few_shots: 
Example:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000674_pos_0

5
standard_description: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
problem_activity: Activity
This activity revisits a context seen previously. Students solved problems like this as early as grade 3 without formulating them in terms of ratios or rates (“If 1 cup of rice serves 3 people, how many people can you serve with 12 cups of rice?”). In this activity, they ultimately find an equation for the proportional relationship.
As students find missing values in the table, they should see that they can always multiply the number of food items by the constant of proportionality. When students see this pattern (MP7) and represent the number of people served by
\(x\)
cups of rice (or
\(s\)
spring rolls) as
\(3x\)
(or
\(\frac{1}{2}s\)
), they are expressing regularity in repeated reasoning (MP8).
Only one row in each table is complete. Based on their experience in the previous lesson, students are more likely to multiply the entries in the left-hand column by 3 (or
\(\frac{1}{2}\)
) than to use scale factors, at least for the first and third rows. If they do, they are more likely to see how to complete the last row in each table.
Some students might use unit rates: If 1 cup of rice can serve 3 people, then
\(x\)
cups of rice can serve
\(3x\)
people. So, students have different ways to generate
\(3x\)
as the expression that represents the number of people served by
\(x\)
cups of rice: completing the table for numerical values and continuing the pattern to the last row; or finding the unit rate and using it in the case of
\(x\)
cups.
Monitor students for different approaches as they are working.
Launch
Tell students that this activity revisits a context they worked on in an earlier lesson.
Student Facing
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 cup of rice serve?
How many people will 3 cups of rice serve? 12 cups? 43 cups?
How many people will
\(x\)
cups of rice serve?
| cups of dry rice | number of people |
|------------------|------------------|
|        1         |                  |
|        2         |        6         |
|        3         |                  |
|        12        |                  |
|        43        |                  |
|      \(x\)       |                  |
A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 spring roll serve?
How many people will 10 spring rolls serve? 16 spring rolls? 25 spring rolls?
How many people will
\(n\)
spring rolls serve?
| number of spring rolls | number of people |
|------------------------|------------------|
|           1            |                  |
|           6            |        3         |
|           10           |                  |
|           16           |                  |
|           25           |                  |
|         \(n\)          |                  |
How was completing this table different from the previous table? How was it the same?
Student Response
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Anticipated Misconceptions
If students have trouble encapsulating the relationship with an expression, encourage them to draw diagrams or to verbalize the relationship in words.
Activity Synthesis
Select students who have used the following approaches in the given order:
Recognize that to move from the first column to the second, you multiply by 3.
Say that 3 is the constant of proportionality.
Recognize that the equation
\(2 \boldcdot  {?} = 6\)
can be used to find the constant of proportionality algebraically.
Say that the 3 can be interpreted as the number of people per 1 cup of rice.
If no student sees that these insights are connected to prior work, explicitly connect them with the lessons of the previous two days. At the end of the discussion, suggest to students that we let
\(y\)
represent the number of people who can be served by
\(x\)
cups of rice. Ask students to write an equation that gives the relationship of
\(x\)
and
\(y\)
. (This builds on work from grade 6, but it may be rusty.) Be sure to write the equation
\(y = 3x\)
where all students can see it and help students interpret its meaning in the context: “To find
\(y\)
, the number of people served, we can multiply the number of cups of rice,
\(x\)
, by 3.”
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students describe their approaches, use color and annotations to scribe their thinking on a display.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR7 Compare and Connect.
As students explain their different approaches to completing the tables, ask “What is similar, what is different?” about their methods. Draw students’ attention to the different ways the constant of proportionality was represented across the different strategies. For instance, when discussing first question, ask “Where do you see the 3 in each approach?” These exchanges strengthen students’ mathematical language use and reasoning based on constant of proportionality.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


aligns: no