Fractions, Ratios & Proportional Thinking

The concept

A fraction names an amount, not two unrelated numbers separated by a line. In 3/4, the denominator 4 tells you that one whole is divided into four equal parts. The numerator 3 tells you to take three of those equal parts. The parts must be the same size; three unequal pieces of a pizza do not necessarily show 3/4.

Models make a fraction visible. On a strip divided into four equal parts, 3/4 fills three parts. On a number line, 3/4 sits between 1/2 and 1. When you know the size of one part, you can also find a fraction of a group: divide the total into the number of equal parts, then take the number of parts named by the numerator.

The concept

Two fractions are equivalent when they name exactly the same amount. For example, 3/4 and 6/8 land in the same place on a fraction strip. To change 3/4 into 6/8, split each fourth into two smaller equal parts. There are now eight parts, and six are shaded, but the total amount has not changed.

Simplifying does not make a fraction smaller; it simply writes it with fewer parts. 12/16 and 3/4 describe the same amount. To simplify, divide the numerator and denominator by the same number. To compare fractions, rewrite them with same-size parts or use a shared model instead of comparing only the top numbers.

The concept

A ratio compares two quantities. If a tray has 2 tomato plants and 3 pepper plants, the ratio of tomatoes to peppers is 2:3. We read this as “two to three.” The order matters: 2:3 does not mean the same thing as 3:2 because each number refers to a different category.

A fraction often compares a part to a whole; a ratio can compare part to part or part to whole. In the same tray, there are 5 plants total, so the ratio of tomatoes to all plants is 2:5. Equivalent ratios grow or shrink by multiplying or dividing both parts by the same factor: 2:3, 4:6, and 6:9 keep the same mixture.

The concept

A proportional relationship keeps the same ratio as quantities change. If 1 centimeter on a map represents 2 kilometers in the real world, then 2 centimeters represents 4 kilometers and 6.5 centimeters represents 13 kilometers. Each quantity is multiplied by the same factor, so the relationship stays constant.

A ratio table makes that pattern visible. Look for multipliers, not only additions. A table can add the same amount and still look organized without keeping the same ratio. Before scaling, name the units, start with one reliable pair of values, and check that every row describes the same situation.