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

Use random sampling to draw inferences about a population.

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

Develop understanding of statistical variability.

Use probability to evaluate outcomes of decisions.

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

Interpret linear models.

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

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

Calculate expected values and use them to solve problems.

Understand and evaluate random processes underlying statistical experiments.

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

Summarize and describe distributions.

Draw informal comparative inferences about two populations.

Investigate patterns of association in bivariate data.
problem_activity: Activity
The mathematical purpose of this activity is for students to understand that approximately normal distributions have a similar proportion of area in regions described by the mean and standard deviations. Students are given a data set including the number of words for 200 submitted short stories. They are asked to determine the proportion of submitted stories that are within certain ranges.
Monitor for students who:
return to the original data set for each question and count the values in each interval
create a frequency table to break the data into natural intervals
create a histogram to visualize the areas in question
use technology to count values in each region
The problem mentions that a page holds approximately 200 words to help students identify natural intervals to summarize the data into a frequency table or histogram.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Arrange students in groups of 2.
Ask, “What do you think the phrase ‘within two standard deviations of the mean’ means?” (It means the interval from the value of the mean minus two times the standard deviation to the value of the mean plus two times the standard deviation.)
If students are using a computer to create a spreadsheet, histogram, or find other information, suggest that they copy and paste the data into a program like Geogebra, Desmos, or other spreadsheet program rather than type all of the data themselves.
Student Facing
A publisher takes submissions for short stories to include in a book. 200 stories are submitted, but the publisher needs to be aware of how long each story is. The way the publisher will put together the collection of stories, a page typically contains 200 words. The mean number of words for each story is 2,600 and the standard deviation is 400 words.
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2643
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2084
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2931
2563
2655
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2465
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2493
2263
2706
2501
2627
2220
2372
2635
3066
2824
2357
2522
2564
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2118
2325
3551
2734
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2763
2867
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2936
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3116
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2946
1849
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2938
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3296
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2973
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1984
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2197
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2018
1927
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2767
If a histogram is created using intervals of 200 words, what would be the area of the bar representing the number of stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of the total area is represented by the bar for stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of stories in this group contain between 2,000 and 2,200 words? Explain or show your reasoning.
How does the proportion of the area you calculated relate to the proportion of stories in the group that contain between 2,000 and 2,200 words?
What proportion of stories in this group are within 1 standard deviation of the mean number of words?
What proportion of stories in this group are within 2 standard deviations of the mean number of words?
What proportion of stories in this group are within 1 standard deviation of 2,400 words?
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
Arrange students in groups of 2. Distribute one copy of the blackline master to each group.
Ask, “What do you think the phrase ‘within two standard deviations of the mean’ means?” (It means the interval from the value of the mean minus two times the standard deviation to the value of the mean plus two times the standard deviation.)
Student Facing
A publisher takes submissions for short stories to include in a book. 200 stories are submitted, but the publisher needs to be aware of how long each story is. The way the publisher will put together the collection of stories, a page typically contains 200 words. The mean number of words for each story is 2,600 and the standard deviation is 400 words.
If a histogram is created using intervals of 200 words, what would be the area of the bar representing the number of stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of the total area is represented by the bar for stories that contain between 2,000 and 2,200 words? Explain or show your reasoning.
What proportion of stories in this group contain between 2,000 and 2,200 words? Explain or show your reasoning.
How does the proportion of the area you calculated relate to the proportion of stories in the group that contain between 2,000 and 2,200 words?
What proportion of stories in this group are within 1 standard deviation of the mean number of words?
What proportion of stories in this group are within 2 standard deviations of the mean number of words?
What proportion of stories in this group are within 1 standard deviation of 2,400 words?
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?
Prove more generally that the proportion of total area taken up by a bar in a histogram is equal to the proportion of all data values that are contained in the interval represented by bar. To begin, let
\(n\)
represent the number of data values in an interval given by one bar,
\(M\)
represent the number of data values in the entire set, and
\(w\)
be the width of the interval in each bar of the histogram. Prove that the proportion of area taken up by the bar is
\(\frac{n}{M}\)
.
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
Some students may have trouble understanding the phrase “within 1 standard deviation of the mean.” Ask students to draw a number line containing the mean value and mark on the number line 1 standard deviation below and 1 standard deviation above the mean.
Activity Synthesis
The goal of this discussion is to get students thinking about histograms as an area model for a distribution. Select previously identified students to share in this order:
return to the original data set for each question and count the values in each interval
create a frequency table to break the data into natural intervals
create a histogram to visualize the areas in question
use technology to sort or count values in each interval
Connect the methods used by asking students about the advantages and disadvantages of the methods they used. (The summary methods, such as a frequency table or histogram, require additional time to set up, but are quicker to find the useful values for questions such as the ones included in the activity. Using technology, such as spreadsheets, to sort or count the data is also quick and accurate, but may take some time to enter the data if it is not already on the computer.)
If possible, display a histogram created by the students. If none of the students created a histogram, display the one here.
[IMG]
Here are some questions for discussion.
“How is this activity related to the previous one?” (In the previous activity, we figured out the area of a shaded region and divided it by the total area under the curve. In this activity, we can think of the bars as having an area.)
“Why would a normal distribution be a good model for the data summarized in the histogram?” (The distribution is approximately symmetric and bell-shaped.)
“Why do you think the area for a bar representing 2,000 to 2,200 words has the same proportion of the whole as the proportion of stories in the same region to the whole?” (Since the area came from
\(\frac{18 \boldcdot 200}{200 \boldcdot 200}\)
, which is equivalent to
\(\frac{18}{200}\)
.)
“Was it surprising to you that more than 95% of the data was within two standard deviations of the mean or that almost 70% of the data was within one standard deviation of the mean? Explain your reasoning.” (Both were surprising to me. I had never really thought about using the standard deviation to create an interval around the mean, so I had never thought about using the standard deviation to estimate the proportion of the data that was within a specific interval. But since most of the data is near the center, I can understand why it is this way.)
“Why do you think it could be useful to look at the proportion of values that are within one standard deviation of the mean?” (In the last lesson, we were introduced to normal curves that were based on the mean and standard deviation. If we modeled a histogram with a standard curve, it would be interesting to know what proportion of values fall within a given interval on normal curves.)


correct_option_index: 7