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: Understand congruence in terms of rigid motions.

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

Define trigonometric ratios and solve problems involving right triangles.

Make geometric constructions.

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

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

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

Prove geometric theorems.

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

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

Reason with shapes and their attributes.

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

Apply trigonometry to general triangles.

Find arc lengths and areas of sectors of circles.

Use coordinates to prove simple geometric theorems algebraically.

Analyze, compare, create, and compose shapes.

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

Understand and apply theorems about circles.

Understand similarity in terms of similarity transformations.

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

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.

Apply geometric concepts in modeling situations.

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

Prove theorems involving similarity.

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

Understand and apply the Pythagorean Theorem.
problem_activity: Activity
The goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transversal connecting two parallel lines) are congruent. This result will be used in a future lesson to establish that the sum of the angles in a triangle is 180 degrees. The second question is optional if time allows. This provides a deeper understanding of the relationship between the angles made by a pair of (not necessarily parallel) lines cut by a transversal.
Expect informal arguments as students are only beginning to develop a formal understanding of parallel lines and rigid motions. They have, however, studied the idea of 180 degree rotations in a previous lesson where they used this technique to show that a pair of vertical angles made by intersecting lines are congruent. Consider recalling this technique especially to students who get stuck and suggesting the use of tracing paper.
Given the diagram, the tracing paper, and what they have learned in this unit, students should be looking for ways to demonstrate that alternate interior angles are congruent using transformations. Make note of the different strategies (including different transformations) students use to show that the angles are congruent and invite them to share their strategies during the discussion. Approaches might include:
A 180 degree rotation about
\(M\)
First translating
\(P\)
to
\(Q\)
and then applying a 180-degree rotation with center
\(Q\)
Launch
Provide access to geometry toolkits. Tell students that in this activity, we will try to figure out why we saw all the matching angles we did in the last activity.
Action and Expression: Develop Expression and Communication
. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “First, I _____ because...”, “I noticed _____ so I...”, “Why did you...?”, “I agree/disagree because….”
Supports accessibility for: Language; Organization
Student Facing
[IMG]
Lines
\(\ell\)
and
\(k\)
are parallel and
\(t\)
is a transversal. Point
\(M\)
is the midpoint of segment
\(PQ\)
.
Find a rigid transformation showing that angles
\(MPA\)
and
\(MQB\)
are congruent.
In this picture, lines
\(\ell\)
and
\(k\)
are no longer parallel.
\(M\)
is still the midpoint of segment
\(PQ\)
.
[IMG]
Does your argument in the earlier problem apply in this situation? Explain.
Student Response
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Activity Synthesis
Select students to share their explanations. Pay close attention to which transformations students use in the first question and make sure to highlight different possibilities if they arise. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent.
Highlight the fact that students are using many of the transformations from earlier sections of this unit. The argument here is especially close to the one used to show that vertical angles made by intersecting lines are congruent. In both cases a 180 degree rotation exchanges pairs of angles. For vertical angles, the center of rotation is the common point of intersecting lines. For alternate interior angles, the center of rotation is the midpoint of the transverse between the two parallel lines. But the structure of these arguments is identical.
Speaking: MLR8 Discussion Supports.
Use this routine to amplify students’ mathematical uses of language when describing and demonstrating transformations used for showing alternate interior angles are congruent. After a student shares their response, invite another student to repeat the reasoning using the following mathematical language: vertical angles, 180 degree rotation, center of rotation, intersecting lines, midpoint, parallel lines. Invite all students to chorally repeat the phrases that include these words in context.
Design Principle(s): Support sense-making, Optimize output (for explanation)


correct_option_index: 4