NUMBER SYSTEMS TWSSP Monday

Welcome Fill out a notecard with the following: Front: Name Back: School, Grade What are two things you know about the structure of our number system? What are two things you wonder about the structure of our number system? Introduce the person to your left by name, school, grade, and share something that person will do this summer

Week Overview A dual focus How is our number system structured and why? What constitutes mathematical proof?

Week Overview Why our number system? The structure is entirely built from necessity Starts easy(ish) and builds in complexity The foundation of arithmetic, algebra, calculus, analysis, topology, … A great platform for understanding proof Why mathematical proof? CCSSM practice 4 The foundation of arithmetic, algebra, calculus, analysis, topology, geometry, data analysis, …

Week Overview Content focused, but transparent in pedagogy Dedicated time every day to consider whole group work in our content area Purple cup time if desired

Monday Agenda Community agreements Preassessment Define the natural numbers, integers, and closure Explore properties and make conjectures about closure

Monday Agenda Questions for today: Under what operations are the natural numbers and subsets of the natural numbers closed? Under what operations are the integers and subsets of the integers closed? Learning targets: The natural numbers are closed under _________ The integers are closed under ____________ Given a subset of the natural numbers or integers and an operation, the subset is either closed under that operation or not closed under that operation. Success criteria: I can determine if a subset of the natural numbers is closed under an operation and justify my conclusion. I can determine if a subset of the integers is closed under an operation and justify my conclusion.

A protocol we will use Think - Go Around – Discuss Private Think Time: Quietly and privately respond to questions. Respect the need for others to process quietly. Go Around: Share your ideas one person at a time without interruptions. Discuss: Come to agreement or consensus that can be shared out with the whole group. Make sure everyone in your group understands the ideas discussed.

Community Agreements What do you need from each other in order to be able to feel safe to explore mathematical ideas, share thinking, and build on and connect with others’ ideas? What do you need to feel respected and valued as part of the mathematical community?

The Natural Numbers The natural numbers are the counting numbers {1, 2, 3, 4, 5, 6, 7, 8, …} We use the symbol ℕ to denote the natural numbers For each of the four arithmetic operations (addition, subtraction, multiplication and division), what happens when you perform those operations on any two natural numbers? What kind of number do you get? Use a think-go around- discuss

Closure A set is closed under an operation if, when we operate on any two elements of the set, the result is also in the set. NOTE: in order to consider closure, we need a set AND an operation Under which of addition, subtraction, multiplication, and division are the natural numbers closed? What about exponentiating (i.e. a b ) or taking square roots? Think of two other things we do to numbers. Are the natural numbers closed under doing those things?

Prove vs. disprove What must we do in order to show that a set is not closed under an operation? If we wanted to disprove the statement, “all math teachers have blue eyes,” what do we have to do? If we wanted to prove the statement, “all math teachers have blue eyes,” what do we have to do?

Finite subsets Each group will be given a finite subset of the natural numbers and two operations. A subset of a set contains only elements of the original set (though may not contain all elements) Is your set closed under your operations? Prepare your whiteboard to share with the rest of the class. Identify your set Identify your operations Determine closure under those operations, with justification. Class: Agree or disagree? Is the justification sufficient?

Primes A prime is a natural number who’s only divisor is itself and 1. 1 is not considered prime. Find all of the primes between 50 and 100 The Fundamental Theorem of Arithmetic Every natural number, other than 1, can be factored into a product of primes in only one way, apart from the order of the factors. Do you see why we don’t want to count 1? Find the prime factorization of your house number.

Primes Under which, if any, of the four operations are the primes closed?

The integers The integers are the natural numbers, plus 0 and all of the negatives of the natural numbers. {…, -3, -2, -1, 0, 1, 2, 3, …} Notation: ℤ Under which of the four arithmetic operations are the integers closed? Conjecture: does closure of a subset imply closure of the larger set?

Choose one Is the set of positive powers of 2, {2 1, 2 2, 2 3, …}, closed under multiplication? Under division? If 3 is a divisor of two numbers, is it a divisor of their sum and their difference? If d is a divisor of two numbers, is it a divisor of their sum and difference?

Exit Ticket (sort of…) Under what operations are the natural numbers closed? Under what operations are the integers closed? Prove without calculating: 7*19*5*23 ≠ 3*13*17*25