Math Prompt: True-False-True

One of the stretch goals for my Charlotte Mason’s Living Math Kickstarter campaign is to add a math journaling prompt to the end of each chapter. So, I’ve been playing around with ideas to get readers writing.

Since the book’s all about how to build mathematical reasoning, I’m looking for ways to prompt creative thinking and flexibility in math calculations.

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I found some fun ideas in Guy Gattegno and Martin Hoffman’s Handbook of Activities for the Teaching of Mathematics (which you can download here), including the following riff off a puzzle created by Lewis Carroll.

Lewis Carroll’s Doublets

During the holiday season of 1877, Charles Dodgson invited his child friends to play a game he called “Word-Links,” later published under his pen name Lewis Carroll in the journal Vanity Fair.

Here’s how he described the rules:

Two words are proposed, of the same length; and the Puzzle consists in linking these together by interposing other words, each of which shall differ from the next word in one letter only.

That is to say, one letter may be changed in one of the given words, then one letter in the word so obtained, and so on, till we arrive at the other given word. The letters must not be interchanged among themselves, but each must keep to its own place.

As an example, the word “head” may be changed into “tail” by interposing the words “heal, teal, tell, tall.” I call the two given words “a Doublet,” the interposed words “Links,” and the entire series “a Chain,” of which I here append an example:—

H E A D
h e a l
t e a l
t e l l
t a l l
T A I L

It is, perhaps, needless to state that it is de rigueur that the links should be English words, such as might be used in good society.

For more puzzles, check out the Lewis Carroll Resources page on Doublets.

True-False-True

You can challenge your children to a similar puzzle with math equations. But we must change the rules slightly:

Transform the given equation into the target by making only one change at a time. The lines of your solution must alternate: true, false, true, etc.

Allowed changes include replacing one number or symbol with a different one, or adding/removing a math operation (symbol + number).

This puzzle pushes students to consider the structure of mathematical expressions. Make it a game by letting your children challenge you, too.

For example, change:

2 + 2 = 4

into…

10 – 2 = 8.

There are many possible solutions. Here’s how I did it:

2 + 2 = 4 (T)

10 + 2 = 4 (F)

10 + 2 = 12 (T)

10 − 2 = 12 (F)

10 − 2 = 8 (T)

Or for a more difficult example, change:

6 ÷ 2 = 3

into…

24 × 6 = 24 × 2 + 24 × 3.

There are many possible solutions. Here’s my method:

6 ÷ 2 = 3 (T)

6 × 2 = 3 (F)

6 × 2 = 12 (T)

24 × 2 = 12 (F)

24 × 2 = 48 (T)

24 × 6 = 48 (F)

24 × 6 = 144 (T)

24 × 6 = 144 × 2 (F)

24 × 6 = 72 × 2 (T)

24 × 6 = 72 × 2 + 24 (F)

24 × 6 = 60 × 2 + 24 (T)

24 × 6 = 60 × 2 + 24 × 3 (F)

24 × 6 = 24 × 2 + 24 × 3 (T)

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