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 properties of rational and irrational numbers.

Reason quantitatively and use units to solve problems.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on vectors.

Represent and model with vector quantities.

Perform arithmetic operations with complex numbers.

Extend the properties of exponents to rational exponents.

Perform operations on matrices and use matrices in applications.

Use complex numbers in polynomial identities and equations.

Represent complex numbers and their operations on the complex plane.

Compute fluently with multi-digit numbers and find common factors and multiples.
problem_activity: Activity
This task helps to transition students from thinking about “how many groups?” to “what fraction of a group?”.
Students compare different lengths of ropes and express their relative lengths in multiplicative terms. Rope B and C are 5 and
\(2\frac12\)
times as long as rope A, respectively, but rope D is shorter than rope A, so then we say that it is
\(\frac34\)
times as long as rope A. They see that it is possible that the answer to a “how many groups?” question is a number less than 1 when the given amount is smaller than the size of a group.
As students work, notice how they go about making multiplicative comparisons. Select students who write clear and concise questions for the equations in the last problem so they can share later.
Digital
Launch
Arrange students in groups of 2. Give students 4–5 minutes of quiet think time and then a couple of minutes to compare their responses with a partner and discuss any disagreements. Clarify that rope A is 4 units long.
Students using the digital materials can use the applet at
ggbm.at/kZUgANCC
to compare the segments. The segments can be moved by dragging endpoints with open circles. The yellow “pins” can help students keep track of the groups.
Student Facing
The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.
Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.
Rope B is _______ times as long as rope A.
Rope C is _______ times as long as rope A.
Rope D is _______ times as long as rope A.
Each equation can be used to answer a question about Ropes C and D. What could each question be?
\({?} \boldcdot 3=9\)
and
\(9 \div 3={?}\)
\({?} \boldcdot 9=3\)
and
\(3 \div 9= {?}\)
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
Arrange students in groups of 2. Give students 4–5 minutes of quiet think time and then a couple of minutes to compare their responses with a partner and discuss any disagreements. Clarify that rope A is 4 units long.
Students using the digital materials can use the applet at
ggbm.at/kZUgANCC
to compare the segments. The segments can be moved by dragging endpoints with open circles. The yellow “pins” can help students keep track of the groups.
Student Facing
Here is a diagram that shows four ropes of different lengths.
[IMG]
Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.
Rope B is _______ times as long as Rope A.
Rope C is _______ times as long as Rope A.
Rope D is _______ times as long as Rope A.
Each equation can be used to answer a question about Ropes C and D. What could each question be?
\({?} \boldcdot 3=9\)
and
\(9 \div 3={?}\)
\({?} \boldcdot 9=3\)
and
\(3 \div 9= {?}\)
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
Some students might associate the wrong lengths with the ropes or confuse the order of comparison (e.g., comparing A to C instead of C to A). Encourage them to put the length of each rope next to the diagram and attend more closely to the ropes being compared.
Activity Synthesis
Display the solutions to the first set of problems for all to see. Give students a minute to check their answers and ask questions. Then, focus class discussion on two key ideas:
The connection between “how many groups?” questions and “how many times as long?” questions. Ask students how these two types of questions are similar and different. Make sure students see that both have the structure of
\(? \boldcdot a =b\)
, where
\(a\)
is the size of 1 group (or the unit we are using for comparison), and
\(b\)
is a given number.
The language commonly used when referring to a situation in which the number of groups is less than 1 whole. Explain that we have seen equal-sized groups where the number of groups is greater than 1, but some situations involve a part of 1 group. So instead of saying “the number of groups” or asking “how many groups?,” we would ask “what fraction of a group?” or “what part of a group?”. For example, in the case of rope D, where the answer is less than 1, we can ask, “What fraction of rope A is rope D?”
Ask 1–2 previously identified students to share the question they wrote for the last pair of equations (
\(? \boldcdot 9=3\)
and
\(3 \div 9= ?\)
). Make sure students see that this pair of equations represent a situation with a fractional group
(i.e., rope D is shorter than rope C, so the length of rope D is a fraction of that of rope C).
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students describe their reasoning about “how many groups?” and “how many times as long?”, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR8 Discussion Supports.
As students compare and contrast these two types of division questions, provide a sentence frame such as: “Something these two types of questions have in common is . . .” and "A difference between these two types of questions is . . . ." This will help students produce and make sense of the language needed to communicate their ideas about the relationship between multiplication and division equations.
Design Principle(s): Support sense-making; Optimize output (for comparison)


correct_option_index: 2