few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Know there is a complex number i such that i^(2) = –1, and every complex number has the form a + bi with a and b real.
problem_activity: Optional activity
The goal of this task is to review the connections between different ways of expressing a relationship: a description, a graph, and likely an equation. Students begin by analyzing different graphs, finding one that matches a given description, and explaining how they know the graph is a correct representation (MP3). They then use their analyses to evaluate the function at a different input value, compare it to another exponential function, and sketch a graph of the second function.
Monitor for students who:
Articulate clearly how the features of the graph enable them to choose the correct graph to represent the depreciation of the first van.
Write an expression to answer questions about the value of either van.
Launch
Clarify the meaning of
depreciation
for students who may not know what it means.
Arrange students in groups of 2. Give them a moment to think quietly about the first question, and then share their response with a partner. They should be ready to explain to each other how they know that certain graphs
cannot
represent the given function.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to highlight each important element in the problem statement in a different color (starting price, year, and percent depreciation). Invite students to use these same colors to identify corresponding elements on the graphs.
Supports accessibility for: Visual-spatial processing
Student Facing
A small business bought a van for $40,000 in 2008. The van depreciates by 15% every year after its purchase.
Which graph correctly represents the value of the van as a function of years since its purchase? Be prepared to explain why each of the other graphs could not represent the function.
Graph A
[IMG]
Graph B
[IMG]
Graph C
[IMG]
Graph D
[IMG]
Find the value of the van 8 years after its purchase. Show your reasoning.
In the same year (2008), the business bought a second van that cost $10,000 less than the first van and depreciates at 10% per year. Would the second van be worth more or less than the first van 8 years after the purchase? Explain or show your reasoning.
On the same coordinate plane as the graph you chose in the first question, sketch a graph that shows the value of the second van, in dollars, as a function of years since its purchase.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may not be sure how to start calculating the value of the van after 8 years. Ask these students to make a table showing years since 2008 and the value of the van each year.
Activity Synthesis
Invite previously identified students to explain how they chose the graph that represents the value of the first van. Make sure students see that:
The $40,000 price of the van appears as the vertical intercept of the graph.
The 15% rate of decay tells you that after 1 year, the van loses 15% of its value, retaining 85% of its original value.
\((0.85) \boldcdot 40,\!000 = 34,\!000\)
, so the point
\((1, 34,\!000)\)
must be on the correct graph.
If students haven’t already shown (in their partner discussions) that they understood why graphs A, C, and D cannot be the right representations, clarify the reasons (as shown in the Student Response).
Conclude the discussion by selecting previously identified students to share how they used expressions to answer questions about the value of the vans over the years. If no students wrote expressions, invite them to do so now. Highlight that:
Calculating the decay factor means subtracting the depreciation from 100%. For example, the first van loses 15% of its value each year, which means each year, it is worth only 85% of the amount of the previous year.
To find the value after 8 years, we could write an expression (
\(40,\!000 \boldcdot (0.85)^t\)
for the first van and
\(30,\!000 \boldcdot (0.9)^t\)
for the second van, where
\(t\)
represents time in years since purchase) and calculate its value when the input is 8.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion. At the appropriate time, invite students to create a visual display that shows how they determined the values of each van 8 years after purchase. Groups should consider how to display their calculations so that another student can interpret them. Some groups may wish to add notes or details to their displays to help communicate their thinking. Give students 3–4 minutes to quietly circulate and observe at least two other visual displays in the room. Invite groups to discuss what they noticed about what is the same and what is different between their display and the others they observed. Listen for and amplify the language students use to interpret the exponential expressions that represent the situation.
Design Principle(s): Cultivate conversation; Support sense-making


aligns: no