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

Perform arithmetic operations with complex numbers.

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

Extend the properties of exponents to rational exponents.

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

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Perform operations on vectors.

Perform operations on matrices and use matrices in applications.

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

Reason quantitatively and use units to solve problems.

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

Represent and model with vector quantities.

Use properties of rational and irrational numbers.
problem_activity: Optional activity
In this Fermi problem, students estimate the total number of tiles needed to cover the Washington Monument.
Monitor for different approaches to solving the problem, and select students to share during the discussion. In particular, look for:
A range of estimates
Different levels of precision in reporting the final estimate
Different choices made to simplify the situation
Launch
Arrange students in groups of 1–4. Provide access to four-function calculators. If students are doing their own research, provide access to internet-enabled devices. If conducting a gallery walk at the end, provide access to tools for making a visual display.
Show a picture of the Washington Monument.
[IMG]
Before starting, ask students to come up with a guess how many tiles it would take to cover the Washington Monument and poll the class. Record all guesses.
Ask students to brainstorm the information they need to answer the question. Give the information provided when students ask for it, or provide access to internet-enabled devices so that students can find the information they need.
Vital information to have on hand includes:
Dimensions of the monument:
[IMG]
Standard sizes for square tiles: side lengths of 1 inch, 6 inches, 8 inches, 1 foot, and
\(1 \frac{1}{2}\)
feet.
Note that 55 feet is the height of the pyramid at the top, not the height of each triangular face. Students would need the Pythagorean Theorem to find the height of these faces, so either tell them it is close to 57.5 feet or they can use the 55 feet for their estimate. This does not have a significant impact on the area calculation, as the pyramid at the top does not account for most of the surface area.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to capture information about the different assumptions they made when calculating the areas of the trapezoidal faces of the Washington Monument building.
Supports accessibility for: Language; Organization
Student Facing
How many tiles would it take to cover the Washington Monument?
[IMG]
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
Students may not account for the faces of the monument that you cannot see in the picture.  Remind these students that some faces are hidden. If three-dimensional building blocks are available, consider having students who have difficulty visualizing the monument build a small model. It is difficult to model the exact shape of the Washington Monument, but the physical model may help students make the needed calculations.
Students may have difficulty converting units, especially if their tiles are measured in inches. Ask these students how many inches are in a foot and how many square inches are in a square foot.
Activity Synthesis
Invite students or groups to share different solution approaches. Consider asking students to create a visual display and conducting a gallery walk. Highlight different methods of calculating the area of the trapezoidal faces of the Washington Monument including:
Putting two trapezoid faces (or all 4 faces) together to make a parallelogram
Decomposing each trapezoid face into a rectangle and two triangles and then rearranging
Discuss why different people have different estimates. Some reasons to highlight include:
Using different-sized tiles to cover the Washington Monument
Thinking about how some tile is wasted, namely the tiles that go on the edge of each face
Rounding
This problem has a different flavor than the two preceding ones. The only place estimation comes into play is with how to cover the top triangles and perhaps thinking about what happens to tiles at the edge of a face. What all the problems share, however, is that we cannot give an
exact
answer. To provide a reasoned estimate, keeping track of all assumptions being made is vital. This tells us not only how accurately we should report our answer but also helps us communicate clearly to others who may have made different assumptions.
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain their strategy for estimating the number of tiles using appropriate mathematical lanague (e.g., face, height, width, surface area, trapezoid, etc.). Be sure to vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 2