8.3Growing Resources
Lesson progress0 of 10 missions
Chapter 8 · Lesson 8.3 · Grade 5 Enrichment

Convert Different Consumers to One Unit

把不同的消耗者换算成同一种单位

A cow and a sheep do not use grass at the same rate. Before combining them, translate every consumer into one common unit. Then the growing-resource model works exactly as before.

Convert first. Add second. Solve third.
30–40 minutesEverything needed is on this page.10 interactive missionsAutosaves in this browser
Mission 1

Headcounts are not consumption rates

Eleven animals do not necessarily consume eleven equal units. We must compare how much each type uses in the same amount of time.

Not complete
1 cowone day of eating
4 sheepone day of eating
Common unit: We will call the grass eaten by one cow in one day one cow-day unit of grass. Each sheep then uses 1 ÷ 4 = 1/4 cow-day unit per day.

Example

Three cows and eight sheep contain 11 animals, but their consumption is:

3 + 8 ÷ 4 = 5 cow-equivalents

Why this matters

A cow-equivalent describes an eating rate: the rate of one cow. Measure grass amounts in cow-days and daily growth or use in cow-day units per day. Multiply a daily rate by days before comparing it with a stock.

Check the idea

Mission 2

Build an equivalence bridge

If one large consumer equals several small consumers, divide when converting small to large and multiply when converting large to small.

Not complete
1 cow = 4 sheep
12 sheep
7 cows
🐄 Cow-equivalents3sheep ÷ ratio
🐑 Sheep-equivalents28cows × ratio
Sheep outside full groups0still count as a fraction of a cow-equivalent
Each complete group contains the selected number of sheep per cowLeftover sheep still eat: include their fractional cow-equivalent
sheep → cow-equivalents: divide by 4cows → sheep-equivalents: multiply by 4

Use this relationship: 1 cow = 4 sheep

Mission 3

Write both pasture scenarios in cow-equivalents

These are separate trials with identical starting grass and constant growth. Each herd eats all the grass in exactly the stated time. Each animal eats at a constant rate, and one cow eats as much per day as four sheep.

Not complete

Scenario ACow unit

16 cows eat for 20 days.

16 × 20 = 320 cow-days
S + 20g = 320

Scenario BConvert first

80 sheep eat for 12 days.

80 sheep ÷ 4 = 20 cow-equivalents20 × 12 = 240 cow-days
S + 12g = 240
Notice: The symbol S means the same starting grass in both equations, and g is measured in cow-day units per day.

Build the common-unit ledger

cow-days
cow-days
Mission 4

Recover daily growth and starting stock

The two equations share the same starting amount. Subtracting them makes that hidden amount cancel.

Not complete
Difference in total use320 − 240 = 80
÷
Difference in time20 − 12 = 8
=
Daily growthg = 10
Both scenarios contain the same starting grasscow-day units

The green part is the shared starting stock. The hatched part is grass grown during each scenario.

Recover the start from Scenario A

S = 320 − 20 × 10 = 120

Check with Scenario B

S = 240 − 12 × 10 = 120

Complete the hidden-quantity calculation

cow-day units/day
cow-day units
Mission 5

Solve a mixed herd

Once cows and sheep are expressed in the same unit, the final step is an ordinary net-rate calculation.

Not complete

Cows

10 cows

Sheep

60 sheep
25cow-day units/day
10daily growth
15net depletion/day
8days until empty
10 + 60 ÷ 4 = 25 cow-equivalents
25 − 10 = 15 units/day reduce the starting stock
120 ÷ 15 = 8 days

Solve the fixed mixed herd: 10 cows and 60 sheep (regardless of the sliders)

Mission 6

Change the measuring unit—not the physical answer

Cow-equivalents are convenient, but sheep-equivalents also work. Every rate and stock must scale together.

Not complete
Measured in cows
Growth = 10 cow-units/day
Start = 120 cow-days
Mixed herd = 25 cow-units/day
Net = 15 cow-units/day
× 4
Measured in sheep
Growth = 40 sheep-units/day
Start = 480 sheep-days
Mixed herd = 100 sheep-units/day
Net = 60 sheep-units/day
Either unit gives8 days120 ÷ 15 = 480 ÷ 60 = 8
Unit consistency rule: Changing units changes the numbers, but not the actual pasture or the number of days.

Verify the sheep-equivalent solution

Mission 7

Guided practice: a second mixed-herd pasture

Work through a complete problem using the same conversion-first routine.

Not complete

Problem

A pasture grows at a constant rate. It can feed 20 cows for 12 days or 60 sheep for 24 days. One cow eats as much per day as four sheep. How long can 12 cows and 88 sheep eat?

Convert60 ÷ 4 = 15
Totals240, 360
Growth10
Start120
New mix34
Time5 days
  1. Convert 60 sheep to 60 ÷ 4 = 15 cow-equivalents.
  2. Compare total use: 20 × 12 = 240 and 15 × 24 = 360.
  3. Find growth: (360 − 240) ÷ (24 − 12) = 10.
  4. Recover the start: 240 − 10 × 12 = 120.
  5. Convert the new herd: 12 + 88 ÷ 4 = 34.
  6. Use the net rate: 120 ÷ (34 − 10) = 5 days.

Enter each key result

days
Mission 8

Use the equivalent-consumer laboratory

Change the ratio, both complete scenarios, and the new mixed group. The laboratory converts everything into large-consumer equivalents before solving.

Not complete

Scenario A

Scenario B

New mixed group

Common-unit rule

large-equivalents = large + small ÷ 4

The lab is valid only when the two times are different and the calculated growth and starting stock are nonnegative.

16Scenario A equivalents
20Scenario B equivalents
10daily growth
120starting stock
25new-group equivalents
8duration
Do not reverse the conversion.If 1 cow = 4 sheep, then 12 sheep = 3 cow-equivalents.
Do not add raw headcounts.5 cows + 20 sheep = 10 cow-equivalents, not 25.
Keep units consistent.If growth is in cow-units, the starting stock and new group must also use cow-units.
The physical answer is unchanged.Switching to sheep-units scales all rates and stocks together.

Error detective

Use the fixed relationship 1 cow = 4 sheep for these questions, regardless of the laboratory settings.

Mission 9

A new mixed herd — independent investigation

Try the eight steps yourself, using a small hint if needed. Check an attempt before opening worked review. Correct all eight to complete this investigation.

Not complete

One cow eats as much as 4 sheep per day. A pasture feeds 18 cows for 10 days or 56 sheep for 15 days. These are separate trials that both end exactly when the pasture runs out. They start with the same amount of grass, have constant growth, and use constant eating rates. How long will 10 cows and 32 sheep last? Use cow-day units.

Write quantities in one consistent unit. Enter decimals or fractions such as 30/7; a decimal within 0.001 is accepted.

Hint

Divide the sheep count by four sheep per cow.

Worked solution

56 ÷ 4 = 14.

Hint

Compare totals only after using the same unit.

Worked solution

18 × 10 = 180.

Hint

Multiply the converted sheep herd’s cow-equivalents by its 15 days.

Worked solution

14 × 15 = 210.

Hint

Divide the difference in total consumption by the difference in trial length.

Worked solution

(210 − 180) ÷ (15 − 10) = 6.

Hint

Subtract all 10 days of growth from the first trial’s total use.

Worked solution

180 − 10 × 6 = 120. Check: 210 − 15 × 6 = 120.

Hint

Convert 32 sheep to cow-equivalents, then add the 10 cows.

Worked solution

10 + 32 ÷ 4 = 18.

Hint

Subtract growth of 6 units/day from the mixed herd’s daily consumption.

Worked solution

18 − 6 = 12.

Hint

Divide remaining stock by its net depletion rate; check the time or capacity requested.

Worked solution

120 ÷ 12 = 10. Converting every quantity to sheep units gives 480 ÷ 48 = 10 too.

Mission 10

Exit ticket

Answer all five questions correctly. The certificate also requires the other core missions and the independent investigation.

Not complete
days

Lesson checkpoints completed

Equivalent-Consumer Unit Builder

You completed the checkpoints on equivalent consumers and consistent units. Revisit any steps for which you needed solution help.

Lesson 8.3 completed

Teaching notes

The converter, alternate-unit proof, dynamic laboratory, workshop, feedback, and exit ticket are additional instructional scaffolds.