# Engaging students: Recognizing equivalent statements

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Tiffany Jones. Her topic, from Geometry: recognizing equivalent statements.

How could you as a teacher create an activity or project that involves your topic?

How has this topic appeared in pop culture?

The topic of recognizing equivalent statements appears in pop culture in Lewis Carroll’s “Alice’s Adventures in Wonderland”.  English mathematician Reverend Charles Lutwidge Dodgson published the work in 1865 under the pseudonym Lewis Carroll.

Reverend Charles Lutwidge Dodgson was a mathematician, so it is not surprising that mathematical concepts appear in his writing. During the tea party scene, examples of logic statements are present. More specifically, how statements and their converse do not have the same meaning or truth-value. The March Hare asks Alice to say what she means, to which she responds with that she means what she says, thinking that to be the same thing. The Hatter disagrees and the conversation continues with three examples to show Alice that the statements are not the same. Here is the text:

Here is the excerpt from the text from chapter seven:

‘Do you mean that you think you can find out the answer to it?’ said the March Hare.

‘Exactly so,’ said Alice.

‘Then you should say what you mean,’ the March Hare went on.

‘I do,’ Alice hastily replied; ‘at least—at least I mean what I say—that’s the same thing, you know.

”Not the same thing a bit!’ said the Hatter. ‘You might just as well say that “I see what I eat” is the same thing as “I eat what I see”!

”You might just as well say,’ added the March Hare, ‘that “I like what I get” is the same thing as “I get what I like”!

”You might just as well say,’ added the Dormouse, who seemed to be talking in his sleep, ‘that “I breathe when I sleep” is the same thing as “I sleep when I breathe”!

”It IS the same thing with you,’ said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing–desks, which wasn’t much.

The lesson on logical statements and truth-values would start with a reading of this section of text or viewing of a clip that keeps the original text.  Take a simple statement and write its converse, inverse, and contrapositive. For example, “I like what I get” becomes “I get what I like”, “I do not like what I do not get”, and “I do not get what I do not like”, respectively.  Discuss the truth-values of each of the statements show that the original and contrapositive are equivalent and that the converse and inverse are equivalent, to help the students see patterns when rewriting a statement.

Then the students will complete a worksheet by MathBits.com to ensure that they understand the process with simple English sentences and to introduce them to the idea with simple mathematical statements. The worksheet includes a portion of the text above for the students’ reference. The worksheet has the students take two simple sentence and write them in the form of “if…, then..”, then the students are to  write their  converse, inverse, and contrapositive. Next, the students compare the truth-values of each statement. Finally, the students are given two mathematical statements and are asked to determine the truth-values.

How can this topic be used in your students’ future courses in mathematics or science?

Recognizing equivalent statements appears in analysis courses and courses which proofs are often used. Being able to recognize equivalent statements adds another tool to the the tool box of proof writing.  With Proof writing, sometimes the contrapositive form of the statement is easier to prove than the original statement itself.

For example, in Math 4050 Advanced Study of Secondary Mathematics Curriculum, the proof of the following theorem is easier to prove by contrapositive than just straight on.

If a prime p divides m*n with m and n composite, then p divides m or        p divides n.

The contrapositive, if p does not divide m and p does not divide n with p, m, and n the same as before, then p does not divide m*n, follows easily (with a little clairvoyance) from another theorem for the class.

Carroll, Lewis. Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/&gt;. October 1, 2015.

Carroll, Lewis. “Chapter VII: A Mad Tea-Party.” Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/17/chapter-vii-a-mad-tea-party/&gt;. October 1, 2015.

Roberts, Frederick, and MathBits.com. Alice in Wonderland Worksheet. S.l.: Commission of the European Communities, 1993. Mathbits. Commission of the European Communities. Web. 1 Oct. 2015

University of North Texas course math 4950 Advanced Study of Secondary Mathematics Curriculum lecture Fall 2015 taught Dr. John Quintanilla

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