Investigation 1: Investigating Fractions and Percents with Manipulatives

New Concept

This investigation uses hands-on fraction pieces to help you see and understand fractions. You will use these tools to solve problems visually.

What’s next

This is just the beginning of our work with fractions. Next, you'll use your manipulatives in guided activities to solve problems involving fraction operations and percents.

Property

To find a part of a fraction, like “one fourth of one half,” you multiply the fractions together. The word “of” is your secret code for multiplication in fraction problems, so this means $\frac{1}{4} \times \frac{1}{2}$.

Examples

What fraction is half of $\frac{1}{3}$? This means you calculate $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$.
If you ate $\frac{1}{2}$ of your $\frac{1}{4}$ share of a pizza, you ate $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$ of the whole pizza.
A quart is $\frac{1}{4}$ gallon. Half a quart is $\frac{1}{2}$ of $\frac{1}{4}$ gallon, which is $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$ of a gallon.

Explanation

Imagine you have a manipulative piece for one-half. Finding a fourth of it means you need an even smaller piece! You are taking a part of an existing part, which is why the final fraction is smaller than what you started with. It’s a piece of a piece!

Property

Equivalent fractions are different fractions that represent the exact same amount. You can often find a simpler equivalent by dividing the numerator and denominator by the same number, which is called simplifying.

Examples

Find a single fraction piece that equals $\frac{4}{8}$. By dividing both parts by 4, you get $\frac{4 \div 4}{8 \div 4} = \frac{1}{2}$.
Find a single fraction piece that equals $\frac{2}{12}$. By dividing both parts by 2, you find $\frac{2 \div 2}{12 \div 2} = \frac{1}{6}$.
How many twelfths equal $\frac{1}{2}$? You can scale up: $\frac{1 \times 6}{2 \times 6} = \frac{6}{12}$. You need six $\frac{1}{12}$ pieces.

Explanation

Think of it like trading coins! One fifty-cent piece is the same as two quarters. The fraction manipulatives show this perfectly: two $\frac{1}{4}$ pieces cover the same space as one $\frac{1}{2}$ piece. They look different but have the same value, which is very useful for simplifying problems.

Property

To add or subtract fractions with different denominators, you must first find a common denominator. This converts them into the same-sized “pieces” so you can combine their numerators fairly.

Examples

To solve for $a$ in $\frac{1}{2} + \frac{1}{3} + a = 1$, first add $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. To make a whole (1), $a$ must be $\frac{1}{6}$.
To solve for $c$ in $\frac{1}{2} + c = \frac{3}{4}$, convert to fourths: $\frac{2}{4} + c = \frac{3}{4}$. So, $c$ must be $\frac{1}{4}$.
To solve $\frac{1}{6} + b = \frac{1}{4}$, convert to twelfths: $\frac{2}{12} + b = \frac{3}{12}$. This shows that $b$ is $\frac{1}{12}$.

Explanation

You can't just add $\frac{1}{2}$ and $\frac{1}{3}$ and get $\frac{2}{5}$! You need to trade them for pieces of the same size, like sixths. Once you have $\frac{3}{6}$ and $\frac{2}{6}$, adding them is a piece of cake. It's all about speaking the same fraction language.

Property

A percent is a special ratio that compares a number to 100. To find what percent a fraction represents, think of the whole circle as 100% and then determine the fraction's equivalent part of that 100.

Examples

What percent of a circle is $\frac{3}{4}$? That's equivalent to $\frac{75}{100}$, which is $75\%$.
What percent of a circle is $\frac{1}{2}$? That's half of $100\%$, which is $50\%$.
What percent is $\frac{1}{4} + \frac{1}{12}$? First add them: $\frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$. This is 33\frac{1}{3}\%.

Explanation

This is like asking, “If this whole pizza is worth 100 points, how many points is my slice worth?” A $\frac{1}{2}$ slice is worth 50 points, or 50%, and a $\frac{1}{4}$ slice is worth 25 points, or 25%. It's a very common way to talk about parts of a whole.