options: Apply and extend previous understandings of numbers to the system of rational numbers.

Reason quantitatively and use units to solve problems.

Perform operations on vectors.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Compute fluently with multi-digit numbers and find common factors and multiples.

Extend the properties of exponents to rational exponents.

Perform operations on matrices and use matrices in applications.

Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform arithmetic operations with complex numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Use complex numbers in polynomial identities and equations.
problem_activity: Warm-up
This warm-up prompts students to review the placement of the digits when subtracting decimal numbers using an efficient algorithm. The first question gives students a chance to think about how different placements of the 5 (the first number) affects the subtraction and the difference. It also gives insight to how students interpret the 5 and its value. For examples:
Would they place the decimal point directly after the 5?
When placing 5 above 7 (in the tenths place), do they see it as 5 ones or as 5 tenths?
Do they wonder if the 5 is missing a decimal point in the second and third case?
In the second and third questions, a context is introduced for the same subtraction. Notice how students interpret the problem now; the equivalence of 5 and 5.00 should become much more apparent in a money context. Identify a couple of students who have correct responses so they could share later.
Launch
Give students up to 1 minute to think about the first question and then ask them to share what they noticed and wondered. Then, give students 1 minute of quiet work time for the remaining questions and follow with a whole-class discussion.
Student Facing
Here are three ways to write a subtraction calculation. What do you notice? What do you wonder?
[IMG]
Clare bought a photo for 17 cents and paid with a
$
5 bill. Look at the previous question. Which way of writing the numbers could Clare use to find the change she should receive? Be prepared to explain how you know.
Find the amount of change that Clare should receive. Show your reasoning, and be prepared to explain how you calculate the difference of 0.17 and 5.
Student Response
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Activity Synthesis
Ask a few students to share which way they think Clare could write the calculation (in the second question) in order to find the amount of change. In this particular case, the first setup is the most conducive to correct computation, but there is not one correct answer. Students could find the answer with any one of the setups as long as they understand that the 5 represents
$
5.00. If any students choose the second or third setup because they can mentally subtract the values without lining them up by place values, invite them to share their reasoning. Ask if they would use the same strategy for dealing with longer decimals (e.g.,
\(5.23 - 0.4879\)
) and, if not, what approach might be more conducive to correct calculation in those cases.
Encourage more students to be involved in the conversation by asking questions such as:
“Do you agree or disagree? Why?”
“Can anyone explain ___’s reasoning in their own words?”
“What is important for us to think about when subtracting this way?”
“How could we have solved this problem mentally?”


correct_option_index: 5