Prealgebra for Two-Year Colleges/Workbook AIE/Problem solving and explaining your reasoning

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Problem solving and explaining your reasoning

The Handshake Problem

Materials Needed: None

Why[edit]

In addition to reviewing procedures for calculation in this class, you will improve your transferable skills of problem solving and communication.

Learning Objectives[edit]

  1. Develop a problem-solving method.
  2. Clearly communicate your problem-solving thought process

Warm-up[edit]

Below are an example problem and two example solutions. Which solution does a better job of communication the problem solver’s thought process? Give three specific reasons for your answer.

Problem:

Oscar collects baseball cards. He has 109 cards for the New York Yankees, 96 for the Chicago White Sox, 79 for the Kansas City Royals, 42 for the Seattle Mariners, 67 for the Oakland Athletics, and 52 for the California Angels. How many cards does he have in total?

Solution #1:

The key word here is “total.” To find the total number of Alan’s baseball cards, we can find the sum of the quantities for each team.

\begin{matrix}
 & & & \color{Red}{33}\ \\
 & \mathrm{Yankees} & \to & 109 \\
 & \mathrm{White sox}  & \to &\ 96 \\
 & \mathrm{Royals}  & \to &\ 79 \\
 & \mathrm{Mariners}  & \to &\ 42 \\
 & \mathrm{Athletics}  & \to &\ 67 \\
+ & \mathrm{Angels} & \to &\ 52 \\
\hline
 & \mathrm{total} & & 445 \end{matrix}

Alan has a total of 445 baseball cards.

Solution #2:


\begin{matrix}
 & 109 \\
 &\ \ 96 \\
 &\ \ 79 \\
 &\ \ 42 \\
 &\ \ 67 \\
+ &\ \ 52 \\
\hline
 & 445 \end{matrix}

Solution #1 is better because is shows steps, e.g. (1) the solution explains why addition was chosen, (2) it shows where each number comes from, (3) it shows the carrying, and (4) it states the answer in context with units.

Activity[edit]

  • Solve the problem below. Clearly communicate your problem-solving thought process. You may work with others in your group of three, but you need to write up your own solutions on your own paper.

Forty-five people attend a party. If every person shook the hand of everyone else just once, how many handshakes were there altogether?

There were 990 handshakes at the party.

  • List the steps of a general problem-solving method, and fill in some details underneath each step.

a) Understand: Identify given information, identify unknown, list assumptions, Try a simpler problem. Draw a diagram or narrate a story.

b) Choose a strategy and do it: Use the perspective of tables (including table of simpler problems), diagrams, or verbal easoning, or formulas. Change perspectives as appropriate.

c) Check & Interpret: Compare two or more methods. State answer in context, with assumptions and units.