Lesson | Loading...

Unit 5, Lesson 20

Rational and Irrational Solutions

$Course Icon$

Integrated Math 2

20.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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 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 number that can be written as a positive or negative fraction 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.)

Activity

None

Numbers like -1.7, , and are known as rational numbers.

Numbers like and are known as irrational numbers.

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

97
-8.2

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Ask students to share their sorting results. Then, ask, “What are some ways you can tell that 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 is rational because . However, 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, is rational because it equals , but the decimal doesn’t start repeating until the seventh digit after the decimal point.

20.2

Activity

Instructional Routines

Graph It MLR8: Discussion Supports

Materials

To Gather

Graphing technology

Activity Narrative

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 and . Students will investigate these kinds of solutions in the next activity.

20.3

Activity

Instructional Routines

None

Materials

None

Activity Narrative

Students ended the previous activity wondering whether solutions that look like and 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 specific 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.

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?” (It is difficult to know. If the solution is a decimal that does not appear to repeat, then it might be irrational, but we cannot be certain from only the limited decimals it provides.)

to access Cool-down.

Student Lesson Summary

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

Rational numbers can be written as positive or negative fractions. Numbers like 12, -3, , -4.79, and are rational. ( can be written as a fraction, because it’s equal to . The number -4.79 is the opposite of 4.79, which is .)
Any number that is not rational is irrational. Some examples are , and . When an irrational number is written as a decimal, its digits go on forever without eventually making a repeating pattern, 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 by graphing a corresponding function (), sometimes we can tell from the -coordinates of the -intercepts. Other times, we can't be sure.

Let's solve and by graphing and .

<p>Graphs of two quadratic functions on a grid.</p>

The graph of crosses the -axis at -0.7 and 0.7. There are no digits after 7, suggesting that the -values are exactly and , which are rational.

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

and , so are exact solutions.

The graph of , created using graphing technology, is shown to cross the -axis at -2.236 and 2.236. It is unclear if the -coordinates stop at three decimal places or if they continue. If they stop or eventually make a repeating pattern, the solutions are rational. If they never stop or make a repeating pattern, the solutions are irrational.

We can tell, though, that 2.236 is not an exact solution to the equation. Substituting 2.236 for in the original equation gives , which we can tell is close to 0 but is not exactly 0. This means 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 are , which are rational.
The solutions to are , which are irrational. (2.236 is an approximation of , not equal to .)

Launch

Give students a moment to look at the equations in the Task 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 two equations while the other solves algebraically. Then they should switch roles for the remaining questions. Students 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.

MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “How do you know that the solution is irrational (or rational)?” This gives both students an opportunity to produce language.
Advances: Conversing

Activity

None

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.

equation	zeros	rational or irrational?

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.

Student Response

to access Student Response.

Building on Student Thinking

When using algebra to solve equations such as , some students may expand 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 correctly. 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.

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?

Display the list of numbers, and ask students to say which are rational and which are irrational. If necessary, tell students that the first 4 numbers are rational and the last 4 are irrational.

Consider keeping students in groups of 2. Provide access to calculators for numerical calculations, if 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

Activity

None

Here is a list of numbers:

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.

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.
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

to access Student Response.

Lesson | Loading...

Settings

Audience

Assessment Access

Lesson 20

Rational and Irrational Solutions

Warm-up

Standards Alignment

Building On

8.NS.A.1

Addressing

Building Toward

HSN-RN.B.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

8.NS.A.1

Addressing

HSA-REI.B.4.b

HSF-IF.C.7.a

Building Toward

HSN-RN.B.3

Standards Alignment

Building On

Addressing

HSN-RN.B.3

Building Toward