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A Competitive Word Problem

Lesson Plans > Mathematics > Competition

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Let a, b, c,...x, y, z be variables such that:

• Every consonant that is preceded by a vowel is two less than twice the value of that vowel.
• Every consonant that is preceded by another consonant is twice the value of that consonant.
• Every vowel is one sixteenth the value of the consonant preceding it.

Write z in terms of a.

b = 2a - 2
c = 2(2a - 2) = 4a - 4
d = 2(4a - 4) = 8a - 8
e =
1
2
a -
1
2

f = 2(
1
2
a -
1
2
) - 2 = a - 3

A Competitive Word Problem

Most days I start my class with a word problem, which I put up on the big screen for students to read. These are often problems that test their problem solving skills, sometimes by utilizing material they are currently studying, and sometimes by remembering things they've learned in previous math courses. I offer bonus points to anyone who can solve it. My students treat it as a sort of competition to see who can get it first, and sometimes I increase the competition level by saying I'll only give bonus to the first student who solves the problem. One of the hard things about this is that it's usually the same two or three students that get the points, and I hate to have everyone else get discouraged!

The problem I did with my students yesterday was a bit different. In order to solve it, no one needed any flashes of insight - they simply had to be able to carry out a repetitive series of algebraic manipulations. The problem is shown in the accompanying slide.

In order to make sure that no one was "left behind" in understanding how to solve the problem, I worked through the first six iterations on the board (the second slide in the slide show). This helped students understand the algebraic process of substitution they were doing.

I told the students that they could have me check their work when they got to the letter "m" and again when they got to the letter "t" - this allowed them an opportunity to fix mistakes. Once they got to "u", however, they were only allowed one final answer, so there were no "second chances."

I liked this problem (even though it ended up taking more class time than I usually allow for the kick-off problem) because it allowed different students to shine; some of my students who rarely get the flash of insight that lets them solve the problems quickly were able to work carefully through the process and get the right answer.

Lesson by Mr. Twitchell

All the Letters

Let a, b, c,...x, y, z be variables such that:

• Every consonant that is preceded by a vowel is two less than twice the value of that vowel.
• Every consonant that is preceded by another consonant is twice the value of that consonant.
• Every vowel is one sixteenth the value of the consonant preceding it.

Write z in terms of a.

Solutions

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