Mathematics I—Semester A

Sample Lesson - Introduction

Lesson 3.17: Recursive Functions Activity

Don't get this stuff on your hands if you have a papercut. Sweet vinegar of Modena, that smarts!

(Source)

One of the beauties of the English language is that the language gives us many different ways to express the same idea. When Elizabeth Barrett Browning wrote How do I love thee? Let me count the ways," she was literally gearing up to list, like, seven different descriptions of her love for Hubby Bob.

Sonnets are great and all, but our favorite kind of re-phrasing takes a different literary form. We like to upcycle overused obscenities into family-friendly exclamations like "Banana shenanigans!" and "Sweet vinegar of Modena!"

We figure that if we're gonna get mad enough about something to start shouting, we might as well try to make the spectacle as funny as possible for any innocent bystanders who happen to be around. It's just our little way of trying to make the world a better place. (Hey, we can't all be poets.)

It just so happens that these lessons are all about expressing something in different ways and about cursing. So put on your thinking cap and get ready to come up with some silly swears.

What's that? The title of the activity says "recursive" not "cursing"? Aw, Fahrvergnügen.

Sample Lesson - Reading

Reading 3.3.17: Standing on the Shoulders of They Might Be Giants

This activity will have us applying what we've spent the past few lessons learning about, the different ways of expressing arithmetic and geometric sequences. If you want to review any of that material, we suggest you do it now. When the chips are down, there'll be no time for hesitation.

To refresh your memory on arithmetic sequences, click here.

To review the basics of geometric sequences, read this.

For an overview of defining functions and sequences recursively, revisit this reading.

This reading puts it all together: arithmetic and geometric sequences, in all their myriad forms. Math is truly a many-splendored thing.

Sample Lesson - Activity

Activity 3.17: The Amazing Recursive Race

In this activity, you'll pit your recursive-rewriting skills against the skills of a partner. Your partner doesn't have to be someone who's taking this course, although if you happen to know someone who will also be doing this activity this week, it would kind of make sense to team up. Otherwise, feel free to rope in a parent, a neighbor, the mailman, or whomever.

If your victim isn't familiar with sequences and recursive notation, guess what? Time to play teacher.You know what they say: "Those who can't do, teach." Wait, wait—wrong quote. The old adage that we meant to cite was "While we teach, we learn."

If all else fails, you can always haul out your old teddy bear, sit him across the table from you, and take turns playing both sides—just like how we used to play "Go Fish" before the stork finally brought us a little brother. Boy, those were lonely years.

Here's what you'll need:

• A partner
• Forty index cards (or quarter-sheets of paper) 
• Two pencils
• A piece of paper to use as a score sheet
• A computer with a scanner and Internet access

Part 1

Step 1: Each player gets ten index cards (or quarter-sheets of paper). Away from the prying eyes of the other player, each person makes up five arithmetic functions and writes each one on its own card. Each player also makes up five arithmetic sequences and writes each one on its own card (list at least four terms).

Step 2: On the same side of the card as the sequence/function, write your initials in one corner and the letter "A" for "arithmetic" in another corner. You can make it scarlet if you're feeling particularly puritanical today.

Step 3: Stack the cards that you've created face-down and trade stacks with your partner.

Step 4: At the same time as your partner, flip the first card over. Rewrite the card's arithmetic function or sequence so that it's defined recursively. When you finish the first card, move on to the second one. You can choose to discard a card and move on to the next one if you get stuck, but you can't come back and finish that card later. It's history.

Step 5: Time to settle the score. Trade stacks again and give your partner a star on each card for which s/he correctly defined the original equation or sequence recursively. Each player gets ten points for each correctly-rewritten card. The first person who got through his/her stack gets an additional 50 points. Use your score sheet to keep track of each player's total.

Step 6: Scan or take photographs of the cards (you can do a few at a time) and upload them below.

Part 2

Step 1: As before, each player gets ten index cards. This time, each person makes up five geometric functions and five geometric sequences. Don't forget to list at least four terms of your sequences.

Step 2: On the same side of the card as the sequence/function, write your initials in one corner and the letter "G" for "geometric" in another. Feel free to add drawings of your most ferocious game face on the backs of some of the cards to intimidate your opponent.

Step 3: Stack the cards that you've created face-down and trade stacks with your partner. Are you having déjà vu yet?

Step 4: At the same time as your partner, flip the first card over. Rewrite the card's geometric function or sequence so that it's defined recursively. When you finish the first card, move on to the second one. You can choose to discard a card and move on to the next one if you get stuck, but you can't come back and finish that card later. What do you mean, we let you do that yesterday? Nice try, bud. We're not that forgetful yet.

Step 5: Score the game as before—50 points to the person who finished their stack first, ten points for every card on which the geometric function or sequence was correctly defined recursively. Add each player's points to yesterday's points to come up with the grand totals for the activity.

Step 6: If you won, gloat. If you lost, try to act like you don't care.

Step 7: Scan or take photographs of the cards and upload them below.