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: Work with addition and subtraction equations.

Solve systems of equations.

Work with equal groups of objects to gain foundations for multiplication.

Understand the connections between proportional relationships, lines, and linear equations.

Generate and analyze patterns.

Perform arithmetic operations on polynomials.

Write expressions in equivalent forms to solve problems.

Use the four operations with whole numbers to solve problems.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Work with radicals and integer exponents.

Solve equations and inequalities in one variable.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Create equations that describe numbers or relationships

Understand properties of multiplication and the relationship between multiplication and division.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and solve problems involving multiplication and division.

Represent and solve problems involving addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Use properties of operations to generate equivalent expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use polynomial identities to solve problems.

Analyze patterns and relationships.

Understand the relationship between zeros and factors of polynomials.

Represent and solve equations and inequalities graphically.

Add and subtract within 20.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Interpret the structure of expressions.
problem_activity: Activity
In this activity, students first encounter exponential expressions with variables.
Launch
Recall with students that when we write
\(6x^2\)
, we mean to multiply 6 by the result of
\(x^2\)
. Remind them that the number part of such a product is called the
coefficient
of the expression, so in this example, 6 is the coefficient of
\(x^2\)
.
Give students 5 minutes of quiet work time to evaluate the expressions. Follow with whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have evaluated the first 2-3 expressions, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
Evaluate each expression for the given value of
\(x\)
.
\(3x^2\)
when
\(x\)
is 10
\(3x^2\)
when
\(x\)
is
\(\frac19\)
\(\frac{x^3}{4}\)
when
\(x\)
is 4
\(\frac{x^3}{4}\)
when
\(x\)
is
\(\frac12\)
\(9+x^7\)
when
\(x\)
is 1
\(9+x^7\)
when
\(x\)
is
\(\frac12\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may use the wrong order when evaluating expressions, such as in multiplying by 3 first in
\(3x^2\)
and then squaring. Have them recall the last lesson where they practiced applying the agreed on order of operations in numeric expressions.
Students may interpret an expression like
\(3x\)
as meaning 3 next to the digit
\(x\)
instead of as multiplication. For example, students may think that
\(3x\)
means 35 instead of 15 when
\(x=5\)
. Discuss how the shorthand notation of coefficients next to variables with no symbols between them tells us to multiply, and that it will simplify future work with expressions and equations.
Activity Synthesis
The purpose of the discussion is to ensure that students understand how to evaluate expressions with variables for a given value of the variable. It is also an opportunity for students to practice interpreting and using vocabulary like
coefficient, variable,
power,
and
exponent
.
Some guiding questions:
"In each expression, what is the coefficient?" (3, 3,
\(\frac14\)
,
\(\frac14\)
, 1, 1.)
"Choose one of the expressions. Describe the steps that you carried out to evaluate the expression." (Sample response: I rewrote
\(9+x^7\)
as
\(9+\left(\frac12\right)^7\)
. I needed to first evaluate the exponent. I knew that
\(\left(\frac12\right)^7\)
means
\(\frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12\)
, which is
\(\frac{1}{128}\)
. So then I wrote
\(9 + \frac{1}{128}\)
, which can also be written
\(9\frac{1}{128}\)
.
"How is evaluating the expressions when
\(x\)
is a fraction similar to when
\(x\)
is a whole number?  How is it different?" (It's similar because you're still just multiplying
\(x\)
by itself a certain number of times. It's different because multiplying a fraction by a fraction is a bit more complicated than multiplying a whole number by itself.)
Speaking, Writing: MLR8 Discussion supports.
Use this to amplify mathematical uses of language to communicate about how to evaluate exponential expressions. Revoice the term “coefficient” during student discussion and press for detail when identifying the coefficients and the order in which they are multiplied in each expression.
Design Principle(s): Support sense-making


correct_option_index: 24