Math Foundations Unit 2 Graphic Organizer

"How do fractions help us represent and work with parts of a whole?"

Weekly Pacing at a Glance

A fraction shows a part of an equal whole. It has two key parts.

\(\frac{3}{4}\)

Part	Description	Example for \(\frac{3}{4}\)
Numerator (Top)	How many parts you have.	We have 3 parts.
Denominator (Bottom)	Total equal parts in the whole.	The whole is cut into 4 parts.

You can change how a fraction looks without changing its value.

To find an equivalent fraction: Multiply the numerator and denominator by the same number.
\(\frac{2}{3} \xrightarrow{\times 4} \frac{8}{12}\)
To simplify a fraction: Divide the numerator and denominator by the same number until you can't anymore.
\(\frac{10}{15} \xrightarrow{\div 5} \frac{2}{3}\)

To see which fraction is bigger, use the Butterfly Method (cross-multiplication).

Step 1: Multiply diagonally: 3 x 7 = 21 and 4 x 5 = 20.
Step 2: Write the products above their fractions: \(\overset{21}{\frac{3}{5}} \quad \overset{20}{\frac{4}{7}}\).
Step 3: Compare the products. Since 21 > 20, the first fraction is greater.
Final Answer: \(\frac{3}{5} > \frac{4}{7}\)

Two ways to write fractions that are greater than 1.

Conversion	Process	Example
Improper to Mixed	Divide the numerator by the denominator. The answer is the whole number, and the remainder is the new numerator.	\(\frac{7}{3} \rightarrow 7 \div 3 = 2 \text{ R}1 \rightarrow 2\frac{1}{3}\)
Mixed to Improper	Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.	\(4\frac{1}{2} \rightarrow (4 \times 2) + 1 = 9 \rightarrow \frac{9}{2}\)

The Golden Rule: If the denominators are the same, just add or subtract the numerators and keep the denominator the same.

Addition Example: \(\frac{2}{9} + \frac{5}{9} = \frac{2+5}{9} = \frac{7}{9}\)
Subtraction Example: \(\frac{7}{10} - \frac{4}{10} = \frac{7-4}{10} = \frac{3}{10}\)