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: 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.

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.
problem_activity: Activity
In this activity, students continue to develop their understanding of what could be considered typical for a group as well as variability in a data set. Students compare distributions with the same mean but different spreads and interpret them in the context of a situation. The context given here (basketball score) prompts them to connect the mean to the notion of how “well” a player plays in general, and deviations from the mean to how “consistently” that player plays.
They encounter the idea of calculating the
average absolute deviation from the mean
as a way to describe variability in data.
Launch
Arrange students in groups of 3–4. Give groups 6–7 minutes to answer the questions and follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Begin with a small-group or whole-class demonstration and think aloud to remind students how calculate mean using data from a dot plot.  Keep the worked-out calculations on display for students to reference as they work.
Supports accessibility for: Memory; Conceptual processing
Listening, Speaking: MLR2 Collect and Display.
Use this routine to collect student ideas and language about playing “well” and “consistently” as they discuss the players’ data in the three dot plots. Record students’ language on a display for all to see. This will help students connect their reasoning about the visual representations of data to statistical terms such as mean, deviation from the mean, and variability in data.
Design Principle(s): Support sense-making
Student Facing
Here are the dot plots showing the number of baskets Elena, Jada, and Lin each made over 12 school days.
On each dot plot, mark the location of the mean with a triangle (
\(\Delta\)
). Then, contrast the dot plot distributions. Write 2–3 sentences to describe the shape and spread of each distribution.
[IMG]
Discuss the following questions with your group. Explain your reasoning.
Would you say that all three students play equally well?
Would you say that all three students play equally consistently?
If you could choose one player to be on your basketball team based on their records, who would you choose?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of the discussion is to highlight that the center of the distribution is not always the only consideration when discussing data. The variability or spread can also influence how we understand the data.
There are many ways to answer the second set of questions. Invite students or groups who have different interpretations of “playing well” and “playing consistently” to share their thinking. Allow as many interpretations to be shared as time permits. Discuss:
“How might we use the given data to quantify ‘playing well’ and ‘playing consistently?’”
“Is there a way to describe variability and consistency in playing precisely and in an objective way (rather than using broad, verbal descriptions)?”
Explain that we can describe variability more formally and precisely—using a number to sum it up; we will look at how to do so in the next activity.


correct_option_index: 2