Unit 5 · Topic 01 · Fractions and Decimals
Anaya marked 3/4 on a number line by splitting the segment from 0 to 1 into quarters and counting three steps.
Kabir marked the same fraction by splitting the segment from 0 to 1 into eighths and counting six steps.
Different number of steps, different-looking grids — yet both pencils landed on the exact same point.
Ms. Rao asked the class to mark 3/4 on a blank number line from 0 to 1. Anaya divided the line into 4 equal parts and counted 3 steps from 0, landing her mark just before 1. Kabir, working independently, divided his line into 8 equal parts instead and counted 6 steps. "You used a different number of divisions," Anaya said, looking at his grid. "Mine can't be the same as yours."
They placed the two number lines directly on top of each other to compare. Anaya's mark at 3/4 and Kabir's mark at 6/8 lined up exactly, on the same point. "They're identical," Kabir said, surprised. Ms. Rao explained: 6/8 simplifies to 3/4 by dividing both the top and bottom by 2 — so despite the different grid, both fractions describe the exact same point on the line.
"Any fraction can be marked on a number line by dividing the space between two whole numbers into equal parts equal to the denominator, then counting parts equal to the numerator," said Ms. Rao. She had them mark 1/3, 2/3, and 3/3 on a fresh line divided into three equal parts, and pointed out that 3/3 landed exactly on the whole number 1 — a fraction with numerator equal to denominator always lands on a whole number.
Kabir then tried marking 5/4 — a fraction bigger than 1. He realized he needed to extend the number line PAST 1, and continue the same quarter-sized steps into the next whole segment. His mark for 5/4 landed one quarter-step past the point marked 1, proving fractions greater than 1 work exactly the same way, just crossing into the next whole-number segment.
To mark a fraction on a number line, divide the space between two whole numbers into as many equal parts as the denominator, then count parts equal to the numerator, starting from 0 (or from whichever whole number the fraction begins at).
Different fractions can land on the exact same point if they are equivalent, like 3/4 and 6/8 — the denominator only decides how many parts the segment is split into, not where the point actually ends up.
A fraction with numerator equal to denominator, like 3/3 or 5/5, always lands exactly on a whole number. Fractions greater than 1, like 5/4, extend past the first whole number, continuing with the same size steps into the next segment.
Draw a number line from 0 to 2 on paper. Mark 1/2, 3/2, and 2/2, then check that 2/2 lands exactly on 1.
Draw two separate number lines from 0 to 1, one split into 4 parts and one split into 6 parts. Mark 2/4 on the first and 3/6 on the second, then place the lines together to check they land on the same point.
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