options: 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.

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.

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.

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.

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.

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.
problem_activity: Optional activity
This activity gives students additional practice in using diagrams and equations to represent division situations involving whole numbers and fractions.
For each problem, many kinds of visual representations are possible, but creating a meaningful representation may be challenging nonetheless. Urge students to use the contexts to generate ideas for useful diagrams, and to start with a draft and modify it as needed. Students may also use the fraction strips in the warm-up as a starting point for drawing diagrams.
As students work and discuss, monitor for effective diagrams or those that can be generalized to different situations (e.g., rectangles, tape diagrams, and number lines). Assign one problem for each group to record on a visual display.
Launch
Arrange students in group of 3–4. Give students 8–10 minutes of quiet work time and a few minutes to share their responses with their group. Tell each group they will be asked to present their solution to one problem. Provide access to geometry toolkits and tools for creating a visual display.
During group discussion, ask students to exchange feedback on each other’s diagrams and to notice any that might be particularly effective, efficient, or easy to understand. Then, they should record the diagram and equations for their assigned problem on a visual display and be prepared to explain them.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to illustrate connections between representations. For example, use the same color to represent the 3 miles in the diagram and the equation,
\(3\div \frac32=2\)
, then label each as “dividend.”
Supports accessibility for: Visual-spatial processing
Student Facing
For each situation, draw a diagram for the relationship of the quantities to help you answer the question. Then write a multiplication equation or a division equation for the relationship. Be prepared to share your reasoning.
The distance around a park is
\(\frac32\)
miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride?
You need
\(\frac34\)
yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?
The water hose fills a bucket at
\(\frac13\)
gallon per minute. How many minutes does it take to fill a 2-gallon bucket?
Student Response
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Student Facing
Are you ready for more?
How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Students may mistake the divisor and the dividend in the problems. Ask students to discuss (in their groups) the number or quantity being divided, and the reasonableness of the different ways of setting up each problem given the context. Representing the situations with objects may also help.
Activity Synthesis
Invite each group to present the solution on their visual display. Ask the rest of the class to think about two things: whether the equations make sense, and how the presented diagram shows the number of groups, the size of each group, and a total amount. Doing so will help students see the structure of the problems in the equations and diagrams.
Make sure students understand how each situation can be expressed by multiplication and division equations. Students should recognize that a question such as “how many batches are in 4 cups if each batch requires
\(\frac23\)
cups?” can be written as both
\(? \boldcdot \frac 23 = 4\)
and
\(4 \div \frac 23 = \,?\)
. With repeated reasoning, they see that a division expression such as
\(5 \div \frac 38\)
can be interpreted as “how many
\(\frac38\)
s are in 5?”
Speaking, Representing: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain their visual display and the relationship between the quantities represented. Prompt groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking. This will help students improve their explanations of their group’s reasoning during the whole-class discussion.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 1