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: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
problem_activity: Activity
The purpose of this activity is for students to make connections between different representations of functions and start transitioning from input-output diagrams to other representations of functions. Students match input-output diagrams to descriptions and come up with equations for each of those matches. Students then calculate an output given a specific input and determine the independent and dependent variables.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and time to share their responses with their partner and come to agreement on their answers. Follow with whole-class discussion.
Student Facing
Record your answers to these questions in the table provided.
Match each of these descriptions with a diagram:
the circumference,
\(C\)
, of a circle with
radius
,
\(r\)
the distance in miles,
\(d\)
, that you would travel in
\(t\)
hours if you drive at 60 miles per hour
the output when you triple the input and subtract 4
the volume of a cube,
\(v\)
given its edge length,
\(s\)
Write an equation for each description that expresses the output as a function of the input.
Find the output when the input is 5 for each equation.
Name the
independent
and
dependent variables
of each equation.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "description"}, {"_attributes": {"scope": "col"}, "_value": "a"}, {"_attributes": {"scope": "col"}, "_value": "b"}, {"_attributes": {"scope": "col"}, "_value": "c"}, {"_attributes": {"scope": "col"}, "_value": "d"}]}]}, {"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "diagram"}], "td": [{}, {}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "equation"}], "td": [{}, {}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "br": [{}], "_values": ["input = 5", "output = ?"]}], "td": [{}, {}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "br": [{}], "_values": ["independent", "variable"]}], "td": [{}, {}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "br": [{}], "_values": ["dependent", "variable"]}], "td": [{}, {}, {}, {}]}]}]}]}
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Choose a 3-digit number as an input.
Apply the following rule to it, one step at a time:
Multiply your number by 7.
Add one to the result.
Multiply the result by 11.
Subtract 5 from the result.
Multiply the result by 13
Subtract 78 from the result to get the output.
Can you describe a simpler way to describe this rule? Why does this work?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The goal of this discussion is for students to describe the connections they see between the different entries for the 4 descriptions. Display the table for all to see and select different groups to share the answers for a column in the table. As groups share their answers, ask:
“How did you know that this diagram matched with this description?” (We remembered the formula for the circumference of a circle, so we knew description A went with diagram D.)
“Where in the equation do you see the rule that is in the diagram?” (The equation is the dependent variable set equal to the rule describing what happens to the independent variable in the diagram.)
“Explain why you chose those quantities for your independent and dependent variables." (We know the independent variable is the input and the dependent variable is the output, so we matched them up with the input and output shown in the diagram.)
Speaking: MLR8 Discussion Supports.
As students describe the connections they noticed in the table across from the different entries for the four descriptors, revoice student ideas to demonstrate mathematical language use. In addition, press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example. This will help students to produce and make sense of the language needed to communicate their own ideas about functions and independent and dependent variables.
Design Principle(s): Support sense-making; Optimize output (for explanation)


aligns: no