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Practice sheet · Biology Path · Lens 1: Math

Scale, chance & sampling

Practice for The Math of Biology: magnification, cell size, doubling, Punnett squares, sampling, food chains and pulse (pp.8–15), plus the Algebra Level Ups (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

Real size = image size ÷ magnification. 1 mm = 1,000 µm. Doubling: after n doublings, 1 cell becomes 2ⁿ. Punnett square: 4 boxes, each a 1 in 4 chance.

Sampling: density = count ÷ area. Mark–recapture: population ≈ marked × second catch ÷ marked in it. Practise it with beans, never by catching animals.

Warm-up

One step

Size, rate and powers.
1

Total zoom

A microscope has a 10× eyepiece and a 40× objective lens. What is the total magnification?

2

Real size

At 400×, a cell’s image is 6 mm across. How big is the real cell, in mm and in µm?

3

Quick pulse

You count 21 beats in 15 seconds. What is the pulse in beats per minute?

4

Doubling

A made-up cell divides every 30 minutes. Starting from 1 cell, how many are there after 3 hours?

Core

Two or three steps

Show the ratio or table you used.
5

Surface and volume

Find the surface area, the volume and the SA : V ratio of a 4 cm cube and a 6 cm cube. Which is better at exchanging heat and gases for its size?

6

Punnett square

B (brown) is dominant over b. Draw Bb × Bb. What chance does a baby have of being bb? Out of 200 offspring, about how many? Repeat for Bb × bb.

Biology Practice · Biology Path · Unizon1/2
Biology Path · Unizon · PracticeScale, chance & sampling
7

Mendel’s flowers

Mendel counted 705 purple and 224 white pea flowers. What is the ratio, as “something : 1”? How close is it to 3 : 1?

8

Quadrats

Five 0.25 m² quadrats hold 7, 9, 6, 10 and 8 dandelions. Find the mean per quadrat, the density per m², and an estimate for a 150 m² lawn.

9

Bean recapture

You mark 25 beans in a jar and mix them back in. A second scoop has 40 beans, 5 of them marked. Estimate the total number of beans.

10

Food chain

Grass stores 30,000 kJ. Using the 10% rule, how much energy reaches a grasshopper, a bird that eats it, and a hawk that eats the bird?

Level Up

Equations

A1–A2 (pp.17–18).
11

Seedling

A seedling is 3 cm tall and grows 1.2 cm a day: h = 3 + 1.2t. When is it 21 cm?

12

Two seedlings

Seedling A: h = 1 + 2.5t. Seedling B: h = 7 + 1.5t. When are they the same height, and how tall?

13

Quadrat side

A square quadrat must cover 0.16 m². Solve s² = 0.16. How long is each side?

14

Garden plot

A plot is 4 m longer than it is wide, with an area of 60 m². Write w(w + 4) = 60, solve it, and give both sides.

Make your own problem

Your turn to be the field biologist

  1. Pick a living thing from the book: a cell, a plant, a bird, a heartbeat.
  2. Write a question with a ratio, a chance or a sample.
  3. Solve it, and say how sure you are: what could make the real answer different?

My problem, and my worked answer:

Biology Practice · Biology Path · Unizon2/2
Biology Path · Unizon · PracticeScale, chance & sampling

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Counts in nature vary; answers near these are fine when you explain them.

Warm-up
1
400× Multiply the two lenses: 10 × 40 = 400.
2
0.015 mm = 15 µm 6 ÷ 400 = 0.015 mm. × 1,000 = 15 µm, about the size of a cheek cell.
3
84 beats per minute 15 s × 4 = 1 minute, so 21 × 4 = 84.
4
64 cells 3 h = 180 min = 6 doublings. 2⁶ = 64.
Core
5
4 cm: 96, 64, 1.5 : 1 · 6 cm: 216, 216, 1 : 1 SA = 6s², V = s³. 96 ÷ 64 = 1.5; 216 ÷ 216 = 1. The smaller cube has more surface for its size, which is why cells stay small (p.9).
6
Bb × Bb: 1/4, about 50 · Bb × bb: 1/2, about 100 Bb × Bb gives BB, Bb, Bb, bb: 1 box in 4. Bb × bb gives Bb, Bb, bb, bb: 2 in 4. Real counts wobble around these.
7
≈ 3.15 : 1 705 ÷ 224 ≈ 3.15. Close to 3 : 1; chance makes real counts drift a little (p.12).
8
8 · 32 per m² · ≈ 4,800 40 ÷ 5 = 8 per quadrat. 4 quadrats make 1 m²: 8 ÷ 0.25 = 32. 32 × 150 = 4,800.
9
≈ 200 beans 25 × 40 ÷ 5 = 200. Count them to check how good the estimate is.
10
3,000 · 300 · 30 kJ Each step keeps about 10%: 30,000 × 0.1 = 3,000, then 300, then 30. That is why there are few top predators.
Level Up
11
day 15 3 + 1.2t = 21 → 1.2t = 18 → t = 15.
12
day 6, both 16 cm 1 + 2.5t = 7 + 1.5t → t = 6. Check: 1 + 15 = 16 and 7 + 9 = 16.
13
0.4 m (40 cm) s = √0.16 = 0.4. The negative root makes no sense for a length.
14
6 m × 10 m w² + 4w − 60 = 0 → (w + 10)(w − 6) = 0 → w = 6 (not −10). Check: 6 × 10 = 60.
Make your own: write the ratio or formula first, keep the units (µm, m², kJ), and remember that chance gives “about”, not “exactly”.
Biology Practice · Biology Path · UnizonAnswer key