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: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
problem_activity: Activity
The last activity in this lesson moves one more step closer to quantifying likelihood of scenarios by ordering them individually rather than into groups. Some of the scenarios have a numerical probability expressed as a percentage, some in decimal form, some as a fraction, and some do not have a numerical probability given (MP7). This gives students the opportunity to work with probabilities expressed as percentages, decimals, and fractions.
You will need the blackline master for this activity.
Launch
Arrange students in groups of 3. Distribute one copy of the blackline master for each group.
Tell students to take turns ordering the cards by beginning with one card, then adding in additional cards one at a time before, after, or between cards as needed.
Give students 5 minutes for group work followed by a whole-class discussion.
Student Facing
Your teacher will give you some cards that describe events. Order the events from least likely to most likely.
After ordering the first set of cards, pause here so your teacher can review your work. Then, your teacher will give you a second set of cards.
Add the new set of cards to the first set so that all of the cards are ordered from least likely to most likely.
Student Response
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Anticipated Misconceptions
Students may have trouble understanding the Rock, Paper, Scissors context. Tell these students that a player randomly chooses one of the three items to play in each round. If students still struggle, tell them that each of the three items are expected to be played with equal likelihood.
Activity Synthesis
The purpose of the discussion is for students to talk about the methods they used to sort the cards and compare likelihood of different situations.
Some questions for discussion:
"How were the numerical values of the likelihoods written?" (Some were written as percentages, some as fractions, and some as decimal values.)
"How did you compare them when there was a mix of percentages, fractions, and decimals?"
"Some of the cards did not have a percentage, fraction, or decimal. How did you determine where those cards would go in the order?"
Speaking, Listening: MLR7 Compare and Connect.
Ask students to prepare a visual display of their sorted cards. Students should consider how to display their cards so that another student can interpret the reasoning behind the order they selected. Some students may wish to add notes or details to their displays to help communicate their thinking. Invite students to investigate each other’s work, ask students to compare the order between displays. Listen for and amplify any comments about the use of percentages, fractions, and decimals to compare the likelihood of different situations. Then encourage students to make connections between the various ways to quantify the likelihood of a situation. Listen for and amplify language students use to explain that the percentage, fraction, and decimal used to quantify the likelihood of the same scenario are all equivalent. This will foster students’ meta-awareness and support constructive conversations as they compare and connect quantities that represent the likelihood of a situation.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 2