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.


Concept/skill: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_neg_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
problem_activity: Activity
In this activity, students continue to compute probabilities for multi-step experiments using the number of outcomes in the sample space. The first problem involves a situation for which students have already seen the sample space. Following this problem, the class will discuss the merits of the different representations for writing out the sample space. The next two problems involve situations in which students may need to write out the sample space on their own. Students are also reminded that some events have a probability of 0, which represents an event that is impossible. In the discussion following the activity, students are asked to think about the probabilities of two events that make up the entire sample space and have no outcomes common to both events (MP2).
Launch
Keep students in groups of 2.
Assign each group a representation for writing out the sample space: a tree, a table, or a list. Tell students that they should write out the sample space for the first problem using the representation they were assigned. (This was done for them in a previous lesson and they are allowed to use those as a guide if they wish.)
Tell students that they should work on the first problem only and then pause for a discussion before proceeding to the next problems.
Give students 2 minutes of partner work time for the first problem followed by a pause for a whole-class discussion centered around the different representations for sample space.
After all groups have completed the first question, select at least one group for each representation and have them explain how they arrived at their answer. As the groups explain, display the appropriate representations for all to see. Ask each of the groups how they counted the number of outcomes in the sample space as well as the number of outcomes in the event using their representation.
List:
Heads 1, heads 2, heads 3, heads 4, heads 5, heads 6, tails 1, tails 2, tails 3, tails 4, tails 5, tails 6
Table:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "1"}, {"_attributes": {"scope": "col"}, "_value": "2"}, {"_attributes": {"scope": "col"}, "_value": "3"}, {"_attributes": {"scope": "col"}, "_value": "4"}, {"_attributes": {"scope": "col"}, "_value": "5"}, {"_attributes": {"scope": "col"}, "_value": "6"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "H1"}, {"_value": "H2"}, {"_value": "H3"}, {"_value": "H4"}, {"_value": "H5"}, {"_value": "H6"}]}, {"td": [{"_value": "T1"}, {"_value": "T2"}, {"_value": "T3"}, {"_value": "T4"}, {"_value": "T5"}, {"_value": "T6"}]}]}]}]}
Tree:
[IMG]
After students have had a chance to explain how they used the representations, ask students to give some pros and cons for using each of the representations. For example, the list method may be easy to write out and interpret, but could be very long and is not the easiest method for keeping track of which outcomes have been written and which still need to be included.
Allow the groups to continue with the remaining problems, telling them they may use any method they choose to work with the sample space for these problems. Give students 10 minutes of partner work time followed by a whole-class discussion about the activity as a whole.
Student Facing
The other day you looked at a list, a table, and a tree that showed the sample space for rolling a number cube and flipping a coin.
Your teacher will assign you one of these three structures to use to answer these questions. Be prepared to explain your reasoning.
What is the probability of getting tails and a 6?
What is the probability of getting heads and an odd number?
Pause here so your teacher can review your work.
Suppose you roll two number cubes. What is the probability of getting:
Both cubes showing the same number?
Exactly
one cube showing an even number?
At least
one cube showing an even number?
Two values that have a sum of 8?
Two values that have a sum of 13?
Jada flips three quarters. What is the probability that all three will land showing the same side?
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may not recognize that rolling a 2 then a 3 is different from rolling a 3 then a 2. Ask students to imagine the number cubes are different colors to help see that there are actually 2 different ways to get these results.
Similarly, some students may think that HHT counts the same as HTH and THH. Ask the student to think about the coins being flipped one at a time rather than all tossed at once. Drawing an entire tree and seeing all the branches may further help.
Activity Synthesis
The purpose of the discussion is for students to explain their methods for solving the problems and to discuss how writing out the sample space aided in their solutions.
Poll the class on how they computed the number of outcomes in the sample space and the number of outcomes in the event for the second set of questions given these options: List, Table, Tree, Computed Outcomes Without Writing Them All Out, Another Method.
Consider these questions for discussion:
“Which representation did you use for each of the problems?”
“Do you think you will always try to use the same representation, or can you think of situations when one representation might be better than another?”
“Did you have a method for finding the number of outcomes in the sample space or event that was more efficient than just counting them?” (The number of outcomes in the sample space for the number cubes could be found using
\(6 \boldsymbol{\cdot} 6 = 36\)
. To find the number of outcomes with at least 1 even number, I knew there would be 6 for each time an even was rolled first and only 3 for each time an odd number was rolled first, so I found the number of outcomes by
\(3 \boldsymbol{\cdot} 6 + 3 \boldsymbol{\cdot} 3 = 27\)
.)
“One of the events had a probability of zero. What does this mean?”
(It is impossible.)
“What would be the probability of an event that was certain?” (1)
“Jada was concerned with having all the coins show the same side. What would be the probability of having at least 1 coin
not
match the others?” (
\(\frac{6}{8}\)
, since there are 6 outcomes where at least 1 coin does not match: HHT, TTH, HTH, THT, HTT, THH.)
“How do the answers to Jada’s question and the one we just answered relate to one another?” (Since every outcome in the sample space has either “at least one heads” or “all tails,” and there is no outcome that applies to both events, together the sum of their probabilities must be 100% or 1.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice 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 will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no