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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Analyze and solve linear equations and pairs of simultaneous linear equations.

Represent and solve equations and inequalities graphically.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Solve systems of equations.

Solve equations and inequalities in one variable.

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

Represent and analyze quantitative relationships between dependent and independent variables.

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

Gain familiarity with factors and multiples.

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

Multiply and divide within 100.

Represent and solve problems involving multiplication and division.

Understand the relationship between zeros and factors of polynomials.

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

Use the four operations with whole numbers to solve problems.

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

Analyze patterns and relationships.

Create equations that describe numbers or relationships

Reason about and solve one-variable equations and inequalities.

Use properties of operations to generate equivalent expressions.

Represent and solve problems involving addition and subtraction.

Work with radicals and integer exponents.

Rewrite rational expressions.

Use polynomial identities to solve problems.

Add and subtract within 20.

Generate and analyze patterns.

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

Write expressions in equivalent forms to solve problems.

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

Write and interpret numerical expressions.

Interpret the structure of expressions.

Perform arithmetic operations on polynomials.

Work with addition and subtraction equations.
problem_activity: Activity
This task is analogous to a previous activity with positive exponents. The number line strongly encourages students to think about how to change expressions so they all take the form
\(b \boldcdot 10^k\)
, where
\(b\)
is between 1 and 10, as in the case of scientific notation. The number line also is a useful representation to show that, for example,
\(29 \boldcdot 10^\text{-7}\)
is about half as much as
\(6 \boldcdot 10^\text{-6}\)
. The last two questions take such a comparison a step further, asking students to estimate relative sizes using numbers expressed with powers of 10.
As students work, look for different strategies they use to compare expressions that are not written as a product of a number and
\(10^{\text-6}\)
. Also look for students who can explain how they estimated in the last two problems. Select them to share their strategies later.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time, followed by partner discussion and whole-class discussion. During partner discussion, ask students to share their responses for the first two questions and reach an agreement about where the numbers should be placed on the number line.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the visual display of rules for exponents.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students as they describe how they compared and estimated the difference in magnitude between the pairs of values
\(29⋅10^{\text-7}\)
and
\(6⋅10^{\text-6}\)
or
\(7⋅10^{\text-8}\)
and
\(3⋅10^{\text-9}\)
. For example, “First I ____ , then I ____ .” or “_____ is (bigger/smaller) because ______ .”
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Label the tick marks on the number line.
Plot the following numbers on the number line:
A.
\(6 \boldcdot 10^{\text -6}\)
B.
\(6 \boldcdot 10^{\text -7}\)
C.
\(29 \boldcdot 10^{\text -7}\)
D.
\((0.7) \boldcdot 10^{\text -5}\)
Which is larger,
\(29 \boldcdot 10^{\text -7}\)
or
\(6 \boldcdot 10^{\text -6}\)
? Estimate how many times larger.
Which is larger,
\(7 \boldcdot 10^{\text -8}\)
or
\(3 \boldcdot 10^{\text -9}\)
? Estimate how many times larger.
Student Response
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Anticipated Misconceptions
Students may have trouble comparing negative powers of 10. Remind these students that, for example,
\(10^\text{-5}\)
is 5 factors that are
\(\frac{1}{10}\)
and
\(10^\text{-6}\)
is 6 factors that are
\(\frac{1}{10}\)
, so
\(10^\text{-5}\)
is 10 times larger than
\(10^\text{-6}\)
.
Students may also have trouble estimating how many times one larger expression is than another. Offer these students an example to illustrate how representing numbers as a single digit times a power of 10 is useful for making rough estimations. We have that
\(9 \boldcdot 10^\text{-12}\)
is roughly 50 times as much as
\(2 \boldcdot 10^\text{-13}\)
because
\(10^\text{-12}\)
is 10 times as much as
\(10^\text{-13}\)
and 9 is roughly 5 times as much as 2. In other words,
\(\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50\)
Activity Synthesis
Select previously identified students to explain how they used the number line and powers of 10 to compare the numbers in the last two problems.
One important concept is that it’s always possible to change an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, we can think of
\(29 \boldcdot 10^\text{-7}\)
as
\((2.9) \boldcdot 10 \boldcdot 10^\text{-7}\)
or
\((2.9) \boldcdot 10^\text{-6}\)
. Another important concept is that powers of 10 can be used to make rough estimates. Make sure these ideas are uncovered during discussion.


correct_option_index: 22