Alberta Grade 8 Math — Unit: Rational & Irrational Numbers
Organizing IdeaNumber — Number sense is developed through understanding the ways numbers describe the world.
Guiding QuestionHow can the density of the number line contribute to a sense of number?
Learning OutcomeStudents interpret rational and irrational numbers.
This lesson covers: the density of rationals on the number line (infinitely many between any two);
irrationals filling the gaps no rational reaches; finding a rational number between any two given rationals;
relating rationals and irrationals to positions on the number line; classifying an irrational expression
as exact or approximate; comparing and ordering real numbers.
Between Any Two Numbers
Pick any two numbers on the number line — say 0.4 and 0.5.
Can you name a number between them? Sure: 0.45. Now name one between 0.4 and 0.45. Easy: 0.42.
Between 0.4 and 0.42? 0.41. You could keep going forever.
The number line is dense. Between any two rationals there are infinitely many more rationals.
And in the spots where no rational lands at all, irrational numbers like
√2 and π live, filling the gaps the rationals cannot reach.
Two big ideas drive this lesson: (1) between any two rationals, infinitely many more rationals fit;
(2) irrational numbers fill the gaps no rational can reach. Together, rationals and irrationals
cover every point on the real number line.
Zoom In Forever
Each click of Zoom in tightens the view onto a smaller slice of the line.
Watch how new rationals keep appearing inside the gap, no matter how narrow it gets.
Range: 0.000 to 1.000Level 1 of 4
Level 1. Five labelled rationals between 0 and 1. Plenty of space — for now.
No matter how far you zoom, the line still has rationals waiting. Density means the rationals
are packed tightly enough that no two of them sit side by side with nothing between them.
Find a Number In the Middle
Given two rationals, how do you find one between them? Three reliable methods.
Pick a pair, then pick a method.
Where the Rationals Don’t Reach
√2 doesn’t equal any fraction. So where on the number line does it sit?
Zoom in toward it. At every level, √2 hides between two rationals
— but it never lands on one.
Range: 1.0 to 2.0Level 1 of 4
Level 1. √2 ≈ 1.41421… sits between 1.4 and 1.5, but isn’t equal to either.
Every irrational has a place on the number line, even though no fraction names it exactly.
Its decimal expansion goes on forever without repeating, so we either write it as an
exact value like √2, or as a rounded approximation like 1.414.
Exact or Approximate?
A number can be written two ways: an exact form (like √2 or
π2),
or a rounded decimal. Click each card to flip between them.
click any card to flip
13 = 0.3
can be written exactly as the fraction 13,
even though its decimal goes forever — the digits repeat. Irrationals like √2 cannot be written
exactly as any fraction or as a terminating or repeating decimal. The exact form is the only exact form.
Round 1 of 3
Rationals on the Line
Click the tiles in order from least to greatest.
Click the smallest number first.
What You Now Know
The real number line is dense. Between any two rationals, infinitely many more rationals exist
— you can always average them, find a common denominator, or work in decimals to land another one
in the gap.
But the rationals don’t cover everything. Irrational numbers fill the gaps no rational can reach.
Numbers like √2 and π have a fixed place on the number line, but their decimal expansions go on forever
without ever repeating. We write them in exact form, or as rounded decimals when we need a number we can use.
Together, rationals and irrationals cover every point on the real number line. There are no gaps. That’s
what mathematicians mean when they say the real numbers are complete.