Number and Estimation

Overview

Number and estimation is the foundation of Foundation Mathematics. This area covers how we read, write, interpret and work with numbers, and how we use estimation to check whether answers make sense in real-world situations.

KEY TAKEAWAY: Number sense — knowing roughly what an answer should be — is just as important as getting the exact answer.

Types of Numbers Used in Foundation Mathematics

Place Value

Understanding place value is essential for reading and writing numbers correctly.

$$\underbrace{3}{\text{thousands}} \underbrace{4}{\text{hundreds}} \underbrace{7}{\text{tens}} \underbrace{2}{\text{ones}} . \underbrace{5}{\text{tenths}} \underbrace{6}{\text{hundredths}}$$

So $3472.56$ means:
- \$3 \times 1000 + 4 \times 100 + 7 \times 10 + 2 \times 1 + 5 \times 0.1 + 6 \times 0.01$

EXAM TIP: In money contexts, always write answers to 2 decimal places: $\$12.50$, not $\$12.5$.

Estimation Strategies

Estimation means finding an approximate answer before (or instead of) calculating exactly. Key strategies:

Front-End Estimation

Use only the leading digit(s):
- \$387 + 512 \approx 400 + 500 = 900$

Rounding Estimation

Round each number to a convenient value first:
- \$48 \times 21 \approx 50 \times 20 = 1000$ (exact: $1008$)

Compatible Numbers

Adjust numbers to make mental arithmetic easy:
- \$197 + 304 \approx 200 + 300 = 500$

Worked Example — Reasonableness Check

A plumber charges $\$85$ per hour and works for $3.5$ hours. Estimate the total charge.

Step 1: Round $\$85 \approx \$90$

Step 2: $\$90 \times 3.5 \approx \$90 \times 4 = \$360$

Step 3: Exact answer: $\$85 \times 3.5 = \$297.50$

Step 4: Reasonableness check — $\$297.50$ is close to $\$360$, so the answer is reasonable.

COMMON MISTAKE: Students often skip the reasonableness check. Always ask: “Does my answer make sense?”

Mental Strategies

• Doubles/halves: \$48 \times 2 = 96$, \$96 \div 2 = 48$
• Multiply by 10/100: Move the decimal point right
• Split method: \$34 \times 6 = (30 \times 6) + (4 \times 6) = 180 + 24 = 204$
• Compensation: \$99 \times 4 = (100 \times 4) - (1 \times 4) = 400 - 4 = 396$

Number Sense in Everyday Contexts

• A $25\%$ discount on $\$80$ should be roughly $\$20$ off → price ≈ $\$60$
• A room of $4.2\text{ m} \times 3.8\text{ m}$ has area ≈ \$4 \times 4 = 16\text{ m}^2$ (exact: $15.96\text{ m}^2$)
• A car travelling at $60\text{ km/h}$ for $2.5\text{ h}$ covers about \$60 \times 2.5 = 150\text{ km}$

VCAA FOCUS: VCAA tasks often ask you to check or verify a given answer. Use estimation to do this — show your working clearly.