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Practice sheet · Criminal Investigation Path · Lens 1: Math

Scales, angles & odds

Practice for The Math of Investigation: scale sketches, fixed points, stain angles, heights, skid marks, cooling, timelines and match odds (pp.8–16), plus A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

At 1:50, paper = real ÷ 50 (same units). Pythagoras: distance = √(a² + b²). sin(angle) = width ÷ length; height = distance × tan(angle). Calculator in degrees.

S = √(254 × f × d) km/h gives a minimum speed. Cooling: halve the difference from room temperature. Speed = distance ÷ time (minutes ÷ 60). Multiply independent chances.

Warm-up

One step

Scales, distances, angles and speeds.
1

Scale

At 1:50, how long is a 2.8 m wall on paper, in cm (p.8)?

2

Fixed point

A made-up object is 5 m along a wall and 12 m straight out from it. How far is it from the corner (p.9)?

3

Stain angle

A coloured-water stain is 8 mm wide and 16 mm long. Find its impact angle (p.10).

4

Timeline

A made-up witness says they biked 6 km in 20 minutes. What speed is that, in km/h (p.14)?

Core

Two or three steps

Say what each number can and can’t show.
5

Both ways

At 1:50, a sketch shows a shelf 7.5 cm from a door. How far is it really, in m? A real table is 1.6 m long: how long is it on the sketch?

6

Two stains

One stain is 9 mm wide and 15 mm long; another is 12 mm wide and 12 mm long. Find both impact angles, to the nearest degree.

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Criminal Investigation Path · Unizon · PracticeScales, angles & odds
7

Height of origin

A stain hit at 40° and is 60 cm from the point of convergence. Estimate the height, to the nearest cm (p.11).

8

Skid marks

A made-up car skids 30 m on dry pavement (f = 0.75). Find the minimum speed, to the nearest km/h. What would the same 30 m skid mean on ice (f = 0.1)?

9

Cooling

A mug of water starts at 84°C in a 20°C room. The difference halves every 10 minutes. What is its temperature after 30 minutes (p.13)?

10

Match odds

Three independent features are shared by 1 in 4, 1 in 10 and 1 in 25 people. What is the chance of all three? About how many people would match in a city of 80,000 (p.15)?

Level Up

Equations

A1–A2 (pp.17–18).
11

Undo the formula

Using d = S² ÷ (254 × f), how long a skid would a car leave from 64 km/h with f = 0.8? Give metres to one decimal place.

12

Two travellers

A made-up walker leaves the rink at 4 km/h. A cyclist leaves the same place 20 minutes later at 12 km/h on the same path. After how many minutes (from when the walker left) does the cyclist catch up, and how far away?

13

Search grid

A search area is a rectangle 5 m longer than it is wide, with an area of 84 m². Find the width and the length.

14

Dropped phone

In a made-up case, a phone falls 78.4 m. Its fall time t satisfies 4.9t² = 78.4. Solve for t in seconds.

Make your own problem

Your turn to be the investigator

  1. Pick a number from the book: a scale, an angle, a friction number, a cooling time or a lineup size.
  2. Write a question that uses it with a made-up, harmless case.
  3. Solve it, and say what the number can’t prove on its own.

My problem, and my worked answer:

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Criminal Investigation Path · Unizon · PracticeScales, angles & odds

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. Cases, measurements and numbers are made-up examples unless the book cites them.

Warm-up
1
5.6 cm 2.8 m = 280 cm; 280 ÷ 50 = 5.6 cm.
2
13 m √(5² + 12²) = √(25 + 144) = √169 = 13.
3
30° sin(angle) = 8 ÷ 16 = 0.5; sin⁻¹(0.5) = 30°.
4
18 km/h 20 min = 20 ÷ 60 h; 6 ÷ (1/3) = 18 km/h.
Core
5
3.75 m · 3.2 cm 7.5 × 50 = 375 cm = 3.75 m. 160 cm ÷ 50 = 3.2 cm.
6
37° · 90° sin⁻¹(9 ÷ 15) = sin⁻¹(0.6) ≈ 37°. 12 ÷ 12 = 1; sin⁻¹(1) = 90°: a round stain fell straight down.
7
about 50 cm 60 × tan(40°) ≈ 60 × 0.839 ≈ 50 cm. Real drops curve, so this is an estimate (p.11).
8
about 76 km/h · about 28 km/h √(254 × 0.75 × 30) = √5,715 ≈ 76. On ice: √(254 × 0.1 × 30) = √762 ≈ 28. The same skid means a much lower speed on ice; both are minimums.
9
28°C Difference 64 → 32 → 16 → 8 in 30 minutes; 20 + 8 = 28°C. Real estimates give ranges (p.13).
10
1 in 1,000 · about 80 4 × 10 × 25 = 1,000. 80,000 ÷ 1,000 = 80 people: a match narrows things but doesn’t pick out one person.
Level Up
11
about 20.2 m 64² = 4,096; 254 × 0.8 = 203.2; 4,096 ÷ 203.2 ≈ 20.2 m.
12
30 minutes · 2 km 4t = 12(t − 1/3) → 4t = 12t − 4 → t = 0.5 h = 30 min; 4 × 0.5 = 2 km.
13
7 m by 12 m w(w + 5) = 84 → w² + 5w − 84 = 0 → (w + 12)(w − 7) = 0 → w = 7; length 12.
14
4 s t² = 78.4 ÷ 4.9 = 16 → t = 4 or −4; time can’t be negative, so t = 4 s.
Make your own: change units before you divide, set your calculator to degrees, and remember every answer here is an estimate about a made-up case, not proof about a real person.
Investigation Practice · Criminal Investigation Path · UnizonAnswer key