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: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
problem_activity: Activity
The purpose of this activity is for students to interact with contexts in which the direction of inequality is the opposite of what they might expect if they try to solve like they would with an equation. For example, in the second problem, the original inequality is
\(9(7-x) \leq 36\)
, but the solution to the inequality is
\(x \geq 3\)
.
Some students might solve the associated equation and then test values
of
\(x\)
to determine the direction of inequality. That method will be introduced in more generality in the next lesson. This activity emphasizes thinking about the context in deciding the direction of inequality.
Launch
Keep students in the same groups. Give 5–10 minutes of quiet work time and partner discussion followed by a whole-class discussion.
Student Facing
Kiran has
$
100 saved in a bank account. (The account doesn’t earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least
$
25 in the account a year from now.
Clare wrote the inequality
\(\text-12x + 100 \geq 25\)
, where
\(x\)
represents the amount Kiran takes out each month. What does
\(\text-12x\)
represent?
Find some values of
\(x\)
that would work for Kiran.
We could express
all
the values that would work using either
\(x \leq \text{__ or } x \geq \text{__}\)
. Which one should we use?
Write the answer to Kiran’s question using mathematical notation.
A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs
$
7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts.
If
\(x\)
represents the dollar amount of the discount, then the amount the teacher will pay can be expressed as
\(9(7-x)\)
. In this expression, what does the quantity
\(7-x\)
represent?
The teacher has
$
36 to spend on the granola bars. The equation
\(9(7-x)=36\)
represents a situation where she spends all
$
36. Solve this equation.
What does the solution mean in this situation?
The teacher does not have to spend all
$
36. Write an inequality relating 36 and
\(9(7-x)\)
representing this situation.
The solution to this inequality must either look like
\(x \geq 3\ \text{or } x \leq 3\)
. Which do you think it is? Explain your reasoning.
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?
Jada and Diego baked a large batch of cookies.
They selected
\(\frac14\)
of the cookies to give to their teachers.
Next, they threw away one burnt cookie.
They delivered
\(\frac25\)
of the remaining cookies to a local nursing home.
Next, they gave 3 cookies to some neighborhood kids.
They wrapped up
\(\frac23\)
of the remaining cookies to save for their friends.
After all this, they had 15 cookies left. How many cookies did they bake?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is to let students voice their reasoning about the direction of the inequality by reasoning about the context. Ask students to share their reasons for choosing the direction of inequality in their solutions. Some students may notice that the algebra in both problems involves multiplying or dividing by a negative number. Honor this observation, but again, the goal is not to turn this observation into a rule for students to memorize and follow. Interpreting the meaning of the solution in the context should remain at the forefront.
As students model real-world situations, questions about the interpretation of the mathematical solution should continue to come up in the conversation. For instance, the amount of the granola bar discount cannot be
$
3.5923, even though this is a solution to the inequality
\(x \geq 3\)
. The value -10 is a solution to Kiran’s inequality, even though he can’t withdraw -10 dollars. Students can argue that negative values for
\(x\)
simply don’t make sense in this context. Some may argue that we should interpret
\(x=\text-10\)
to mean that Kiran deposits
$
10 every month.
Engagement: Develop Effort and Persistence
. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention
Representing, Writing, Conversing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their reasons for the direction of the inequality in their solution, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. This will help students understand how context determines the direction of inequality.
Design Principle(s): Support sense-making


aligns: no