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

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

Calculate expected values and use them to solve problems.

Summarize and describe distributions.

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

Draw informal comparative inferences about two populations.

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

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

Use random sampling to draw inferences about a population.

Understand and evaluate random processes underlying statistical experiments.

Use probability to evaluate outcomes of decisions.

Investigate patterns of association in bivariate data.

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

Develop understanding of statistical variability.

Interpret linear models.
problem_activity: Activity
The mathematical goal of this activity is for students to use the area under a normal curve to find the proportion of values in certain intervals. While the values in this activity are chosen to highlight the usual areas within 1, 2, and 3 standard deviations of the mean, it is not essential for students in this course to memorize those values.
Digital
Launch
Remind students that the phrase “area under the curve” means the area between the
\(x\)
-axis and the curve. Demonstrate how to use graphing technology to display the normal curve from the previous activity (a normal curve with mean of 40 and standard deviation of 2) and to find the area under the curve for values less than 39. See
ggbm.at/w9tzezej
.
If you are using the GeoGebra Probability Calculator, available in the toolkit or at
geogebra.org/classic#probability
, select Normal distribution and customize it by adjusting the mean and standard deviation. Choose the distribution you want to calculate from the three buttons labeled as shown below. Adjust the interval limits by either replacing their values in the formula, or by dragging them in the graph.﻿ The area under the curve is given by the calculated probability.
It may be helpful for students to open two GeoGebra windows—one for each of the situations in the problem statement.
[IMG]
Calculate the probability of a distribution.
[IMG]
Calculate the probability of a left-sided distribution.
[IMG]
Calculate the probability of a right-sided distribution. ﻿
[IMG]
Toggle between the graphs of the probability density function and the cumulative distribution function.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies, such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to provide students with an opportunity to produce the language of mathematical questions. Display only the context of the situation, including the statistics about lifespans of two different types of lightbulbs, without revealing the questions that follow. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that involve estimates of proportion. This will build student understanding of the language used to describe characteristics and applications of the normal curve.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
The life span of light bulbs is approximately normally distributed. Some statistics about life spans of two different types of light bulbs are listed.
LED bulbs: mean = 2,300 days, standard deviation = 230 days
incandescent bulbs: mean = 100 days, standard deviation = 10 days
To estimate the proportion of bulbs that burn out in a certain interval of time, use technology to find the area under the normal curve and above the appropriate interval.
Estimate the proportion of LED bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 2,070 days).
Estimate the proportion of incandescent bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 90 days).
Estimate the proportion of LED bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 2,530 days).
Estimate the proportion of incandescent bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 2,070 and 2,530 days).
Estimate the proportion of incandescent bulbs expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 90 and 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 2 standard deviations less than the mean and 2 standard deviations greater than the mean (between 1,840 and 2,760 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1,900 days and 2,100 days.
Estimate the proportion of incandescent bulbs that are expected to burn out in the interval between 107 and 118 days.
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
Remind students that the phrase “area under the curve” means the area between the
\(x\)
-axis and the curve. Demonstrate how to use graphing technology to display the normal curve from the previous activity (a normal curve with mean of 40 and standard deviation of 2) and to find the area under the curve for values less than 39.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies, such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to provide students with an opportunity to produce the language of mathematical questions. Display only the context of the situation, including the statistics about lifespans of two different types of lightbulbs, without revealing the questions that follow. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that involve estimates of proportion. This will build student understanding of the language used to describe characteristics and applications of the normal curve.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
The life span of light bulbs is approximately normally distributed. Some statistics about life spans of two different types of light bulbs are listed.
LED bulbs: mean: 2,300 days, standard deviation: 230 days
incandescent bulbs: mean: 100 days, standard deviation: 10 days
To estimate the proportion of bulbs that burn out in a certain interval of time, use technology to find the area under the normal curve and above the appropriate interval.
Estimate the proportion of LED bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 2,070 days).
Estimate the proportion of incandescent bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 90 days).
Estimate the proportion of LED bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 2,530 days).
Estimate the proportion of incandescent bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 2,070 and 2,530 days).
Estimate the proportion of incandescent bulbs expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 90 and 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 2 standard deviations less than the mean and 2 standard deviations greater than the mean (between 1,840 and 2,760 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1,900 days and 2,100 days.
Estimate the proportion of incandescent bulbs that are expected to burn out in the interval between 107 and 118 days.
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
Students should be careful to use the correct mean and standard deviation for the type of bulb in the question. For students who struggle, ask them to more clearly identify the questions related to LED (or incandescent) bulbs to ensure they use the correct values in their technology.
Activity Synthesis
The goal of this discussion is for students to understand that a normal curve can be used to estimate the proportion of data that lie within a certain interval when the data is approximately normal. Ask,
“Why is it important that we are told that the life spans are approximately normally distributed?” (If the data had a different type of distribution, modeling with a normal distribution would not accurately describe the situation. Compare this to modeling quadratic data with a line.)
“Why are the answers for the first 4 questions the same?” (Since the normal curve is determined by the standard deviation and mean, the area to the left of the position one standard deviation below the mean is the same regardless of the values. Since the normal curve is symmetric, the area less than one standard deviation below the mean is equal to the area greater than one standard deviation above the mean.)
“Since all normal curves have the same areas for the regions relative to the mean and number of standard deviations away, approximately what percentage of values are within 1 standard deviation of the mean? (About 68% are within 1 standard deviation of the mean.)
“Approximately what percentage of values are within 2 standard deviations of the mean?” (About 95% are within 2 standard deviations of the mean.)
“Approximately what percentage of values are within 3 standard deviations of the mean?” (About 99.7% are within 3 standard deviations of the mean.)


correct_option_index: 6