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: 
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.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

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_pos_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: 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?
problem_activity: Activity
The mathematical purpose of this activity is to use a two-way table as a sample space to decide if events are independent and to estimate conditional probabilities.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Give students 10 minutes of quiet time to work the problems and then pause for a brief whole-class discussion.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies. Some students may benefit from a checklist or list of steps to be able to use the spreadsheet technology.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
A pharmaceutical company is testing a new medicine for a disease using 115 test subjects. Some of the test subjects are given the new medicine and others are given a placebo. The results of their tests are summarized in the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "no more symptoms"}, {"_attributes": {"scope": "col"}, "_value": "symptoms persist"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "given medicine"}], "td": [{"_value": "31"}, {"_value": "26"}, {"_value": "57"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given placebo"}], "td": [{"_value": "16"}, {"_value": "42"}, {"_value": "58"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{"_value": "47"}, {"_value": "68"}, {"_value": "115"}]}]}]}]}
Divide the value in each cell by the total number of test subjects to find each probability to two decimal places. Some of the values have been completed for you.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "no more symptoms"}, {"_attributes": {"scope": "col"}, "_value": "symptoms persist"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given medicine"}], "td": [{"_value": "0.27"}, {}, {"_value": "0.50"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given placebo"}], "td": [{}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{}, {}, {"_value": "1"}]}]}]}]}
If one of these test subjects is selected at random, find each probability:
\(P(\text{symptoms persist})\)
\(P(\text{given medicine and symptoms persist})\)
\(P(\text{given placebo or symptoms persist})\)
From the original table, divide each cell by the total for the row to find the probabilities with row conditions. Some of the values have been completed for you.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "no more symptoms"}, {"_attributes": {"scope": "col"}, "_value": "symptoms persist"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "given medicine"}], "td": [{"_value": "0.54"}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given placebo"}], "td": [{}, {}, {"_value": "1"}]}]}]}]}
\(P(\text{symptoms persist | given medicine})\)
\(P(\text{no more symptoms | given placebo})\)
Jada didn’t read the instructions for the previous problem well and used the table she created on the first problem to divide each cell by the probability total for each row. For example, in the top left cell she calculated
\(0.27 \div 0.5\)
. Complete the table using Jada’s method.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "no more symptoms"}, {"_attributes": {"scope": "col"}, "_value": "symptoms persist"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given medicine"}], "td": [{}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given placebo"}], "td": [{}, {}, {}]}]}]}]}
What do you notice about this table?
From the original table, divide each cell by the total for the column to find the probabilities with column conditions. Some of the values have been completed for you.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "no more symptoms"}, {"_attributes": {"scope": "col"}, "_value": "symptoms persist"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given medicine"}], "td": [{"_value": "0.66"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "given placebo"}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{}, {"_value": "1"}]}]}]}]}
\(P(\text{given medicine | symptoms persist})\)
\(P(\text{given placebo | no more symptoms})\)
Are the events “symptoms persist” and “given medicine” independent events? Explain or show your reasoning.
Based on your work, does being given this medicine have an impact on whether symptoms persist or not?
Student Response
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Student Facing
Are you ready for more?
Consider the data collected from all students in grades 11 and 12 and their intentions of going to prom.
65% of the students who are going to prom are from grade 12. Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "going to prom"}, {"_attributes": {"scope": "col"}, "_value": "not going to prom"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "grade 11"}], "td": [{}, {}, {"_value": "127"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "grade 12"}], "td": [{}, {}, {"_value": "116"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{"_value": "124"}, {"_value": "119"}, {"_value": "243"}]}]}]}]}
Based on your results, what is the probability that a student from this group, selected at random, is a student in grade 11 that did not go to prom?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students might struggle with understanding the notation for conditional probability. Tell these students that
\(P(\text{A | B})\)
means the probability of A happening under the condition that B happens.
Activity Synthesis
The purpose of this discussion is for students to interpret probabilities in a two-way table.
Ask students to interpret each of the probabilities they found in questions 1, 2, and 4. For example, “What does the probability in question 1a mean in the context of the problem?”
Here are sample interpretations of the probabilities for questions 1, 2, and 4.
The probability that a test subject chosen at random from all of the subjects still had symptoms after the test.
The probability that a test subject chosen at random from all of the subjects was among those given medicine and still had symptoms after the test.
The probability that a test subject chosen at random from all of the subjects still had symptoms after the test or was given a placebo. 1
The probability that a test subject chosen at random from those who were given medicine had symptoms after the test.
The probability that a test subject chosen at random from those who were given placebo had no more symptoms after the test
The probability that a test subject chosen at random from those who had symptoms after the test was given medicine.
The probability that a test subject chosen at random from those who had no more symptoms after the test was given placebo.
Display what Jada did when creating the top left cell in table for all to see:
\(\frac{P(\text{no more symptoms and given medicine})}{P(\text{given medicine})}\)
Here are some questions for discussion.
“What do you notice about what Jada did and
\(P(\text{no more symptoms | given medicine})\)
?” (I noticed that they are equal.)
“What is an equation that represents this situation using A and B for the events?” (
\(P(\text{A | B}) = \frac{P(\text{A and B})}{P(\text{B})}\)
)
"What is the relationship between this equation and the multiplication rule?" (From the multiplication rule, we know that
\(P(\text{A and B}) = P(\text{A | B}) \boldcdot P(\text{B})\)
. If we divide both sides by
\(P(\text{B})\)
, we get the equation that Jada used.)
“How did you determine whether the events were dependent or independent?” (I determined that they were dependent events because
\(P(\text{symptoms persist}) = 0.6\)
and
\(P(\text{symptoms persist | given medicine}) = 0.46\)
were not equal. If the events were independent, these probabilities would be equal.)
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


aligns: no