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: 
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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Summarize numerical data sets in relation to their context, such as by:
- Reporting the number of observations.
- Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
problem_activity: Activity
In the previous activity, students evaluated the performance of three students based on the mean and variability. Here they learn the term
mean absolute deviation (MAD)
as a way to quantify variability and calculate it by finding distances between the mean and each data value. Students compare data sets with the same mean but different MADs and interpret the variations in context.
While this process of calculating MAD involves taking the absolute value of the difference between each data point and the mean, this formal language is downplayed here. Instead, the idea of “finding the distance,” which is always positive, is used. This is done for a couple of reasons. One reason is to focus students' attention on the statistical work rather than on terminology or symbolic work. Another reason is that finding these differences may involve operations with signed numbers, which are not expected in grade 6.
Teacher Notes for IM 6–8 Accelerated
The activity Shooting Hoops (Part 2) and the activity referred to in the launch are not included in IM 6–8 Math Accelerated.
During the launch, ask students what they think will be true about distances between data points and the mean, knowing that the mean can be considered a balance point. (The sum of the distances to the left of the mean is the same as the sum of the distances to the right of the mean.) Then tell students that the distances can also tell us something else about a distribution that will be explored in this activity.
Launch
Remind students in the last lesson they found distance between each data point and the mean, and found that the sum of those distances on the left and the sum on the right were equal, which allows us to think of the mean as the balancing point or the center of the data. Explain that the distance between each point and the mean can tell us something else about a distribution.
Arrange students in groups of 2. Give students 4–5 minutes to complete the first set of questions with their partner, and then 4–5 minutes of quiet time to complete the remaining questions. Follow with a whole-class discussion.
Student Facing
The tables show Elena, Jada, and Lin’s basketball data from an earlier activity. Recall that the mean of Elena’s data, as well as that of Jada and Lin’s data, was 5.
Record the distance between each of Elena’s scores and the mean.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Elena"}]}, {"_value": "4"}, {"_value": "5"}, {"_value": "1"}, {"_value": "6"}, {"_value": "9"}, {"_value": "7"}, {"_value": "2"}, {"_value": "8"}, {"_value": "3"}, {"_value": "3"}, {"_value": "5"}, {"_value": "7"}]}, {"td": [{"_attributes": {"style": "width:150px"}, "strong": [{"_value": "distance from\u00a05"}]}, {"_value": "1"}, {}, {}, {"_value": "1"}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Now find
the average of the distances
in the table. Show your reasoning and round your answer to the nearest tenth.
This value is the
mean absolute deviation (MAD)
of Elena’s data.
Elena’s MAD: _________
Find the mean absolute deviation of Jada’s data. Round it to the nearest tenth.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Jada"}]}, {"_value": "2"}, {"_value": "4"}, {"_value": "5"}, {"_value": "4"}, {"_value": "6"}, {"_value": "6"}, {"_value": "4"}, {"_value": "7"}, {"_value": "3"}, {"_value": "4"}, {"_value": "8"}, {"_value": "7"}]}, {"td": [{"strong": [{"_value": "distance from\u00a05"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Jada’s MAD: _________
Find the mean absolute deviation of Lin’s data. Round it to the nearest tenth.
{"table": [{"_attributes": {"border": "1", "style": "width:500px"}, "tbody": [{"tr": [{"td": [{"strong": [{"_value": "Lin"}]}, {"_value": "3"}, {"_value": "6"}, {"_value": "6"}, {"_value": "4"}, {"_value": "5"}, {"_value": "5"}, {"_value": "3"}, {"_value": "5"}, {"_value": "4"}, {"_value": "6"}, {"_value": "6"}, {"_value": "7"}]}, {"td": [{"strong": [{"_value": "distance from\u00a05"}]}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}]}]}]}]}
Lin’s MAD: _________
Compare the MADs and dot plots of the three students’ data. Do you see a relationship between each student’s MAD and the distribution on her dot plot? Explain your reasoning.
[IMG]
Student Response
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Student Facing
Are you ready for more?
Invent another data set that also has a mean of 5 but has a MAD greater than 2. Remember, the values in the data set must be whole numbers from 0 to 10.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may recall the previous lesson about thinking of the mean as a balance point and think that the MAD should always be zero since the left and right distances should be equal. Remind them that distances are always positive, so finding the average of these distances to the mean can only be zero if all the data points are exactly at the mean.
Activity Synthesis
During discussion, highlight that finding how far away, on average, the data points are from the mean is a way to describe the variability of a distribution. Discuss:
“What can we say about a data set whose data points have very small distances from the mean?”
“What about a data set with points that show large distances from the mean?”
“Does a data set with smaller distances (and therefore smaller average distances) show less or more variability?”
“What do MAD values of 2, 1.5, and 1 mean in this context?”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary.  Include the following term and maintain the display for reference throughout the unit: mean absolute deviation (MAD). Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Listening, Speaking, Representing: MLR8 Discussion Supports.
To provide a support as students explain their reasoning, invite students to use the sentence frame: “The largest (or smallest) mean absolute deviation value matches with the dot plot with the _____ spread because . . .”. This will help students produce explanations connecting visual and numeric representations of data distribution.
Design Principle(s): Support sense-making


aligns: yes