## Saturday, September 05, 2009

### Original Thinking about Number Theory

Last week, I explained my rationale for splitting Pre-Algebra into three tracks (twenty minutes a day) and showed my plan for geometry and blogged our first week of geometry lessons. The second track for this year is number theory, a branch of pure--the opposite of practical and hands-on--mathematics. It includes topics like negative numbers, prime numbers, integers, whole numbers, rational numbers, etc. Number theory includes finding the greatest common factor and least common multiple.

Of the three tracks (geometry, algebra/arithmetic, and number theory), the latter worries me the most because number theory is the most abstract. Last June, after I split it all up into three tracks, I began experimenting with how to introduce negative numbers, which my plan covers in fourteen weeks. The most difficult aspect of using negative numbers is knowing how to add and subtract them. The concept of negative exists everywhere, even though we do not always use negative signs. I developed a series of games that addressed going from a number in one state of being, crossing 0, an going into another state of being. Rather than getting wrapped up in abstract terms like positive and negative, I found concrete situations to which Pamela can relate:
• Crossing the year 0 to study either the years B.C. or A.D.
• Crossing 0 degrees Celsius to temperatures above or below zero.
• Crossing the equator to go either North or South.
• Crossing 0 cents to have money or go into debt.
• Crossing 0 points in a card game to have points on the table or in the hole.
What makes the games fun is how we use context and imagination. We used to live in Minnesota and know full well what going above and below freezing temperatures feels like. So, when we play Minnesota Winter, we get upset at dropping temperatures and cheer when the thermometer rises. Pamela's favorite game is Time Travelers because we get to visit certain people (Mozart if we land on 1700 A.D. or Jesus if we arrive in 0). The most exciting moment for us is hitting 1900 A.D. because we talk about electricity, televisions, and computers but know we can't have HDTV until we hit 2000 A.D.

Time Traveler - Pamela and I both play this game, which starts at 0. We pick a card: red means to go backwards in time, while black means to go forward. The number of centuries jumped is the value of the card. If the value takes us out of range of the board (past the lowest year B.C. or highest year A.D.), we reject the card and keep picking until a value works. Pamela keeps track of her years on paper to help her see the pattern. When we first started playing the game, we counted each and every move. Now, she can do all aspects of movement in her head with few mistakes!

Minnesota Winter - This game does not have players because our imagination makes it fun. Pamela rolls the dice and keeps score. We place the red paper clip at 5 below to allow more opportunities to straddle 0. If she rolls an even number, the temperature increases by that amount. If odd, it decreases by that amount. She keeps track of the temperatures on her sheet, while I move the paper clip up or down. Over many rolls, the temperature will rise and once it hits the highest temperature on the paper, the game ends.

Going to Kenya - I already have a link for the map game, and the video shows Pamela and I playing it last week.

Cards - I do not have video of this game because it is not all that exciting. Pamela picks a card. If it's red, she loses pennies; if it's black, she gains pennies. The number of pennies are the value of the card. "On the table" is for a positive number of pennies left, while "in the hole" represents how many pennies she owes. She keeps track of the pennies on a sheet like the ones above.

What are the results of playing these games? Pamela understands the following concrete concepts about the idea of using negative numbers:
• If you go forward and are in positive territory, you add the numbers and stay in positive territory: 500 A.D. + 300 years = 800 A.D.
• If you go backward and are in negative territory, you add the numbers and stay in negative territory: 300 B.C. - 1000 years = 1300 B.C.
• If you go backward and are in positive territory, you subtract the numbers and stay in positive territory only if the number going backward is less than the starting number: 800 A.D. - 400 years = 400 A.D.
• If you go forward and are in negative territory, you subtract the numbers and stay in negative territory only if the number going forward is less than the starting number: 800 B.C. - 400 years = 400 B.C.
• If you go backward and are in positive territory, you subtract the numbers and end up in negative territory only if the number going backward is greater than the starting number: 500 A.D. - 900 years = 400 B.C.
• If you go forward and are in negative territory, you subtract the numbers and end up in negative territory only if the number going forward is greater than the starting number: 800 B.C. - 1200 years = 400 A.D.

Number Theory Track
I picked out Math-U-See lessons that fit into the category of number theory and spreaded them out over thirty-six weeks.

Week 1 - Practice crossing back and forth zero in games (money, map, time, temperatures, cards, etc.
Week 2 - Practice crossing back and forth zero in games (money, map, time, temperatures, cards, etc.
Week 3 - Transition to doing number line problems and introduce the concept of negative and positive.
Week 4 - Transition to word problems involving situations encountered in the games plus the number line to focus on the idea of negative versus positive.
Week 5 - Do selected problems for adding negative numbers from Lesson 1 of MUS Pre-Algebra.
Week 6 - Add a twist to the game where you can draw a card that will subtract some or all negative points before you make a move. Transition to doing subtracting negative numbers on a number line.
Week 7 - Transition from subtracting negative numbers on a number line to word problems.
Week 8 - Do selected problems for subtracting negative numbers from Lesson 2 of MUS Pre-Algebra.
Week 9 - Demonstrate multiplying positives and negatives through the concept of adding or subtracting multiple times.
Week 10 - Transition to multiplying negative numbers on a number line to word problems.
Week 11 - Do selected problems for multiplying negative numbers from Lesson 3 of MUS Pre-Algebra.
Week 12 - Demonstrate dividing positives and negatives through splitting up a negative or positive number into groups.
Week 13 - Transition to dividing negative numbers on a number line to word problems.
Week 14 - Do selected problems for dividing negative numbers from Lesson 4 of MUS Pre-Algebra.
Week 15 - Demonstrate additive inverse with concrete objects and a balance and Math-U-See blocks.
Week 16 - Transition to additive inverse in stick diagrams and word problems.
Week 17 - Do selected problems for additive inverse from Lesson 9 of MUS Pre-Algebra.
Week 18 - Demonstrate commutative and associative properties with concrete objects and a balance and Math-U-See blocks.
Week 19 - Transition to commutative and associative properties in stick diagrams and word problems.
Week 20 - Do selected problems for commutative and associative properties from Lesson 11 of MUS Pre-Algebra.
Week 21 - Demonstrate multiplicative inverse with concrete objects and a balance and Math-U-See blocks.
Week 22 - Transition to multiplicative inverse in stick diagrams and word problems.
Week 23 - Do selected problems for multiplicative inverse from Lesson 13 of MUS Pre-Algebra.
Week 24 - Demonstrate distributive properties with concrete objects and Math-U-See blocks.
Week 25 - Transition to distributive properties in stick diagrams and word problems.
Week 26 - Do selected problems for distributive properties from Lesson 12 of MUS Pre-Algebra.
Week 27 - Set up a chart of prime numbers and divisible numbers and see if Pamela can discover the pattern. Work past the chart to determine if other numbers are prime. Explore multiples with Math-U-See blocks.
Week 28 - Transition to factorization and visual ways of doing that. Explore common multiples with Math-U-See blocks.
Week 29 - Explore least common multiples with Math-U-See blocks. Factor the multiples to see if there is a pattern with primes.
Week 30 - Transition to pictures and fractions problems.
Week 31 - Do selected problems for prime numbers and least common multiples from Lesson 21 of MUS Pre-Algebra.
Week 32 - Explore common factors with Math-U-See blocks. Transiton to exploring the greatest common factor with the blocks.
Week 33 - Transition to pictures and fraction problems.
Week 34 - Do selected problems for greatest common factor from Lesson 22 of MUS Pre-Algebra.
Week 35 - Describe kinds of numbers and group them (counting numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers). Develop a Venn diagram and identification ideas.
Week 36 - Do selected problems for irrational and real numbers from Lesson 30 of MUS Pre-Algebra.

walking said...

On Sunday morning, way outside of "school hours", Pamela was thinking about our games and making connections of her own. She said that even dice is like the black cards (moving forward) and odd is like the red cards (moving backward.

To give her a little preview for next week when I introduce negative versus positive numbers, I said, "There is a special word for that and I am going to tell you what it is in math next week."

Isn't that cool? By playing concrete games to spotlight an abstract idea, Pamela already sees the concept of negative versus positive without knowing these abstract words.

JEMD1966 said...

Hello, Don Delzer here, Jennifer's husband. I like your examples very much, how to teach this abstract idea using these real examples. I had to think for a moment about the dice game. Since the even numbers add to 12, and the odd numbers add to 9, your game would trend in that upward direction as it does.

There is a problem with the calendar game, and that is, AD 1 is preceeded by 1 BC. (http://en.wikipedia.org/wiki/Anno_Domini) There is no year zero. So it doesn't follow the number-theroy rules. You will get a one-year error crossing the boundary. Fixing the error is easy with a one-year offset, but it does make the example complicated. I understand Pamela is great with calendars, so she may know this already, or may be very interested in the problem if she didn't know it.

Stephanie said...

This is fabulous!

walking said...

Don, we only use one dice, so it would trend upwards, giving us the hope that spring will come and providing for an endpoint to the game.

Thanks for teaching me something new!!! I'm going to look ahead to Augustus Caesar's World and see if they address it.

I think I will keep the game as is because the true goal is to teach mathematics and not history, which is icing on the cake . . .

Donna said...

Wow, Tammy. I wish I had your brain. This is something I have struggled to explain!

walking said...

Nature of Reading, I think that is the problem. We explain too much. What the games did for Pamela was to let her explore the idea of negative. She began seeing patterns and making her own connections so that, when I finally introduce the term negative, she will have a known to which she can link this unknown . . .