B1. Number Sense

Example

Write 870215 in words and in expanded form.

1. Split into groups of three from the right; say the left group, then "thousand", then the right group.
2. Expanded: each digit times its place value, added.
3. Check: the expanded parts sum back to 870215.

Example

Place -3 on the line and describe it two ways.

1. Count from 0 toward the left: it is that many steps below zero.
2. As a temperature: -3 degrees is below freezing. As an elevation: below sea level.
3. Its opposite is the same distance on the other side of 0.

Example

Order **−2, 0.75, $\frac{1}{2}$, −5, 1.2** from least to greatest.

1. Negatives first, further from zero is smaller: −5 < −2.
2. Then the positives by size: $\frac{1}{2}$ = 0.5 < 0.75 < 1.2.
3. Least to greatest: −5, −2, $\frac{1}{2}$, 0.75, 1.2.

Example

Read 0.640 and compare it with 0.5.

1. 0.640 is 640 thousandths: tenths, hundredths, thousandths places.
2. Compare place by place from the left: tenths first.
3. Line up the decimal points; add zeros to match lengths (0.5 = 0.500) before comparing.

Example

Round 2.468 to the nearest hundredth and tenth. Round **$0.\overline{3}$** (0.333…) to the nearest hundredth.

1. Hundredth: look at thousandths, 8 → up: 2.47. Tenth: look at hundredths, 6 → up: 2.5.
2. 0.333…: hundredths 3, next digit 3 → 0.33.
3. A repeating decimal never ends; rounding gives a usable value.

Example

Write $\frac{1}{8}$ and $\frac{3}{8}$ as decimals, and 0.640 as a fraction.

1. $\frac{1}{8}$ = 0.125 (1 ÷ 8). $\frac{3}{8}$ = 3 × 0.125 = 0.375.
2. 0.640 = $\frac{640}{1000}$; simplify by common factors.
3. Denominator 1000 ↔ three decimal places.

Summary

• Groups of three digits to a million.
• Integers: negatives left of or below zero.
• Order mixed numbers by converting to decimals.
• Thousandths: line up points, add zeros.
• Eighths are thousandths: 0.125.