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: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
problem_activity: Activity
In the previous lesson, students analyzed the graph of a linear, non-proportional relationship (number of cups in a stack versus the height of the stack). This task focuses on interpreting the slope of a graph and where it crosses the
\(y\)
-axis in context. Students are given cards describing situations with a given rate of change and cards with graphs. Students match each graph with a situation it could represent, and then use the context to interpret the meaning of the slope. They find where the line crosses the vertical axis, i.e., the
vertical intercept,
and interpret its meaning in each situation. They also decide if the two quantities in each situation are in a proportional relationship.
Make sure that students draw the triangle they use to compute the slope. There are strategic choices that can be made to make the computation easier and more precise. Watch for students who use different triangles for the same slope computation, and ask them to share their reasoning during the whole-group discussion.
In the whole-group discussion at the end of the task, discuss and emphasize the meaning of the terms slope and vertical intercept or
\(y\)
-intercept (in situations where the name of the variable graphed on the vertical axis is
\(y\)
).
You will need the Slopes and Graphs blackline master for this lesson.
Launch
Tell students that they will match a set of cards describing different relationships with a set of cards showing graphs of lines. The axes on the graphs are not labeled (since this could be used as an aid in the matching). Instruct students to add labels to the axes as they make their matches.
Arrange students in groups of 2. Distribute a set of 12 cards to each group. 10 minutes of group work and then whole-class discussion.
Student Facing
Your teacher will give you 6 cards describing different situations and 6 cards with graphs.
Match each situation to a graph.
Pick one proportional relationship and one non-proportional relationship and answer the following questions about them.
How can you find the slope from the graph? Explain or show your reasoning.
Explain what the slope means in the situation.
Find the point where the line crosses the vertical axis. What does that point tell you about the situation?
Student Response
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Activity Synthesis
The slopes of the 6 lines given on the situation card are all different, so the matching part of the task can be accomplished by examining the slopes of the different lines. Invite students who have made strategic choices of slope triangles for calculating the slopes (for example, Graph 6 contains the points
\((0,0)\)
and
\((5,20)\)
, which give a value of
\(\frac{20}{5}\)
for the slope), and ask them to share.
Next, focus the discussion on the interpretation of the point where the line crosses (or touches) the
\(y\)
-axis. For some situations, the contextual meaning of this point is abstract. For example, in Situation B, a square with side length 0 is just a point that has no perimeter and so it “makes sense” that
\((0,0)\)
is on the graph. Some students may argue that a point is not a square at all but if we consider it to be a square then it definitely has 0 perimeter. In other situations, the point where the line touches the
\(y\)
-axis has a very natural meaning. For example, in Situation A, it is the amount Lin’s dad spent on the tablet and 0 months of service: so this is the cost of the tablet.
Define the
vertical intercept
or
\(y\)
-intercept as the point where a line crosses the
\(y\)
-axis. Note that sometimes “
\(y\)
-intercept” refers to the numerical value of the
\(y\)
-coordinate where the line crosses the
\(y\)
-axis. Go over each of the situations and ask students for the meaning of the vertical intercept in the situation (A: cost of the device, B: perimeter of a square with 0 side length, C: amount of money in Diego’s piggy bank before he started adding \$5 each week, D: money Noah saved helping his neighbor, E: amount of money in Elena's piggy bank before she started adding money, F: amount Lin’s mom has paid for internet service before her service begins.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following terms and maintain the display for reference throughout the unit: vertical intercept, y-intercept. Invite students to suggest language or diagrams to include on the display that will support their understanding of these terms.
Supports accessibility for: Memory; Language
Representing, Conversing: MLR7 Compare and Connect.
As students share their matches with the class, call students’ attention to the different ways the vertical intercept is represented graphically and within the context of each situation. Take a close look at Graphs 2 and 3 to distinguish what the 40 represents in each corresponding situation. Wherever possible, amplify student words and actions that describe the correspondence between specific features of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: yes