Mean Master Key
Teacher Master Key
U8, L6: Mean — Master Key
Annotated Student Pages & Pedagogical Facilitation Guide
6th Grade Mathematics
Unit 8: Statistics
Uncommon Schools
Do Now (Student Page 1)
1
A two-gallon jug of apple cider costs $7.88. What is the price per gallon?
A. $1.98
B. $3.94 Correct
C. $7.88
D. $15.76
Pedagogical Note: Unit rate review: $7.88 ÷ 2 gallons = $3.94 per gallon. Ensure students do not multiply by 2 (Choice D).
2
Karin trains for a race (biking + running). Training details are below:
| Activity | Distance (miles) | Time (minutes) | Speed (Annotated) |
|---|
| Biking | 75 miles | 300 minutes | 0.25 miles/minute |
| Running | 30 miles | 240 minutes | 0.125 miles/minute |
Karin predicts she will spend 48 minutes biking and 24 minutes running. What is the total distance of the race?
A. 14 miles
B. 15 miles Correct
C. 16 miles
D. 22 miles
Annotated Work Steps:
1. Speeds: Biking = 75 ÷ 300 = 0.25 mi/min; Running = 30 ÷ 240 = 0.125 mi/min
2. Predicted Distances: Biking = 48 min × 0.25 = 12 miles
3. Running = 24 min × 0.125 = 3 miles
4. Total Distance: 12 + 3 = 15 miles (Choice B)
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 1 of 8
Do Now Cont. & Excellence (Student Pages 2 & 3)
Equation Skill-Build
3
Solve the equation:
45.5 + x = 60
Choice A. 14.5
Work: x = 60 - 45.5 = 14.5
4
Solve the equation:
x + 23.45 = 85
Choice B. 61.55
Work: x = 85 - 23.45 = 61.55
Excellence: One-Step Equations (Solve for x)
1
x + 27 = 48
x = 21
48 - 27 = 21
2
x + 10 = 17
x = 7
17 - 10 = 7
3
x - 73 = 17
x = 90
17 + 73 = 90
4
x - 73 = 144
x = 217
144 + 73 = 217
5
10 - x = 77
x = -67
Or x=67 if 10+x=77
6
x + 1 = 9
x = 8
9 - 1 = 8
7
5 - x = 12
x = -7
Or x=7 if 5+x=12
8
2 = x + 12
x = -10
2 - 12 = -10
9
x - 8 = 19
x = 27
19 + 8 = 27
10
x - 22 = 30
x = 52
30 + 22 = 52
11
13 = x + 4
x = 9
13 - 4 = 9
12
x - 4 = 5
x = 9
5 + 4 = 9
Differentiated Support Note: Questions 5, 7, and 8 may yield negative solutions. In 6th-grade math classes where negative integers have not been formally taught, explain that these questions represent typo variations of standard addition/subtraction. Teach the standard absolute value of balancing where applicable.
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 2 of 8
Warm-Up (8.9.2) - Part 1 (Student Page 4)
Fair Share Concept
1. The kittens in a room at an animal shelter are placed in 5 crates.
Crate 1 🐱🐱 2 Kittens
Crate 2 🐱 1 Kitten
Crate 3 🐱🐱🐱🐱 4 Kittens
Crate 4 🐱🐱🐱 3 Kittens
Crate 5 Empty 0 Kittens
a. The manager wants the kittens distributed equally. How might that be done? How many kittens end up in each crate?
Sample Answer & Explanation: "We can find the total number of kittens and divide them equally. There are 2 + 1 + 4 + 3 + 0 = 10 kittens in total. Since there are 5 crates, we divide 10 by 5.
Each crate will end up with exactly 2 kittens.
Physically, we could move 1 kitten from Crate 3 to Crate 2, then move 1 kitten from Crate 3 and 2 kittens from Crate 4 into Crate 5. This levels out all crates to 2."
b. This equal distribution is called the average. Explain how the expression 10 ÷ 5 is related to the average.
Sample Student Explanation: "The 10 in the expression represents the total sum of all the items in our data set (the 10 kittens combined). The 5 represents the number of categories (the 5 crates we are sharing them among). Dividing the total by the number of groups calculates the fair share, which is the average of 2."
Key Lesson Takeaway: Before introducing formulas, use this warm-up to solidify the conceptual meaning of "mean" as fair share redistribution. Help students realize that the total count stays invariant (always 10 kittens), but leveling out spreads them out uniformly.
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 3 of 8
Warm-Up (8.9.2) - Part 2 (Student Page 5)
Graphical Redistribution
2. Five servers share workload equally. Scheduled: A: 3 hrs • B: 6 hrs • C: 11 hrs • D: 7 hrs • E: 4 hrs
a. Scheduled Work Hours (Original)
12—9—6—3—0—
3
A
6
B
11
C
7
D
4
E
Sum of Hours = 31 hours
b. Rearranged Hours (Equal Share)
12—9—6—3—0—
6.2
A
6.2
B
6.2
C
6.2
D
6.2
E
Mean = 6.2 hours each
c. Based on your second drawing, what is the average or mean number of hours?
Answer: 6.2 hours (or 6 1/5 hours).
Calculated as: (3 + 6 + 11 + 7 + 4) ÷ 5 = 31 ÷ 5 = 6.2 hours.
d. Which server will see the biggest change? Which server will see the least change?
Sample Student Explanation: "Server C sees the biggest change. They went from 11 hours down to 6.2 hours, which is a reduction of 4.8 hours. Server B sees the least change. They went from 6 hours up to 6.2 hours, which is an increase of only 0.2 hours."
Alert on decimal answers: Students may initially struggle drawing 6.2. Explain that since 31 cannot be split into integers, some parts are shared. 0.2 hours is equivalent to 12 minutes.
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 4 of 8
Task 1 (8.9.3) - Part 1 (Student Page 6)
Determining the Mean
Mai's recorded bus ride times to school over 12 days (in minutes):
9869107 61298108
1. Find the mean for Mai's data. Show your reasoning.
Complete Mathematical Proof:
- Sum of all data: 9+8+6+9+10+7+6+12+9+8+10+8 = 102 minutes
- Number of days recorded (data values count): 12
- Calculated Mean: 102 ÷ 12 = 8.5 minutes
The mean commute time is 8.5 minutes.
2. In this situation, what does the mean tell us about Mai's trip?
Sample Student Explanation: "The mean of 8.5 minutes represents the typical length of time the bus trip took. It means that if all 12 trips were perfectly equal, they would have each taken exactly 8.5 minutes."
3. Tyler's 5-day walk to school has a mean of 11 minutes. Without calculating, predict if each of the data sets shown could be Tyler's. Explain your reasoning.
| Data Set | Could it be? | Explain reasoning (Annotated Answer) |
|---|
| A: 11, 8, 7, 9, 8 | No | All numbers except one are below 11. There are no numbers above 11 to balance them out, so the average must be less than 11. |
| B: 12, 7, 13, 9, 14 | Yes | The values are spread above and below 11. 12, 13, 14 are above; 7, 9 are below. This balance makes 11 highly possible as the center. |
| C: 11, 20, 6, 9, 10 | No | Although 20 balances out 6, 9, and 10, when calculated, the sum is 56, giving a mean of 11.2 (not exactly 11). |
| D: 8, 10, 9, 11, 11 | No | Every single number is either 11 or lower than 11. Without any values above 11 to pull the average up, the mean has to be below 11. |
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 5 of 8
Task 1 Cont. & Practice (Student Page 7 & 8)
Independent Practice
4. Determine which data set is Tyler's. Explain how you know.
Correct Answer: "Tyler's data set is Data Set B."
Mathematical proof: Sum of B = 12 + 7 + 13 + 9 + 14 = 55. Dividing the total by 5 values: 55 ÷ 5 = 11 minutes exactly.
Fill in the blank definition:
We can find the mean by adding all the values in the data set and dividing the sum by the total number of values.
Independent Practice
1
Box 1: 32 blocks | Box 2: 18 blocks | Box 3: 41 blocks | Box 4: 9 blocks.
Select all ways to make each box have equal blocks (Mean = 25):
A. Remove all blocks, make 4 piles of 25, and redistribute. Correct
E. Remove 7 from Box 1 to Box 2, and 16 from Box 3 to Box 4. Correct
Explanation: Box 1 becomes 32-7=25; Box 2 becomes 18+7=25; Box 3 becomes 41-16=25; Box 4 becomes 9+16=25.
2. Golf strokes: 2, 3, 1, 4, 5, 2, 3, 4, 3. Explain how Clare's average is 3.
Sample Student Explanation: "Clare is correct because the sum of her strokes is 27 (2+3+1+4+5+2+3+4+3=27). Since there are 9 holes in total, dividing the total strokes by the holes gives: 27 ÷ 9 = 3. This represents the 'fair share' or average number of strokes per hole."
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 6 of 8
Exit Ticket & Homework Answers (Pages 10 & 11)
Daily Assessments
Exit Ticket: Finding Means
1. City low temperatures: 5, 8, 6, 5, 10, 7, 1. What was the average temperature? Show reasoning.
Answer: 6 degrees Celsius
Reasoning: Sum the values: 5 + 8 + 6 + 5 + 10 + 7 + 1 = 42. Divide by the count of days (7): 42 ÷ 7 = 6.
2. Mean of 4 numbers is 7. Three of the numbers are 5, 7, and 7. What is the fourth number?
Answer: 9
Reasoning: If the average of 4 numbers is 7, their total sum must equal 4 × 7 = 28. Sum of known three: 5 + 7 + 7 = 19. To find the fourth, subtract known from total: 28 - 19 = 9.
Homework: Mean Practice
1. Money raised: $25.50, $49.75, and $37.25. Equal share? Show reasoning.
Answer: $37.50
Reasoning: Total raised: $25.50 + $49.75 + $37.25 = $112.50. Divided by 3 classes: $112.50 ÷ 3 = $37.50.
2. Mai's English quiz scores: 4, 3, 4 (worth 5 pts). What score does she need on the 4th quiz to average 4?
Answer: 5
Reasoning: To get an average of 4 on 4 quizzes, total points must be 4 × 4 = 16. Current points = 4 + 3 + 4 = 11. Score needed = 16 - 11 = 5.
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 7 of 8
Mixed Review (Student Page 12)
Daily Spiral Review
1. Earthworm Length Histograms (Containers A vs B)
Container A Histogram Shape
Centered around 40 mm (Symmetric/Spread)
Container B Histogram Shape
Skewed Right (Heavily concentrated 5-20 mm)
a. Which container tends to have longer worms than the other container?
Container A. Because Container A's distribution is shifted further right (centered around 40 mm) compared to Container B (centered around 15 mm).
b. For which container would 15 mm be a reasonable description of a typical length?
Container B. Most of Container B's data is clustered closely between 5 to 20 mm. 15 mm sits right in the middle of this high-frequency cluster.
c. If length is related to age, which container had the most young worms?
Container B. Container B has a high frequency of extremely short worms (5-15 mm), representing a significantly younger population.
2. Diego's Equation Claim
Diego thinks that x = 3 is a solution to the equation x2 = 16. Do you agree? Explain.
Disagree.
Proof of check: Substitute 3 into the equation: 32 = 3 × 3 = 9. Since 9 ≠ 16, 3 is not a solution. The correct positive solution is x = 4 because 42 = 16, and the negative solution is x = -4.
Adapted from Illustrative Math © 2025-26 Uncommon Schools Page 8 of 8
