Thirty-Six

Unit 2 · NumbersN2 · Real Numbers · Propulsion

Fractions, Rational & Irrational Numbers

Compute with fractions and decide what number system an expression belongs to.

  • 29 min
  • Unlocks: Fuel Injectors
  • Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

Rational number
A number expressible as a ratio of integers.
Irrational number
A real number not expressible as a ratio of integers.
Common denominator
A shared denominator used to add or subtract fractions.

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Try it first

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

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The big idea

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See the structure

A fraction is division. Add fractions only after building a common denominator; multiply straight across and cancel common factors before multiplying.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

A rational number can be written as a ratio of integers. Terminating and repeating decimals are rational; a nonzero rational times an irrational number is irrational.

Explore in Desmos

Compare exact and decimal forms

Line 1 evaluates to the exact fraction 7/127/12. Line 2 is irrational and displays a nonterminating decimal approximation. Use the fraction key or convert the first decimal back to verify its exact form.

The calculator is live: change anything and see what happens.

Compare exact and decimal forms

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Stretch to a 36

Compare exact forms before rounding. Squaring nonnegative quantities or locating them between nearby perfect squares often settles the order quickly.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Use a common denominator

Problem

What is the value of 13+14\frac13+\frac14?
Use twelfths: 13=412\frac13=\frac4{12} and 14=312\frac14=\frac3{12}.

Which exact-number operation preserves the value?

Add numerators: 712\frac7{12}.

Which size or reverse-operation check confirms the exact value?

Worked example

Example 2: Track a fraction of a fraction

Problem

A tank is 34\frac34 full; then 13\frac13 of its water is used. If 40 L remain, what is capacity?
The fraction remaining is 34(1−13)=12\frac34(1-\frac13)=\frac12.

Which exact-number operation preserves the value?

If half the capacity is 40 L, capacity is 40/(1/2)=8040/(1/2)=80 L.

Which size or reverse-operation check confirms the exact value?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

What is the value of 56−14\frac56-\frac14?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Keep a fraction calculation exact

Type 5/6, 1/4, and 5/6-1/4 on separate lines. Use the fraction display key if Desmos first shows decimals.
Read 5/65/6, 1/41/4, and 7/127/12. The result is positive and smaller than 5/65/6, which checks the direction and size of the subtraction.

Keep a fraction calculation exact

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.