standard_description: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
problem_activity: 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.
Sums:
The sum of two rational numbers is rational.
The sum of a rational number and an irrational number is irrational.
The sum of two irrational numbers is irrational.
Products:
The product of two rational numbers is rational.
The product of a rational number and an irrational number is irrational.
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
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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.
If
\(\sqrt{2}=\frac{a}{b}\)
, then
\(2 = \frac {\boxed{\phantom{300}}}{\boxed{\phantom{300}}}\)
.
Explain why
\(a^2\)
must be an even number.
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.)
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
\(b^2\)
.
Explain why the resulting equation shows that
\(b^2\)
, and therefore
\(b\)
, are also even numbers.
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
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
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


aligns: no