B1. Number Sense

Example

Write 24649 in expanded form and in words.

1. Expanded: 20000 + 4000 + 600 + 40 + 9.
2. Words: 24 thousand, 6 hundred 49 — say the thousands group, then the rest.
3. Decompose another way: 24000 + 649.

Example

Order 33450, 23450, 25950 from greatest to least.

1. Same number of digits, so compare from the left: ten-thousands first.
2. 33450 has the largest ten-thousands digit.
3. 23450 and 25950 tie there; compare thousands: 25950 is larger. Order: 33450, 25950, 23450.

Example

Write $\frac{7}{4}$ as a mixed number, and find a fraction equivalent to $\frac{3}{4}$.

1. How many wholes? 4 pieces make 1 whole, with 3 left: $1\frac{3}{4}$.
2. Equivalent: multiply top and bottom by 2 → $\frac{6}{8}$.
3. Multiplying or dividing both parts by the same number keeps the amount.

Example

Order $\frac{5}{8}$, $\frac{1}{3}$ and $\frac{7}{6}$ from least to greatest.

1. Benchmarks: $\frac{1}{3}$ is under one half; $\frac{5}{8}$ is over one half; $\frac{7}{6}$ is over 1.
2. So: $\frac{1}{3} < \frac{5}{8} < \frac{7}{6}$.
3. For close fractions, use a common denominator: $\frac{5}{8} = \frac{15}{24}$ and $\frac{2}{3} = \frac{16}{24}$.

Example

Read the grid as a decimal. Then order 1.05, 1.5, 0.60 from least to greatest.

1. 60 of 100 shaded → 0.60, read "60 hundredths".
2. Compare whole parts first: 0.60 is least. Then tenths: 1.05 has 0 tenths, 1.5 has 5 → 1.05 < 1.5.
3. Least to greatest: 0.60, 1.05, 1.5. More digits does not mean bigger.

Example

Round 4.76 and 4.72 to the nearest tenth.

1. Look at the hundredths digit: 5 or more rounds the tenths up.
2. 4.76 → hundredths 6 → 4.8.
3. 4.72 → hundredths 2 → 4.7.

Example

Write 60% as a fraction in lowest terms and as a decimal. Then write $\frac{1}{5}$ as a percent.

1. 60% = $\frac{60}{100}$ = 0.60; simplify the fraction by dividing top and bottom by a common factor.
2. $\frac{1}{5}$ = $\frac{20}{100}$ = 0.20 = 20%.
3. To turn any fraction into a percent, make its denominator 100.

Summary

• Ten-thousands place; group digits in threes.
• Improper ↔ mixed; equivalent by scaling both parts.
• Compare fractions with benchmarks or common denominators.
• Hundredths: compare whole, tenths, then hundredths.
• Fraction = decimal = percent, all out of 100.