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: 
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.

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:
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 and division and of fractions to multiply and divide rational numbers.

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:
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: 
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.

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, 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
problem_activity: Activity
The purpose of this activity is for students to find the areas of regions involving different-sized circles and compare the strategies used. The first question introduces subtraction as a strategy to find the area around the outside of a circle. The second question introduces division to find the area of fractions of a circle.
Monitor for students who use different strategies for finding the area, including:
Calculating 30.96 square units for Figure A,
\(4 \boldcdot 7.74\)
for Figure B, and
\(9 \boldcdot 3.44\)
for Figure C
Realizing that all 3 figures end up being
\(144 - 113.04\)
so their areas had to be equal
Noticing that Figure B is composed of 4 scaled copies of Figure A, each with a scale factor of
\(\frac12\)
and therefore
\(\frac14\)
as much area. Figure C is composed of 9 scaled copies of Figure A, each with a scale factor of
\(\frac13\)
and therefore
\(\frac19\)
as much area. (If a student does not make this realization, it is
not
necessary for the teacher to bring it up during the discussion.)
Calculating 8.28 square units for Figure D and 8.71 square units for Figure E
Realizing that Figures D has 2 fewer squares and 2 more quarter-circles (which are smaller than the squares) so it must have a smaller area
The task affords an opportunity for students to engage in MP6. Some students may leave the area expressions in terms of
\(\pi\)
, but others may use an approximation for
\(\pi\)
. It is very important, particularly for the first problem, that they use the
same
approximation for
\(\pi\)
in all 3 figures. Otherwise they will end up with numerically different answers.
Launch
Arrange students in groups of 2. Display the image in the first question and ask students to make a prediction before calculating. Give 30 seconds of quiet think time before sharing with their partner. Give students quiet work time followed by a whole-class discussion.
Student Facing
Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.
[IMG]
Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Which figure has a longer perimeter, Figure D or Figure E? How much longer?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
In the first question, students may not know how to find the radius of the circles. Suggest having them cut off the shaded regions and rearrange them to show that the length of each side fits half way across the circle (marking the radius).
In the second question, students might benefit from cutting and rearranging the figures. Some students might assume, based on previous activities, that the areas of both figures are equal. However, Figure D has more pieces that are parts of a circle, and Figure E has more units that are a full square. Ask students whether the fourth of the circle has the same area as the square.
Activity Synthesis
There are two main goals for this discussion: for students to notice ways to be more efficient when comparing the areas of the regions and to be introduced to expressing answers in terms of pi.
Display Figures A, B, and C for all to see. Ask selected students to share their reasoning. Sequence the strategies from most calculations to most efficient.
If there were selected students who determined the areas were equal before calculating, ask them to share how they could tell. If there were no selected students, ask the class how we could determine that the area of the shaded regions in Figures A, B, and C were equal before calculating the answer of 30.96.
Next, focus the discussion on leaving answers in terms of
\(\pi\)
for each figure. Explain to students that in Figure A, the radius of the circle is 6, so the area of the circular region is
\(\pi \boldcdot 6^2\)
. Instead of multiplying by an approximation of
\(\pi\)
, we can express this answer as
\(36\pi\)
. This is called answering
in terms of
\(\pi\)
. Consider writing “
\(36\pi\)
” inside the large circle of Figure A.
Discuss:
In terms of
\(\pi\)
, what is the area of one of the circular regions in Figure B? (
\(9\pi\)
)
What is the combined area of all four circles in Figure B? (
\(4 \boldcdot 9\pi\)
, or
\(36\pi\)
)
What is the area of one of the circular regions in Figure C? (
\(4\pi\)
)
What is the combined area of all nine circles in Figure C? (
\(9 \boldcdot 4\pi\)
, or
\(36\pi\)
)
Consider writing “
\(9\pi\)
” and “
\(4\pi\)
” inside some of the circles in Figures B and C. Explain that the area of the shaded region for each of these figures is
\(144 - 36\pi\)
.
[IMG]
Discuss how students' strategies differed between the first problem (about Figures A, B, and C) and the second problem (about Figures D and E) and why.
Ask students to express the area of Figures D and E in terms of
\(\pi\)
. Record and display their answers of
\(2 + 2\pi\)
and
\(4 + 1.5\pi\)
for all to see. Ask students to discuss how they can tell Figure E’s area is larger than Figure D’s area when they are both written in terms of
\(\pi\)
.
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
After students have determined which figure has the largest shaded region, ask students to show their work and provide a brief explanation of their reasoning. Ask each student to meet with 2–3 other partners for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “How did you find the radius of each circle in the figure?”, “Why did you . . . ?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own explanation and learn about different strategies to find area.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness


aligns: no