A Problem Solving Approach to Mathematics for Elementary School Teachers (12th Edition)

12th Edition

ISBN: 9780321987297

Author: Rick Billstein, Shlomo Libeskind, Johnny Lott

Publisher: PEARSON

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Textbook Question

Chapter 8.2, Problem 1MC

Student were asked to write an algebraic expression for the sum of five consecutive natural numbers .One student wrote x + ( x + 1 ) + ( x + 2 ) + ( x + 3 ) + ( x + 4 ) = 5 x + 10 . Another wrote ( x − 2 ) + ( x − 1 ) + x + ( x + 1 ) + ( x + 2 ) = 5 x .

a. Explain who is correct and why.

b. Does either approach prove that the sum of five consecutive integers equals 5 times the middle term?

c. Does the statement in part (b) generalize to the sum of consecutive term in every arithmetic sequence? Justify answer.

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Chapter 8.1B, Problem 20A

Chapter 8.2, Problem 2MC

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T2.3: Prove that there exists a connected graph with degrees d₁ ≥ d₂ >> dn if and only if d1, d2,..., dn is graphic, d ≥ 1 and di≥2n2. That is, some graph having degree sequence with these conditions is connected. Hint - Do not attempt to directly prove this using Erdos-Gallai conditions. Instead work with a realization and show that 2-switches can be used to make a connected graph with the same degree sequence. Facts that can be useful: a component (i.e., connected) with n₁ vertices and at least n₁ edges has a cycle. Note also that a 2-switch using edges from different components of a forest will not necessarily reduce the number of components. Make sure that you justify that your proof has a 2-switch that does decrease the number of components.

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Chapter 8 Solutions

A Problem Solving Approach to Mathematics for Elementary School Teachers (12th Edition)

Ch. 8.1 - Use the fact that each prime in the prime...Ch. 8.1 - Find the value of 3 on a calculator. Explain why...Ch. 8.1 - Is it true that a+b=a+b for all a and b? If not,...Ch. 8.1 - The number 227 can be written as an infinite...Ch. 8.1 - a. Use the fact that if a prime p appears in the...Ch. 8.1 - Without using a calculator, order the following....Ch. 8.1 - a. To prove that 2335 a student wrote the...Ch. 8.1 - The sequence 1,1.01,1.001,1.0001,... is an...Ch. 8.1 - Let each number of a group choose a number between...Ch. 8.1 - Prob. 11MC

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