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: Fluently divide multi-digit numbers using the standard algorithm.
problem_activity: Activity
This activity introduces students to the term
median
. They learn that the median describes the middle value in an ordered list of data, and that it can capture what we consider typical for the data in some cases.
Students learn about the median through a kinesthetic activity. They line up in order of the number of letters in their name. Then, those at both ends of the line count off and sit down simultaneously until one or two people in the middle remain standing. If one person remains standing, that person has the median number of letters. If two people remain standing, the median is the mean or the average of their two values.
Students then practice identifying the median of other data sets, by analyzing both tables of values and dot plots.
Teacher Notes for IM 6–8 Accelerated
The earlier activity mentioned in the second question is not included in IM 6–8 Math Accelerated.
Launch
Explain to students that, instead of using the mean, sometimes we use the “middle” value in an ordered list of data set as a measure of center. We call this the
median
. Guide students through the activity:
Give each student an index card. Ask them to write their first and last names on the card and record the total number of letters in their name. Display an example for all to see.
Ask students to stand up, holding their index cards in front of them, and arrange themselves in order based on the number of letters in their name. (Consider asking students to do so without speaking at all.) Look for the student whose name has the fewest letters and ask him or her to be the left end of the line. Ask the student with the longest full name to be the right end of the line. Students who have the same number of letters should stand side-by-side.
Tell students that, to find the median or the middle number, we will count off from both ends at the same time. Ask the students at the two ends of the line say “1” at the same time and sit on the floor, and the students next to them to say “2” and then sit down, and so on. Have students count off in this fashion until only one or two students are standing.
If the class has an odd number of students, one student will remain standing. Tell the class that this student’s number is the median. Give this student a sign that says “median” If the class has an even number of students, two students will remain standing. The median will be the mean or average of their numbers. Ask both students to hold the sign that says “median.” Explain that the median is also called the “50th percentile,” because half of the data values are the same size or less than it and fall to the left of it on the number line, and half are the same size or greater than it and fall to the right.
Ask students to find the median a couple more times by changing the data set (e.g., asking a few students to leave the line or adding new people who are not members of the class with extremely long or short names). Make sure that students have a chance to work with both odd and even numbers of values.
Collect the index cards and save them; they will be used again in the lesson on box plots.
Ask students to complete the rest of the questions on the task statement.
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: median. 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
Student Facing
Your teacher will give you an index card. Write your first and last names on the card. Then record the total number of letters in your name. After that, pause for additional instructions from your teacher.
Here is the data set on numbers of siblings from an earlier activity.
1
0
2
1
7
0
2
0
1
10
Sort the data from least to greatest, and then find the
median
.
In this situation, do you think the median is a good measure of a typical number of siblings for this group? Explain your reasoning.
Here is the dot plot showing the travel time, in minutes, of Elena’s bus rides to school.
[IMG]
Find the median travel time. Be prepared to explain your reasoning.
What does the median tell us in this context?
Student Response
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Anticipated Misconceptions
When determining the median, students might group multiple data points that have the same value and treat it as a single point, instead of counting each one separately. Remind them that when they lined up to find the median number of letters in their names, every student counted off, even their name has the same number of letters as their neighbors.
Activity Synthesis
Select a few students to share their responses to the questions about number of siblings and Elena's travel times. Focus the discussion on the median as another measure of the center of a data set and whether it captures what students would estimate to be a typical value for each data set.
Emphasize to students that the median is a
value
and not an
individual
. For example, if the last person standing in the class has 5 letters in their first name, the median is the number 5 and not the person standing. If there is another student who had 5 letters in their name, they might have switched places with the last person standing when lining up initially. Although the person standing changed, the median remains the same value of 5.
At this point, it is unnecessary to compare the mean and the median. Students will have many more opportunities to explore the median and think about how it differs from the mean in the upcoming activities.
Writing, Conversing: MLR3 Clarify, Critique, Correct.
Before students share their responses to the questions about number of siblings and Elena's travel times, present an incorrect response. For example, display the statement: “There is no median for this data set because two students remained standing.” Demonstrate the process of interpreting the statements to uncover possible errors by thinking aloud. Voice the questions you ask yourself to call students' attention to the possible types of errors. Invite students to work with a partner to clarify the reasoning, and create an improved statement. Select 2–3 groups to share with the whole class. This helps students evaluate, and improve upon the written mathematical arguments of others.
Design Principle(s): Cultivate conversation; Maximize meta-awarenes


aligns: no