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: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
problem_activity: Optional activity
This activity examines how measurement errors behave when they are added together. In other words, if I have a measurement
\(m\)
with a maximum error of 1% and a measurement
\(n\)
with a maximum error of 1%, what percent error can
\(m + n\)
have? In addition to examining accuracy of measurements carefully (MP6), students work through examples and look for patterns (MP8) in order to hypothesize, and eventually show, how percent error behaves when measurements with error are added to one another.
Monitor for students who look for patterns, recognize the usefulness of the distributive property, or formulate the problem abstractly with variables.
Launch
Read the problem out loud and ask students what information they would need to know to be able to solve the problem. Students may say that they need to know what length the boards are supposed to be, because it is likely that they haven't realized that they can solve the problem without this information. Explain that floor boards come in many possible lengths, that 18-inch and 36-inch lengths are both common, but the boards can be anywhere between 12 and 84 inches. Ask students to pick values for two actual lengths and figure out the error in that case. Then they can pick two different examples, make the calculations again, and look for patterns.
Provide access to calculators.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. If students are unsure where to begin, suggest that they draw a diagram to help illustrate the information provided.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A wood floor is made by laying multiple boards end to end. Each board is measured with a maximum percent error of 5%. What is the maximum percent error for the total length of the floor?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may pick some example lengths but then struggle with knowing what to do with them. Ask them “what would be the maximum measured lengths? The minimum? What would be the error if both measurements were maximum? What if they were both minimum?”
Some students may pick numbers that make the calculations more complicated, leading to arithmetic errors. Suggest that they choose simple, round numbers for lengths, like 50 inches or 100 centimeters.
Activity Synthesis
The goal of this discussion is for students to generalize from their specific examples of measurements to understand the general pattern and express it algebraically.
Poll the class on the measurements they tried and the maximum percent error they calculated. Invite students to share any patterns they noticed, especially students who recognized the usefulness of the distributive property for making sense of the general pattern.
Guide students to use variables to talk about the patterns more generally.
If a board is supposed to have length
\(x\)
with a maximum percent error of 5%, then the shortest it could be is
\(0.95x\)
and the longest it could be is
\(1.05x\)
.
If another board is supposed to have length
\(y\)
, it could be between
\(0.95y\)
and
\(1.05y\)
.
When the boards are laid end-to-end, the shortest the total length could be is
\(0.95x + 0.95y\)
, which is equivalent to
\(0.95(x + y)\)
.
The longest the total length could be is
\(1.05x + 1.05y\)
, or
\(1.05(x + y)\)
.
Because of the distributive property, we can see that the maximum percent error is still 5% after the board lengths are added together.
One interesting point to make, if students have also done the previous activity about measuring pencils, is that you could
measure
the sum of the board lengths with a lower percent error than you could measure each individual board (assuming your tape measure is long enough), just like an error of 1 mm was a smaller percentage of the length of the longer pencil.


aligns: no