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: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
problem_activity: Activity
In a previous lesson, students had a primer on histograms—how they are drawn, how they differ from dot plots, and what information they can tell us. In this activity, students practice drawing a histogram for a given data set and using it to answer statistical questions. To help students understand the lengths involved in the data set, students are asked to draw various lengths used to group the worms in the first histogram.
As students organize the data set and draw their histogram, notice any challenges or questions students come across. If a question is raised by multiple groups, consider discussing it with the whole class. Also pay attention to how students use the histogram to identify a “typical” length—some might describe it in terms of the size of the bins (e.g., “a typical length is between 20 and 40 mm”); others might choose a value within a bin or a boundary between bins. Invite them to share their reasonings later.
Launch
Arrange students in groups of 2. Provide access to centimeter rulers.
Consider giving students a brief overview of the context for the problems in the activity. Tell students that there are nearly 6,000 species of earthworms in the world.  Some earthworms help the environment, while others (generally not native to the region in which they are found) may harm the environment.  Earthworms that are native to a particular region of the world are often raised, by farmers, in terrariums (a container or bin similar to an aquarium but it contains soil and leaves). The terrarium-raised earthworms provide bait for people who fish, provide food for various wildlife, and decompose food waste into soil.  Food waste and water are added to the terrariums as food for raising and growing worms.  Soil produced by the worms as they eat the food waste is often used as fertilizer.
Explain that the lengths of the worms in the bins provide information about the ages of the worms, which can be useful for the farmer. In this activity, students will organize the lengths of the earthworms in several terrariums or bins.
Give students 8–10 minutes of quiet work time, and then 3–4 minutes to discuss their work and complete the activity with a partner.
Action and Expression: Internalize Executive Functions.
Activate or supply background knowledge by demonstrating how to use a ruler to draw a line segment for a given length.
Supports accessibility for: Memory; Conceptual processing
Student Facing
An earthworm farmer set up several containers of a certain species of earthworms so that he could learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measured the lengths of 25 earthworms in one of the containers. Each length was measured in millimeters.
[IMG]
Using a ruler, draw a line segment for each length:
20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters
Here are the lengths, in millimeters, of the 25 earthworms.
6
11
18
19
20
23
23
25
25
26
27
27
28
29
32
33
41
42
48
52
54
59
60
77
93
Complete the table for the lengths of the 25 earthworms.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "length"}, {"_attributes": {"scope": "col"}, "_value": "frequency"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0 millimeters to less than 20 millimeters"}, {}]}, {"td": [{"_value": "20 millimeters to less than 40 millimeters"}, {}]}, {"td": [{"_value": "40 millimeters to less than 60 millimeters"}, {}]}, {"td": [{"_value": "60 millimeters to less than 80 millimeters"}, {}]}, {"td": [{"_value": "80 millimeters to less than 100 millimeters"}, {}]}]}]}]}
Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.
[IMG]
Based on the histogram, what is a typical length for these 25 earthworms? Explain how you know.
Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?
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?
Here is another histogram for the earthworm measurement data. In this histogram, the measurements are in different groupings.
[IMG]
Based on this histogram, what is your estimate of a typical length for the 25 earthworms?
Compare this histogram with the one you drew. How are the distributions of data summarized in the two histograms the same? How are they different?
Compare your estimates of a typical earthworm length for the two histograms. Did you reach different conclusions about a typical earthworm length from the two histograms?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
When determining frequencies of data values, students might lose track of their counting. Suggest that they use tally marks to keep track of the number of occurrences for each bin.
When drawing the histogram, students might mistakenly use bar graphs as a reference and leave spaces between the bars. Ask them to look at the bars in other histograms they have seen so far and to think about what the gaps might mean considering that the bars are built on a number line.
Activity Synthesis
Ask one or two students to display their completed histograms for all to see and briefly describe the overall distribution. Then, select a few other previously identified students to share their responses and explanations for the last two questions.
Focus the discussion on how identifying center and spread using a histogram is different than doing so using a dot plot. Discuss:
“In a histogram, are we able to see clusters of values in the distribution?”
“Can we see the largest and smallest values? Can we tell the overall spread?”
“How do we identify the center of a distribution?”
From the various estimates that students give for a typical earthworm length (and from earlier exercises), students should begin to see that identifying a typical value of a distribution is not a straightforward or precise process so far. Explain that in upcoming lessons they will explore how to describe a typical value and characterize a distribution more systematically.
Representing, Conversing: MLR7 Compare and Connect.
As the selected students show and describe their histograms, invite pairs to discuss: “What is the same and what is different?” about their own histograms and reasoning. This will help students better understand there are multiple valid ways for describing center and spread.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: no