As a informal diagnostic, and a way of mixing up the seats, I have students draw a problem from a cube. They find a desk with the solution to their problem. (Worked well for when I didn't have a roster for the class.)

I kept the problems simple--most could be answered from the multiplication chart on the wall. I also created problems in pairs, so that students could choose between two desks with their answer.

This is the day that I traditionally do the "Missionaries and Cannibals" problem. I present the problem, break the students into small groups (a couple of pairs of table partners in each group, max) and about halfway through, pass out some colored chips for them to use as symbolic people to (hopefully) make their thinking a little bit easier. We talk about it, and only very few, if any, figure it out during class. Thus, it effectively runs up to the end of class -- and I leave them hanging on until the following class. A few more will figure it out overnight (one of my 8th graders last year went home and played with his peas and carrots at dinner, using them as the surrogate missionaries and cannibals).

Middle school/pre-algebra

M&M Count (first week)

I give a small bag (1.69 oz) to each member. Before opening students determine how many of each color are in the bag. Record. Open the bag and get the actual count. Record. Make a conjecture about all 1.69 oz. bags. Then I pull out the extra-large size bag from Costco. Using the preliminary information, they mathematically estimate how many are in the bag, and estimate the number of each color. (The count of M&Ms are pretty accurate, however, the color ratios are much different in the big bag.) We analyze our predictions and speculate as to why the color ratios are different. Students created pie charts with data are hung on the bulletin board. (Day 1 and Day 2)

Day 3. Start with 4 M&Ms. Toss. For each candy that shows an M, add another candy. Do this 6 or 7 times. Record results. Graph. Is it linear? Based on the graph, how many M&Ms are there after 10 flips? 15 flips? etc. (Obviously an exponential graph.) Now start all the M&Ms, eat all M&Ms that show the M. Record results. Do this until there are no candies left. Graph results. (Hint: give out baggies to keep the candy in, some students don't like chocolate!) Graphs are hung on the bulletin board.

Day 4. Discuss math behind exponential graphs, possibly introduce equation. Talk about the famous allowance problem. (For one month, would you want $2 a day, or one penny the first day, doubling each day after.)

Day 5. Class rules? My number one class rule is : Don't divide by zero. (I tell the students I assume they will follow the school rules that are conveniently posted in their planners.)

I start with the Devil and the bridge problem. (You may want to change the devil into a genie.) Students spend some time solving alone, and then in groups. Ask for problem solving strategies. Then I change the parameters of the problem in several ways. (The number of crossings, the amount the devil takes, tripling the amount.) We come back to this problem several times during the year. After equation solving, we solve the problem with algebra. Later, we will graph the amount of money in the pocket after each crossing. And finally, student rewrite the problem after we talk about SCAMPER.

You buy some styrofoam cups. Lots of 'em. After you have the class organized so you and they are comfortable (after seating arrangements but before any syllabus discussion) you get 'em in groups with those around them.

You say, how many stacked cups would it take to reach the top of my head? You hold one up.

You take bets from the groups. Betting is fun. You write down the bets. You make sport of the groups who wager only one above the previous group's wager.

You say, alright, we're gonna figure it out now and if anyone gets close to the answer, we'll cancel homework for the first night. Of course, you weren't planning any homework anyway, but they go nuts.

(Depending on the age, there's also a great discussion to be had here about how "close" is close enough. 5% error? 10% error? What does x% error even mean?)

You pass out a ruler and three cups to each group and you facilitate. You wander around. Ask them how they'll do it.

They'll ask you how tall you are. (Big helper: use centimeters.)

Many will find the height of the cup and then divide it into your height. Have them stack that many cups and watch as it doesn't come close. Have them discuss why.

The question to ask is: if you add one cup to the stack what happens to the height of the stack.

You should hover your group interaction around the idea of slope and y-intercept. The slope here is the lip of the cup: how much the height increases every time you add a cup. The y-intercept is everything that isn't the lip (the base). Your equation is:

height of stack of cups = number of cups * lip height + base height

You don't need to take them into detail on the equation. This project has been done — first day! — with younger crowds.

At the end, you actually stack the cups high and see who was closest. Maybe pass out candy. Cancel the imaginary homework assignment. Maybe hold up a different brand of cup, one with a thinner lip and ask what would happen.

So you've collaborated, done some project-based learning, tackled a challenging problem together, joked around, become acquainted with some students. The syllabus, the rules, the standards, you can always go over those another day.

## Seating

I kept the problems simple--most could be answered from the multiplication chart on the wall. I also created problems in pairs, so that students could choose between two desks with their answer.

## Middle School/Pre-Algebra

This is the day that I traditionally do the "Missionaries and Cannibals" problem. I present the problem, break the students into small groups (a couple of pairs of table partners in each group, max) and about halfway through, pass out some colored chips for them to use as symbolic people to (hopefully) make their thinking a little bit easier. We talk about it, and only very few, if any, figure it out during class. Thus, it effectively runs up to the end of class -- and I leave them hanging on until the following class. A few more will figure it out overnight (one of my 8th graders last year went home and played with his peas and carrots at dinner, using them as the surrogate missionaries and cannibals).

Middle school/pre-algebra

I give a small bag (1.69 oz) to each member. Before opening students determine how many of each color are in the bag. Record. Open the bag and get the actual count. Record. Make a conjecture about all 1.69 oz. bags. Then I pull out the extra-large size bag from Costco. Using the preliminary information, they mathematically estimate how many are in the bag, and estimate the number of each color. (The count of M&Ms are pretty accurate, however, the color ratios are much different in the big bag.) We analyze our predictions and speculate as to why the color ratios are different. Students created pie charts with data are hung on the bulletin board. (Day 1 and Day 2)M&M Count (first week)Day 3. Start with 4 M&Ms. Toss. For each candy that shows an M, add another candy. Do this 6 or 7 times. Record results. Graph. Is it linear? Based on the graph, how many M&Ms are there after 10 flips? 15 flips? etc. (Obviously an exponential graph.) Now start all the M&Ms, eat all M&Ms that show the M. Record results. Do this until there are no candies left. Graph results. (Hint: give out baggies to keep the candy in, some students don't like chocolate!) Graphs are hung on the bulletin board.

Day 4. Discuss math behind exponential graphs, possibly introduce equation. Talk about the famous allowance problem. (For one month, would you want $2 a day, or one penny the first day, doubling each day after.)

Day 5. Class rules? My number one class rule is : Don't divide by zero. (I tell the students I assume they will follow the school rules that are conveniently posted in their planners.)

Algebra IFree Math Videos Online - An excellent collection of animated math videos.

I start with the Devil and the bridge problem. (You may want to change the devil into a genie.) Students spend some time solving alone, and then in groups. Ask for problem solving strategies. Then I change the parameters of the problem in several ways. (The number of crossings, the amount the devil takes, tripling the amount.) We come back to this problem several times during the year. After equation solving, we solve the problem with algebra. Later, we will graph the amount of money in the pocket after each crossing. And finally, student rewrite the problem after we talk about SCAMPER.

Surrogacy law TorontoAlgebra IIGeometryYou buy some styrofoam cups. Lots of 'em. After you have the class organized so you and they are comfortable (after seating arrangements but

beforeany syllabus discussion) you get 'em in groups with those around them.You say,

how many stacked cups would it take to reach the top of my head?You hold one up.You take bets from the groups. Betting is fun. You write down the bets. You make sport of the groups who wager only one above the previous group's wager.

You say, alright, we're gonna figure it out now and if anyone gets close to the answer, we'll cancel homework for the first night. Of course, you weren't planning any homework anyway, but they go nuts.

(Depending on the age, there's also a great discussion to be had here about how "close" is close enough. 5% error? 10% error? What does x% error even

mean?)You pass out a ruler and three cups to each group and you facilitate. You wander around. Ask them how they'll do it.

They'll ask you how tall you are. (Big helper: use centimeters.)

Many will find the height of the cup and then divide it into your height. Have them stack that many cups and watch as it doesn't come close. Have them discuss why.

The question to ask is: if you add one cup to the stack what happens to the height of the stack.

You should hover your group interaction around the idea of slope and y-intercept. The slope here is the lip of the cup: how much the height increases every time you add a cup. The y-intercept is everything that isn't the lip (the base). Your equation is:

height of stack of cups = number of cups * lip height + base heightYou don't need to take them into detail on the equation. This project has been done — first day! — with younger crowds.

At the end, you actually stack the cups high and see who was closest. Maybe pass out candy. Cancel the imaginary homework assignment. Maybe hold up a different brand of cup, one with a thinner lip and ask what would happen.

So you've collaborated, done some project-based learning, tackled a challenging problem together, joked around, become acquainted with some students. The syllabus, the rules, the standards, you can always go over those another day.

PrecalculusCalculus