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: 
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_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Optional activity
This activity is optional. It gives students an opportunity to reason about and to graph the rules of piecewise functions without a context.
Students are given the equations that define two piecewise functions, along with strips of paper, each containing a part of a graph and a portion of the horizontal axis (no scale is shown). Their job is to arrange the strips, apply a scale on each axis, and add open and closed circles to the graph to accurately represent the function values at each interval of input.
Unlike the piecewise functions students saw earlier in the lesson, each function here contains intervals defined by non-constant linear expressions. Support students as needed in reasoning about those intervals. If desired, this activity could be done over two class periods. (Students could piece together the graph of the first function in one class period and do the same for the second function at another time.)
To create an accurate graph, students need to make sense of the values, expressions, and inequality symbols in each equation and persevere in discerning their connections to the graph (MP1).
Here are images of the graphs on the blackline master for reference and planning. Cut each graph along the dashed lines and the vertical axis before giving it to students.
Function
\(f\)
[IMG]
Function
\(g\)
[IMG]
Launch
Arrange students in groups of 2–3. Give each group the strips for function
\(f\)
(pre-cut from the blackline master), glue or glue stick, and a sheet of paper on which to adhere the pieces.
Tell students that they are to arrange the pieces according to the rules that define
\(f\)
, as shown in the activity statement, and to add a label and a scale to each axis after the pieces are arranged. Explain that each piece contains a part of the horizontal axis, and the gridlines are 1 unit apart. Encourage students to double check their graph before gluing the pieces on another sheet of paper.
Consider discussing students' graphs of function
\(f\)
before giving students the pieces for function
\(g\)
and asking them to do another round.
Student Facing
Your teacher will give your group strips of paper with parts of a graph of a function. Gridlines are 1 unit apart.
Arrange the strips of paper to create a graph for each of the following functions.
\(\displaystyle f(x)=\begin{cases} \begin{align} \text-5,&\qquad \text-10< x< \text-5 \\ x,&\qquad \text-5\leq x< 0 \\ 1, &\qquad 0\leq x< 3\\ x-2, &\qquad 3\leq x< 8\\6, &\qquad 8\leq x < 1 0\ \end{align} \end{cases}\)
\(\displaystyle g(x)=\begin{cases} \begin{align} 5.5,&\qquad \text-10< x\leq \text-8 \\ 4,&\qquad \text-8< x\leq \text-3 \\ \text-x, &\qquad \text-3< x\leq 2\\ \text-3.5, &\qquad 2< x\leq 5 \\ x-5, &\qquad 5< x\leq 10 \end{align} \end{cases}\)
To accurately represent each function, be sure to include a scale on each axis and add open and closed circles on the graph where appropriate.
Student Response
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Activity Synthesis
Invite students to display their completed graphs and their reasoning. In particular, discuss how students determined the appearance of the intervals defined by non-constant linear expressions such as
\(x\)
,
\(x-4\)
, or
\(\text-x\)
.
If different groups created different graphs for the same function, ask them to analyze one another's work and try to reach an agreement.
If time permits, ask students to identify the domain and range of each function.
Speaking, Representing: MLR8 Discussion Supports.
Make sure students can explain their visual display and how they reasoned about the rules for different intervals. If needed, give them additional time to prepare their explanation. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion.
Design Principle(s): Support sense-making; Cultivate conversation


aligns: no