B1. Number Sense

Example

Compare Canada's population, about 40 000 000, with Earth's, about 8 000 000 000.

1. Group digits: 40 million vs 8 billion.
2. Powers of ten: $4 \times 10^{7}$ vs $8 \times 10^{9}$.
3. 8 billion ÷ 40 million = 200 — Earth has about 200 times as many people.

Example

Is 16 a perfect square? Find its square root, and explain what a square root means.

1. 4 × 4 = 16, so yes — a square with side 4 has area 16.
2. $\sqrt{16} = 4$: the side length of a square with that area.
3. Perfect squares to 144: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Example

Order **$-\frac{3}{4}$, 0.375, $-\frac{1}{8}$, −0.5** from least to greatest.

1. Decimals: −0.75, 0.375, −0.125, −0.5.
2. Negatives, most negative first: −0.75, −0.5, −0.125.
3. Then the positive: −0.75 < −0.5 < −0.125 < 0.375.

Example

Simplify $\frac{24}{36}$ and $\frac{45}{60}$.

1. Find a common factor of top and bottom; divide both. 24 and 36 share 12: $\frac{24}{36} = \frac{2}{3}$.
2. 45 and 60 share 15: $\frac{45}{60} = \frac{3}{4}$.
3. Lowest terms when the only common factor left is 1.

Example

Name a decimal between 0.3 and 0.4, and a fraction between $\frac{1}{2}$ and $\frac{3}{4}$.

1. Add a decimal place: 0.35 sits between 0.3 and 0.4 — so do 0.31 and 0.399.
2. Common denominator 8: $\frac{4}{8}$ and $\frac{6}{8}$; between them, $\frac{5}{8}$.
3. There is always another number between two numbers.

Example

Round 3.456 to the nearest hundredth, tenth and whole number.

1. Hundredth: thousandths digit 6 → up: 3.46.
2. Tenth: hundredths digit 5 → up: 3.5.
3. Whole: tenths digit 4 → down: 3. Round from the original each time, not from a rounded value.

Example

Write 75% as a decimal and a fraction in lowest terms. Then convert $\frac{3}{5}$ to a percent.

1. 75% = 0.75 = $\frac{75}{100}$, then simplify.
2. $\frac{3}{5}$: 3 ÷ 5 = 0.6 = 60%. Or scale to 100: $\frac{60}{100}$.
3. Percent ↔ decimal: move two places. Fraction → decimal: divide.

Summary

• Groups of three to a billion; powers of ten.
• √ of a perfect square is its side.
• Negative rationals: further left is smaller.
• Simplify by common factors; between-numbers always exist.
• Round from the original; convert by dividing or scaling.