Making Predictions

Making predictions is an important step in all data collection activities.  After a brief explanation of the problem or experiment, students are asked to think about what they expect the results to be and make a prediction.  The teacher leads a discussion of the predictions and the reasoning behind them, often posting these on chart paper for future reference during the data analysis phase.


I recently read a study on this very subject.  It found that having students make predictions about new math situations or problems did indeed foster deeper mathematical thinking and understanding in students.  Additionally, this practice also increased student engagement in the task, as they were invested in the results.


One data collection activity I have used with students from kindergarten through middle school, is the Cheerios Investigation.  Students are told about an advertising campaign in which Cheerios randomly includes one of six toys in each box of Cheerios.  They hope that this will prompt families to buy more boxes of Cheerios trying to get all six toys.  Students are asked to think about this and predict how many boxes of Cheerios the average family would need to buy to get all six toys.


The teacher then leads a discussion about the predictions, including the reasoning behind them.   In my experience, there's always the eternal optimist (or naive person) who says six.  Then there's the off-the-cuff response of 100 or some other large number.  The most popular response is 36, although I've yet to hear an adequate mathematical reason for that number.  Nonetheless, the teacher accepts all predictions and reasoning, posting them on a chart for later reference.


Students then conduct the simulation, using a die and tallying the results until they have indeed gotten each of the six toys (each toy represented by a number 1-6 on the die).  They post their results on the class line plot, then repeat the experiment as time allows.


The class line plot is the basis for the data analysis.  Students may compare their small sample to the larger class sample.  Discussion may include mean, median, mode, range, outliers, etc., as appropriate for the class level.  Finally, students are asked to write an analysis of the data, this time in a letter to Mr. (or Mrs.) Oats, explaining why the plan to include toys will increase the number of boxes of Cheerios families will buy.


It's a simple activity but it leads to rich mathematical discussion and students are actively involved and engaged in the results from beginning to end.  Additionally, if several classes in the school conduct this experiment, classes can share results to generate an even larger sample.

See  Mathwire's Cereal Toy Investigation to view a lesson plan and download handouts.


Mathwire also offers the Cereal Toy Investigation applet, designed to allow students to quickly generate larger samples, extending the die-toss experience.  Students simply click the Next Box button to buy another cereal box.  The applet is designed to stop when the student has accumulated all six toys so that students may record this number before running another trial.   [Note:  This app requires Java.]

Bat Probability

There Was An Old Lady Who Swallowed A Bat!
by Lucille Colandro

After reading the book, investigate Batty Old Lady Probability.  Students spin to collect all of the items the Batty Old Lady swallowed, and tally each spin on the recording sheet. They then calculate the total spins it took them to get all 7 items, and add that figure to the class data. Teachers may help students analyze the class data and learn about probability in the process. The pdf document includes directions, game mat, picture cards, spinner, recording sheet and writing to learn handout.

Heads & Tails Data Collection

In this activity, students toss a penny and a dime and record whether the outcome is HH, TT, or HT and which coin is heads or tails.   Students cut apart their results and add them to the class results to create a class pictograph which can be analyzed and compared to the expected outcomes.   Students analyze the results to determine if the game is fair or unfair. 

It is important to use two different coins so that students see that the coins matching (HH or TT) and the coins not matching (HT or TH) are equally likely outcomes. Students often respond initially that the game is unfair because the player who wins when the coins match has two chances to win while the other player only has one chance. Using two different coins and recording the results of both coins helps students dispel this initial misconception as they analyze the graph results and create a tree diagram for the event. 

Download Heads & Tails Data Collection which includes the student recording sheet and directions for creating a class graph.

The Game of Pig

Pig is one of my favorite probability games.  I have played the game with kindergarten through college level students and used it in teacher training sessions.  Everyone LOVES the game and it's fun to watch who plays it safe and who's the risk-taker.  

In my own classroom experience, I have found that 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! 

I devised a method that uses only one die tossed by the teacher, so this is a great transition activity that quiets a class as they strain to hear and record the results of the die toss AND decide if they will stop or continue to play.   It's a win-win because students feel that they're playing, but they're actually learning a lot about the probability of a one-die toss.

• 
  Download Pig directions.
• Download Pig template or Pig Tally template, and place in a sheet protector to record die tosses.  Students may use dry erase markers for a reusable recording sheet.  The Pig Tally template requires students to use tally marks to record each toss of the die.  This recording sheet is a very visual presentation of the results and proves extremely useful when students devise winning strategies, as described below.

Results:  Younger students may use calculators to total each column.  Older students should use mental math to find the sum of the die tosses.  Students should write the total of all 3 columns in the upper right hand corner of the template and circle it.

Extending the Data Collection & Analysis:

• Ask pairs or small groups of students to talk about the game and come up with a winning strategy.  For example, some groups decide to stop once they have 20 points.  Another group decides to stop when they get 3 of any number.
• Have each group share their winning strategy and explain why they think it is a winner.
• Play the game again.  Groups MUST play according to their winning strategy.
• Discuss the results and allow groups to refine their strategies, if desired, before playing 1-2 more times.
• Finally, ask students to write about what they learned from this game.  Is there really a winning strategy that works all the time?  Explain your thinking.

Two of Everything

Use the book Two of Everything to introduce students to the doubling pattern.  Students will enjoy the story and predict what will happen each time something is put into the magic pot.

Make the math connection even more concrete by using a pot and linking cubes to illustrate the story.  Students may use a Magic Pot Workmat to record the input and output, then write a rule.

Extension:  Extend the activity by using the magic pot and simple rules to create data sets.  Students should guess the rule by giving an input number and telling the output that would result.  By not stating the rule, students provide more time for other class members to generalize the rule.  Each guess simply provides another data set for students to enter in their table.

Enrichment:  After reading Two of Everything, challenge students to solve NCTM's 5 Coins problem:  Would you rather have 1000 coins or 5 coins and a Magic Doubling Pot that works 10 times?  Students might work in pairs and use in/out charts to record their solutions.