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 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_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
problem_activity: Optional activity
In the previous lesson, students built trundle wheels using three different wheel sizes, and they tested the functionality of their wheels in the classroom. In this activity, they use their trundle wheels to measure a longer path of about 50–100 meters. This is the same path that they measured during a previous lesson with a different method. If several groups are sharing a trundle wheel, then they each measure the given path once and share their data with each other.
After students measure, they spend the remainder of the lesson on computations and sharing results. Students get a chance to connect the mathematical formulas and computations with the aspects of the hands-on experience they had in making and using the wheels. They attend to precision when they are deciding on how to report their results and when they are comparing results with other groups (MP6).
Digital
Launch
Keep students in the same groups from the previous lesson. Remind students of the path they should measure. Instruct them to come back to the classroom to finish their calculations as soon as they have recorded their measurements.
Give students 10–20 minutes to take turns measuring and 10 minutes of group work time to finish their calculations, followed by whole-class discussion.
There is also a digital version of a trundle wheel available. Students can be assigned a path, numbered from 1 to 5, and collect data on different-sized trundle wheels. The applet is programmed to stop automatically at one of the five distances. The students must keep track of the number of rotations the wheel makes before it stops.
Action and Expression: Provide Access for Physical Action.
Support access to tools and assistive technologies. Monifor for students who may need an additional demonstration, or assistance using their trundle wheel to measure a longer path. Consider modeling a systematic way to record the number of clicks to calculate an example distance as students complete this task.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Student Facing
Earlier you made trundle wheels so that you can measure long distances.
If you do not have a trundle wheel, you can use an applet to simulate measuring. Your teacher will tell you a path whose distance you will measure.
Measure the path with your trundle wheel three times and calculate the distance. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "diameter"}, {"_attributes": {"scope": "col"}, "_value": "number of rotations"}, {"_attributes": {"scope": "col"}, "_value": "computation"}, {"_attributes": {"scope": "col"}, "_value": "distance"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "trial 1"}, {}], "td": [{}, {}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "trial 2"}, {}], "td": [{}, {}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "trial 3"}, {}], "td": [{}, {}, {}]}]}]}]}
Decide what distance you will report to the class. Be prepared to explain your reasoning.
Compare this distance with the distance you measured the other day for this same path.
Compare your results with the results of two other groups. Express the differences between the measurements in terms of percentages.
To use the applet:
Choose a diameter for the wheel, and enter a number 1 -- 5 for the path you will measure.
Watch carefully to keep track of the number of rotations that the wheel makes before it stops.
Try three different diameters for the same path, and compare your results.
Answer the questions.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in the same groups from the previous lesson. Remind students of the path they should measure. Instruct them to come back to the classroom to finish their calculations as soon as they have recorded their measurements.
Give students 10–20 minutes to take turns measuring and 10 minutes of group work time to finish their calculations, followed by whole-class discussion.
There is also a digital version of a trundle wheel available. Students can be assigned a path, numbered from 1 to 5, and collect data on different-sized trundle wheels. The applet is programmed to stop automatically at one of the five distances. The students must keep track of the number of rotations the wheel makes before it stops.
Action and Expression: Provide Access for Physical Action.
Support access to tools and assistive technologies. Monitor for students who may need an additional demonstration, or assistance using their trundle wheel to measure a longer path. Consider modeling a systematic way to record the number of clicks to calculate an example distance as students complete this task.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Student Facing
Earlier you made trundle wheels so that you can measure long distances. Your teacher will show you a path to measure.
Measure the path with your trundle wheel three times and calculate the distance. Record your results in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "trial number"}, {"_attributes": {"scope": "col"}, "_value": "number of clicks"}, {"_attributes": {"scope": "col"}, "_value": "computation"}, {"_attributes": {"scope": "col"}, "_value": "distance"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "1"}], "td": [{}, {}, {}]}, {"th": [{"_value": "2"}], "td": [{}, {}, {}]}, {"th": [{"_value": "3"}], "td": [{}, {}, {}]}]}]}]}
Decide what distance you will report to the class. Be prepared to explain your reasoning.
Compare this distance with the distance you measured the other day for this same path.
Compare your results with the results of two other groups. Express the differences between the measurements in terms of percentages.
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 lose count when counting the number of rotations along the path. Encourage them find a systematic way to record the number of clicks while their group member is walking along the path.
Activity Synthesis
Ask each group to report their measurement for the length of the path and record their answers for all to see. Guide students to compare these answers by asking questions like these:
“Do all of these answers seem reasonable? Do any of these answers seem unreasonable? Why?”
“Why are these answers not all exactly the same? What are some sources of error?” (Not going in a straight line, the wheel wobbles, the ground is uneven, only counting number of clicks but not parts of rotations etc.)
“What units did you use? What units would be most convenient for designing the course of a 5K walk-a-thon?” (Metric, since we are designing a 5 km course.)
“What degree of precision is appropriate to report?" (To the closest 1 meter at most. Reporting cm or mm on such long distances with a tool like a trundle wheel would be implying a degree of precision that would not be appropriate.)
If time permits, consider asking “If you could choose your own diameter for a trundle wheel, what would it be?” (A diameter that creates a circumference of 1 m would be convenient, i.e. about 32 cm.)
Collect and store students’ trundle wheels so they will have access to them again in the next lesson.
Speaking: MLR8 Discussion Supports.
To support students’ discussion of their measurements comparisons, invite them to use a sentence frame such as: “Our measurement is _____% (higher/lower) than group _____’s measurement because . . . ." Call on another student or group to restate what they heard using mathematical language. Encourage students to press each other for clear explanations and precise use of language.
Design Principle(s): Optimize output (for comparison)


aligns: yes