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.


Your response:
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.


Your response:
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.


Your response:
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: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
problem_activity: Activity
In this activity, students draw a bar graph and histogram, then describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.
Launch
Students will need the data on their travel methods and times, collected at the beginning of the unit. Distribute or display the data collected for these questions from the survey given earlier in the unit. Alternatively, complete the tables in the blackline master ahead of time.
Arrange students in groups of 2. Give one copy of the blackline master to each group of students. Display the data from the prior survey or the completed frequency tables for all to see or give a copy to each group of 2 students. Give students 5–6 minutes to complete the activity. Ask one partner to create a bar graph to represent the data on the class’s travel methods and the other to create a histogram to represent the data on travel times, and then answer the questions together.
Student Facing
Your teacher will provide you with some data that your class collected the other day.
Use the data to draw a histogram that shows your class’s travel times.
[IMG]
Describe the distribution of travel times. Comment on the center and spread of the data, as well as the shape and features.
Use the data on methods of travel to draw a bar graph. Include labels for the horizontal axis.
[IMG]
Describe what you learned about your class’s methods of transportation to school. Comment on any patterns you noticed.
Compare the histogram and the bar graph that you drew. How are they the same? How are they different?
Student Response
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Student Facing
Are you ready for more?
Use one of these suggestions (or make up your own). Research data to create a histogram. Then, describe the distribution.
Heights of 30 athletes from multiple sports
Heights of 30 athletes from the same sport
High temperatures for each day of the last month in a city you would like to visit
Prices for all the menu items at a local restaurant
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.
Select one student to show a completed histogram and another to show a completed bar graph. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:
“How are the bar graphs and histograms alike? How are they different?”
“Can we use a bar graph to display the data on travel times? Why or why not?”
“Can we use a histogram to display the data on methods of travel? Why or why not?”
Next, select a few other students to share their descriptions of the distributions shown on each type of display. Then, ask questions such as:
“How are your descriptions of the distribution for travel methods different than those for travel times?”
“Can you talk about the shape of a distribution shown on a bar graph? Can you talk about its center and spread? Why or why not?”
Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of a categorical data on a bar graph because a bar graph does
not
use a number line. This means the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to record differences between histograms and bar graphs.  Consider creating a shared display and keeping it visible throughout the unit.
Supports accessibility for: Language; Organization
Representing, Conversing: MLR2 Collect and Display.
As students discuss the similarities and differences between histograms and bar graphs, record and display phrases you hear in a chart, such as a Venn diagram. Use this as a reference when highlighting or revoicing academic vocabulary from this unit (frequency, bins, numerical data, categorical data). This will help students connect unique and shared characteristics of histograms and bar graphs while reinforcing mathematical language.
Design Principle(s): Maximize meta-awareness


correct_option_index: 0