Equivalent Fractions on Grids and Number Lines

Three twelfths, one quarter and twenty-five hundredths look like different fractions and are the same amount. Shade three of twelve pieces on a strip, one of four on a strip the same length, twenty-five of a hundred cells on a grid — the shaded part ends in the same place every time. These are equivalent fractions: one amount, cut into different-sized pieces, with a different name for each cut. When a test shows a shaded strip and asks which grid shows the same fraction, you are matching the amount shaded, not the number of cells.

Here is why equivalent fractions work. Take one quarter and cut every piece in two: now there are eight pieces and two are shaded, two eighths. Cut every piece in three instead: twelve pieces, three shaded, three twelfths. Cutting every piece the same way multiplies the number of pieces and the number of shaded pieces by the same amount — so multiply the top and the bottom by the same number and the fraction does not change. Going the other way, dividing top and bottom by the same number, is simplifying: eighteen twenty-fourths, divide both by six, three quarters.

On a number line a fraction is a position. One third sits one third of the way from zero to one, whether or not a mark is drawn there, and the test question is often which of several marks is closest to it. One third is about zero point three three. A mark at one quarter is zero point two five, about eight hundredths short; a mark at three eighths is zero point three seven five, about four hundredths past. So three eighths is closer. Turning each fraction into a decimal, or into a common denominator, makes closest a subtraction instead of a guess.

To compare fractions with different denominators, rename them so the pieces match. Three eighths against two fifths: eighths and fifths do not line up, so use fortieths, the lowest common multiple of eight and five. Three eighths is fifteen fortieths; two fifths is sixteen fortieths. Two fifths is larger, by one fortieth. There are shortcuts when they fit — same numerator, the smaller denominator wins; or compare each fraction to a half — but renaming always works and tells you exactly how far apart they are.

The test's favourite version gives a picture with a number of pieces and fractions that do not match it. Jack paints two fifths of a ten-section wall yellow and half of it purple — how many sections are green? Rename each fraction into tenths, because the wall is in tenths: two fifths is four tenths, four sections; a half is five tenths, five sections. Nine painted, one green. A hundred-cell grid is renaming to hundredths; a thirty-six-cell strip is renaming to thirty-sixths. Match the fraction to the pieces in front of you. Quiz and exercises now.