Probability+and+Statistics

3.13 Probability and statistics. //The student solves problems by collecting, organizing, displaying, and interpreting sets of data.// 3.13A Collect, organize, record, and display data in pictographs and bar graphs where each picture or cell might represent more than one piece of data 3.13B Interpret information from pictographs and bar graphs 3.13C Use data to describe events as more likely than, less likely than, or equally likely as.
 * TEKS:**

Uses spinners, coins, dice, drawing a card or choosing an object from a bag to determine the idea of chance through experiments. Records collected data in tables Analyzes previously recorded data to decide the probability of an event.
 * Core Components:**


 * I decided to do the plans a little different for this unit since it is easily taught through games. Be sure to focus on the vocabulary: more likely, less likely, or equally likely. Below are a ton of activities you can use for teaching probability. I have carefully looked at each one to make sure they are good, so browse and see what looks good to you. Have fun with this unit!

We will do probability the first week and statistics (graphing) the second week.

Week One:**

Students place M&M markers on the numbers 2-12. Students may place one M&M marker on each number or place several on some numbers and leave other numbers blank. Next, students toss two 6-sided dice, find the sum, and remove an M&M marker from that number, if there is still one. The first player to remove all markers wins the game.
 * M 'n M Probability:**



This is best introduced as a class activity. Students collect points for each toss of the die unless a ONE is tossed, which means they lose all of the points they have collected in the round. To prevent losing their points, students may elect to stop at any point in the game before a ONE is tossed and they get to keep the points they collected but get no further points. Students love the game and begin to appreciate that theoretical probability and experimental probability are often quite different!
 * PIG - A Probability Experiment:**




 * Heads and Tails Game:**

One student is heads and one student is tails. Students start their markers on the star in the middle of the snake. If the coin lands on heads, the heads person moves his/her marker one space toward the head of snake. If the coin lands on tails, the tails person moves his/her marker one space toward the tail of the snake. The first person to reach the head or tail of the snake wins the game. Students should enter the winner (Heads or Tails) on the class tally chart as a data collection exercise. Class discussion should focus on analyzing the data to determine if the game is fair or not.




 * Coin Flipping:**


 * What's the Chance:**


 * Wheel of Choice:**


 * Valentine Probability:**


 * Heart Probability:**

 Materials: 20 pennies per two students 5 dimes per two students "Penny Flip" Ten Frame for each child
 * Penny Flipping Game:**

Instructions:  Extensions:
 * Demonstrate flipping a coin to the class. Explain the the front of the coin is called "heads" and the back is called "tails".
 * Pair students in groups of two. Provide one student with a red ten frame and one with a blue ten frame.
 * Explain to the children that they will take turns being the first to choose either "heads" or "tails". ( Each time the coin is flipped one child will choose and the other will take the opposite choice by default.)
 * Both partners then flip a penny.
 * The partner who guessed right takes the penny and places it on his/her ten frame.
 * When one student has a full ten frame he/she will exchange the pennies for a dime.
 * Both children clear their ten frames and start over.
 * The first student to win three dimes is the winner.

Initiate a class discussion following the game to introduce the concept of probability. Ask the children if either "heads" or "tails" seemed to come up more frequently in the games they played. Did anyone choose "heads" or "tails" because their partner seemed to be lucky with that choice? Have the partners play one more round of the game...this time placing their pennies with the winning side of the coin up on the ten frame as they go. Did one side come up more often for each child? Did one side come up more often in each group? Did one side come up more often than the other for the entire class? Count and graph the answers from these questions as a whole class.

(Grades 1–8) Math concepts: Students of all ages can play this game, as long as they're able to add the numbers that come up on two dice. While younger children benefit from the practice of adding, older students have the opportunity to think about the probability of the sums from rolling two dice. The object: to remove all the counters in the fewest rolls possible. How to play: Two or more players can play. Each player needs 11 counters, a game strip that lists the numbers from 2 to 12 spaced far enough apart so the counters can fit on top of each number, and a recording sheet. Here are the rules for playing: 1. Each player arranges 11 counters on the game strip and records the arrangement. 2. Once the counters are arranged, players take turns rolling the dice. 3. For each roll, all players can remove one counter if it is on the sum rolled. Players keep track of the number of rolls of the dice it takes to clear their game board. After students have had the chance to play the game for several days or so, have a class discussion about the different ways they arranged the counters and the number of rolls it took. Have them write about the arrangements that are best for removing the counters in the fewest number of rolls. For an extension, try Which Number Wins?
 * Two-Dice Sums:**

(Grades 1–8) Math concepts: In this individual activity, students roll two dice and record the results. Make a recording sheet that is an 11 x 12 block grid with the numbers 2 through 12 across the top. While young children gain practice with addition facts, older children can examine the data, compare results with other classmates, and think about why some sums are more likely than others. To do the activity, students need two dice and a recording sheet. The object: to roll the dice and record the number fact in the correct column, stopping when one number gets to the finish line. How to play: Post a class chart that lists the numbers from 2 to 12 and have students make a tally mark to show the winning sum. Have each child do the experiment at least twice. After you've collected the data, discuss with the class why it seems that some sums "win" more than others. Young children may not be able to explain it, but older students often figure out that there is only one way to get the sums of 2 and 12, and six ways to get a sum of 7. After discussing the data, return to the game of Two-Dice Sums and see if students revise their strategies. You may want to ask students to write about the game and the likelihood of two-dice sums.
 * Which Number Wins?:**

(Grades 3–5) Math skills: This two-person game involves probability and strategy, and gives children experience with multiplication in a geometric context. The object: to make rectangular arrays with Cuisenaire Rods and place them on 10-by-10-centimeter grids until no more space is available. The game encourages students to think strategically as they consider where to place their rectangles to avoid being blocked. How to play: students need Cuisenaire Rods, one die, and a grid sheet for each (Make a 10cm x 10cm grid. Also leave space for students to record how many of their squares are covered and uncovered.) The rules are: 1. On his or her turn, a player rolls the die twice to determine which Cuisenaire Rods to take. The first roll tells "how long" a rod to use. The second roll tells "how many" rods to take. 2. Players arrange their rods into a rectangle, place it on their grid, and trace it. They write the multiplication sentence inside. 3. The game is over when one player can't place a rectangle because there's no room on the grid. Then players figure out how many of their squares are covered and how many are uncovered and check each other's answers. After students have had experience playing the game, talk with them about strategies for placing rectangles and figuring out their final scores.
 * How Long? How Many?:**

Objective: The student will explore probability, practice addition, develop number sense and use probability terms as they play a game with color tiles. Materials:
 * Simple Probability:**

Small bowls (or paper plates) large enough to hold 12 color tiles. ( One bowl for each pair of students Color tiles, 12 per pair of students. Dice one die per pair of students. One sheet of chartpaper titled "Rolls to Empty the Bowl' and listing the numbers 2 - 12 down the left side.

Anticipatory Set:

Read a story from the book, __Math Fun: Test Your Luck.__ Talk about probability and probability words used in the story. Wnte a list of probability words on the chalkboard such as, chance, perhaps, likely, unlikely, probably, possible, perhaps, maybe and could be. Tell the students they get to play a probability game today and after they play a few times, you'll look at what's likely or probable to happen. Play an example round with one of the students to demonstrate the game. Ask the student to predict the number of rolls it will take to empty the bowl.

Activity:

Post the class chart paper with the heading "Rolls to Empty the Bowl" in a location where students can add to it. Divide the students into pairs. Each pair gets 1 die, 12 color tiles, a bowl, and a paper and pencil for recording. To play they will roll the die, note the number that comes up, and take out that many color tiles. While one person rolls to see how many times it takes to empty the bowl, the other person will record each of the other student's rolls on a sheet of paper. They will continue with one rolling and the other recording until the bowl is empty. For example, one student rolls a 3, then a 4, then a 6. The other student writes 3 + 4 + 5 = 12. Rolls to empty the bowl = 3. Explain that it is not necessary to go out exactly in this game. For example, if there are two tiles in the bowl, and a five is roiled, you may remove the tiles.

Practice:

Have each pair play the game five times. After each game, have the pair record the number of rolls it took to empty the bowl on the class chart paper entitled, "Rolls to Empty the Bowl."

Closure:

As a class examine the chart to identify the most likely number of roils it took to empty the bowl. Also identify the most and fewest rolls it took t'or anyone to empty the bowl. Ask them it' they notice anything else. Discuss the fewest and the greatest number of rolls it could take to empty the bowl and how likely or unlikely that would be. What amount of rolls could never empty the bowl? Why? Point to words on the probability words chart as they are used in discussion.

Extension:

Play the game using 20 tiles. Have the students use subtraction skills instead of addition by subtracting the numbers rolled on the dice trom the 12 tiles.


 * Online Probabilty Game:**

@http://pbskids.org/cyberchase/games/probability/


 * More Online Probability Games:**

I had to actually sit down and play the games myself in order to understand how they work or how I could use these with my whole class. This is defintiely NOT something you want to do on "the fly" but they are excellent if you take the time to see how they work. There are games with dice and spinners and many more.

@http://jmathpage.com/JIMSProbabilitypage.html


 * Fish Tank:**

@http://www.bbc.co.uk/education/mathsfile/shockwave/games/fish.html


 * Design Your Own Probability Game:**

I think this would work best in groups or with your higher level students who need a challenge.



The kids are familiar with this site. There are a couple of videos introducing probability and some games to play.
 * Brain POP:**

@http://jmathpage.com/JIMSProbabilitypage.html


 * Balloon Bonanza:**

Simple but gets the point across.

@http://www.hbschool.com/activity/balloon_bonanza/


 * Spinning Probability:**

This is a MUST PLAY game! Save it for later in the week, when the kids have a good grasp on probability. This uses spinners that you have to match to a probability (fraction).

@http://www.hbschool.com/activity/probability_circus/


 * Additional Activities:**




 * Probability Common Assessment:



Week 2:**

Begin the unit by asking students to bring in graphs they find in the newspaper of magazines at home. I am gathering a collection that we can use as well. Create a large class chart with the following headings: bar graphs, picture graphs (pictographs), line graphs, pie graphs. Do a sort with the graphs the students bring in. Have the students discuss the different charateristics of each type of graph and list these on the chart.

We will do a differnt type of graph each day. For each type of graph students need to read premade graphs as well as create graphs of their own. I have given you a packet of graphs to use for the "reading graphs" portion of each day. As for the "making graphs" portion, I have listed some ideas below.


 * Day One: Bar Graphs**

Have students choose something they would like to know about their fellow students: favorite ice cream, favorite sport, favorite color, etc. The students can create individual graphs (see the example survey sheet and graphing sheet below) to show the information they collect.

It is also easy to have a simple bar graph posted each day when the kids come in, similar to the "Ticket to Learn", they would put their response on the graph with a sticker or check mark or post-it as they came in the room. You could use some of the samples listed below. Then you could simply ask questions as they refer to the graph. Be sure to have them compare responses (how many more than, how many fewer _ __than__ _, etc) as well as obvious questions (how many students chose ).



Questions to ask about class graphs:




 * Day Two: Pictographs**

Amber did this really cool activity last year with butcher paper and premade cutout shapes. She had watermelons, shoes, book, apples, etc. The students created a question they could ask their fellow students that related to their cut-out. For example, if they had a sunflower cutout, they could ask, "Do you like sunflower seeds?" After they polled the class, they grouped their responses and graphed them on butcher paper. They used the cutouts to represent each "vote" on the graph. Each group had to make sure and have a "key" on their graph. Some of the kids had their cut-outs stand for more than one vote. For example, one sunflower could really stand for 2 votes. This made they key very important. They looked great in the hall and I think it taught the lesson perfectly.

This gave me an idea, we should take pictures of some of these projects as we do them and post them on the wiki to use as examples from year to year. Just a thought. I would be more than happy to do this, I love pictures!

Here are some printable cutouts you could use as well as the graphing lesson that goes with each one.




 * Day Three: Line Graphs**

Last year several of us created large graphs and tracked the temperature each day for a couple of weeks. I have large graph paper that is perfect for this. Create one large line graph that can be posted in the room. Discuss with the kids what information would need to be put on the bottom (dates) and along the side (temperatures). As you graph the temperature each day, be sure to connect them with a line and discuss how you can see patterns in the temperature by following the lines.


 * Day Four: Pie Graphs**

Cut several circles from construction paper, then divide the circles into sections according to how many students you have. Next, poll the students on a question that is easily graphed (do you live in a house or apartment, do you have a cat, a dog, or neither). Try not to ask a question that gives too many choices. Students "vote" by choosing a section of circle that represents their choice. For example, if I have a dog and the color for dog is blue, then I would choose a blue piece of the circle and connect it with everyone else's piece. If everyone votes, then a complete circle would be made. If you need me to show this to you in more detail, I will be more than happy to!


 * Day Five: Review and Test**