Math Education · Grade 5
Lesson progress0 of 10 missions
Chapter 3 · Lesson 3.1

Average, Total, and Number of Values

平均数、总和与个数

Learn what an average means, connect it to the total and the number of values, and use the same idea to understand average speed.

An average hides a total. Reveal the total first.
Grade 5 enrichment 35–45 minutes 10 missions Autosaves on this device

By the end of this lesson, I can…

  • explain an average as a fair share;
  • use average = total ÷ count;
  • find a total or count from the same relationship;
  • match every total with the correct number of values;
  • calculate average speed from total distance and total time.
Total = Average × Count Match the total with its count. To find a count by division, the average must be nonzero.
Open the mission map
1. Average as a fair share 2. The formula family 3. Find an average 4. Reveal a total 5. Find the count 6. Units and reasonableness 7. Average speed 8. Error detective 9. Practice workshop 10. Exit ticket
Mission 1

Average means “share the total equally”

Not completed

Four jars hold different numbers of counters. An average tells us how many counters each jar would have if the same total were shared equally.

6 + 8 + 10 + 12 = 36 36 ÷ 4 = 9

The total stays 36. Only the way it is shared changes.

Hint

Add all four jar values first. Then divide that unchanged total by the number of jars.

Mission 2

Three quantities, one connected family

Not completed

Average problems connect a total, a count, and an average. The total and count must describe the same group, with at least one value. Here, average means the arithmetic mean: total ÷ number of values. Treat given averages as exact unless rounding is explicitly requested.

TOTAL
AVERAGE
COUNT
Average = Total ÷ Count
Matching rule: Never divide a total from one group by a count from another group. First ask, “How many values produced this total?”
Mission 3

Find an average from a total and a count

Not completed

A team earns 48 points across 6 games. Sharing the 48 points equally across the 6 games reveals the average.

Average = Total ÷ Count
48 points÷6 games=8 points per game
Average
8

Change the total or count above. Notice that the average changes because it is the total shared across the number of values.

Mission 4

An average hides a total

Not completed

In the chapter’s first worked problem, four mathematics tests have an average of 68. Before asking about a future test, the book first reveals the current total.

68
68
68
68
Total = Average × Count 68 × 4 = 272 points

The four actual scores do not all need to be 68. The equal boxes represent the same total in a fair-share model.

Mission 5

Find how many values produced a total

Not completed

The worked example states the average and total formulas. From the same relationship, we can divide the total by the average to recover the count.

Count = Total ÷ Average

This division requires a nonzero average. If the total and average are both 0, they do not tell us how many values there are.

420 points ÷ 70 points per game = 6 games
Count check: When the count means people, tests, jars, or games, it must be a whole number. If division gives a decimal, recheck the data and units, and whether the given average was rounded; do not round the count to force an answer.
Mission 6

Keep the units—and check whether the answer makes sense

Not completed

An average has a meaning and a unit. It should also lie between the smallest and largest values in the group.

Test scoresAverage unit: points
Pages readAverage unit: pages per day
TravelAverage unit: kilometres per hour
Bounds check: If all values are between 12 and 20, their average cannot be 25. The average may be a decimal even when every original value is a whole number.
Mission 7

Average speed uses total distance and total time

Not completed

A bus travels for the first 3 hours at 35 km/h, then covers another 80 km during the next 2 hours. This is the chapter’s opening journey example.

Stage 1 distance
105 km
Stage 2 distance
80 km

Stage 1

3 hours at 35 km/h

Distance3 × 35 = 105 km
Time3 hours
Key ideaBuild distance first
Total distance = 3 × 35 + 80 = 185 km Total time = 3 + 2 = 5 h Average speed = 185 ÷ 5 = 37 km/h
Mission 8

Error detective: protect the total–count relationship

Not completed

Strong problem solvers do more than calculate—they diagnose why a method fails.

A
Wrong operation

A student should divide the total of 6 numbers by 6, but multiplies the total by 6 and gets 1800.

B
Unweighted speeds

A student averages 35 km/h and 40 km/h directly, even though they lasted 3 hours and 2 hours.

C
Average must be one of the data values

A student rejects an average of 7.5 because every original value was a whole number.

Hint for Error A

The wrong result says: total × 6 = 1800. First undo that multiplication to find the total. Then use total ÷ 6.

Mission 9

Average workshop

Not completed

Decide which quantity is missing, choose the matching formula, and keep the total paired with the correct count.

1Find the total

Nine rounds average 7 points.

2Find the average

156 pages are read in 6 days.

3Find the count

A total of 360 has an average of 45.

4Highest possible new average

Four English tests average 85. The next test is scored out of 100. What is the highest possible five-test average?

5Missing score challenge

Three subjects average 74. After a fourth subject is added, the average rises by 3 points. What is the fourth score?

6Another journey

A vehicle travels 2 hours at 45 km/h, then covers 60 km in 3 hours. Find the whole-trip average speed.

7Check the bounds

The values are 20, 24, 28, and 32.

8Choose the right statement

A total of 525 has an average of 75.

Hints for Questions 4–6

4: Find the current total, add the maximum possible next score, then divide by 5.
5: Compare the old total for 3 subjects with the new total for 4 subjects.
6: Build both stage distances, then use total distance ÷ total time.

Mission 10

Exit ticket

Not completed

Complete these five questions without opening the hints. A perfect score unlocks your certificate.

Certificate of Mastery

Average–Total–Count Navigator

This certifies that

can explain an average as a fair share, connect average with total and count, and calculate average speed from total distance and total time.

Teaching notes

The first-step total of 4 × 68 = 272 comes from the worked test-score example on the same page.