few_shots: Example:
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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Your response:
Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K
options: Interpret linear models.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Develop understanding of statistical variability.

Use random sampling to draw inferences about a population.

Investigate chance processes and develop, use, and evaluate probability models.

Understand independence and conditional probability and use them to interpret data.

Use probability to evaluate outcomes of decisions.

Calculate expected values and use them to solve problems.

Summarize and describe distributions.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Summarize, represent, and interpret data on a single count or measurement variable.

Investigate patterns of association in bivariate data.
problem_activity: Optional activity
The mathematical purpose of this activity is for students to apply conditional probability in the context of a problem. It is important that students use the ruler to draw the X on each card so that slight differences in the way the Xs are drawn are not recognizable.
Launch
Arrange students in groups of 2. Give 3 cards and one ruler to each group.
Demonstrate the game by playing one round with the students.
Tell students, "The first card will be blank on both sides, the second card will have an X on one side, and the third card will have an X on both sides. It is important to use the ruler when drawing the Xs on the cards since you should not be able to guess the card based on small differences in how the X is drawn."
Place all three cards in the bag. Remove one and hold it against the wall so that only one side can be seen by the students. Ask them which card they think this card is based on what they are seeing. Is it the blank card, the card with one X on one side, or the card with an X on both sides?
Engagement: Provide Access by Recruiting Interest.
Begin with a small-group or whole-class demonstration of how to play the game. Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Your teacher will give you 3 index cards.
Leave one card blank.
Use a ruler to draw lines connecting opposite corners to make an X on one side of the second card.
Use a ruler to draw lines connecting opposite corners to make an X on both sides of the third card.
Put all three cards in the bag.
One partner will remove a card from the bag and place it on the desk immediately so that only one side of the card can be seen. The goal is to guess correctly which card is on the desk: the blank card, the card with an X only on one side, or the card with an X on both sides.
Noah is playing the game and is looking at a card that shows an X. He says, “I have a fifty-fifty chance of correctly guessing which card it is.” Do you agree with Noah? Explain your reasoning.
Play the game many times with your partner, taking turns for who takes a card out of the bag. Record whether the side that shows has an X or is blank, which card you guess, and which card it actually is when you check. Continue to play until your teacher tells you to move on.
Use the results from your games to estimate
\(P(\text{the card had an X on both sides | the side showing had an X on it})\)
. Explain or show your reasoning.
Since all of the outcomes are known, find the actual probability
\(P(\text{the card had an X on both sides | the side showing had an X on it})\)
. Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may have difficulty figuring out the answer to question 4. Prompt students to think about the total number of sides rather than the number of cards. Emphasize that the number of sides are the sample space, not the number of cards.
Activity Synthesis
The purpose of this discussion is for students to communicate their reasoning with each other.
Poll the class to collect information on about what was shown and which card it turned out to be.
|                  | both sides blank | one X | two Xs |
|------------------|------------------|-------|--------|
| blank side shown |                  |       |        |
|     X shown      |                  |       |        |
Students should notice that two of the positions (top right and bottom left) must contain zeros.
Here are some questions for discussion.
“How did you figure out the answer to question 4?” (I figured out that it was
\(\frac{2}{3}\)
since
\(P(\text{X on both sides | saw an X}) = \frac{P(\text{X on both sides and saw an X})}{P(\text{saw an X})}\)
. We know
\(P(\text{the card had an X on both sides and the side showing had an X on it}) = \frac{2}{6}\)
(there are 6 sides that could be showing and 2 of them are Xs from the card with an X on both sides) and
\(P(\text{the side showing had an X on it}) = \frac{3}{6}\)
(there are 6 sides that could be showing and 3 of them have an X). Then we can find
\(P(\text{the card had an X on both sides | the side showing had an X on it}) = \frac{ \left( \frac{2}{6} \right)}{\left( \frac{3}{6} \right)}\)
which is
\(\frac{2}{3}\)
.
“Do you think that playing the game 100 times in question 2 would have helped you arrive at the answer to question 4? Explain your reasoning.” (I think it might have made me feel more confident in my answer to question 4. Since my group only played the game 12 times, it really did not give me much insight. I wonder what would have happened if we played it 100 times. Maybe we could combine the data from the whole class to get a better ideas of what would happen in the long run.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 12