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 ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Activity
Students design districts in three towns to “gerrymander” the results of elections. In two of the towns, the election results can be skewed to either color. In the third, it isn’t possible to skew the results.
Launch
Arrange students in groups of 2–4.
Explain the history of gerrymandering: Sometimes people in charge of designing districts make them in strange shapes to give the election results they want. One of the first was Elbridge Gerry (governor of Massachusetts in 1812), whose party designed a district that many people thought looked like a salamander. They called a Gerrymander, and the name stuck. It means a very strangely shaped, spread-out district designed to produce a certain result.
Usually districts are required to be connected: a person traveling to all parts of the district should be able to stay inside the district. There should be no “islands” that are separated by parts of other districts.
Engagement: Internalize Self-Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to four out of six task statements.
Supports accessibility for: Organization; Attention
Student Facing
Smallville’s map is shown, with opinions shown by block in green and gold. Decompose the map to create three connected, equal-area districts in two ways:
Design three districts where
green
will win at least two of the three districts. Record results in Table 1.
[IMG]
Table 1:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Design three districts where
gold
will win at least two of the three districts. Record results in Table 2.
[IMG]
Table 2:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Squaretown’s map is shown, with opinions by block shown in green and gold. Decompose the map to create five connected, equal-area districts in two ways:
Design five districts where
green
will win at least three of the five districts. Record the results in Table 3.
[IMG]
Table 3:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
|    4     |                                                      |                                                     |                                                                                    |                                                 |
|    5     |                                                      |                                                     |                                                                                    |                                                 |
Design five districts where
gold
will win at least three of the five districts. Record the results in Table 4.
[IMG]
Table 4:
| district | number of blocks                       for green | number of blocks                       for gold | percentage                       of blocks                       for green | representative’s                       vote |
|----------|--------------------------------------------------|-------------------------------------------------|----------------------------------------------------------------------------|---------------------------------------------|
|    1     |                                                  |                                                 |                                                                            |                                             |
|    2     |                                                  |                                                 |                                                                            |                                             |
|    3     |                                                  |                                                 |                                                                            |                                             |
|    4     |                                                  |                                                 |                                                                            |                                             |
|    5     |                                                  |                                                 |                                                                            |                                             |
Mountain Valley’s map is shown, with opinions by block shown in green and gold. (This is a town in a narrow valley in the mountains.) Can you decompose the map to create three connected, equal-area districts in the two ways described here?
Design three districts where
green
will win at least 2 of the 3 districts. Record the results in Table 5.
[IMG]
Table 5:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Design three districts where
gold
will win at least 2 of the 3 districts. Record the results in Table 6.
[IMG]
Table 6:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
The white streets on the maps do not disconnect the districts.
However, probably squares that are only connected at their corners should not be considered connected to each other. (But you or the students can make their own rule, since they are in charge of making districts.)
Activity Synthesis
This set of questions is similar to the previous activity, except that the patterns are irregular, so more planning and computation is needed to design districts.
Ask several students with different maps to show and explain their work.
The important result for this problem is that it’s possible to design both fair districts (where the result of the vote is similar to the vote if all individual votes were counted) and unfair districts. According to the rules (equal area and connected), these sets of districts are legal. Encourage students to try to devise some rules for drawing districts that would be more fair.


correct_option_index: 0