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: Distinguish between correlation and causation.

Compute (using technology) and interpret the correlation coefficient of a linear fit.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
problem_activity: Activity
The mathematical purpose of this activity is for students to:
distinguish between linear and nonlinear relationships in bivariate, numerical data
informally assess the fit of a linear model
compare the slope and the vertical intercepts of different linear models
describe the relationship between two variables.
Students are given cards showing scatter plots and a linear model. They sort the cards based on how well the lines fit the data as well as by slope and intercept in increasing order.
A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
Launch
Arrange students in groups of 2. Give students a chance to familiarize themselves with what is on the cards. For example, you might ask them to sort the cards into categories of their choosing, and explain their categories to their partner.
Conversing: MLR8 Discussion Supports.
In pairs, ask students to take turns sorting the cards and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ _____ should be before _____ because . . . .”, and “I noticed _____ , so I . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about linear models.
Design Principle(s): Support sense-making; Maximize meta-awareness
Action and Expression: Internalize Executive Functions.
Provide students with a template for organizing their observations. Provide a template or invite students to fold a blank piece of paper in thirds, and label with three headers of “
\(x\)
,” “
\(y\)
” and “linear model fits?” to collect their answers. Explain that in the first column, they will always write increasing, since they will be reading each graph from left to right, then have them fill in the behavior of the y values in the next column, and in the last column, their conclusions about whether a linear model fits well.
Supports accessibility for: Language; Organization
Student Facing
Your teacher will give you a set of cards that show scatter plots.
Arrange all the cards in three different ways. Ensure that you and your partner agree on the arrangement before moving on to the next one. Sort all the cards in order from:
best to worst for representing with a linear model
least to greatest slope of a linear model that fits the data well
least to greatest vertical intercept of a linear model that fits the data well
For each card, write a sentence that describes how
\(y\)
changes as
\(x\)
increases and whether the linear model is a good fit for the data or not.
Student Response
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Activity Synthesis
The purpose of this discussion is for students to discuss the goodness of fit for linear models.
Here are some questions for discussion.
“How are scatter plots of A and F the same? How are they different?” (They have the same slope and the linear model for each scatter plot are equally well fit. They are different because they have a different vertical intercept.)
“How do you know if a linear model is a good fit?” (You need to look at the scatter plot and the line of best fit and make a decision about whether or not the data follows a linear trend.)
“Why is the goodness fit for the linear model in scatter plot B better than the fit for the linear model in scatter plot A?” (The data in B falls on or very close to the linear model. The data in A is scattered around the line of best fit and has roughly the same number of values below that line of best fit as it does above the line of best fit.)


correct_option_index: 2