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 to multiply a fraction or whole number by a fraction.

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: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: 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: Optional activity
In this activity, students measure lengths and determine possibilities for actual lengths. There are two layers of attending to precision (MP6) involved in this task:
Deciding how accurately the pencils can be measured, probably to the nearest mm or to the nearest 2 mm, but this depends on the eyesight and confidence of the student
Finding the possible percent error in the measurement chosen
Launch
Arrange students in groups of 2. Provide access to calculators. Give students 4–5 minutes of quiet work time, followed by partner and whole-class discussion.
Writing, Speaking: MLR8 Discussion Supports.
To support students as they respond to “How accurate are your estimates?”, provide a sentence frame such as: “My estimate is within ___ mm of the actual length because . . . ." Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. This will help students use mathematical language as they justify the accuracy of their estimates.
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Estimate the length of each pencil.
How accurate are your estimates?
For each estimate, what is the largest possible percent error?
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
Some students may think that they can find an exact value for the length of each pencil. Because the pictures of the pencils are far enough away from the ruler, it requires a lot of care just to identify the “nearest” millimeter (or which two millimeter markings the length lies between). Prompt them to consider the error in their measurements by asking questions like, “Assuming you measured the pencil accurately to the nearest millimeter, what is the longest the actual length of the pencil could be? What is the shortest it could be?”
Some students may not remember how to calculate percent error. Ask them, “What is the biggest difference possible between the estimated and actual lengths? What percentage of the actual length would that be when the difference is as big as possible?”
Activity Synthesis
The goal of this discussion is for students to practice how they talk about precision.
Discussion questions include:
“How did you decide how accurately you can measure the pencils?” (I looked for a value that I was certain was less than the length of the pencil and a value that I was certain was bigger. My estimate was halfway in between.)
“Were you sure which mm measurement the length is closest to?” (Answers vary. Possible responses: Yes, I could tell that the short pencil is closest to 5.4 cm. No, the long pencil looks to be closest to 17.7 mm, but I’m not sure. I am sure it is between 17.6 cm and 17.8 cm.)
“Were the percent errors the same for the small pencil and for the long pencil? Why or why not?” (No. I was able to measure each pencil to within 1 mm. This is a
smaller
percentage of the longer pencil length than it is of the smaller pencil length.)
Other possible topics of conversation include noting that the level of accuracy of a measurement depends on the measuring device. If the ruler were marked in sixteenths of an inch, we would only be able to measure to the nearest sixteenth of an inch. If it were only marked in cm, we would only be able to measure to the nearest cm.


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