L1 · Rational Numbers on the Number Line (including Repeating Decimals)

Learning goal

“I can recognize a rational number in any of its forms - integer, fraction, terminating decimal, repeating decimal - place it on the number line, and find another rational number between any two rationals.”

Rational number — any number that can be written as a quotient a/b of integers with b not equal to 0. Every integer is rational (-4 = -4/1), and so is every terminating or repeating decimal.

Terminating decimal — a decimal that ends: 5/8 = 0.625 and then stops.

Repeating decimal — a decimal whose digits fall into a block that repeats forever, written with a bar over the block: 1/3 = 0.333... The bar is what makes it rational - the pattern never breaks.

Density — between any two rational numbers there is always another rational number, so there are infinitely many rationals between any two. The number line has no gaps you can point to between rationals.

Worked example

Classify and place four numbers: 0.75, -3, 2/5, 0.333... Step 1: Test each against the definition - can it be written as a/b of integers, with b not 0? 0.75 = 3/4, rational. -3 = -3/1, rational (every integer is). 2/5 is already a quotient. 0.333... = 1/3 - its digits repeat, so it is a fraction in disguise. Step 2: Place them on a number line pre-segmented from -4 to 1 in quarters: -3 sits on the integer mark -3; 2/5 = 0.4 sits just left of the 1/2 mark; 0.333... = 1/3 sits just left of 2/5 (0.333... < 0.4); 0.75 sits on the 3/4 mark. Order on the line: -3, 1/3, 2/5, 0.75. Bar model alongside: one whole bar cut into quarters shows 3/4 = 0.75 as the same point the line shows - two representations, one number. Say it plainly: "rational" is not a form, it is a family test - if a number can be written as a quotient of integers, it belongs, whatever it looks like.

Check yourself — explain it

Why is 0.333... rational even though its decimal never ends?

Paired problem — example still visible

Write 9/10 as a decimal, and write -12 as a quotient. -> 0.9; -12/1, so -12 is rational.

Faded completion — last step is yours

"0.6 as a fraction: 0.6 = 6/10, and dividing numerator and denominator by 2 gives ___." -> 3/5

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