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: Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

Solve systems of equations.

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

Work with addition and subtraction equations.

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

Represent and solve problems involving addition and subtraction.

Analyze patterns and relationships.

Generate and analyze patterns.

Multiply and divide within 100.

Write and interpret numerical expressions.

Write expressions in equivalent forms to solve problems.

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

Use the four operations with whole numbers to solve problems.

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

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

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

Reason about and solve one-variable equations and inequalities.

Use properties of operations to generate equivalent expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Represent and solve equations and inequalities graphically.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

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

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

Work with radicals and integer exponents.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Gain familiarity with factors and multiples.

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

Rewrite rational expressions.

Use polynomial identities to solve problems.
problem_activity: Activity
This activity allows students to formulate a mathematical model around the framing task they saw earlier (MP4). Students are prompted to write an equation but are not expected to solve it at this point. In writing an equation and interpreting the solution in context, students practice reasoning quantitatively and abstractly (MP2).
The prompt to write an equation is left relatively open to allow for different approaches. For instance, students may think in terms of:
The length and width of the framed picture. The length is 7 inches plus the thickness of the frame,
\(x\)
, on either side, which gives
\(7 + 2x\)
. The height is 4 inches plus the same thickness,
\(x\)
, for the top and bottom, which gives
\(4 + 2x\)
. The total area is
\(7 \boldcdot 4 + 4 \boldcdot (2.5)\)
or 38 square inches, so the equation is
\((7+2x)(4+2x) = 38\)
.
The area of the picture plus the area of the frame. The area of the picture is 28 square inches. The area of the frame, in square inches, can be found by decomposing it into four squares (in the corners) that are
\(x\)
by
\(x\)
or
\(x^2\)
each, two rectangles that are
\(7x\)
each (top and bottom), and two rectangles that are
\(4x\)
each (left and right). The equation is
\(4x^2 + 2(7x) + 2(4x) + 28 = 38\)
or
\(4x^2 + 22x + 28= 38\)
.
As students work, monitor for different strategies students use. Invite students with contrasting approaches to share later.
Launch
Tell students that they are now to write an equation to represent the quantities in the framing problem. Consider arranging students in groups of 2 and asking them to think quietly about the questions before conferring with their partner.
If students have trouble getting started, suggest that they start by labeling the diagram with relevant lengths. Then, if needed, use scaffolding questions such as:
“How could we show that the frame is the same thickness all the way around?” (Use the same variable, for example
\(x\)
, to label the thickness of the strip of framing on all four sides.)
“What is the combined area of the picture and the framing material?” (38 square inches, because
\(7 \boldcdot 4 + 4 \boldcdot (2.5)=38\)
)
“How could we express the width of the framed picture?” (
\(7+2x\)
) “What about the height of the framed picture?” (
\(4+2x\)
)
“Once the framing material is cut up and arranged around the picture, how could we express its area?” (Two rectangles with area
\(x(7+2x)\)
and two rectangles with area
\(4x\)
, or
\(2x(7+2x) + 2(4x)\)
)
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (creating a frame for a rectangular picture). If students understand the situation based on their work in the prior activity, skip to the next read. After the second read, students list any quantities that can be counted or measured, without focusing on specific values (dimensions of the picture and framing materials). During the third read, the question or prompt is revealed. Invite students to brainstorm possible strategies, referencing the relevant quantities named after the second read. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams. Provide students with graph paper and suggest that they draw and label their diagrams with everything they know so far. Invite students to suggest types of diagrams that might be helpful to draw, such as the strips of framing material, and the picture both with and without the frame.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Here is a diagram that shows the picture with a frame that is the same thickness all the way around. The picture is 7 inches by 4 inches. The frame is created from 10 square inches of framing material (in the form of a rectangle measuring 4 inches by 2.5 inches).
[IMG]
Write an equation to represent the relationship between the measurements of the picture and of the frame, and the area of the framed picture. Be prepared to explain what each part of your equation represents.
What would a solution to this equation mean in this situation?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may struggle to express the overall length and width of the framed picture because of trouble combining numbers and variables. Consider drawing a segment composed of 3 pieces of length 1.5 inches, 10 inches, and 1.5 inches, and prompting students to find the length of the entire segment. Then, change each 1.5 to an
\(x\)
and ask for an expression for the length of the entire segment (
\(10+2x\)
).
If students mistake the result of adding
\(x\)
and
\(x\)
as
\(x^2\)
, consider drawing a square with side length
\(x\)
and ask students to write expressions for the perimeter and area. Then, ask them to point out the difference between
multiplying
to find area and
adding
to find total length. If needed, draw a few more segments that are decomposed into parts, with each part labeled with a number or a variable expression, and ask students to write expressions for the total length.
Activity Synthesis
Select students to present their equations and reasoning. If no students bring up one of the strategies shown in the Activity Narrative, bring it up. Consider asking students whether
\((7+2x)(4+2x)=38\)
and
\(4x^2 + 22x + 28 = 38\)
(or other correct equations that come up) are equivalent and how to check if they are.
Discuss what a solution to these equations represents. Make sure students understand that a solution reveals the thickness of the frame when all of the framing material (10 square inches) is used.
Solicit some ideas on how they might go about finding the solution(s). Students may mention:
graphing the equation
\(y=(7+2x)(4+2x)\)
and finding a point on the graph with a
\(y\)
-coordinate of 38
evaluating the expression on one side of the equation at different values of
\(x\)
until it has a value of 38
using a spreadsheet to evaluate the expression at different values of
\(x\)
and see what values produce an output of 38
We can see that the equations
\((7+2x)(4+2x)=38\)
and
\(4x^2 + 22x + 28 = 38\)
are each composed of a
quadratic expression
. Tell students that these are examples of
quadratic equations
. They will learn several techniques for using algebra to solve equations like this.
Formally, a quadratic equation is defined as one that can be written in the form of
\(ax^2+bx+c=0\)
where
\(a\)
is not 0. This formal definition will be introduced in the next lesson, when students work with equations of that form and make sense of them in context.


correct_option_index: 16