Real-world fraction multiplication

How do you multiply fractions and whole numbers with a picture? You have lots of choices: area models, groups of objects, tape diagrams, or number lines. They all show the same idea.

Draw a number line from 0 to 10 and split it into 5 equal parts. Each part is 2, because 10 divided by 5 is 2. Now jump 4 of those parts: 2, 4, 6, 8. So 4/5 times 10 is 8.

You can also use an area model. To see 2/3 times 3/4, draw a rectangle, shade 3/4 of it in one direction, then shade 2/3 of it in the other direction. The part with both shadings is the answer. Count the small squares and you get 6/12, which simplifies to 1/2.

Why does this work? Because multiplying by a fraction means taking part of something. When you see a word problem that says 2/3 of a batch or 3/4 of a distance, that of is your clue to multiply.

You can find the area of a rectangle with fraction sides in two ways.

Way one: tile it with little squares. Picture a rectangle that is 2/3 unit by 3/4 unit. Tile it with squares that are 1/12 unit on each side. When you count the tiles, you get 6 of them, so the area is 6/12, which simplifies to 1/2 square unit.

Way two: just multiply the side lengths. Multiply 2/3 times 3/4. The top gives 6 and the bottom gives 12, so the area is 6/12, which is 1/2. Both ways agree, because area really is length times width.

What if the sides are mixed numbers, like 2 1/2 feet by 1 3/4 feet? Change them to improper fractions first: 5/2 and 7/4. Then multiply to get 35/8, which is 4 3/8 square feet. For big shapes made of joined rectangles, split the shape into pieces, find each area, and add them up.

Some shapes look hard until you split them into rectangles. A shape made of two joined rectangles is called a composite figure, and here is how to handle one when the sides are mixed numbers.

Step one: split the figure into rectangles whose sides you can measure. Draw a dotted line if it helps.

Step two: for each rectangle, change the mixed-number sides into improper fractions. For example, 2 1/2 becomes 5/2 and 1 3/4 becomes 7/4.

Step three: multiply the length and width of each rectangle to get its area. Remember to simplify your answer.

Step four: add the areas of all the rectangles to get the total area.

Quick check: the total should be a little less than if you rounded every side up. If your total came out bigger than that, look for a spot where a side was read wrong or a fraction was not simplified.