Overview for Exam 4
· Don’t forget about divisibility from Section 5.3: Understand still what a|b means (there is a unique integer c with b = ac)
· Be able to find the prime factorization of a number using either the ladder method (being able to check simple divisors as you go through, generally 2 as many times as it goes in, 3 as well, then 5, etc. Also know how to use a tree for it.
· Know the Fundamental Theorem of Arithmetic, that each number has exactly one prime factorization. Know how many total divisors a number has based on its prime factorization.
· Know that a prime number has two divisors, a perfect square has an even number and a composite number has an even number of divisors but more than 2.
· Be able to solve a simple word problem using factorization, such as the bookstore/notebook problem we worked from Section 5.3 using divisors.
· Understand the difference between a prime factorization compared to the list of all divisors. These are very different from each other! (Why?)
· Be able to use especially intersection of the sets of divisors and the powers from prime factorizations of two/three numbers or powers of a, b, c,… to find the GCD/GCF of them.
· Be able to demonstrate LCM of two/three numbers using rods, using the intersection of sets of multiples and using powers from prime factorizations of numbers or powers of a, b, c,…
· Be able to use divisibility (see first bullet in Section 5.4) to make claims about the GCD/LCM.
· Be able to solve simple word problems like those in HW for these two concepts.
· Know that GCD(a,b) x LCM(a,b) = ab.
Sections 6.1-6.3/Dr. Lunt’s Materials
· Concentrate on being able to successfully do operations with fractions and exponents as per the HW; that is, at a higher level than using pictures and manipulatives.
· Be able to demonstrate by picture how a student would argue that 3/5 = 6/10, for example. Be able to order rational numbers smallest to largest using reasoning as per Dr. Lunt’s materials.
· Be able to explain through pictures fraction addition/subtraction and multiplication/division. For example, what does a common denominator mean in pictures or with manipulatives?
· Understand that a x b means a groups of size b (i.e., repeated addition). Be able to demonstrate in drawings how a product of fractions comes from this and that multiplication (of fractions or of a fraction and whole number, too) is commutative.
· Know and be able to demonstrate how common errors can be eliminated, such as canceling terms (i.e., why is (a+b)/b = a not true?)
· Understand mixed fractions and “improper” fractions (numerator exceeds denominator) and their operations as well. Be able to work with these at the level of demonstration to students and at a higher level as well.
· Understand division of fractions as well and how to explain them. Be able to write simple word problems for all operations including division.