Lesson 20

Rational and Irrational Solutions

20.1: Rational or Irrational? (5 minutes)

Warm-up

This warm-up refreshes students’ memory about rational and irrational numbers. Students think about the characteristics of each type of number and ways to tell that a number is rational. This review prepares students for the work in this lesson: identifying solutions to quadratic equations as rational or irrational, and thinking about what kinds of numbers are produced when rational and irrational numbers are combined in different ways.

Launch

Remind students that a rational number is “a fraction or its opposite” and that numbers that are not rational are called irrational. (The term rational is derived from the word ratio, as ratios and fractions are closely related ideas.)

Student Facing

Numbers like -1.7, \(\sqrt{16}\), and \(\frac53\) are known as rational numbers.

Numbers like \(\sqrt{12} \text{ and } \sqrt{\frac59}\) are known as irrational numbers.

Here is a list of numbers. Sort them into rational and irrational.

  • 97
  • -8.2
  • \(\sqrt5\)
  • \(\text-\frac{3}{7}\)
  • \(\sqrt{100}\)
  • \(\sqrt{\frac94}\)
  • \(\text-\sqrt{18}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their sorting results. Then, ask, “What are some ways you can tell a number is irrational?” After each student offers an idea, ask others whether they agree or disagree. Ask students who disagree for an explanation or a counterexample.

Point out some ways to tell that a number might be irrational:

  • It is written with a square root symbol and the number under the symbol is not a perfect square, or not the square of a recognizable fraction. For example, it is possible to tell that \(\sqrt{\frac94}\) is rational because \(\frac94=\left(\frac32\right)^2\). However, \(\sqrt{18}\) is irrational because 18 is not a perfect square.
  • If we use a calculator to approximate the value of a square root, the digits in the decimal expansion do not appear to stop or repeat. It is possible, however, that the repetition doesn’t happen within the number of digits we see. For example, \(\sqrt{\frac{1}{49}}\) is rational because it equals \(\frac17\), but the decimal doesn’t start repeating until the seventh digit after the decimal point.

20.2: Suspected Irrational Solutions (15 minutes)

Activity

In this activity, students use algebraic reasoning to solve quadratic equations and to identify the solutions as rational or irrational. But first, they inspect the zeros of a corresponding function for hints about whether the solutions might be rational. They notice that it is impossible to know whether a solution is irrational simply by looking at the decimal approximation of a point shown on a graph.

While solutions obtained by algebraic solving can better show the types of number they are, some solutions are difficult to classify because they are combinations of rational and irrational numbers, such as \(\sqrt{2} + 7\) and \(\pm 4 \sqrt{5}\). Students will investigate these kinds of solutions in the next activity.

Launch

Give students a moment to look at the equations in the activity statement and be prepared to share what they noticed about the two sets of equations. If not mentioned by students, point out that the equations in the two questions describe the same set of functions. Ask students how finding the zeros of the equations in the first question relates to solving the equations in the second question. (Both are ways to solve the second set of equations.).

Arrange students in groups of 2. Provide access to devices that can run Desmos or other graphing technology. Ask students to take turns graphing and solving algebraically. One partner should graph the first 2 equations while the other solves algebraically, and then switch roles for the remaining questions. Each student should decide whether they think the zeros or the solutions they found—by graphing or by algebraic solving—are rational or irrational and then discuss their thinking.

Conversing: MLR8 Discussion Supports. Use this routine to help students describe their reasoning for determining whether a solution is rational or irrational. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “_____ is rational/irrational, because _____.” Invite the listener, Partner B, to press for additional details referring to specific features of the solution, such as whether the square root is a rational number. Students should switch roles for each equation. This will help students justify how to determine whether solutions are rational or irrational.
Design Principle(s): Support sense-making; Cultivate conversation

Student Facing

  1. Graph each quadratic equation using graphing technology. Identify the zeros of the function that the graph represents, and say whether you think they might be rational or irrational. Be prepared to explain your reasoning.
    equations      zeros      rational or irrational?
    \(y=x^2 - 8\)    
    \(y=(x-5)^2 - 1\)    
    \(y=(x - 7)^2 - 2\)    
    \(y=\left( \frac{x}{4} \right)^2 -5\)    
  2. Find exact solutions (not approximate solutions) to each equation and show your reasoning. Then, say whether you think each solution is rational or irrational. Be prepared to explain your reasoning.

    1. \(x^2 - 8 = 0\)
    2. \((x - 5)^2 = 1\)
    3. \((x - 7)^2 = 2\)
    4. \(\left( \frac{x}{4} \right)^2 -5=0\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

When solving equations such as \((x-5)^2=1\) algebraically, some students may expand \((x-5)^2\) rather than use square roots (perhaps to use the quadratic formula to solve). For students who use this approach, check to see if they expanded \((x-5)^2\) correctly and remind them, if needed, that \((x-5)^2=(x-5)(x-5)=x^2-10x+25\). These students may benefit from continued use of a rectangle diagram to expand. Emphasize the need to check their answers against the graphical solution. When their solutions don’t agree, encourage them to find the error in their work.

Activity Synthesis

Invite students to share their responses and explanations. Ask students to compare the zeros they found by graphing and the solutions they found algebraically. Discuss with students:

  • “How do the zeros of the functions \(y=x^2 - 8\) and \(y=(x - 7)^2 - 2\), which you found by graphing, compare to the solutions to \(x^2 = 8\) and \((x - 7)^2 = 2\)?” (The zeros found by graphing are decimal approximations. They show decimal approximations up to 3 decimal places, or more, depending on the tool or setting used. The solutions found algebraically can be written as expressions that use the square root symbol.)
  • “Is it easier to tell whether the solutions to an equation are irrational by graphing and finding the zeros, or by solving algebraically?” (By solving algebraically. Observing the coordinates on a graph doesn’t always help, because it is impossible to tell whether a number is irrational by only looking at a few digits of its decimal approximation.)

Draw students’ attention to the numerical expressions they encountered while solving the equations:

  • \(\sqrt{1}+5\): This expression is rational because it equals 6.
  • \(\sqrt8\): This is irrational because 8 is not a perfect square.
  • \(\sqrt{2} + 7\): \(\sqrt2\) is irrational and 7 is rational. Is the sum rational or irrational?
  • \(\pm 4 \sqrt{5}\): 4 is rational and \(\sqrt5\) is irrational. Is the product rational or irrational?

Tell students that they will now experiment with the sums and products of rational and irrational numbers and investigate what kinds of numbers the sums and products are.

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Make a large two-column display, reserving one side for examples of rational numbers and another side for examples of irrational numbers. Invite students to suggest language to include that will support their understanding of rational and irrational numbers. Keep the chart visible throughout the lesson, and as examples are discussed, add them to the chart.
Supports accessibility for: Conceptual processing; Language

20.3: Experimenting with Rational and Irrational Numbers (15 minutes)

Activity

Students ended the previous activity wondering whether solutions that look like \(7 \pm \sqrt2\) and \(4\sqrt5\) are rational or irrational. In this activity, they pursue that question and experiment with adding and multiplying different types of numbers. The goal here is to make conjectures about the sums and products by noticing regularity in repeated reasoning with concrete numbers (MP8). In an upcoming lesson, students will reason logically and abstractly about what the sums and products of rational and irrational numbers must be.

Launch

Tell students that they will now further investigate what happens when different types of numbers are combined by addition and multiplication. Are the results rational or irrational? Can we come up with general rules about what types of numbers the sums and products will be?

Consider keeping students in groups of 2. Provide access to calculators for numerical calculations, in case requested. Students who choose to use technology to help them analyze patterns practice choosing a tool strategically (MP5).

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to begin by sorting and labeling the initial numbers into rational or irrational. Invite each partner group to choose and respond to either the sum or the product statements. Provide sentence frames for students to use in their discussion. For example, “I agree/disagree with _____ because . . . .” and “Here is an example that shows . . . .” Then, have partners switch or select students to share with the whole class, so that each student has examples and conclusions for all truth statements.
Supports accessibility for: Organization; Attention

Student Facing

Here is a list of numbers:

  • \(\displaystyle 2\)
  • \(\displaystyle 3\)
  • \(\displaystyle \frac13\)
  • \(\displaystyle 0\)
  • \(\displaystyle \sqrt2\)
  • \(\displaystyle \sqrt3\)
  • \(\displaystyle \text-\sqrt3\)
  • \(\displaystyle \frac{1}{\sqrt3}\)

Here are some statements about the sums and products of numbers. For each statement, decide whether it is always true, true for some numbers but not others, or never true.

  1. Sums:

    1. The sum of two rational numbers is rational.
    2. The sum of a rational number and an irrational number is irrational.
    3. The sum of two irrational numbers is irrational.
  2. Products:

    1. The product of two rational numbers is rational.
    2. The product of a rational number and an irrational number is irrational.
    3. The product of two irrational numbers is irrational.

Experiment with sums and products of two numbers in the given list to help you decide.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

It can be quite difficult to show that a number is irrational. To do so, we have to explain why the number is impossible to write as a ratio of two integers. It took mathematicians thousands of years before they were finally able to show that \(\pi\) is irrational, and they still don’t know whether or not \(\pi^{\pi}\) is irrational.

Here is a way we could show that \(\sqrt{2}\) can’t be rational, and is therefore irrational.

  • Let's assume that \(\sqrt{2}\) were rational and could be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are non-zero integers.
  • Let’s also assume that \(a\) and \(b\) are integers that no longer have any common factors. For example, to express 0.4 as \(\frac {a}{b}\), we write \(\frac25\) instead of \(\frac{4}{10}\) or \(\frac {200}{500}\). That is, we assume that \(a\) and \(b\) are 2 and 5, rather than 4 and 10, or 200 and 500.
  1. If \(\sqrt{2}=\frac{a}{b}\), then \(2 = \frac {\boxed{\phantom{300}}}{\boxed{\phantom{300}}}\).
  2. Explain why \(a^2\) must be an even number.
  3. Explain why if \(a^2\) is an even number, then \(a\) itself is also an even number. (If you get stuck, consider squaring a few different integers.)
  4. Because \(a\) is an even number, then \(a\) is 2 times another integer, say, \(k\). We can write \(a=2k\). Substitute \(2k\) for \(a\) in the equation you wrote in the first question. Then, solve for \(b^2\).
  5. Explain why the resulting equation shows that \(b^2\), and therefore \(b\), are also even numbers.
  6. We just arrived at the conclusion that \(a\) and \(b\) are even numbers, but given our assumption about \(a\) and \(b\), it is impossible for this to be true. Explain why this is.

If \(a\) and \(b\) cannot both be even,\(\sqrt{2}\) must be equal to some number other than \(\frac{a}{b}\).

Because our original assumption that we could write \(\sqrt{2}\) as a fraction \(\frac{a}{b}\) led to a false conclusion, that assumption must be wrong. In other words, we must not be able to write \(\sqrt{2}\) as a fraction. This means \(\sqrt{2}\) is irrational!

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the six statements for all to see. Invite students to share their responses. After each student shares, ask others whether they agree or disagree. Ask students who disagree for an explanation or a counterexample. Develop a consensus on what the class thinks is true when we combine numbers by addition and multiplication. Record and display the consensus for all to see, for instance:

  • \(\text {rational} + \text {rational} = \)
  • \(\text {rational} + \text {irrational} =\)
  • \(\text {irrational} + \text {irrational} =\)
  • \(\text {rational} \boldcdot \text {rational} =\)
  • \(\text {rational} \boldcdot \text {irrational} =\)
  • \(\text {irrational} \boldcdot \text {irrational} =\)

Explain to students that in an upcoming lesson, they will have a chance to test their conjectures.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language, and invite students to chorally repeat phrases that include relevant vocabulary in context.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

To help students consolidate the insights from this lesson, ask students questions such as:

  • “How would you explain to a classmate who is absent today how to tell if a number is rational or irrational?”
  • “How can we tell from a graph created using graphing technology whether the solutions to a quadratic equation could be rational or irrational?”
  • “Why is the \(x\)-intercept information given by a graphing tool not a sure way to tell if those numbers are irrational?”

20.4: Cool-down - What Kind of Solutions? (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

The solutions to quadratic equations can be rational or irrational. Recall that:

  • Rational numbers are fractions and their opposites. Numbers like 12, -3, \(\frac53, \sqrt{25}\), -4.79, and \(\sqrt\frac{9}{16}\) are rational. (\(\sqrt{25}\) is a fraction, because it’s equal to \(\frac51\). The number -4.79 is the opposite of 4.79, which is \(\frac{479}{100}\).)
  • Any number that is not rational is irrational. Some examples are \(\sqrt2, \pi, \text-\sqrt5\), and \(\sqrt{\frac72}\). When an irrational number is written as a decimal, its digits do not stop or repeat, so a decimal can only approximate the value of the number.

How do we know if the solutions to a quadratic equation are rational or irrational?

If we solve a quadratic equation \(ax^2+bx+c=0\) by graphing a corresponding function (\(y=ax^2+bx+c\)), sometimes we can tell from the \(x\)-coordinates of the \(x\)-intercepts. Other times, we can't be sure.

Let's solve \(x^2-\frac{49}{100}=0\) and \(x^2-5=0\) by graphing \(y=x^2-\frac{49}{100}\) and \(y=x^2-5\).

Graphs of two quadratic functions on a grid.

The graph of \(y=x^2-\frac{49}{100}\) crosses the \(x\)-axis at -0.7 and 0.7. There are no digits after the 7, suggesting that the \(x\)-values are exactly \(\text-\frac{7}{10}\) and \(\frac{7}{10}\), which are rational.

To verify that these numbers are exact solutions to the equation, we can see if they make the original equation true.

\((0.7)^2-\frac{49}{100}=0\) and \((\text-0.7)^2-\frac{49}{100}=0\), so \(\pm 0.7\) are exact solutions.

The graph of \(y=x^2-5\), created using graphing technology, is shown to cross the \(x\)-axis at -2.236 and 2.236. It is unclear if the \(x\)-coordinates stop at three decimal places or if they continue. If they stop or eventually make a repeating pattern, the solutions would be rational. If they never stop or make a repeating pattern, the solutions would be irrational.

We can tell, though, that 2.236 is not an exact solution to the equation. Substituting 2.236 for \(x\) in the original equation gives \(2.236^2-5\), which we can tell is close to 0 but is not exactly 0. This means \(\pm2.236\) are not exact solutions, and the solutions may be irrational.

To be certain whether the solutions are rational or irrational, we can solve the equations.

  • The solutions to \(x^2-\frac{49}{100}=0\) are \(\pm 0.7\), which are rational.
  • The solutions to \(x^2-5=0\) are \(\pm \sqrt5\), which are irrational. (2.236 is an approximation of\(\sqrt5\), not equal to \(\sqrt5\).)

What about a solution like \(\text-4 + \sqrt 6\), which is a sum of a rational number and an irrational one? Or a solution like \(\frac15 \sqrt3\), which is a product of a rational number and an irrational number? Are they rational or irrational?

We will investigate solutions that are sums and products of different types of numbers in an upcoming lesson.