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.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
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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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: 2