counted
by finding the product of the numerators.

By the end of this lesson students should be able to reason that the product of the numerators
determine the number of pieces that you are interested in and the product of the denominators
is the number of pieces in which the unit is partitioned.

Kathy’s Cake

Determine the fraction of the square that is colored

Fraction: Fraction:
Fraction: Fraction:

Multiplying Fractions

1. Fill in the table below.

Picture # of darkly shaded pieces
# of total pieces in square
Fraction of the square
shaded dark

2. Look for patterns in each row. Explain the patterns you see.

3. Use the pattern you describe in question 2 to the answers to the problems below.

4. Rewrite the pattern you used as a rule for multiplying fractions. Explain why your rule

Post Lesson Reflection


1) Number of class periods allocated to this lesson: ______________

2) Student Pages used: __________________

3) Adaptations made to lesson: (For example: added extra examples, eliminated
certain problems, changed fractions used)

4) Adaptations made on Student Pages:

5) To improve the lesson I suggest:

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