options: Apply and extend previous understandings of numbers to the system of rational numbers.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform operations on vectors.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform arithmetic operations with complex numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Represent complex numbers and their operations on the complex plane.

Use properties of rational and irrational numbers.

Perform operations on matrices and use matrices in applications.

Extend the properties of exponents to rational exponents.

Compute fluently with multi-digit numbers and find common factors and multiples.

Reason quantitatively and use units to solve problems.
problem_activity: Activity
In this task, students explore division situations (in the context of baking cookies) where the number of groups and a total amount are given, but the size of 1 group is unknown. They write multiplication equations in which the missing factor answers the question “how much in each group?” instead of “how many groups?”
Students continue to use equations, diagrams, and the connection between multiplication and division in their reasoning. No grid is provided here, but allow students to use graph paper to support their diagram drawing if desired or needed.
As students work, monitor how they start their diagrams and which quantity they represent first. If they are not quite sure how to show a particular quantity, ask them to refer to earlier diagrams and notice how the number of groups, the size of each group, and the total amount were represented on a single diagram. Select students whose diagrams would be instructive to others to share later.
Launch
Arrange students in groups of 3–4. Give groups 5–6 minutes of quiet work time and 2 minutes to share their responses in their group. Provide access to geometry toolkits (especially graph paper and colored pencils).
Student Facing
To make 5 batches of cookies, 10 cups of flour are required. Consider the question: How many cups of flour does each batch require?
We can write equations and draw a diagram to represent this situation.
\(\displaystyle 5\, \boldcdot {?} = 10\)
\(\displaystyle 10 \div 5 = {?}\)
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This helps us see that each batch requires 2 cups of flour.
For each question, write a multiplication equation and a division equation, draw a diagram, and find the answer.
To make 4 batches of cupcakes, it takes 6 cups of flour. How many cups of flour are needed for 1 batch?
To make
\(\frac12\)
batch of rolls, it takes
\(\frac54\)
cups of flour. How many cups of flour are needed for 1 batch?
Two cups of flour make
\(\frac23\)
batch of bread. How many cups of flour make 1 batch?
Student Response
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Anticipated Misconceptions
When writing multiplication equations, some students might simply use the smaller number as a factor and the larger number as the product. Or, when writing division equations, they might simply divide the larger number by the smaller number without attending to what the numbers mean. Encourage them to think about what the numbers represent. To help them see the connection between verbal statements and mathematical equations, support their reasoning with simple examples, e.g., “To make 2 batches of cookies, we need 6 cups of flour. How many cups are needed for 1 batch?” This can be interpreted as “2 groups of what equals 6?” and then
\(2 \boldcdot {?} = 6\)
.
Activity Synthesis
Select previously identified students to share their diagrams and reasoning, or display the ones here for all to see. If other students used alternative approaches, invite a couple of them to share.
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To highlight some key ideas in this lesson and in the unit, discuss:
“How are these diagrams like those in previous lessons? How are they different?” (Alike: We can use them to show a total amount, the number of groups, the size of 1 group. Different: We don't know the value of 1 group here. In past diagrams, we were given the size of 1 group, but needed to find the number of groups.)
“How are these equations and the ones in previous lessons alike and different?” (Alike: The multiplication equations have one unknown factor. The division equations have the same setup. Different: The unknown factor here represents a different quantity than in past lessons.)
“How are the diagrams for the second and third questions different from the one in the first question?” (In the first one, the given amount is greater than the size of 1 batch. In the other two, the given amount is less than the size of 1 batch or is a fraction of a batch.)​
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 3