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.


Your response:
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.


Your response:
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.


Your response:
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.

Apply geometric concepts in modeling situations.

Understand and apply the Pythagorean Theorem.

Analyze, compare, create, and compose shapes.

Use coordinates to prove simple geometric theorems algebraically.

Understand congruence in terms of rigid motions.

Find arc lengths and areas of sectors of circles.

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

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

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

Experiment with transformations in the plane.

Apply trigonometry to general triangles.

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

Reason with shapes and their attributes.

Prove geometric theorems.

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).

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

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

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

Prove theorems involving similarity.

Understand similarity in terms of similarity transformations.

Make geometric constructions.

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply theorems about circles.

Explain volume formulas and use them to solve problems.

Solve real-world and mathematical problems involving area, surface area, and volume.
problem_activity: Activity
In this activity, students work together to investigate lines of symmetry and communicate their thinking in a visual display. As you monitor, ask groups if they have found all possible lines of symmetry and to explain how they know all shapes of that type have that same symmetry.
Launch
Arrange students in groups of 2–4. Provide each group with tools for creating a visual display.
Assign a different shape from the blackline master to each group and provide enough copies of that shape for each student in each group. Note that the cool-down uses a rectangle, so you may choose to assign the rectangle to a group of students who need more processing time.
Give students 5 minutes of work time followed by 5 minutes to put together their visual display. Explain that they will not have enough time to make the visual display perfect, so the purpose is to get their ideas down in an organized way.
Student Facing
Determine all the
lines of symmetry
for the shape your teacher assigns you. Create a visual display about your shape. Include these parts in your display:
the name of your shape
the definition of your shape
drawings of each line of symmetry
a description in words of each line of symmetry
one non-example in a different color (a description and drawing of a reflection
not
over a line of symmetry)
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?
Look at all of the shapes the class explored and focus on those which had more than one line of symmetry.
What is true for all the lines of symmetry in these shapes?
Give an example of a shape that has two or more lines of symmetry that do not intersect at the same point.
What would happen if you did a sequence of two different reflections across lines of symmetry for the shapes you explored in class?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students might think that any line that divides the area of a figure in half, like a diagonal of a non-square rectangle, is a line of symmetry. Ask students to reflect across these lines to see that the shapes are not taken onto themselves.
Activity Synthesis
The purpose of this discussion is to identify what types of shapes have reflection symmetry.
Ask groups to display their visual displays in the classroom for all to see in order of the number of lines of symmetry their shape has. They will have to communicate with other groups to accomplish this.
Invite students to do a “gallery walk” in which they leave written feedback on sticky notes for the other groups. Here is guidance for the kind of feedback students should aim to give each other:
Was there anything about the organization of the visual display that made the ideas especially clear? Was there anything about the organization that could be improved?
Was there anything about the way the ideas are explained that made the ideas especially clear? Was there anything about the explanations that could be improved?
Are there any lines of symmetry that the group missed?
Are there any lines of symmetry that don’t work for all shapes of the given type?
After the gallery walk, ask students to share their observations about reflection symmetries for the different shapes. Consider asking:
“Which shapes had the fewest lines of symmetry? Which had the most?” (The parallelogram that is not a rhombus didn’t have any lines of symmetry. The circle has the most since it has infinite.)
“Were there any lines that you thought would be lines of symmetry, but when you tried to reflect, they turned out not to be?” (The diagonal of a parallelogram that is not a rhombus seemed like it would be a line of symmetry, but it’s not.)
“Describe a shape that would have no lines of symmetry.” (A scalene triangle has no lines of symmetry.)
Save the visual displays if possible for comparison with displays of angles of rotation that create symmetry in a subsequent lesson.
Engagement: Develop Effort and Persistence.
Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention


correct_option_index: 10