1. Teacher Note: What to print
For this lesson, you can print:
Homework
Two in class practice problems that align with the slides (good for interpreting tables)
and/or
A group of 9 slides starting around slide 29 that you can cut up and give to students for group practice constructing tables

2. Two Way Frequency Tables

3. Warm Up: Trig Review 𝟏.
3. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33˚. How tall is the tree? 𝟐.

4. Warm Up: Trig Review cos 47.2 = 3 𝑥 𝑥∗ cos 47.2 =3 𝑥= 3 cos⁡(47.2)
SOH-CAH-TOA
Warm Up: Trig Review 𝟏.
cos = 3 𝑥
𝑥∗ cos =3
𝑥= 3 cos⁡(47.2)
𝑥=4.4
cos 𝜃 = 7 10
θ= cos −1 ( )
𝑥=45.6˚
Adjacent
Hypotenuse 𝟐.

5. SOH-CAH-TOA
Warm Up: Trig Review
3. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33˚. How tall is the tree?
𝟑𝟑˚
𝟏𝟎𝟎 𝐟𝐞𝐞𝐭
tan 33 = ℎ 100
100∗ tan 33 =ℎ
ℎ=64.94 feet

6. Two Way Frequency Tables
Objective
DOL
SWBAT construct and analyze two-way tables.
Given 2 CR problems, SW construct and analyze two-way tables with 80% accuracy.
EQ: How can we represent information?

7. What’s the best kind of cake?
Let them eat cake
_________________
What’s the best kind of cake?
Everything else
Record the number that choose the dominant flavor and track of each student that chooses the dominate flavor. You may have them come to the board to initial which they prefer.

8. What’s better: Cake or cupcakes?
Let them eat cake
_________________
What’s better: Cake or cupcakes?
Everything else
Record the preference of every student. You may have them come to the board to initial their preference.
Cake
Cupcakes

9. Two Way Frequency Tables
_________________
Everything else
Complete the table based on our data.
Cake
Cupcakes
Total
Everything else
Begin by adding the favorite flavor
Cake
Cupcakes

10. Two Way Frequency Tables
Two way frequency tables: Show data in a way that lets us make comparisons
For example…
Cake
Cupcakes
Total
Everything else
How many students preferred our most popular flavor?
How many students preferred cupcakes?
If we’re just making dessert for the students who preferred our most popular flavor, should we make cake or cupcakes?

11. Two Way Frequency Tables
Two way frequency tables: Show data in a way that lets us make comparisons
For example…
Cake
Cupcakes
Total
Everything else
Of the four options, what was the most common preference?
Was there a relationship between the flavor we preferred and whether we prefer cake or cupcakes?

12. Example: Rainy Day Fun
The Summer Camp for Kids staff is planning an indoor activity for the campers to do on rainy days.
They are considering a Lego activity and a finger painting activity.
They sent out a survey to all the kids who will be coming to the camp to find out their preferences.

13. Rainy Day Fun
The staff organized the results of the survey into this table
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓

14. (the raw data for each combination)
Notes
Two way frequency tables: Show data in a way that lets us make comparisons
1st variable
(type of cake)
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓
2nd variable
(cake flavor) } }
Joint Frequencies
(the raw data for each combination)
Marginal Frequencies
(the totals)

15. Rainy Day Fun Seeing the results How many boys preferred Legos?
Think-Write-Share
Seeing the results
How many boys preferred Legos?
How many girls preferred Legos?
How many campers preferred painting?
How many girls responded to the survey?
How many campers responded to the survey?
Fist to Five
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓

16. Write your response on your whiteboard.
Rainy Day Fun
What’s wrong with this interpretation?
More boys prefer Legos (37 to 26)
More girls prefer painting (14 to 8)
Therefore, they should let boys to Legos and girls paint
Write your response on your whiteboard.
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓

17. Write your response on your whiteboard.
Rainy Day Fun
What’s wrong with this interpretation?
More boys prefer Legos (37 to 26)
More girls prefer painting (14 to 8)
Therefore, they should let boys to Legos and girls paint
Write your response on your whiteboard.
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓

18. Finding Missing Information

19. Key Club Fundraising
The Key Club will sell candy as a fundraiser. They surveyed 80 students about their favorite candy. The results are shown in the frequency table. Fill in the missing information.
Hint: The joint frequencies always add up to the marginal frequencies
(37+43=80 and 19+18=37)
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗
Girls 𝟒𝟑 𝟑𝟓 𝟏𝟖 𝟑𝟕 𝟖𝟎
First, what do you know about the grand total?
Whiteboards: Cell by cell

20. Hint: The joint frequencies always add up to the marginal frequencies
Key Club Fundraising
The Key Club will sell candy as a fundraiser. They surveyed 80 students about their favorite candy. The results are shown in the frequency table. Fill in the missing information.
Hint: The joint frequencies always add up to the marginal frequencies
(37+43=80 and 19+18=37)
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗 𝟏𝟖 𝟑𝟕
Girls 𝟐𝟔 𝟏𝟕 𝟒𝟑 𝟒𝟓 𝟑𝟓 𝟖𝟎

21. Two Way Relative Frequency Tables
Back to summer camp

22. Two Way Relative Frequency Tables
With one change to the table, we can make it easy to see what proportion of respondents preferred each option.
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓
Two Way Frequency Table (raw data)
Legos
Painting
Total
Boys
𝟎.𝟒𝟒
𝟎.𝟎𝟗
𝟎.𝟓𝟑
Girls
𝟎.𝟑𝟏
𝟎.𝟏𝟔
𝟎.𝟒𝟕
𝟎.𝟕𝟒
𝟎.𝟐𝟔
𝟏.𝟎𝟎
Two Way Relative Frequency Table (proportions)

23. Notes: Two Way Relative Frequency Tables
Two way relative frequency table: A type of table that shows relative data in a way that lets us make proportional comparisons
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓
Two Way Frequency Table (raw data)
Instead of total numbers, this shows the proportion of respondents with that preference.
For example, only 16% of respondents were girls who prefer painting to Legos.
Legos
Painting
Total
Boys
𝟎.𝟒𝟒
𝟎.𝟎𝟗
𝟎.𝟓𝟑
Girls
𝟎.𝟑𝟏
𝟎.𝟏𝟔
𝟎.𝟒𝟕
𝟎.𝟕𝟒
𝟎.𝟐𝟔
𝟏.𝟎𝟎
Two Way Relative Frequency Table (proportions)

24. Notes: Two Way Relative Frequency Tables
To make two way relative frequency tables…
Divide the number in that cell by the total number of respondents
Legos
Painting
Total
Boys 𝟑𝟕 𝟖 𝟒𝟓
Girls 𝟐𝟔 𝟏𝟒 𝟒𝟎 𝟔𝟑 𝟐𝟐 𝟖𝟓
Two Way Frequency Table (raw data)
Legos
Painting
Total
Boys
𝟎.𝟒𝟒
𝟎.𝟎𝟗
𝟎.𝟓𝟑
Girls
𝟎.𝟑𝟏
𝟎.𝟏𝟔
𝟎.𝟒𝟕
𝟎.𝟕𝟒
𝟎.𝟐𝟔
𝟏.𝟎𝟎
Two Way Relative Frequency Table (proportions)
𝟑𝟕/𝟖𝟓
𝟖/𝟖𝟓
𝟒𝟓/𝟖𝟓
The grand total cell will always be 1.00.
𝟐𝟔/𝟖𝟓
𝟏𝟒/𝟖𝟓
𝟒𝟎/𝟖𝟓
𝟔𝟑/𝟖𝟓
𝟐𝟐/𝟖𝟓
𝟖𝟓/𝟖𝟓

25. Whiteboards: Cell by cell
Key Club Fundraising
In your notes for the Key Club Fundraiser, draw a relative frequency table next to the frequency table you already have.
The first value is given to you below. Finish the first row, then move to the second.
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗 𝟏𝟖 𝟑𝟕
Girls 𝟐𝟔 𝟏𝟕 𝟒𝟑 𝟒𝟓 𝟑𝟓 𝟖𝟎
Two Way Frequency Table (raw data)
Lollipop
Peanut Butter Cup
Total
Boys
𝟎.𝟐𝟒
𝟎.𝟐𝟑
𝟎.𝟒𝟔
Girls
𝟎.𝟑𝟑
𝟎.𝟐𝟏
𝟎.𝟓𝟒
𝟎.𝟓𝟔
𝟎.𝟒𝟒
𝟏.𝟎𝟎
Two Way Relative Frequency Table (proportions)

26. Constructing Tables

27. Chocolates and Caramels
Lena has a box of 18 chocolate and caramel candies.
10 candies have chocolate and caramel
3 candies have neither chocolate nor caramel
12 candies have chocolate
Complete the table given this information.
Caramel
No caramel
Total
Chocolate 𝟏𝟎 𝟐 𝟏𝟐
No chocolate 𝟑 𝟔 𝟏𝟑 𝟗 𝟏𝟖
Step 1. Make a table
Step 2. Fill in the given values
Step 3. Calculate the missing values

28. Teacher Note
You can print the next 9 slides in a 3 by 3 grid, cut them into rows and distribute them to students to work in groups. You could also give only one piece of information (rather than 3) to each student and then require them to find others with the same topic to gather enough information to complete the tables. Note: I’ve given no answers to the questions. Students will be responsible for giving the answers to the class.

29. Soccer vs. Football
ESPN surveyed 50 respondents from the United States and Mexico about whether they prefer soccer or football.
26 people from the United States responded
8 people from the United States preferred soccer
23 people from Mexico preferred soccer
Given this information, complete the two way frequency table and the two way relative frequency table.
Step 1. Make a table
Step 2. Fill in the given values
Step 3. Calculate the missing values

30. Soccer vs. Football Soccer Football Total USA 𝟖 𝟏𝟖 𝟐𝟔 Mexico 𝟐𝟑 𝟏 𝟐𝟒𝟑𝟏 𝟏𝟗 𝟓𝟎

31. Soccer vs. Football How many people from Mexico responded?
Were both countries evenly represented? Why?
What does the number 18 represent?
Which sport is more popular overall?
What patterns do you see in the data?
What percent of U.S. respondents prefer soccer?
What is the meaning of each marginal frequency?

32. Cats and Dogs
A new Wal-Mart surveyed 75 people in the neighborhood about their pets.
25 people have a dog
20 people have a cat
9 people have a cat and a dog
Given this information, complete the two way frequency table and the two way relative frequency table.
Step 1. Make a table
Step 2. Fill in the given values
Step 3. Calculate the missing values

33. Cats and Dogs
Dog
No dog
Total
Cat 𝟗 𝟏𝟏 𝟐𝟎
No cat 𝟏𝟔 𝟑𝟗 𝟓𝟓 𝟐𝟓 𝟓𝟎 𝟕𝟓

34. Cats and Dogs How many respondents have no dog?
What percent of respondents have no pet?
How common is it to have a cat and dog?
What does the number 55 represent?
Which animal is more popular? Why?
What patterns do you see in the data?

35. Ribs vs. Tacos Two food trucks work an event with 100 people.
40 people ate at the rib truck
50 people skipped the taco truck
35 people ate at neither truck
Given this information, complete the two way frequency table and the two way relative frequency table.
Step 1. Make a table
Step 2. Fill in the given values
Step 3. Calculate the missing values

36. Ribs vs. Tacos
Ribs
No Ribs
Total
Tacos 𝟐𝟓 𝟓𝟎
No tacos 𝟏𝟓 𝟑𝟓 𝟒𝟎 𝟔𝟎
𝟏𝟎𝟎

37. Ribs vs. Tacos
Looking at all four joint frequencies, what was the most popular option?
What was the second most popular option?
What does the number 15 represent?
What percent of people had no ribs?
What do the marginal frequencies represent?
What patterns do you see in the data?

38. Practice

39. Key Club Fundraising How many students preferred lollipops?
WB checks
How many students preferred lollipops?
How many girls preferred peanut butter cups?
How many boys answered the survey?
Were boys and girls both evenly represented? Why? 45 17 37
Boys and girls were represented fairly evenly because
about the same number of boys and girls responded.
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗 𝟏𝟖 𝟑𝟕
Girls 𝟐𝟔 𝟏𝟕 𝟒𝟑 𝟒𝟓 𝟑𝟓 𝟖𝟎

40. Key Club Fundraising Explain what the number 19 means in this table.
WB checks
Explain what the number 19 means in this table.
Explain what the number 35 means in this table.
19 represents the number of boys who preferred lollipops.
35 represents the total number of people who preferred peanut butter cups.
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗 𝟏𝟖 𝟑𝟕
Girls 𝟐𝟔 𝟏𝟕 𝟒𝟑 𝟒𝟓 𝟑𝟓 𝟖𝟎

41. Key Club Fundraising What is the meaning of each marginal frequency?
WB checks
What is the meaning of each marginal frequency?
What is the meaning of each joint frequency?
The right column represents the total number of boys and girls who responded while the bottom row represents the total number of respondents who prefer each candy.
Each joint frequency represents the preference for each candy of boys or girls.
Lollipop
Peanut Butter Cup
Total
Boys 𝟏𝟗 𝟏𝟖 𝟑𝟕
Girls 𝟐𝟔 𝟏𝟕 𝟒𝟑 𝟒𝟓 𝟑𝟓 𝟖𝟎

42. Practice: What’s for lunch?
WB checks
A total of 247 students were surveyed about what they liked best for lunch. The results can be shown in a two-way frequency table. Fill in the table.
Begin with the grand total, then total juniors, then freshmen pizza.
Pizza
Salad
Chicken Sandwich
Total
Freshmen 𝟏𝟖 𝟑𝟗
𝟏𝟎𝟐
Sophomores 𝟒𝟐 𝟖𝟕
Juniors 𝟐𝟖 𝟖𝟓 𝟔𝟓 𝟒𝟓 𝟓𝟖
𝟐𝟒𝟕

43. Practice: What’s for lunch?
WB checks
How many juniors preferred pizza?
What was preferred by most freshmen?
What was preferred least by sophomores?
Overall, what was the most popular choice? 14
Pizza
Salad
Chicken sandwich
Pizza
Salad
Chicken Sandwich
Total
Freshmen 𝟒𝟓 𝟏𝟖 𝟑𝟗
𝟏𝟎𝟐
Sophomores 𝟐𝟔 𝟏𝟗 𝟒𝟐 𝟖𝟕
Juniors 𝟏𝟒 𝟐𝟖 𝟏𝟔 𝟓𝟖 𝟖𝟓 𝟔𝟓 𝟗𝟕
𝟐𝟒𝟕

44. Practice: What’s for lunch?
WB checks
Were the different grade levels equally represented?
What was the meaning of the number 28 in the table?
No, there were 100 freshman and only 58 juniors.
28 juniors preferred chicken sandwiches.
Pizza
Chicken Sandwich
Salad
Total
Freshmen 𝟒𝟓 𝟏𝟖 𝟑𝟗
𝟏𝟎𝟐
Sophomores 𝟐𝟔 𝟏𝟗 𝟒𝟐 𝟖𝟕
Juniors 𝟏𝟒 𝟐𝟖 𝟏𝟔 𝟓𝟖 𝟖𝟓 𝟔𝟓 𝟗𝟕
𝟐𝟒𝟕

45. Practice: What’s for lunch?
WB checks
What is the meaning of the number 85 in the table?
Explain the meaning of the marginal frequencies in this table.
A total of 85 students across all grades preferred pizza.
The marginal frequencies (totals) on the right represent the number of respondents per grade and the ones on the bottom represent the total number of students with that preference.
Pizza
Chicken Sandwich
Salad
Total
Freshmen 𝟒𝟓 𝟏𝟖 𝟑𝟗
𝟏𝟎𝟐
Sophomores 𝟐𝟔 𝟏𝟗 𝟒𝟐 𝟖𝟕
Juniors 𝟏𝟒 𝟐𝟖 𝟏𝟔 𝟓𝟖 𝟖𝟓 𝟔𝟓 𝟗𝟕
𝟐𝟒𝟕

46. Practice: What’s for lunch?
WB checks
Explain the meaning of the joint frequencies in this table
How can you check your work to be sure you have filled in the table correctly?
The joint frequencies represent how many students in each grade preferred each lunch option.
Make sure the totals add up in each row and column.
Pizza
Chicken Sandwich
Salad
Total
Freshmen 𝟒𝟓 𝟏𝟖 𝟑𝟗
𝟏𝟎𝟐
Sophomores 𝟐𝟔 𝟏𝟗 𝟒𝟐 𝟖𝟕
Juniors 𝟏𝟒 𝟐𝟖 𝟏𝟔 𝟓𝟖 𝟖𝟓 𝟔𝟓 𝟗𝟕
𝟐𝟒𝟕

47. Practice: What’s for lunch?
WB checks
Make a two way relative frequency table of the data
Pizza
Chicken Sandwich
Salad
Total
Freshmen
𝟎.𝟏𝟖
𝟎.𝟎𝟕
𝟎.𝟏𝟔
𝟎.𝟒𝟏
Sophomores
𝟎.𝟏𝟏
𝟎.𝟎𝟖
𝟎.𝟏𝟕
𝟎.𝟑𝟓
Juniors
𝟎.𝟎𝟔
𝟎.𝟐𝟑
𝟎.𝟑𝟒
𝟎.𝟐𝟔
𝟎.𝟑𝟗
𝟏.𝟎𝟎

48. How can we represent information?
Summary
How can we represent information?

49. DOL
A total of 176 people were surveyed about what outdoor fitness activity they preferred to do in the summer. The results can be shown in a two-way frequency table. Complete the table and answer the questions.
Which sports get more popular with age?
What was most popular overall? What percent of people preferred it?
Were the different age groups represented evenly? Why?
What do the marginal frequencies tell us?
Soccer
Basketball
Jogging
Total
Teenagers 𝟖 𝟏𝟔 𝟔𝟐
College age 𝟏𝟕 𝟓𝟕
Adults 𝟏𝟖 𝟕𝟖 𝟒𝟑

50. DOL Which sports get more popular with age?
Basketball and jogging get more popular with age.
What was most popular overall? What percent of people preferred it?
Soccer is most popular overall. 44% of respondents prefer it.
Soccer
Basketball
Jogging
Total
Teenagers 𝟑𝟖 𝟖 𝟏𝟔 𝟔𝟐
College age 𝟐𝟑 𝟏𝟕 𝟓𝟕
Adults 𝟏𝟖 𝟐𝟐 𝟕𝟖 𝟒𝟑 𝟓𝟓
𝟏𝟕𝟔

51. DOL Were the different age groups represented evenly? Why?
The age groups were represented about evenly. College age and adult respondents had 57 each and teenagers were only 5 away.
What do the marginal frequencies tell us?
The right column shows the total number of respondents for each age and the bottom row shows the total number of preferences each sport received.
Soccer
Basketball
Jogging
Total
Teenagers 𝟑𝟖 𝟖 𝟏𝟔 𝟔𝟐
College age 𝟐𝟑 𝟏𝟕 𝟓𝟕
Adults 𝟏𝟖 𝟐𝟐 𝟕𝟖 𝟒𝟑 𝟓𝟓
𝟏𝟕𝟔