standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Warm-up
This Math Talk encourages students to think about the fact that subtracting a number is equivalent to adding the opposite of that number (that is,
\(100-5=100+\text-5\)
) and to rely on the structure of the expressions on each side of the equal sign to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students reason that
\((x-1)(x-4)\)
is equivalent to
\((x+\text-1)(x+\text-4)\)
, which in turn helps them expand the expression using diagrams and the distributive property.
Some students may reason about the value of
\(n\)
by reasoning. For example, to find
\(n\)
in
\(40-8=40+n\)
, they see that the left side of the equation is 32 and reason that the number being added to 40 must be negative, and that it must be -8. Others may carry out the steps for solving equations, for example, by subtracting 40 from both sides, or by adding 8 to both sides and then subtracting 48 from both sides.
Any of these strategies are fine. The key point is to recognize that the number being added on one side is the opposite of the value being subtracted from the other side. If after a couple of questions students show an understanding of this, and if time is limited, it is not essential for students to complete all questions.
As students solve these equations mentally, they practice looking for and making use of structure (MP7).
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Solve each equation mentally.
\(40-8=40+n\)
\(25+\text-100=25-n\)
\(3-\frac12=3+n\)
\(72-n=72+6\)
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
“Who can restate
\(\underline{\hspace{.5in}}\)
’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
\(\underline{\hspace{.5in}}\)
’s strategy?”
“Do you agree or disagree? Why?”
If not already made clear in students’ explanations, highlight that subtracting a number is gives the same outcome as adding the opposite of that number.
Tell students that remembering that subtraction can be thought of in terms of addition can help us rewrite quadratic expressions such as
\((x-5)(x+2)\)
or
\((x-9)(x-3)\)
, where one or both factors are differences.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)


aligns: no