few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
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

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Solve real-world and mathematical problems involving area, surface area, and volume.

Understand and apply the Pythagorean Theorem.

Use coordinates to prove simple geometric theorems algebraically.

Apply trigonometry to general triangles.

Prove theorems involving similarity.

Visualize relationships between two-dimensional and three-dimensional objects.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Classify two-dimensional figures into categories based on their properties.

Analyze, compare, create, and compose shapes.

Explain volume formulas and use them to solve problems.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Apply geometric concepts in modeling situations.

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Find arc lengths and areas of sectors of circles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Understand similarity in terms of similarity transformations.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Prove geometric theorems.

Translate between the geometric description and the equation for a conic section.

Make geometric constructions.

Define trigonometric ratios and solve problems involving right triangles.
problem_activity: Optional activity
In this activity, students are given pictures and descriptions of planes cutting prisms and pyramids. Students are asked to draw cross sections freehand but this is not a skill that is required in order for students to be able to describe two-dimensional shapes created from cross sections, which is why this is an optional activity. Some pictures are of a moving plane. Students describe how the cross section changes as the plane moves.
If students have access to the digital activity there are applets to explore the cross sections.
Adapted from applets created in GeoGebra by
Anthony C.M. OR
.
Digital
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time to discuss shapes of cross sections with a partner. Follow with a whole-class discussion.
Student Facing
Use the applet to draw each cross section and describe it in words. Start each one by dragging the gray bar on the left side across the screen until you see the whole 3D image on the left and the controls on the right.
Here is an applet with a rectangular prism, 4 units by 2 units by 3 units.
A plane cuts the prism parallel to the bottom and top faces.
The plane moves up and cuts the prism at a different height.
A vertical plane cuts the prism diagonally.
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.
A plane cuts the pyramid parallel to the base.
A vertical plane cuts the prism.
A cube has an edge of length 4.
A plane cuts off the corner of the cube.
The plane moves farther from the corner and makes a cut through the middle of the cube.
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 3–5 minutes of quiet work time followed by time to discuss shapes of cross sections with a partner. Follow with a whole-class discussion.
Student Facing
Draw and describe each cross section.
Here is a picture of a rectangular prism, 4 units by 2 units by 3 units.
[IMG]
A plane cuts the prism parallel to the bottom and top faces.
[IMG]
The plane moves up and cuts the prism at a different height.
[IMG]
A vertical plane cuts the prism diagonally.
[IMG]
A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units.
A plane cuts the pyramid parallel to the base.
[IMG]
A vertical plane cuts the prism.
[IMG]
A cube has an edge of length 4.
A plane cuts off the corner of the cube.
[IMG]
The plane moves farther from the corner and makes a cut through the middle of the cube.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select students to share their drawings and descriptions. Consider asking some of the following questions:
“How did you figure out the shape of the cross section?”
“What helped you visualize the shape?”
“Were any of the shapes you drew here similar to the shapes you described in the previous activity?”
Conversing: MLR8: Discussion Supports.
To support a rich discussion while students are describing two-dimensional shapes created from cross sections, invite students to include details in their descriptions and drawings of each cross section. Provide sentence frames such as: “The cross section is ____ because . . .”, “A plane cuts _____ as shown by . . .”, or “The dimensions of a cross section are ______ since . . . .” This will help students make sense of complex language as they draw and describe cross sections shown in each figure.
Design Principle(s): Cultivate conversation


correct_option_index: 6