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: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.
problem_activity: Narrative
The purpose of this activity is for students to begin to distinguish triangles from other shapes. By working with variants and non-examples of triangles, students begin to develop their understanding of what makes a shape a triangle. Students may more easily identify equilateral triangles as triangles but may also identify other examples as triangles. When students discuss with their group which group a shape should be placed in, students informally describe the attributes of the shape (MP6).
Required Materials
Materials to Copy
Triangle Sort Cards
Required Preparation
Cut out triangle cards from the blackline master. Each group of 4 needs 1 set of cards.
Launch
Groups of 4
Give each group a set of cards.
“Work with your group to sort the shapes into 2 groups. Put the shapes that are triangles on the left side of your page. Put the shapes that are not triangles on the right side of your page. When you place a shape, tell your group why you think the shape belongs in that group.”
Activity
4 minutes: small-group work time
Monitor for students who discuss attributes of triangles when sorting.
“Write a number to show how many shapes are in each group.”
1 minute: independent work time
“Walk around to see how the other groups organized their shapes. Did they organize them the same way that your group did?”
6 minutes: work time
Student Facing
Let’s put the shapes into 2 groups.
Triangle
[IMG]
[IMG]
[IMG]
Not a Triangle
[IMG]
[IMG]
Student Response
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Advancing Student Thinking
I
f students put shapes other than triangles into the group of triangles, consider asking:
“How did you choose which group to put this shape in?”
“What is alike, or the same, about all of the shapes in this group?”
Activity Synthesis
Display cards O, K, and G next to each other.
“Noah says that the shape in the middle is not a triangle because it is pointing down and triangles have to point up. Do you agree with Noah? Why or why not?“


aligns: no