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: 
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_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
problem_activity: Activity
In this lesson, the meaning of the words
model
and
modeling
are explained in terms of a linear model. Students are not expected to define the words, but should be comfortable understanding and using them. A model is used to predict prices for diamonds not included in the data as well as compare existing data points to the model.
Digital
Launch
Keep students in groups of 2. Allow students 5 minutes quiet work time followed by partner and whole-class discussion.
[IMG]
Show the first image for all to see. Ask students, “Which diamond do you think has the highest price? Which diamond do you think has the lowest price?” Most likely, they will guess that the largest has the highest price and the smallest has the lowest price. Validate this intuition and let them know that the size of a diamond is
one
thing that its price is based on. Tell students that the size of a diamond is commonly measured in
carats
, which is a measure of weight. (One carat is equivalent to 200 milligrams.)
[IMG]
Show the second image for all to see. Ask students, “Which diamond do you think has the higher price?” Poll the class, and ask a few students to share their reasoning. Tell students that these diamonds are the same size, but they are not the same price. The price of a diamond is not just based on size, but also on how many flaws are inside the stone. Since the diamond on the right has fewer flaws (which makes it more sparkly), it has a higher price than the diamond on the left.
Tell students, “You will look at some data for the prices of diamonds of different sizes. You have used mathematics to analyze real-world situations, identifying variables in a situation and describing their relationships mathematically. This process is called
modeling,
and the mathematical description is called a
model
. Sometimes you made assumptions about the situation or ignored some features so that the model would be simpler.”
Action and Expression: Develop Expression and Communication
. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “It looks like…” and “We are trying to…”
Supports accessibility for: Language; Organization
Speaking: MLR8 Discussion Supports.
Provide sentence frames to support students as they work with their partner and explain their reasoning. For example, “I think ____ because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Here is a table that shows weights and prices of 20 different diamonds.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "weight (carats)"}, {"_attributes": {"scope": "col"}, "_value": "actual price (dollars)"}, {"_attributes": {"scope": "col"}, "_value": "predicted price (dollars)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "3,772"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "4,221"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "4,032"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "5,385"}, {"_value": "4,429"}]}, {"td": [{"_value": "1.05"}, {"_value": "3,942"}, {"_value": "4,705"}]}, {"td": [{"_value": "1.05"}, {"_value": "4,480"}, {"_value": "4,705"}]}, {"td": [{"_value": "1.06"}, {"_value": "4,511"}, {"_value": "4,760"}]}, {"td": [{"_value": "1.2"}, {"_value": "5,544"}, {"_value": "5,533"}]}, {"td": [{"_value": "1.3"}, {"_value": "6,131"}, {"_value": "6,085"}]}, {"td": [{"_value": "1.32"}, {"_value": "5,872"}, {"_value": "6,195"}]}, {"td": [{"_value": "1.41"}, {"_value": "7,122"}, {"_value": "6,692"}]}, {"td": [{"_value": "1.5"}, {"_value": "7,474"}, {"_value": "7,189"}]}, {"td": [{"_value": "1.5"}, {"_value": "5,904"}, {"_value": "7,189"}]}, {"td": [{"_value": "1.59"}, {"_value": "8,706"}, {"_value": "7,686"}]}, {"td": [{"_value": "1.61"}, {"_value": "8,252"}, {"_value": "7,796"}]}, {"td": [{"_value": "1.73"}, {"_value": "9,530"}, {"_value": "8,459"}]}, {"td": [{"_value": "1.77"}, {"_value": "9,374"}, {"_value": "8,679"}]}, {"td": [{"_value": "1.85"}, {"_value": "8,169"}, {"_value": "9,121"}]}, {"td": [{"_value": "1.9"}, {"_value": "9,541"}, {"_value": "9,397"}]}, {"td": [{"_value": "2.04"}, {"_value": "9,125"}, {"_value": "10,170"}]}]}]}]}
The scatter plot shows the prices and weights of the 20 diamonds together with the graph of
\(y = 5,\!520x- 1,\!091\)
.
The function described by the equation
\(y = 5,\!520x- 1,\!091\)
is a
model
of the relationship between a diamond’s weight and its price.
This model
predicts
the price of a diamond from its weight. These predicted prices are shown in the third column of the table.
Two diamonds that both weigh 1.5 carats have different prices. What are their prices? How can you see this in the table? How can you see this in the graph?
The
model
predicts that when the weight is 1.5 carats, the price will be
$
7,189. How can you see this in the graph? How can you see this using the equation?
One of the diamonds weighs 1.9 carats. What does the model predict for its price? How does that compare to the actual price?
Find a diamond for which the model makes a very good prediction of the actual price. How can you see this in the table? In the graph?
Find a diamond for which the model’s prediction is not very close to the actual price. How can you see this in the table? In the graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in groups of 2. Allow students 5 minutes quiet work time followed by partner and whole-class discussion.
[IMG]
Show the first image for all to see. Ask students, “Which diamond do you think has the highest price? Which diamond do you think has the lowest price?” Most likely, they will guess that the largest has the highest price and the smallest has the lowest price. Validate this intuition and let them know that the size of a diamond is
one
thing that its price is based on. Tell students that the size of a diamond is commonly measured in
carats
, which is a measure of weight. (One carat is equivalent to 200 milligrams.)
[IMG]
Show the second image for all to see. Ask students, “Which diamond do you think has the higher price?” Poll the class, and ask a few students to share their reasoning. Tell students that these diamonds are the same size, but they are not the same price. The price of a diamond is not just based on size, but also on how many flaws are inside the stone. Since the diamond on the right has fewer flaws (which makes it more sparkly), it has a higher price than the diamond on the left.
Tell students, “You will look at some data for the prices of diamonds of different sizes. You have used mathematics to analyze real-world situations, identifying variables in a situation and describing their relationships mathematically. This process is called
modeling,
and the mathematical description is called a
model
. Sometimes you made assumptions about the situation or ignored some features so that the model would be simpler.”
Action and Expression: Develop Expression and Communication
. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “It looks like…” and “We are trying to…”
Supports accessibility for: Language; Organization
Speaking: MLR8 Discussion Supports.
Provide sentence frames to support students as they work with their partner and explain their reasoning. For example, “I think ____ because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Here is a table that shows weights and prices of 20 different diamonds.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "weight (carats)"}, {"_attributes": {"scope": "col"}, "_value": "actual price (dollars)"}, {"_attributes": {"scope": "col"}, "_value": "predicted price (dollars)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {"_value": "3,772"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "4,221"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "4,032"}, {"_value": "4,429"}]}, {"td": [{"_value": "1"}, {"_value": "5,385"}, {"_value": "4,429"}]}, {"td": [{"_value": "1.05"}, {"_value": "3,942"}, {"_value": "4,705"}]}, {"td": [{"_value": "1.05"}, {"_value": "4,480"}, {"_value": "4,705"}]}, {"td": [{"_value": "1.06"}, {"_value": "4,511"}, {"_value": "4,760"}]}, {"td": [{"_value": "1.2"}, {"_value": "5,544"}, {"_value": "5,533"}]}, {"td": [{"_value": "1.3"}, {"_value": "6,131"}, {"_value": "6,085"}]}, {"td": [{"_value": "1.32"}, {"_value": "5,872"}, {"_value": "6,195"}]}, {"td": [{"_value": "1.41"}, {"_value": "7,122"}, {"_value": "6,692"}]}, {"td": [{"_value": "1.5"}, {"_value": "7,474"}, {"_value": "7,189"}]}, {"td": [{"_value": "1.5"}, {"_value": "5,904"}, {"_value": "7,189"}]}, {"td": [{"_value": "1.59"}, {"_value": "8,706"}, {"_value": "7,686"}]}, {"td": [{"_value": "1.61"}, {"_value": "8,252"}, {"_value": "7,796"}]}, {"td": [{"_value": "1.73"}, {"_value": "9,530"}, {"_value": "8,459"}]}, {"td": [{"_value": "1.77"}, {"_value": "9,374"}, {"_value": "8,679"}]}, {"td": [{"_value": "1.85"}, {"_value": "8,169"}, {"_value": "9,121"}]}, {"td": [{"_value": "1.9"}, {"_value": "9,541"}, {"_value": "9,397"}]}, {"td": [{"_value": "2.04"}, {"_value": "9,125"}, {"_value": "10,170"}]}]}]}]}
The scatter plot shows the prices and weights of the 20 diamonds together with the graph of
\(y = 5,\!520x- 1,\!091\)
.
[IMG]
The function described by the equation
\(y = 5,\!520x- 1,\!091\)
is a
model
of the relationship between a diamond’s weight and its price.
This model
predicts
the price of a diamond from its weight. These predicted prices are shown in the third column of the table.
Two diamonds that both weigh 1.5 carats have different prices. What are their prices? How can you see this in the table? How can you see this in the graph?
The model predicts that when the weight is 1.5 carats, the price will be
$
7,189. How can you see this in the graph? How can you see this using the equation?
One of the diamonds weighs 1.9 carats. What does the model predict for its price? How does that compare to the actual price?
Find a diamond for which the model makes a very good prediction of the actual price. How can you see this in the table? In the graph?
Find a diamond for which the model’s prediction is not very close to the actual price. How can you see this in the table? In the graph?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is to help students understand the relationship between the data and a linear model of the data.
To highlight there relationship, ask:
"What does a point in the scatter plot represent?" (The
actual
weight and price of a diamond.)
"What does the line represent?" (The
predicted
price of a diamond based on its weight.)
"What does it mean when a point is close to the line? When it is far away from the line?" (When it is close, the model predicts the price well. The farther away in the vertical direction, the worse the prediction.)
"How can you use the graph to predict the price of a diamond that weighs 1.1 carats? How can you use the equation?" (Find the point on the line that lines up with
\(x = 1.1\)
or substitute
\(x = 1.1\)
into the model equation.)


aligns: no