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

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

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.

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.

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.

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

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

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.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Narrative
The purpose of this activity is for students to draw two diagrams that represent a unit fraction of a unit fraction. Students work with the same unit fractions in both diagrams. The directions were intentionally written to encourage students to partition a unit square in different ways. Students initially partition the square into thirds in the first problem and into fourths in the second problem. Students may complete the diagrams in a way that makes sense to them. During the synthesis, students will connect both of the diagrams to the expressions
\(\frac {1}{4} \times \frac {1}{3}\)
and
\(\frac {1}{3} \times \frac {1}{4}\)
.
Launch
Groups of 2
Activity
2 minutes: Independent work time
10 minutes: partner work time
Monitor for students who:
draw diagrams like those in the student responses
recognize that both diagrams have the same amount shaded
can explain the different ways they partitioned each diagram
Student Facing
Show
\(\frac{1}{3}\)
of the square.
Shade
\(\frac{1}{4}\)
of
\(\frac{1}{3}\)
of the square.
How much of the whole square is shaded?
[IMG]
Show
\(\frac{1}{4}\)
of the square.
Shade
\(\frac{1}{3}\)
of
\(\frac{1}{4}\)
of the square.
How much of the whole square is shaded?
[IMG]
How are the diagrams the same and how are they different?
Student Response
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Advancing Student Thinking
If students do not identify how much of the whole square is shaded, ask: “How can you adapt your diagram to show how many pieces would be in the whole square?”
Activity Synthesis
Ask previously selected students to share their responses in the order given.
“How are the diagrams the same?” (They both have
\(\frac {1}{4}\)
and
\(\frac {1}{3}\)
in them. They both have
\(\frac {1}{12}\)
of the whole square shaded.)
“How are the diagrams different?” (In one diagram, I started by showing
\(\frac {1}{3}\)
of the square and then shaded in
\(\frac {1}{4}\)
of
\(\frac {1}{3}\)
. In the other diagram, I started by showing
\(\frac {1}{4}\)
of the square and then shaded in
\(\frac {1}{3}\)
of
\(\frac {1}{4}\)
.)
Display:
\(\frac{1}{12}\)
,
\(\frac{1}{4}\times\frac{1}{3}\)
,
\(\frac{1}{3}\times\frac{1}{4}\)
“How do the diagrams represent these expressions?”
Adapt diagrams as needed in order to show all three expressions.
“How do we know that both squares have the same amount shaded?” (They both have
\(\frac {1}{12}\)
shaded.)


correct_option_index: 1