Activity: Even + Even + Odd =? USE THEORETICAL PROBABILITIES AND EXPERIMENTAL RESULTS TO MAKE PREDICTION & DECISIONS FIND THE PROBABILITIES OF DEPENDENT AND INDEPENDENT EVENTS VALIDATE CONCLUSIONS USING MATHEMATICAL PROPERTIES AND RELATIONSHIPS Prerequisites some experience with probability Preparation Color Tiles or Substitutes 2 Dice for each pair Overhead #1 and #2 Worksheet #1 for each student Have students get in pairs. I want you to represent the numbers 1 through 10 using color tiles in the following way. I drew the following on the board: 1 2 3 4 5 6 7 8 9 10 Can anyone see a pattern to what I am doing? I called on Kevin, You are building up in rows of 2. You just keeping adding the next block on top, offered Courtney. I m going to pass out bags of colors tiles. I want you to work with you partner to build what you think 6 would look like. I circulated through out the room to make sure everyone understood the pattern and had created a rectangle that is 2 color tiles wide and 3 color tiles tall. I d like you to build the numbers up to and including ten. I m going to pass out a sheet for you to draw what your answers look like. I passed out Worksheet #1 and observed as pairs completed the task. I selected students who had finished early draw the next figures on the overhead. After everyone had time to finish, I called the class to attention. What do you notice about the figures? They look like skyscrapers that just keep getting taller and taller, Art suggested. Yeah, all the even numbers are like buildings and the odd numbers are like buildings with chimneys, Alex added. What do you think 23 would look like? I asked. Lequita raised her hand, Like two long columns with an extra one on top it would be 11 tiles tall and 2 tiles wide plus an extra on top. If I took an odd number and added another odd number to it, do you think the answer would be even or odd? I wrote odd + odd =? on the board. Discuss with your partner and raise your hand when you think you have the answer. The answer will be even, said Katherine. You turn the second odd number upside down and they fit together to make an even number. Katherine came to the board and drew : odd odd : www.mathlessonbank.com pg. 1 of 7
ACTIVITY: EVEN + EVEN + ODD =? LESSON PAGES In part 2, solve the following problems. Then choose one to explain using complete sentences. I wrote on the board: Even + Even = Odd + Even = Odd + Odd + Odd = *Bonus: Odd + Odd + Odd + Even + Even = I gave students ample time to answer part 2 and write their explanation. As a class we discussed the solution to the bonus problem. Many students had drawn pictures to show that the answer would be odd. I presented an additional way to think about the problem. I wrote Odd + Odd + Odd + Even + Even = on the board. I then covered up the last part so that only Odd + Odd + Odd was showing. We ve already solved this problem, haven t we? Colby answered that odd + odd + odd was odd. I recorded that on the board like this: Odd + Odd + Odd + Even + Even = I then covered up the beginning with my arm and only left Even + Even showing. And we already did this problem too, right? Courtney said that even + even was even. I recorded that as well. Odd Even Odd + Odd + Odd + Even + Even = So the problem is just odd plus even, which we have also figured out is odd. This is just another way to think about the problem. Now I have a multiplication question for you. I wrote the following on the board 2 x Odd = I called on Jake, That s the same as odd + odd which is even. I recorded: 2 x Odd same as Odd + Odd = Even What about 3 x odd? I asked. Maria answered that the solution was odd. I asked her to explain. The first two fit together to make an even number and the last one still has an extra square on top. So that means the answer is odd. Once again 3 x odd is the same thing as odd + odd + odd. I had students work through Part 3 of Worksheet #1. Next I displayed Overhead #1 and read the rules for Dice Odds aloud. After answering questions about directions, I had students get in pairs, passed out dice, and let them get to work. Some pairs instantly had the feeling from everything we had already been discussing that Version 1 was unfair and Version 2 was fair. Here s some of the logic that was used: For Version 2, the possibilities were even + even = even odd + odd = even even + odd = odd odd + even = odd : www.mathlessonbank.com pg. 2 of 7
So each player has an equal chance of winning and the game is fair. Some pairs drew pictures to illustrate the point, some described the odds in words, while others wrote out all the possible combinations. Version 1 was a bit more challenging for students. The possibilities turn out like this: even x even = even even x odd = even odd x even = even odd x odd = odd It is 3 times more likely for player A to score a point than player B. To correct the unfairness of the game, Player B must earn 3 points each time the dice produces an odd product. Of course, this is all theoretical probability. What happens when you play the game can vary quite a bit. This was an important topic of our class discussion after pairs were finished playing both versions of the game. ACTIVITY: EVEN + EVEN + ODD =? LESSON PAGES Sometimes I like to extend student s learning with another game shown on Overhead #2. Though it is not a game of probability, it does require logical thinking and goes along with the even & odd theme for the lesson. Students seem to really get into this game, and I like to see how they go about creating strategies for winning. : www.mathlessonbank.com pg. 3 of 7
Name: Date: Odds & Evens Part 1: Record your answers here. 1 2 3 4 5 6 7 8 9 10 ACTIVITY: EVEN + EVEN + ODD =? WORKSHEET #1 Part 2: 1.) 2.) 3.) 4.) Bonus: Choose one of the equations in questions #1-#4 and explain your answer using complete sentences: pg. 4 of 7
ACTIVITY: EVEN + EVEN + ODD =? WORKSHEET #1 Part 3: Tell whether each answer would be even or odd. 2 x Odd = 3 x Odd = 4 x Odd = 5 x Odd = 6 x Odd = 7 x Odd = 8 x Odd = 9 x Odd = 2 x Even = 3 x Even = 4 x Even = 5 x Even = 6 x Even = 7 x Even = 8 x Even = 9 x Even = What patterns do you notice? Is that what you expected? Why or why not? What patterns do you notice? Is that what you expected? Why or why not? pg. 5 of 7
Game: Dice Odds Version 1: Two players take turns rolling 2 dice and multiplying the numbers shown. If the answer is even, then Player A earns a point. If the answer is odd, then Player B earns a point. The game ends after 25 rolls. ACTIVITY: EVEN + EVEN + ODD =? OVERHEAD #1 Version 2: Two players take turns rolling 2 dice and adding the numbers shown. If the answer is even, then Player A earns a point. If the answer is odd, then Player B earns a point. The game ends after 25 rolls. for each version of the game answer: 1. Is the game fair? How do you know? 2. If the game is not fair, reassign the points so the game is fair. 3. Play each version of Dice Odds with a partner using the point system you decided on and record the results. 4. Are the actual results what you expected? pg. 6 of 7
Game: 13 Tiles Directions: Begin with 13 color tiles. Two players take turns removing 1, 2 or 3 tiles from the pile. The player who finishes with an odd number of tiles wins. (Remember on each turn a player must remove 1, 2 or 3 tiles you cannot skip your turn). ACTIVITY: EVEN + EVEN + ODD =? OVERHEAD #2 1. Can you come up with a winning strategy? 2. Does it matter who goes first? 3. How would the game change if there were 14 tiles? 15 tiles? pg. 7 of 7