few_shots: 
Example 1:
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.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
problem_activity: Activity
In this activity, students create two-way tables displaying relative frequency. The relative frequency table converts the actual frequency data to percentages which can be useful when comparing groups that include different totals. Finally, students use the relative frequencies to look for a pattern in the data. In the following lesson, students will work with associations in categorical data more explicitly. In this activity, students should use an informal understanding of association to think about whether one is present in the data based on the relative frequency table.
Launch
Keep students in groups of 2–3. After an introduction to relative frequency tables, allow students 3 minutes quiet work time followed by partner discussion and whole-class discussion.
Display the table for all to see.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "watch the news daily"}, {"_attributes": {"scope": "col"}, "_value": "does not watch the news daily"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "younger than 18"}], "td": [{"_value": "30"}, {"_value": "80"}, {"_value": "110"}]}, {"th": [{"_value": "18 or older"}], "td": [{"_value": "10"}, {"_value": "5"}, {"_value": "15"}]}, {"th": [{"_value": "total"}], "td": [{"_value": "40"}, {"_value": "85"}, {"_value": "125"}]}]}]}]}
Ask students, "Based on this data, who is more likely to watch the news daily: someone who is younger than 18 or someone who is 18 or older?"
Tell students that, based on the numbers in the table, there are more younger people who watch the news (30) than older (10). On the other hand, the survey reached out to 110 young people and only 15 older people. Without looking at the whole table, that information may have been missed.
In cases like this, finding a relative frequency including percentages can be more helpful than looking at the actual frequency, which is what they are going to do now.
Use MLR 2 (Collect and Display) to co-create a graphic organizer with students comparing and contrasting the definitions and characteristics of “actual frequency” and “relative frequency” to be referenced and revised throughout this and later activities.
Conversing, Representing, Writing: MLR2 Collect and Display.
As students work in groups, capture the vocabulary and phrases students use when comparing and contrasting the definitions and characteristics of “actual frequency” and “relative frequency.” Listen for students who identify pros and cons for each representation and when one is preferable compared to the other. Record their language on a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a two-way table that shows data about cell phone usage among children aged 10 to 18.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "has cell phone"}, {"_attributes": {"scope": "col"}, "_value": "does not have cell phone"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "10 to 12 years old"}], "td": [{"_value": "25"}, {"_value": "35"}, {"_value": "60"}]}, {"th": [{"_value": "13 to 15 years old"}], "td": [{"_value": "40"}, {"_value": "10"}, {"_value": "50"}]}, {"th": [{"_value": "16 to 18 years old"}], "td": [{"_value": "50"}, {"_value": "10"}, {"_value": "60"}]}, {"th": [{"_value": "total"}], "td": [{"_value": "115"}, {"_value": "55"}, {"_value": "170"}]}]}]}]}
Complete the table. In each row, the entries for “has cell phone” and “does not have cell phone” should have the total 100%. Round entries to the nearest percentage point.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "has cell phone"}, {"_attributes": {"scope": "col"}, "_value": "does not have cell phone"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "10 to 12 years old"}], "td": [{"_value": "42%"}, {}, {}]}, {"th": [{"_value": "13 to 15 years old"}], "td": [{}, {}, {"_value": "100%"}]}, {"th": [{"_value": "16 to 18 years old"}], "td": [{}, {"_value": "17%"}, {}]}]}]}]}
This is still a two-way table. Instead of showing
frequency
, this table shows
relative frequency
.
Two-way tables that show relative frequencies often don’t include a “total” row at the bottom. Why?
Is there an association between age and cell phone use? How does the two-way table of relative frequencies help to illustrate this?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
A pollster attends a rally and surveys many of the participants about whether they associate with political Party A or political Party B and whether they are for or against Proposition 3.14 going up for vote soon. The results are sorted into the table shown.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "for"}, {"_attributes": {"scope": "col"}, "_value": "against"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "party A"}], "td": [{"_value": "832"}, {"_value": "165"}]}, {"th": [{"_value": "party B"}], "td": [{"_value": "80"}, {"_value": "160"}]}]}]}]}
A news station reports these results by saying, “A poll shows that about the same number of people from both parties are voting against Proposition 3.14.”
A second news station shows this graphic.
[IMG]
Are any of the news reports misleading? Explain your reasoning.
Create a headline, graphic, and short description that more accurately represents the data in the table.
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 may not calculate relative frequencies correctly. Look to see if they are dividing into the total for each row, instead of some other number in the row, or the total for the entire table.
Activity Synthesis
The purpose of this discussion is to help students see the usefulness of two-way tables that display relative frequency.
Some questions for discussion:
“What does the 42% in table mean?” (42% of people 10 to 12 years old have a cell phone.)
“What is the total of the percentages in the first column? What's wrong with that answer?” (205%. It doesn't make sense since it is greater than 100%. It is combining percentages of different wholes, so it doesn't make sense to add them.)
“Did you say there was or was not an association between age and cell-phone use? Explain your reasoning.” (There is an association since the percentages for the 10 to 12 year olds is very different from the other two age groups.)
“What did the relative frequency table show you that was harder to see in the original two-way table?” (It was much easier to see the association with the percentages than the numbers. For example, the number of people without a cell phone for the older 2 age groups was the same, but the percentages were a little different.)


aligns: no