options: Rewrite rational expressions.

Solve equations and inequalities in one variable.

Solve systems of equations.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Use polynomial identities to solve problems.

Represent and solve problems involving addition and subtraction.

Understand the connections between proportional relationships, lines, and linear equations.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

Understand and apply properties of operations and the relationship between addition and subtraction.

Perform arithmetic operations on polynomials.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand the relationship between zeros and factors of polynomials.

Understand solving equations as a process of reasoning and explain the reasoning.

Work with addition and subtraction equations.

Interpret the structure of expressions.

Generate and analyze patterns.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand properties of multiplication and the relationship between multiplication and division.

Work with equal groups of objects to gain foundations for multiplication.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Create equations that describe numbers or relationships

Use the four operations with whole numbers to solve problems.

Represent and solve equations and inequalities graphically.

Represent and analyze quantitative relationships between dependent and independent variables.

Add and subtract within 20.

Write expressions in equivalent forms to solve problems.

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Represent and solve problems involving multiplication and division.

Reason about and solve one-variable equations and inequalities.
problem_activity: Activity
The purpose of this task is for students to rewrite rational numbers with terminating decimal expansions in fraction form and fractions with terminating decimal expansions as decimals. This activity is the first of a series of three in which students rewrite numbers in different ways, supporting their understanding of what rational and irrational numbers are and how they can be represented.
Monitor for students who write
\(\frac15\)
and
\(\frac{2}{10}\)
for 0.2.
Launch
Arrange students in groups of 2. Do not provide access to calculators.
Remind students that all rational numbers are fractions and their opposites, and they can all be written in the form
\(\frac{a}{b}\)
where
\(a\)
and
\(b\)
are integers (with
\(b\neq 0\)
). In fact, there are many equivalent fractions that represent a single rational number. For example, 5 is equivalent to
\(\frac{10}{2}\)
and
\(\frac{15}{3}\)
.
Students should complete the problems individually, then compare with their partners and come to resolutions over any differences. Repeat this process for the second problem.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Look for students who find equivalent ways to represent fractions and decimals. Check for precision in calculations and encourage partners to seek clarification in their discussions.
Supports accessibility for: Memory; Organization
Writing: MLR3 Clarify, Critique, Correct.
Present an incorrect statement that reflects a possible misunderstanding from the class. For example, finding the decimal representations of rational numbers, an incorrect statement is: “For finding the decimal representation of
\(\frac38\)
, divide 8 by 3 using long division ”. Prompt students to identify the error, and then write a correct version. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness
Student Facing
Rational numbers are fractions and their opposites. All of these numbers are rational numbers. Show that they are rational by writing them in the form
\(\frac{a}{b}\)
or
\(\text -\frac{a}{b}\)
.
0.2
\(\text -\sqrt{4}\)
0.333
\(\sqrt[3]{1000}\)
-1.000001
\(\sqrt{\frac19}\)
All rational numbers have decimal representations, too. Find the decimal representation of each of these rational numbers.
\(\frac38\)
\(\frac75\)
\(\frac{999}{1000}\)
\(\frac{111}{2}\)
\(\sqrt[3]{\frac18}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may write that
\(0.333=\frac13\)
. If so, ask them to check their work by calculating the decimal representation of
\(\frac13\)
.
Activity Synthesis
The purpose of this discussion is to highlight different strategies for rewriting rational numbers in different forms. Select previously identified students to share their solutions, including one student who wrote
\(\frac15\)
for 0.2 and another who wrote
\(\frac{2}{10}\)
. For the problems with roots, the values were purposefully chosen to emphasize to students that just because a number is written with a square or cube root does not mean it is not rational.
For 0.2, draw a number line with the numbers 0, 1, and 2 with plenty of space between the integers. Subdivide the segment from 0 to 1 into 5 equal pieces, and then ask where to plot
\(\frac15\)
. Then label
\(\frac15\)
and ask how we can see that this is
\(\frac{2}{10}\)
. (Subdivide each fifth into two equal pieces—now each one is
\(\frac{1}{10}\)
.) Then label the point
\(\frac{2}{10}\)
and 0.2.
[IMG]
Remind students that the point corresponds to a rational number, and we have a lot of different ways we can represent that number.


correct_option_index: 31