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

Work with addition and subtraction equations.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.

Use polynomial identities to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve systems of equations.

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

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships

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

Analyze patterns and relationships.

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

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Add and subtract within 20.

Represent and solve problems involving multiplication and division.

Write and interpret numerical expressions.

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

Represent and solve problems involving addition and subtraction.

Use the four operations with whole numbers to solve problems.

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

Perform arithmetic operations on polynomials.

Represent and solve equations and inequalities graphically.

Interpret the structure of expressions.

Solve equations and inequalities in one variable.

Generate and analyze patterns.

Multiply and divide within 100.

Reason about and solve one-variable equations and inequalities.

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

Gain familiarity with factors and multiples.
problem_activity: Narrative
The purpose of this activity is for students to see how, when multiplying a number larger than ten, the distributive property can be used to decompose the factor into tens and ones, creating two smaller products. Base-ten blocks are used to help students visualize what is happening when a factor is decomposed to make two more easily known products. Factors slightly larger than ten can be naturally decomposed into a ten and some ones using place value. This will be useful in subsequent lessons as students progress towards fluent multiplication and division within 100.
When students see that you can decompose a teen number into tens and ones and use this to multiply teen numbers, they look for and make use of structure (MP7).
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students base-ten blocks.
Activity
“Take a few minutes to look at Tyler’s strategy and decide if you agree or disagree with it.”
2–3 minutes: independent work time
3–4 minutes: partner discussion
Monitor for students who connect the expressions
\(7 \times 10\)
and
\(7 \times 3\)
to the tens and ones portion of the place value diagram and
\(7 \times 13\)
to the entire diagram.
Have students share why they agreed or disagreed with Tyler’s strategy with a focus on using the place value diagram as a justification.
Consider asking:
“Where do we see the
\(7 \times 13\)
in the place value diagram?”
“Where do the 10 and the 3 come from?”
“How could you use the place value diagram to figure out how to find the value of
\(7 \times 13\)
?”
“Now use Tyler’s method on your own to find the value of
\(3 \times 14\)
.”
2–3 minutes: independent work time
“Share your solution and your reasoning with your partner.”
2–3 minutes: partner discussion
Student Facing
Tyler says he can use base-ten blocks to find the value of
\(7 \times 13\)
because he knows
\(7 \times 10\)
and
\(7 \times 3\)
. He says this diagram proves his thinking.
Do you agree or disagree? Explain your reasoning.
[IMG]
Use Tyler’s method to find the value of
\(3 \times 14\)
. Explain or show your reasoning.
Student Response
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Advancing Student Thinking
If students say they don’t see
\(7\times10\)
and
\(7\times3\)
in Tyler's diagram, consider asking:
“Where do you see
\(7\times13\)
in the diagram?”
“If we separate the tens and ones, what expression could we use to describe the tens? The ones?”
Activity Synthesis
Display base-ten blocks or place value diagrams that students used to solve. As students explain their work, write multiplication expressions to represent them.
“How does this diagram (or the base-ten blocks) show how Tyler’s method could be used to multiply
\(3 \times 14\)
?” (We can see there are 3 tens which is 30. We can see that there are 3 groups of 4 ones which is 12.
\(30 + 12\)
is 42. The whole diagram represents
\(3 \times 14\)
.)
If there is time, ask students to find the value of
\(4 \times 12\)
and
\(5 \times 16\)
using the base-ten blocks and Tyler’s strategy.


correct_option_index: 14