Lesson 9.1 — Assume One Type: Baseline and Difference
假设全是一类：基准与差量
Some problems give two totals at once: how many objects there are, and how much all of those objects contribute together. We will solve them by pretending every object is one type, comparing with the real total, and repairing the difference one replacement at a time.
Two totals tell two different stories
There are 40 chickens and rabbits altogether. Together they have 130 legs. Before calculating, identify what each total measures.
Object total
This total counts heads or animals. Every animal contributes exactly one to this count.
Weighted total
This total depends on type: a chicken contributes 2 legs, while a rabbit contributes 4.
| Type | Count contribution | Leg contribution |
|---|---|---|
| Chicken | 1 animal | 2 legs |
| Rabbit | 1 animal | 4 legs |
Check the quantities
Create an all-rabbit baseline
Pretend all 40 animals are rabbits. Then replace rabbits with chickens. Every replacement removes exactly two legs.
Move the slider to replace rabbits with chickens.
Replace rabbits one at a time
rabbit → chicken changes the total by 4 − 2 = 2 legs.
Read the baseline
Repair the baseline and solve the problem
The number of replacements tells us the number of chickens, because every replacement changes one rabbit into one chicken.
1 · Baseline
2 · Difference
3 · Replacements
The two checks below must both be true.
Count check
Leg check
Complete the solution
Reverse the assumption: start with all chickens
A correct method should also work from the lower baseline.
All-chicken baseline
The real group has 50 more legs than this baseline.
Upgrade one animal
Every chicken changed into a rabbit repairs 2 missing legs.
Then:
Use the all-chicken route
Guided practice: 40 animals and 90 legs
Choose either baseline. The final counts must be the same.
Find both animal counts
Build a general baseline-and-difference machine
The method works whenever every object is one of two types with different contributions.
Enter a two-type problem
Here a is the contribution of each low-type object and b of each high-type object, with b > a. Replacing one low-type object by one high-type object keeps the object count fixed and increases the total in steps of b − a.
All-low formula
Then find the rest
Use the formula without skipping meaning
There are 18 objects. The low type contributes 2 units, the high type contributes 5 units, and the total contribution is 54.
Verify the answer — and detect impossible data
A division result is not enough. A valid count must pass four checks.
Verification checklist
- Counts are whole numbers.
- Counts are not negative.
- Counts add to the object total.
- Weighted contributions add to the stated total.
Quick feasibility range
The real weighted total must lie between the all-low and all-high baselines.
Error detective
Sometimes the object count is hidden
A zoo has ostriches and giraffes. Together they have 30 eyes and 44 legs. Every animal has two eyes; each ostrich has two legs and each giraffe four. Find the number of each type.
First recover the number of animals
Only after finding 15 animals should we use the leg contributions.
Then use a baseline
Assume all 15 animals are ostriches with 2 legs:
The real group has 14 extra legs. Each giraffe adds 2 extra legs.
Complete the hidden-count problem
Extra transfer check: cars and three-wheeled motorcycles
There are 21 vehicles and 68 wheels. Each car has four wheels and each motorcycle three; count only these wheels. An all-motorcycle baseline has 63 wheels. The five extra wheels reveal five cars and sixteen motorcycles.
Baseline-and-difference workshop
Solve at least six of the eight problems. Enter numbers only unless a choice is provided.
Try a fresh challenge
Try these new questions before opening help. Earn your certificate by completing Missions 2, 4, and 9 and answering all five exit questions correctly. Optional reflections and extra exploration do not affect your score.
Hint
Start with 28 animals, each with 2 legs.
Worked solution — open after trying
28 × 2 = 56 legs.
Hint
Compare the actual total with your baseline.
Worked solution — open after trying
80 − 56 = 24 legs.
Hint
Each replacement adds 4 − 2 legs.
Worked solution — open after trying
24 ÷ 2 = 12 rabbits.
Hint
The two counts must add to 28.
Worked solution — open after trying
28 − 12 = 16 chickens; 16 × 2 + 12 × 4 = 80.
Hint
Compare 48 with 10 × 3, then test divisibility by 4.
Worked solution — open after trying
48 − 30 = 18. Since 18 ÷ 4 = 4.5, this would need half an object; no whole-count solution exists.
Baseline-and-Difference Model Builder
Lesson checkpoints completed
You practised creating a baseline, measuring replacement differences and checking both totals. Revisit any steps for which you needed solution help.
Lesson 9.1 complete
Teaching notes
The interactive sliders, generalized machine, error-detection tasks, added practice, and exit ticket are new instructional scaffolds.