few_shots: 
Example 1:
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?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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

Example 3:
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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: In Class Launch
Use after Unit 3, Lesson 11
In this task, students will experiment with cups in order to see what shape they seem to trace out when they are rolled on a flat surface, and then refine their thinking in order to prove what the shape is. If possible, provide cups with a variety of sizes and shapes. Note that cylindrical cups will not curve as they are rolled, and so will not make the same kind of shape as cups with slanted sides.
Ask students to roll some cups around. Then ask them to guess what shape the cup traces out on the table or floor. Sketch the suggested shapes for all to see. Tell students their task is to try to prove what the shape is.
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
After students have completed the task, either invite them to share how they knew what shape the cup would trace out, or have a gallery walk so that students can see and comment on each other’s arguments.
Then invite them to share their responses to the last question: how did they make a cup that would trace out the largest possible area? If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
What shape does a cup trace out when it is rolled on a flat surface? Use words and pictures to explain how you know.
How can you locate the center and calculate the area of the shape that is traced out?
Using one sheet of 8.5 inch by 11 inch paper, how could you make a cup that will trace out the largest possible area?
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
If students see the questions in Part 2 before they have finished Part 1, it will give away the shape that the cup traces out, so hand out Part 2 after the discussion of Part 1. Part 1 can be discussed before all students have finished their explanations, as long as they all have some developed ideas about what the shape is. For the discussion, ask students to share what shape they think the cup traces out and why they think so. Students do not need to have fully developed proofs, but take the opportunity to press them to clarify their terminology and justify their assumptions.
After students have completed the task, invite them to share their responses to the last question: how did they make a cup that would trace out a larger area? (The bottom radius should be closer to the top radius so that the radius of the ring will be larger, or the cup should be taller so that the ring is thicker.) Then ask how they would make a shape that traces out the largest possible area. If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
Part 1:
What shape does a cup trace out when it is rolled on a flat surface? Explain or show your reasoning.
Pause for discussion.
Part 2:
The outside of this ring has a radius of 55 cm. The inside of the ring has a radius of 38 cm. Find the area of the ring.
[IMG]
Triangle
\(ABC\)
is isosceles. Find the length of side
\(AC\)
using the information given. The two vertical lines are parallel.
[IMG]
Here is the cross-section of a cup. It shows the width of the cup at the top and at the bottom, and the slant height of the cup. If we extend the sides of the cup down until they meet at a point, we will make a triangle. What would the slant height of that triangle be?
[IMG]
Measure the widths of the top and bottom of your cup, and measure its slant height. If you drew the cross-section of the cup and then extended the sides to make a triangle, what would the slant height of the triangle be?
If you rolled your cup in a circle to make a ring, what would the area of the ring be?
Give an example of a cup that would trace out a larger area than your cup would.


aligns: no