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Rational Number Classification Interactive

Run the Rational Number Classification Interactive MicroSim Fullscreen

About This MicroSim

A number is rational if it can be written as \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not 0. That one sentence is easy to memorize and surprisingly tricky to use: is \(0.666...\) rational? What about \(\sqrt{9}\), or \(\frac{22}{7}\), which looks so much like \(\pi\)?

This MicroSim gives you 16 numbers, one at a time and in random order. For each one you:

  1. Classify it. Select Yes, it's rational or No, it's not rational. The number card turns blue for rational or orange for not rational, and you get immediate feedback in green (correct) or red (not quite) with a short reason.
  2. Write it as p/q. When a rational number is not already shown as a fraction, type an equivalent fraction such as 2/3 and select Check Fraction. Any correct fraction counts, so 6/9 and -10/2 are fine.
  3. See why. Select Show me why to reveal the reasoning one step at a time, including the "let \(x\) equal the repeating decimal" method and a short proof that \(\sqrt{2}\) cannot be a fraction.
  4. Select Next Number to continue.

The right side shows the number three ways. The number line marks where it sits between \(-6\) and \(6\). The zoomed-in line magnifies the surrounding interval 10 times; move your pointer over it to zoom 100 times, which is enough to see that \(\frac{22}{7}\) and \(\pi\) are different points. The fraction bar appears for rational numbers once you have written the fraction, showing \(p\) shaded parts out of \(q\) equal parts.

The score tracker counts first-try results separately for classifying and for writing fractions.

Iframe Embed Code

Copy this iframe to your website:

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<iframe src="https://dmccreary.github.io/algebra-1/sims/rational-number-classification-interactive/main.html"
        height="602px" width="100%" scrolling="no"></iframe>

Lesson Plan

Learning Objective

Students will classify numbers as rational or not rational and express rational numbers in the form \(\frac{p}{q}\), justifying each decision with the definition. (Bloom's level: Apply)

Audience and Prerequisites

Designed for Algebra I students in grade 9. Students should know what integers are, be able to convert a terminating decimal to a fraction, and have seen the definition of a rational number.

Suggested Activity

  1. Predict (3 minutes). Before opening the MicroSim, write \(0.666...\), \(\sqrt{9}\), \(\frac{22}{7}\), and \(\pi\) on the board. Have students vote rational or not rational for each and keep the tally visible.
  2. Practice (8 minutes). Students work through at least ten numbers. For every "Not quite," they select Show me why and read all of the steps before moving on.
  3. Zoom in (3 minutes). When \(\frac{22}{7}\) or \(\pi\) comes up, have students hover over the zoomed-in line and describe what they see. Ask, "How close is close enough to be equal?"
  4. Sort and explain (5 minutes). With a partner, students sort the numbers they saw into three groups: decimals that end, decimals that repeat, and decimals that do neither. They write one sentence explaining which groups are rational.
  5. Revisit the vote. Return to the board tally and correct it together.

Assessment

Students demonstrate mastery by classifying at least 8 of 10 numbers correctly on the first try, writing a correct fraction for a terminating decimal and for a repeating decimal, and explaining in their own words why \(\sqrt{9}\) is rational but \(\sqrt{2}\) is not.

Common Misconceptions

  • "A decimal that goes on forever is never rational." Repeating decimals such as \(0.666...\) go on forever and are rational.
  • "Every square root is irrational." \(\sqrt{9} = 3\).
  • "\(\frac{22}{7}\) equals \(\pi\)." It is a rational approximation of an irrational number.
  • "A decimal with a pattern is rational." \(0.1010010001...\) has a pattern but no repeating block.

References

  1. Related Algebra I chapter - Course explanation and diagram specification.
  2. Rational number - Wikipedia - Definition and properties.
  3. Irrational number - Wikipedia - Examples and proofs, including the square root of 2.
  4. Repeating decimal - Wikipedia - Why repeating decimals are rational and how to convert them to fractions.
  5. p5.js Reference - Documentation for the interactive graphics library.