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 volume of cylinders, cones, and spheres.

Understand congruence in terms of rigid motions.

Understand similarity in terms of similarity transformations.

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

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

Use coordinates to prove simple geometric theorems algebraically.

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

Understand and apply the Pythagorean Theorem.

Make geometric constructions.

Explain volume formulas and use them to solve problems.

Analyze, compare, create, and compose shapes.

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

Understand and apply theorems about circles.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Find arc lengths and areas of sectors of circles.

Reason with shapes and their attributes.

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

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

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

Apply trigonometry to general triangles.

Apply geometric concepts in modeling situations.

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

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

Prove geometric theorems.

Experiment with transformations in the plane.

Classify two-dimensional figures into categories based on their properties.
problem_activity: Optional activity
The goal of this activity is for students to start to develop a systematic, point-by-point sequence of transformations that will work to take
any
pair of congruent polygons onto one another. This is especially important because when transformations are used in triangle congruence proofs in a subsequent lesson, students will be justifying how they know that a certain transformation will take one triangle onto another. Finding a single transformation that takes each vertex of one triangle onto each corresponding vertex of another cannot be generalized because it is specific to one pair of congruent triangles. In a proof, students must define sequences of rigid transformations that take a given triangle to any triangle with congruent corresponding parts, and justify why that sequence is guaranteed to take each vertex to its corresponding vertex.
Monitor for students who:
Find a single, separate transformation that takes
\(A\)
to
\(A’\)
and
\(B\)
to
\(B’\)
, so that when
\(B\)
goes to
\(B’\)
,
\(A\)
no longer remains on
\(A’\)
.
Look for a single transformation that takes quadrilateral
\(ABCD\)
onto quadrilateral
\(A’B’C’D’\)
all at once.
Line up one or two points and then get stuck.
Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges. You might find students who use translation, then rotation, then reflection, as well as students who use reflections across perpendicular bisectors and then edges.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take
\(ABC\)
to
\(DEF\)
in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram
\(ABCD\)
to
\(A’B’C’D’\)
.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A'B'C'D'\)
.
1.
2.
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
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take
\(ABC\)
to
\(DEF\)
in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram
\(ABCD\)
to
\(A’B’C’D’\)
.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A'B'C'D'\)
.
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
In this unit, we have been focusing on rigid transformations in two dimensions. By thinking carefully about precise definitions, we can extend many of these ideas into three dimensions. How could you define rotations, reflections, and translations in three dimensions?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select previously identified students to share in this order:
Look for a single transformation that moves all the points at once.
Line up one or two points and then get stuck.
Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges.
For each strategy shared, focus on what question that strategy helps to answer or what is unique and valuable about that strategy.
If a student who shares looked for a single transformation, focus on the elegance of finding a single rigid motion, as well as inviting the student to share what was hard about their strategy.
If a student who shares got stuck, emphasize the strategic choice to try to line up just one or two points at a time, and why that might be easier than finding one or two more complicated transformations.
If a student who shares systematically lined up one point at a time using transformations with fixed points to ensure that previously lined-up points did not move, emphasize why this strategy could be used for lots of different starting conditions.
Speaking: MLR 7 Compare and Connect
. Use this routine when students present their sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A’B’C’D’\)
. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students’ attention to the way that transformations are described precisely (that is, order, numbers, words). These exchanges strengthen students’ mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness


correct_option_index: 25