Unit 2: Understanding Fractions

Vocabulary

Concept

A fraction is a way to express a value that is not a whole number. Think of a pizza cut into equal slices. The fraction tells you how many slices you have compared to the total number of slices the pizza was cut into. The key idea is "parts of an equal whole."

I Do: Examples

Example 1: In the fraction \(\frac{3}{8}\), the numerator is 3 and the denominator is 8. This means we have 3 parts out of a total of 8 equal parts.

Example 2: If a class has 10 girls out of 25 total students, the fraction of girls is \(\frac{10}{25}\). The numerator (10) is the part (girls), and the denominator (25) is the whole (total students).

Vocabulary

Concept

The main idea is that you can change how a fraction looks without changing its value. You do this by multiplying or dividing the numerator and denominator by the same number. This is like cutting a pizza into more, smaller slices—you have more slices, but the total amount of pizza is the same.

I Do: Examples

Example 1 (Equivalents): To make an equivalent fraction for \(\frac{2}{3}\), I can multiply the top and bottom by 2: \( \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \). I can also multiply by 5: \( \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \). So, \(\frac{2}{3}\), \(\frac{4}{6}\), and \(\frac{10}{15}\) are all equivalent.

Example 2 (Simplifying): To simplify \(\frac{6}{8}\), I can see that both numbers are divisible by 2: \( \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \). Since 3 and 4 don't share any common factors, \(\frac{3}{4}\) is the simplest form.

Vocabulary

Concept

When fractions have different denominators, it's hard to tell which is bigger just by looking. We need a method to compare them. A quick way is the Butterfly Method (or cross-multiplication). You multiply diagonally to see which side produces a bigger number. That side has the bigger fraction.

I Do: Examples

Example 1: To compare \(\frac{2}{3}\) and \(\frac{3}{4}\), I cross-multiply. \(2 \times 4 = 8\) and \(3 \times 3 = 9\). Since 9 is bigger than 8, \(\frac{3}{4}\) is the bigger fraction. \(\overset{8}{\frac{2}{3}} < \overset{9}{\frac{3}{4}}\)

Example 2: To compare \(\frac{5}{6}\) and \(\frac{3}{4}\), I cross-multiply. \(5 \times 4 = 20\) and \(3 \times 6 = 18\). Since 20 is bigger than 18, \(\frac{5}{6}\) is the bigger fraction. \(\overset{20}{\frac{5}{6}} > \overset{18}{\frac{3}{4}}\)

Vocabulary

Concept

Improper fractions and mixed numbers are two different ways to write the same value when that value is more than one. Think of having 5 halves of a pizza, which is \(\frac{5}{2}\). That's the same as having 2 whole pizzas and 1 half left over, which is \(2\frac{1}{2}\).

I Do: Examples

Improper to Mixed: To convert \(\frac{7}{3}\), I ask "How many times does 3 go into 7?" It goes in 2 times (\(3 \times 2 = 6\)) with a remainder of 1. So, the mixed number is \(2\frac{1}{3}\).

Mixed to Improper: To convert \(3\frac{1}{4}\), I multiply the whole number (3) by the denominator (4), which is 12. Then I add the numerator (1) to get 13. The denominator stays the same. So, the improper fraction is \(\frac{13}{4}\).

Vocabulary

Concept

The golden rule of adding and subtracting fractions is: you can only do it if the denominators are the same! If they are, you simply add or subtract the numerators and keep the denominator the same. Think about it: if you have 2 eighth-slices of pizza and someone gives you 3 more eighth-slices, you now have 5 eighth-slices. The size of the slice (the denominator) didn't change.

I Do: Examples

Addition: \(\frac{2}{8} + \frac{3}{8}\). The denominators are the same (8). So, I add the numerators: \(2 + 3 = 5\). The denominator stays the same. The answer is \(\frac{5}{8}\).

Subtraction: \(\frac{7}{10} - \frac{4}{10}\). The denominators are the same (10). So, I subtract the numerators: \(7 - 4 = 3\). The denominator stays the same. The answer is \(\frac{3}{10}\).