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

Work with addition and subtraction equations.

Use properties of operations to generate equivalent expressions.

Generate and analyze patterns.

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

Perform arithmetic operations on polynomials.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Represent and solve problems involving multiplication and division.

Solve equations and inequalities in one variable.

Interpret the structure of expressions.

Work with radicals and integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Solve systems of equations.

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

Represent and solve equations and inequalities graphically.

Reason about and solve one-variable equations and inequalities.

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

Analyze patterns and relationships.

Represent and solve problems involving addition and subtraction.

Use the four operations with whole numbers to solve problems.

Use polynomial identities to solve problems.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

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

Understand the relationship between zeros and factors of polynomials.

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

Rewrite rational expressions.

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

Add and subtract within 20.
problem_activity: Optional activity
This activity is optional. It shows three different methods for solving an equation: by rewriting it in factored form, by transforming it into an expression in which the squared term has a coefficient 1 and then rewriting in factored form, and by completing the square. The second method involves temporarily substituting a part of the expression with another variable so that the squared term has a coefficient of 1, which makes it easier to rewrite in factored form. Students encountered this strategy in an optional activity (“Making It Simpler”) in an earlier lesson. If students did not do that optional activity, consider doing it first before using this activity.
The numbers in the equations here do not lend themselves to be easily rewritten in factored form (without guessing and checking or substitution), or to be easily rewritten into perfect squares (because the leading coefficient and the constant terms are not perfect squares), but students see that these equations can be solved with these strategies. The activity further illustrates that any of these methods can be rather tedious, providing further motivation to solve using the quadratic formula, which students will learn in a future lesson.
Launch
Display the perfect squares students saw in the preceding activity. Ask them what they notice about the coefficient of the square term in each expression.
\(\displaystyle 100x^2 + 80x + 16\)
\(\displaystyle 36x^2 - 60x + 25\)
\(\displaystyle 25x^2 + 40x + 16\)
\(\displaystyle 0.25x^2- 14x + 196\)
Students are likely to notice that the coefficient of every
\(x^2\)
is a perfect square, which made it easier to rewrite the completed square as squared factors. Solicit some ideas about whether we could still complete the square if a quadratic expression does not have a perfect square for the coefficient of
\(x^2\)
. Then, tell students that they will explore a few methods for solving such equations.
Arrange students in groups of 2. Ask one partner to study the first two methods and the other partner to study the third method, and then take turns explaining their understanding to each other. Discuss the methods, especially the third one, before students begin using them to solve equations.
Make sure students see that, in both the second and third methods, the first step involves multiplying both sides of the equation by 3 to make the coefficient of
\(x^2\)
a perfect square. Each term on the left side of the equation changes by a factor of 3, but the right side of the equation remains 0 because multiplying 0 by any number results in 0.
If time is limited, consider asking students to solve only three equations, using each method once.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, cut the three methods apart into separate pieces. Begin by presenting one method at a time and invite students to annotate the examples as they analyze the steps. After students annotate the first method, invite students who have great clarity in their annotations to share their work, and display for all to see. Encourage students to refer to their annotations as they approach the last problems themselves.
Supports accessibility for: Memory; Organization
Student Facing
Here are three methods for solving
\(3x^2 + 8x + 5 = 0\)
.
Try to make sense of each method.
Method 1:
\(\displaystyle \begin {align}3x^2 + 8x + 5 &= 0\\ (3x + 5)(x + 1) &= 0 \end{align}\)
\(\displaystyle \begin {align} x = \text- \frac53 \quad \text{or} \quad x = \text-1\end {align}\)
Method 2:
\(\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\ (3x)^2 + 8(3x) + 15 &= 0\\ U^2 + 8U + 15 &= 0\\ (U+5)(U+3) &= 0 \end{align}\)
\(\displaystyle \begin {align} U = \text-5 \quad &\text{or} \quad U = \text-3\\3x = \text-5 \quad &\text{or} \quad 3x = \text-3\\ x = \text- \frac53 \quad &\text{or} \quad x = \text-1 \end{align}\)
Method 3:
\(\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\9x^2 + 24x + 16 &= 1\\(3x + 4)^2 &= 1 \end{align}\)
\(\displaystyle \begin {align} 3x+4 = 1 \quad & \text{or} \quad 3x+4 = \text-1\\x = \text-1 \quad & \text{or} \quad x = \text- \frac53 \end {align}\)
Once you understand the methods, use each method at least one time to solve these equations.
\(5x^2 + 17x + 6 = 0\)
\(6x^2 + 19x = \text-10\)
\(8x^2 - 33x + 4 = 0\)
\(8x^2 - 26x = \text-21\)
\(10x^2 + 37x = 36\)
\(12x^2 + 20x - 77=0\)
Student Response
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Student Facing
Are you ready for more?
Find the solutions to
\(3x^2 -6x + \frac{9}{4} = 0\)
. Explain your reasoning.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Consider asking students who solve the same equation using different methods to compare and contrast their solution strategies. Then, invite them to reflect on the solving process:
“Did the numbers in certain equations make it easier to solve by one method versus the others?”
“What are the advantages of each method? What about its drawbacks?”
“Is there a strategy you prefer or find reliable? Which one and why?”
Tell students that in upcoming lessons, they will look at a more efficient method for solving quadratic equations.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Before addressing the two questions, “What are the advantages of each method? What about its drawbacks?”, give students 2–3 minutes to plan what they will say when they present their responses to the class. Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. If needed, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making


correct_option_index: 9