few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Understand independence and conditional probability and use them to interpret data.

Summarize, represent, and interpret data on a single count or measurement variable.

Develop understanding of statistical variability.

Investigate chance processes and develop, use, and evaluate probability models.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use probability to evaluate outcomes of decisions.

Calculate expected values and use them to solve problems.

Interpret linear models.

Summarize and describe distributions.

Understand and evaluate random processes underlying statistical experiments.

Use random sampling to draw inferences about a population.

Investigate patterns of association in bivariate data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Draw informal comparative inferences about two populations.
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.
| weight (carats) | actual price (dollars) | predicted price (dollars) |
|-----------------|------------------------|---------------------------|
|        1        |         3,772          |           4,429           |
|        1        |         4,221          |           4,429           |
|        1        |         4,032          |           4,429           |
|        1        |         5,385          |           4,429           |
|      1.05       |         3,942          |           4,705           |
|      1.05       |         4,480          |           4,705           |
|      1.06       |         4,511          |           4,760           |
|       1.2       |         5,544          |           5,533           |
|       1.3       |         6,131          |           6,085           |
|      1.32       |         5,872          |           6,195           |
|      1.41       |         7,122          |           6,692           |
|       1.5       |         7,474          |           7,189           |
|       1.5       |         5,904          |           7,189           |
|      1.59       |         8,706          |           7,686           |
|      1.61       |         8,252          |           7,796           |
|      1.73       |         9,530          |           8,459           |
|      1.77       |         9,374          |           8,679           |
|      1.85       |         8,169          |           9,121           |
|       1.9       |         9,541          |           9,397           |
|      2.04       |         9,125          |          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.
| weight (carats) | actual price (dollars) | predicted price (dollars) |
|-----------------|------------------------|---------------------------|
|        1        |         3,772          |           4,429           |
|        1        |         4,221          |           4,429           |
|        1        |         4,032          |           4,429           |
|        1        |         5,385          |           4,429           |
|      1.05       |         3,942          |           4,705           |
|      1.05       |         4,480          |           4,705           |
|      1.06       |         4,511          |           4,760           |
|       1.2       |         5,544          |           5,533           |
|       1.3       |         6,131          |           6,085           |
|      1.32       |         5,872          |           6,195           |
|      1.41       |         7,122          |           6,692           |
|       1.5       |         7,474          |           7,189           |
|       1.5       |         5,904          |           7,189           |
|      1.59       |         8,706          |           7,686           |
|      1.61       |         8,252          |           7,796           |
|      1.73       |         9,530          |           8,459           |
|      1.77       |         9,374          |           8,679           |
|      1.85       |         8,169          |           9,121           |
|       1.9       |         9,541          |           9,397           |
|      2.04       |         9,125          |          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.)


correct_option_index: 11