few_shots: Example 1:
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

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


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


Thought: This task asks students to solve an equation in one variable. Since the problem asks for many different solution methods, students are also encouraged to solve the problem graphically.
Answer: P, U

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Mathematical concept/skill options: 
A. Apply trigonometry to general triangles.
B. Understand similarity in terms of similarity transformations.
C. Analyze, compare, create, and compose shapes.
D. Prove geometric theorems.
E. Make geometric constructions.
F. Graph points on the coordinate plane to solve real-world and mathematical problems.
G. Understand congruence and similarity using physical models, transparencies, or geometry software.
H. Understand congruence in terms of rigid motions.
I. Draw, construct, and describe geometrical figures and describe the relationships between them.
J. Translate between the geometric description and the equation for a conic section.
K. Reason with shapes and their attributes.
L. Prove theorems involving similarity.
M. Explain volume formulas and use them to solve problems.
N. Find arc lengths and areas of sectors of circles.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Use coordinates to prove simple geometric theorems algebraically.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
S. Understand and apply theorems about circles.
T. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
U. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
V. Classify two-dimensional figures into categories based on their properties.
W. Apply geometric concepts in modeling situations.
X. Solve real-world and mathematical problems involving area, surface area, and volume.
Y. Experiment with transformations in the plane.
Z. Understand and apply the Pythagorean Theorem.
AA. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.


Thought: This task asks students to make geometric constructions, such as bisecting an angle.
Answer: E

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Use probability to evaluate outcomes of decisions.

Calculate expected values and use them to solve problems.

Draw informal comparative inferences about two populations.

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

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

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

Summarize and describe distributions.

Understand and evaluate random processes underlying statistical experiments.

Interpret linear models.

Investigate patterns of association in bivariate data.

Develop understanding of statistical variability.

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

Understand independence and conditional probability and use them to interpret data.

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

Use random sampling to draw inferences about a population.
problem_activity: Activity
The mathematical purpose of this activity is to use a two-way table as a sample space to decide if events are independent and to estimate conditional probabilities.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Give students 10 minutes of quiet time to work the problems and then pause for a brief whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies. Some students may benefit from a checklist or list of steps to be able to use the spreadsheet technology.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
A pharmaceutical company is testing a new medicine for a disease using 115 test subjects. Some of the test subjects are given the new medicine and others are given a placebo. The results of their tests are summarized in the table.
|                | no more symptoms | symptoms persist | total |
|----------------|------------------|------------------|-------|
| given medicine |                  |                  |       |
| given placebo  |                  |                  |       |
|     total      |                  |                  |       |
Divide the value in each cell by the total number of test subjects to find each probability to two decimal places. Some of the values have been completed for you.
|                | no more symptoms | symptoms persist | total |
|----------------|------------------|------------------|-------|
| given medicine |                  |                  |       |
| given placebo  |                  |                  |       |
|     total      |                  |                  |       |
If one of these test subjects is selected at random, find each probability:
\(P(\text{symptoms persist})\)
\(P(\text{given medicine and symptoms persist})\)
\(P(\text{given placebo or symptoms persist})\)
From the original table, divide each cell by the total for the row to find the probabilities with row conditions. Some of the values have been completed for you.
|                | no more symptoms | symptoms persist | total |
|----------------|------------------|------------------|-------|
| given medicine |                  |                  |       |
| given placebo  |                  |                  |       |
\(P(\text{symptoms persist | given medicine})\)
\(P(\text{no more symptoms | given placebo})\)
Jada didn’t read the instructions for the previous problem well and used the table she created on the first problem to divide each cell by the probability total for each row. For example, in the top left cell she calculated
\(0.27 \div 0.5\)
. Complete the table using Jada’s method.
|                | no more symptoms | symptoms persist | total |
|----------------|------------------|------------------|-------|
| given medicine |                  |                  |       |
| given placebo  |                  |                  |       |
What do you notice about this table?
From the original table, divide each cell by the total for the column to find the probabilities with column conditions. Some of the values have been completed for you.
|                | no more symptoms | symptoms persist |
|----------------|------------------|------------------|
| given medicine |                  |                  |
| given placebo  |                  |                  |
|     total      |                  |                  |
\(P(\text{given medicine | symptoms persist})\)
\(P(\text{given placebo | no more symptoms})\)
Are the events “symptoms persist” and “given medicine” independent events? Explain or show your reasoning.
Based on your work, does being given this medicine have an impact on whether symptoms persist or not?
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?
Consider the data collected from all students in grades 11 and 12 and their intentions of going to prom.
65% of the students who are going to prom are from grade 12. Complete the table.
|          | going to prom | not going to prom | total |
|----------|---------------|-------------------|-------|
| grade 11 |               |                   |       |
| grade 12 |               |                   |       |
|  total   |               |                   |       |
Based on your results, what is the probability that a student from this group, selected at random, is a student in grade 11 that did not go to prom?
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
Students might struggle with understanding the notation for conditional probability. Tell these students that
\(P(\text{A | B})\)
means the probability of A happening under the condition that B happens.
Activity Synthesis
The purpose of this discussion is for students to interpret probabilities in a two-way table.
Ask students to interpret each of the probabilities they found in questions 1, 2, and 4. For example, “What does the probability in question 1a mean in the context of the problem?”
Here are sample interpretations of the probabilities for questions 1, 2, and 4.
The probability that a test subject chosen at random from all of the subjects still had symptoms after the test.
The probability that a test subject chosen at random from all of the subjects was among those given medicine and still had symptoms after the test.
The probability that a test subject chosen at random from all of the subjects still had symptoms after the test or was given a placebo. 1
The probability that a test subject chosen at random from those who were given medicine had symptoms after the test.
The probability that a test subject chosen at random from those who were given placebo had no more symptoms after the test
The probability that a test subject chosen at random from those who had symptoms after the test was given medicine.
The probability that a test subject chosen at random from those who had no more symptoms after the test was given placebo.
Display what Jada did when creating the top left cell in table for all to see:
\(\frac{P(\text{no more symptoms and given medicine})}{P(\text{given medicine})}\)
Here are some questions for discussion.
“What do you notice about what Jada did and
\(P(\text{no more symptoms | given medicine})\)
?” (I noticed that they are equal.)
“What is an equation that represents this situation using A and B for the events?” (
\(P(\text{A | B}) = \frac{P(\text{A and B})}{P(\text{B})}\)
)
"What is the relationship between this equation and the multiplication rule?" (From the multiplication rule, we know that
\(P(\text{A and B}) = P(\text{A | B}) \boldcdot P(\text{B})\)
. If we divide both sides by
\(P(\text{B})\)
, we get the equation that Jada used.)
“How did you determine whether the events were dependent or independent?” (I determined that they were dependent events because
\(P(\text{symptoms persist}) = 0.6\)
and
\(P(\text{symptoms persist | given medicine}) = 0.46\)
were not equal. If the events were independent, these probabilities would be equal.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 12