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: 
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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category counts to be less than or equal to 10.)
problem_activity: Narrative
The purpose of this activity is to introduce the foot as a length unit in the U.S. customary system. Students learn that a foot is longer than an inch and is the same length as 12 inches. They use the length of 12-in rulers as a tool to measure lengths in feet by iterating the length of a ruler, or, more precisely, the length of 12 inches shown on the ruler. The experience iterating lengths of a ruler to measure a length in feet helps students build a mental benchmark for the approximate length of a foot that they can use to estimate lengths.
Students measure lengths of tape that represent realistic lengths of different types of fish. They measure each length to the nearest inch and to the nearest foot. In the synthesis, students discuss the difference in their measurements and relate the differences to the size of the length units (MP7).
MLR2 Collect and Displa
y. Circulate, listen for, and collect the language students use as they measure the length of each animal. Display measurement words and comparative language such as: “inches,” “feet,” “ruler,” “more than,” “less than,” “longer,” “shorter,” “smaller,” and so on. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Required Materials
Materials to Gather
Inch tiles
Rulers (inches)
Tape (painter's or masking)
Required Preparation
Tape strips of these lengths on the floor. Label each strip with the letter and fish name. It may be helpful to make multiple sets of these lines to keep the groups small.
Strip A, largemouth bass: 2 feet
Strip B, spiny dogfish shark: 4 feet
Strip C, catfish: 3 feet
Strip D, koi: 2 feet and 6 inches
Launch
Groups of 3–4
Give each student an inch ruler and access to inch tiles.
“You have measured the length of objects and the sides of shapes using inches. If you are measuring longer objects, like the fish in the warm-up, you might want to use a different unit.”
“A
foot
is a longer length unit in the U.S. Customary Measurement System. A foot is the same length as 12 inches.”
“When we measure a length that starts at 0 on the ruler and ends at the 12, we can say the length is 12 inches or we can say the length is 1 foot.”
“What are some things you see around the classroom that are about 1 foot long?"
30 seconds: quiet think time
Share responses.
Activity
Display the images of the fish.
“In previous lessons, we measured the lengths of different kinds of reptiles. In this lesson, you will be measuring the lengths of fish. What do you know about fish?”
Share responses.
“Let’s imagine we work at the aquarium and need to measure the lengths of these fish. The tape on the floor represents their lengths.”
“Look at the tape strips on the floor. Use your rulers to measure each length first in feet. Then measure the length in inches using the rulers or your inch tiles.”
10 minutes: group work time
As needed, pause after students have measured their first fish in feet and invite students to share how they used their rulers to measure in feet.
Student Facing
Work with your group to measure the tape strips around the classroom in feet. Then measure the length in inches.
Tape A
largemouth bass
[IMG]
Length in feet: __________
Length in inches: __________
Tape B
spiny dogfish shark
[IMG]
Length in feet: __________
Length in inches: __________
Tape C
catfish
[IMG]
Length in feet: __________
Length in inches: __________
Tape D
koi
[IMG]
Length in feet: __________
Length in inches: __________
What did you notice about the number of feet compared to the number of inches when you measured the tape strips?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students measure lengths in feet by lining up multiple rulers or iterating the same ruler without considering gaps that may be caused by the lengths of the ruler before the 0 in or after the 12 in mark, point out these gaps on the ruler and consider asking:
“Can you show me how you measured the length of the ______ in feet?”
“Did you measure the length without any gaps? How do you know?”
“How would you revise the way you measure to make sure you have no gaps or overlaps?”
Activity Synthesis
“What did you notice about your measurements when you switched from measuring in feet to measuring in inches?” (The numbers are smaller for the number of feet. I had to count more inches than feet.)
“Why are our measurements in feet so much smaller than our measurements in inches?” (Feet are longer, so you don't need as many to measure the length of the tape. It takes more inches to cover the same length.)


aligns: no