Fractions of a whole (age 9+)

Multiplying a fraction by a whole number can sound tricky, but pictures make it friendly. You can use an area model, groups of objects, a tape diagram, or a number line. They all show the same idea. Try a number line for 4/5 times 10. Draw a line from 0 to 10 and split it into 5 equal parts, so each part is 2. Now 4/5 means 4 of those parts. Jump 2, 2, 2, 2 and you land on 8. So 4/5 times 10 is 8. The same trick works for word problems. If each friend eats 3/8 of a pizza and 5 friends eat, draw 5 groups of 3 eighths. Count the shaded parts: 15 eighths, which is 1 whole pizza and 7/8 more. Drawing first and using the shortcut second is a great habit. The shortcut says: multiply the top number by the whole number and keep the bottom number the same.

Why do number lines and fraction models help when you multiply a fraction by a whole number? A picture turns a hard calculation into small, easy steps. Say you want 3/4 times 6. Draw a line from 0 to 6, split it into 4 equal jumps, so each jump is 6 divided by 4, which is 1 1/2. Three of those jumps take you to 4 1/2. The number line also helps you answer questions like: between which two whole numbers does the answer lie? Because you can see the answer land halfway between 4 and 5. Models are like a step by step guide. They slow the problem down so your eyes can follow it. Once the picture makes sense, the shortcut feels easy: times the top number by the whole number, keep the bottom number, then simplify if you can.

Comparing fractions means asking which one takes up more of the same sized whole. Models make it easy to see. Try comparing 4/6 and 6/9. Draw two bars of the exact same size. Split the first bar into 6 equal pieces and shade 4 of them. Split the second bar into 9 equal pieces and shade 6 of them. Now look: the shaded parts cover the same amount of each bar. So 4/6 and 6/9 are equal, even though they look different. This skill helps when a multiplication answer is bigger than 1. If 6 times 3/4 makes 18/4, you can compare that amount to whole numbers and see it lands between 4 and 5. Same sized wholes are the golden rule. If the wholes are different sizes, the comparison does not count.

Equivalent fractions are fractions that name the same part of a whole, even when they look different. One half and 2/4 are equivalent, because both cover half of the same bar. You can find equivalent fractions with a model. Shade 1/2 of a pie chart, then split every piece into two. The shaded amount did not change, but now 2 out of 4 pieces are shaded. The numbers doubled, and the fraction is worth the same. This idea is a big help when you multiply. Six times 3/4 gives 18/4, and simplifying 18/4 by halving gives 9/2, which is 4 1/2. Equivalent fractions let you rewrite the answer in its friendliest form, so you can spot whole numbers it sits between and compare it with other amounts.