few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001097

7
options: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
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
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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.


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