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: Develop understanding of statistical variability.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Calculate expected values and use them to solve problems.

Investigate patterns of association in bivariate data.

Use probability to evaluate outcomes of decisions.

Understand independence and conditional probability and use them to interpret data.

Understand and evaluate random processes underlying statistical experiments.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Interpret linear models.

Investigate chance processes and develop, use, and evaluate probability models.

Summarize and describe distributions.

Summarize, represent, and interpret data on a single count or measurement variable.

Use random sampling to draw inferences about a population.

Draw informal comparative inferences about two populations.
problem_activity: Activity
The mathematical purpose of this activity is to introduce students to curves representing normal distributions. Students compare normal curves with different means and standard deviations to determine how those statistics affect the curve. In particular, ensure students understand that normal curves are entirely defined by the mean and standard deviation, that the mean defines the center of the distribution, and the standard deviation affects the spread of the distribution.
Student Facing
These curves represent
normal distributions
with different means and standard deviations. What do you notice?
mean = 10. standard deviation = 1
[IMG]
mean = 10. standard deviation = 2
[IMG]
mean = 8. standard deviation = 2
[IMG]
mean = 12. standard deviation = 1
[IMG]
mean = 10. standard deviation = 0.8
[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?
The equation for the curve for a normal distribution is
\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \boldcdot e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
where
\(\sigma\)
is the standard deviation,
\(\mu\)
is the mean, and
\(e\)
is a particular number close to 2.718.
Notice the part of the function that is
\((x - \mu)\)
. Using your understanding of transformations of functions, how does changing
\(\mu\)
affect the graph of the function?
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
The important things for students to notice are:
The curves are entirely defined by the mean and standard deviation.
The curve is symmetric around the mean.
The standard deviation affects the width and height of the curve. A larger standard deviation is wider and shorter. A smaller standard deviation is thinner and taller.
Here are some questions for discussion:
“Why do you think the normal curve for a larger standard deviation is wider and shorter than one with a smaller standard deviation?” (We would expect a larger standard deviation to mean that data is more spread out, so it makes sense that is wider. Since the area must remain 1 under the curve, it has to get shorter to get wider.)
“What is the shape of the normal curve?” (Bell-shaped and symmetric)
“Where is the line of symmetry for each curve?” (
\(x = 10\)
,
\(x = 8\)
,
\(x = 12\)
, and
\(x = 10\)
, respectively)
Tell students,
“Although it is difficult to see in these graphs, the
\(x\)
-axis is a horizontal asymptote for normal curves. As you follow the curve to the right (or left), the curve gets closer and closer to the
\(x\)
-axis without ever actually reaching it.”
“Normal curves are bell-shaped and are most reliably used to model bell-shaped or approximately bell-shaped distributions.”
“The area under the normal curve (between the curve and the
\(x\)
-axis) is always 1. When the standard deviation increases, the curve spreads out to the right and left. To maintain the same area, though, the height of the peak must decrease.”


correct_option_index: 12