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: Extend the properties of exponents to rational exponents.

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

Perform operations on matrices and use matrices in applications.

Represent and model with vector quantities.

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

Use complex numbers in polynomial identities and equations.

Perform arithmetic operations with complex numbers.

Represent complex numbers and their operations on the complex plane.

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

Compute fluently with multi-digit numbers and find common factors and multiples.

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

Reason quantitatively and use units to solve problems.

Perform operations on vectors.

Use properties of rational and irrational numbers.
problem_activity: Activity
The purpose of this activity is for students to draw their own axes for different sets of coordinates. They must decide which of the four quadrants they need to use and how to scale the axes. Some students may use logic such as “the largest/smallest point is this, so my axes must go at least that far.” Identify these strategies for the discussion. Monitor for differences in scales and axes where the points were still able to be plotted correctly to highlight during discussion.
Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 15 minutes. Skip the part of the activity synthesis that references making connections back to the warm-up.
Launch
Arrange students in groups of 2. Allow 10 minutes for students to construct their graphs and discuss them with their partners. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time. Invite students to share their reasoning about where they placed each axis and how they determined an appropriate scale. Some students may benefit from access to partially-created graphs with varying degrees of completion—for example, axes with or without unlabeled tick marks.
Supports accessibility for: Memory; Organization
Listening, Speaking: MLR2 Collect and Display.
Listen for and display vocabulary and phrases students use to justify their choice of axes (e.g., “minimum/maximum
\(x\)
- or
\(y\)
-coordinate” or “appropriate units”). Continue to update collected language students used to explain their reasoning to their peers. Remind students to borrow language from the display during paired and whole-class discussions.
Design Principle(s): Maximize meta-awareness
Student Facing
Here are three sets of coordinates. For each set, draw and label an appropriate pair of axes and plot the points.
\((1, 2), (3, \text-4), (\text-5, \text-2), (0, 2.5)\)
[IMG]
\((50, 50), (0, 0), (\text-10, \text-30), (\text-35, 40)\)
[IMG]
\(\left(\frac14, \frac34\right), \left(\frac {\text{-}5}{4}, \frac12\right), \left(\text-1\frac14, \frac {\text{-}3}{4}\right), \left(\frac14, \frac {\text{-}1}{2}\right)\)
[IMG]
Discuss with a partner:
How are the axes and labels of your three drawings different?
How did the coordinates affect the way you drew the axes and label the numbers?
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
Make sure that students label distances on their axes consistently. For example, if the first tick mark after 0 is 3, then the next must be 6 in order for the spacing to be consistent.
Activity Synthesis
The key takeaway from this discussion is that defining axes and scale is a process of reasoning, not an exact science. Ask students to share their strategies about how to place and scale their axes. First, display previously selected student responses that capture the same data on their axes, but with slightly different origins or scales. Ask students which axes they think better represent the data. If not mentioned by students, point out that axes with a lot of empty space probably could benefit from either a different scale or a different origin. Select previously identified students to demonstrate a reliable strategy for finding the needed maximum or minimum. Link back to the warm-up when talking about how to scale the axes: if there are larger numbers, then a bigger scale makes more sense. Also draw attention to the fractional coordinates and how using decimal equivalents might make it easier to scale.


correct_option_index: 4