options: Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
problem_activity: Narrative
The purpose of this activity is for students to write expressions for and find the area of shaded regions whose side lengths are a unit fraction and a non-unit fraction. Students look at a series of diagrams with an increasingly large shaded region so they can look for and make use of structure (MP7).
It is important to relate the work here to what students have already learned about the product of a unit fraction and a unit fraction and this is the goal of the synthesis. For example, students have seen that
\(\frac{1}{5} \times \frac{1}{3} = \frac{1}{15}\)
. Since
\(\frac{6}{5}\)
is 6
\(\frac{1}{5}\)
s, the product
\(\frac{6}{5} \times \frac{1}{3}\)
will be 6
\(\frac{1}{15}\)
s or
\(\frac{6}{15}\)
.
MLR8 Discussion Supports.
Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Advances: Listening, Conversing
Launch
Groups of 2
Display image A from the student workbook.
“What is a multiplication expression that represents the shaded region?”(
\(\frac{2}{5} \times \frac{1}{3}\)
,
\(2 \times \frac{1}{15}\)
)
“How does the diagram represent your expression?”(There is
\(\frac{2}{5}\)
of the first row shaded and that row is
\(\frac{1}{3}\)
of the square. There are 2 shaded pieces and each is
\(\frac{1}{15}\)
of the whole square.)
Activity
3–5 minutes: independent work time
3–5 minutes: partner discussion
Monitor for students who:
notice that the product of the numerators represents how many pieces of the square are shaded.
notice that the product of the denominators represents the number of pieces in the whole square.
Student Facing
Write a multiplication expression that represents the shaded region in each diagram.
A
[IMG]
B
[IMG]
C
[IMG]
What patterns do you notice in the multiplication expressions?
Han wrote this equation to represent the area of the shaded region. Explain how the diagram represents the equation.
\(\phantom{2.5cm} \\ \frac {6}{5} \times \frac {1}{3} = \frac {6}{15}\)
[IMG]
Student Response
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Advancing Student Thinking
If students write mathematically correct multiplication expressions that do not represent the product of a unit fraction and a non-unit fraction, write this type of multiplication expression that represents the diagram and ask “How does this expression represent the area of the shaded region in the diagram?”
Activity Synthesis
Ask previously selected students to share the patterns they noticed in the table.
Display the expression from the last problem:
\(\frac {6}{5} \times \frac {1}{3} = \frac {6}{15}\)
.
“How does the diagram show
\(\frac {1}{3}\)
?” (The first row of pieces in a square is
\(\frac{1}{3}\)
of the square.)
“How does the diagram show
\(\frac {6}{5}\)
?” (There are 6 pieces shaded and each one is
\(\frac{1}{5}\)
of the row.)
“How does the diagram show
\(\frac {6}{5} \times \frac {1}{3}\)
?” (There is
\(\frac{6}{5}\)
of a row shaded and that row is
\(\frac{1}{3}\)
of a square unit.)


correct_option_index: 6