options: 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.

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.

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

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

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

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

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.

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.

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

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

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.
problem_activity: Narrative
The purpose of this activity is for students to use diagrams to reason about products of a whole number and a non-unit fraction with diagrams, building on their work with diagrams that represent products of a whole number an a unit fraction. They begin to generalize that the number of shaded parts in a diagram that represents
\(n \times \frac{a}{b}\)
is
\(n \times a\)
and to explain that generalization (MP8).
Launch
Groups of 2
“Let's represent some other products of a whole number and a fraction and find their values.”
Activity
“Take a few quiet minutes to work on the activity. Afterwards, share your responses with your partner.”
5–7 minutes: independent work
2–3 minutes: partner discussion
Monitor for the strategies students use to reason about the last two problems.
Identify students who reason visually (using diagrams), additively, and multiplicatively to share in the synthesis.
Student Facing
This diagram represents
\(\frac{2}{5}\)
.
[IMG]
Show how you would use or adjust the diagram to represent
\(4 \times \frac{2}{5}\)
.
What is the value of the shaded parts in your diagram?
This diagram represents
\(\frac{5}{8}\)
.
[IMG]
Show how you would use or adjust the diagram to represent
\(3 \times \frac{5}{8}\)
.
What is the value of the shaded parts in your diagram?
Find the value of each expression. Draw a diagram if you find it helpful. Be prepared to explain your reasoning.
\(2 \times \frac {1}{6}\)
\(2 \times \frac{4}{6}\)
\(2 \times \frac {5}{6}\)
\(4 \times \frac{5}{6}\)
Mai said that to multiply any fraction by a whole number, she would multiply the whole number and the numerator of the fraction and keep the same denominator. Do you agree with Mai? Explain your reasoning.
Student Response
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Activity Synthesis
Discuss the four multiplication expressions in the third problem.
Select 1–2 students who might have drawn diagrams for all expressions.
“How does your diagram show the value of
\(2 \times \frac{4}{6}\)
?” (There are 2 groups of
\(\frac{4}{6}\)
, so there are 8 sixths shaded, which is
\(\frac{8}{6}\)
.)
Select 1–2 students who drew a diagram for some expressions and reason numerically for others.
“Why did you choose to draw a diagram for some expressions and to do something else for others?” (After drawing the first two diagrams, I realized that I'd have to draw a lot of groups or parts, so I thought about the numbers instead.)
Select 1–2 students who reasoned about all expressions numerically.
“How did you find the value of the expressions without drawing diagrams at all?” (I saw a pattern in earlier problems, that we can multiply the whole number and the numerator of the fraction and keep the denominator.)
Discuss the last problem in the lesson synthesis.


correct_option_index: 4