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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_pos_0

8
standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
problem_activity: Activity
Students continue to compare quadratic and exponential functions in this activity. This time, they decide how to compare the functions.
Monitor for different strategies students may use to compare the functions. Identify students who:
Create one or more tables of values and compare the values of
\(p(x)\)
and
\(q(x)\)
at increasingly large values of
\(x\)
.
Graph the functions defined by
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
and compare the graphs.
Create one or more tables of values, calculate the growth factors at equal intervals of input, and then compare the growth factors.
If students choose graphing or spreadsheet tools strategically to perform comparisons, they practice MP5.
Launch
Ask students to observe the equations representing the two functions and determine which function is exponential and which is quadratic. Invite students to share how they know. Make sure students recognize that
\(p\)
is quadratic and
\(q\)
is exponential before they begin the activity.
Provide access to graphing technology and spreadsheet tools. This may be a good opportunity for students to experiment with the graphing window. If the horizontal dimension is very small (for example,
\(0<x<5\)
) or the vertical dimension is very large (for example,
\(0<y<3,\!000\)
), the two graphs will be hard to distinguish. As needed, remind students to think about adjusting the graphing window to make the graphs more informative.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to adjust the graphing window to experiment with the horizontal and vertical dimensions.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Here are two functions:
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
.
Investigate the output of
\(p\)
and
\(q\)
for different values of
\(x\)
. For large enough values of
\(x\)
, one function will have a greater value than the other. Which function will have a greater value as
\(x\)
increases?
Support your answer with tables, graphs, or other representations.
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 says that some exponential functions grow more slowly than the quadratic function
\(f(x)=x^2\)
as
\(x\)
increases. Do you agree with Jada? Explain your reasoning.
Let
\(f(x) = x^2\)
. Could you have an exponential function
\(g(x)=b^x\)
so that
\(g(x) < f(x)\)
for all values of
\(x\)
?
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
Select students to present their strategies in the sequence listed in the Activity Narrative. If no students chose to graph the functions, consider displaying the graphs for all to see.  By presenting strategies that involved calculation of growth factors last, students will deepen their understanding of why exponential values will eventually overtake the quadratic values.
Again, one way to make sense of why exponential functions eventually grow faster than quadratic functions is by thinking of the growth factors. The output of an exponential function always increases by the same factor when the input increases by 1 (for example, 3 for the exponential function
\(q\)
studied here). The output of a quadratic function, on the other hand, increases by smaller and smaller amounts when the input increases by 1. So even though a quadratic function may take larger values than an exponential function for many inputs, the values of the exponential function will eventually overtake the quadratic.
Conversing: MLR2 Collect and Display.
As students share their analysis with the class, listen for and collect the language students use to identify and describe how the output of the exponential function eventually outgrows that of the quadratic function. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness


aligns: yes