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: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
problem_activity: Optional activity
To close the unit, students practice using the methods they have learned. Students begin by collecting data in small groups and then analyzing the data for the entire class. They create a scatter plot and two-way tables to look for patterns in the data (MP2). Although it is expected there is a very weak, if any, association in the data, the discussion asks students to think about what it would mean if there were a positive or negative association in context.
Launch
Arrange students in groups of 4. Select one student from each group to submit the data from the group when they finish recording their data. Prepare a table for all to see that is large enough to include the data from all of the students in the class.
Tell students that the same data is treated as both numerical and categorical data at different times within this activity. This is sometimes done when there is a threshold that makes sense for the context. In this case, it makes sense to look at data for which the second time is greater or less than the first time. Although the actual values may be lost when viewing the data as categorical, it can provide a different type of insight. In this example, we can more easily see if there was an improvement in reaction time for the second attempt.
Engagement: Internalize Self Regulation.
Provide a project checklist that chunks the various steps of the project into a set of manageable tasks.
Supports accessibility for: Organization; Attention
Representing, Conversing: MLR8 Discussion Supports.
Demonstrate the process for one round of the experiment, in which a group of students and staff play an example round while the rest of the class observes. This will help invite more student participation, conversations, and meta-awareness of language involving collecting and analyzing the data.
Design Principle(s): Support sense-making; Maximize meta-awareness.
Student Facing
In this activity, you’ll get two chances to guess at how long 30 seconds is, then look for an association between the two guesses of all students.
Work with a partner. Follow the instructions listed here to gather your data.
One of you will hold a stopwatch where the other person cannot see it.
The person holding the stopwatch says “go” and starts the timer.
The other person says “stop” when they think 30 seconds have passed.
The person holding the stopwatch will stop the timer, then report and record the time to the nearest second.
The person holding the stopwatch will give a second chance, repeating the experiment.
After
both
times are recorded, switch roles.
Record the group data in this table. When you finish, a group member should give the data to the teacher.
| name | time 1 | time 2 |
|------|--------|--------|
|      |        |        |
|      |        |        |
|      |        |        |
|      |        |        |
Look at your data. Comparing Time 1 to Time 2, do you think there is a positive association, a negative association, or no association? Discuss your thinking with your group.
What are some ways you could organize and represent the entire class's data?
Make a scatter plot of the entire class’s data and look for patterns. Identify any outliers and the type of any association you observe.
[IMG]
Draw two lines on your scatter plot: a vertical line and a horizontal line, each representing 30 seconds for one trial. Use the table for the class’s data to complete this two-way table.
|                 | time 2 < 30 sec | time 2 = 30 sec | time 2 > 30 sec | total |
|-----------------|-----------------|-----------------|-----------------|-------|
| time 1 < 30 sec |                 |                 |                 |       |
| time 1 = 30 sec |                 |                 |                 |       |
| time 1 > 30 sec |                 |                 |                 |       |
|      total      |                 |                 |                 |       |
Use the two-way table to decide whether there is an association between Time 1 and Time 2. Explain how you know.
Student Response
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Activity Synthesis
The purpose of the discussion is to solidify understanding of the strategies and tools from the unit. Specifically, how to organize and analyze categorical data to determine if there is an association between two variables.
Some questions for discussion:
“What representation (data table, scatter plot, two-way table, or something else) helped the most in determining any relationships between time 1 and time 2?”
“If someone's time was slower on the second attempt, how would that appear on the scatter plot?” (The point would be above and to the left of the line
\(y = x\)
, since the value for the
\(y\)
-coordinate would be greater than the value for the
\(x\)
-coordinate like the point
\((1.2, 1.4)\)
.)
“If there was a positive association between the variables, what would that mean?” (In general, the slower the first time, the slower the second time. The faster the first time, the faster the second time as well.)
“If there was a negative association between the variables, what would that mean?” (In general, the slower the first time, the faster the second time. The faster the first time, the slower the second time.)
"If the class showed a tendency to overcompensate by going from a time that was longer than 30 seconds to one that was shorter or the other way around, how would that appear in the two-way table?" (The numbers in the table would be greatest on the bottom left and top right of the table with smaller numbers elsewhere.)


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