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.]

Fish out of Water Game

This is a simple counting game, designed by a Monmouth University student.  It's a great game for Dr. Seuss day, accompanying One Fish, Two Fish, Red Fish, Blue Fish.  Students begin with 20 fish out of water, toss a die, and place that many fish back in the bowl.


While the file provides a game mat, teachers might choose to use a bowl and fish manipulatives.  It's fairly easy to find plastic fish in dollar stores.  Plastic fruit containers from the grocery store simulate a fish bowl and the lid can be used to store the pieces in the fish bowl for use in the math center.


Although the game was designed as counting practice for young students, it may be enriched to provide a data collection activity for older students.  Given the premise of the game, students are asked to estimate how many times they will have to toss the die to get all of the fish back in the bowl.  Students then play the game several times, collect data and add it to class results.  Students analyze the data in small groups and draw conclusions backed up by the data.  The teacher then leads a class analysis of the data.


Download Fish out of Water, which includes game mats for a single player, two players, center icons, and directions.  Add this game to the math center for independent practice.

Snowman Probability

This activity is designed as a fun, seasonal data collection activity that introduces students to the probability of a spinner.  Although each piece of the snowman is equally likely to be spun, the actual game results are often quite different.


In this game, students spin to win all of the pieces to create the snowman.  Students may place a transparent spinner over the spinner template, or use a pencil and paper clip to create a spinner.


Before playing, ask students to estimate how many spins they think they will need to collect all of the pieces for their snowman.  Call on students to share both their number of spins and why they think that is a reasonable number.  Record these estimates for later discussion.


Give each student a copy of the game mat and insert the mat into a clear sheet protector.  Students may then use dry erase pens to draw the pieces as they collect them.


Have students use the Snowman Probability Recording Sheet to collect data on how many spins it takes to get all of the pieces.  Insert this sheet in a clear sheet protector for a renewable activity.  Be sure to collate the class data and analyze this larger sample to get an approximate number of spins required.


Download the Snowman Probability Game file which includes the game mat, spinner, directions and recording sheet.  

Dice in the Classroom

There are so many effective math games and data investigations that involve tossing dice.  One way to help young students toss dice is to place the dice in a clear jar.  Students simply shake the jar, place it down on its lid, and read the dice toss.  Teachers may place 1-4 dice in the jar and they're ready to go.  HINT:  This also works for coin tosses!

Some teachers prefer to use the small trays in which vegetables and fruits are sometimes packed or small boxes or box lids.  The sides stop the dice from rolling off the student desk.  If noise is a concern in the classroom, place a sheet of craft foam in the bottom of the tray and the dice tosses will be silent.

Now check out some activities from the Mathwire collections:

• Data Analysis:  One Die Toss
• Data Analysis:  Two Dice Toss

Fat Bat Game

Students roll a die to see how many insects their bat eats. Students may continue rolling, in this Bat version of Pig, until they elect to stop, or until they roll a 1. But be careful! If your bat is still eating (collecting points) when a one is tossed, you are a Fat Bat and lose all of your points for that round.

This game is designed to provide a fun experience in the experimental probability of a single die toss. However, students get lots of practice adding a string of single digit numbers, as they total up their winning points for each round. A data analysis option is included to formally extend the analysis of the game's probability for older students.

Download Fat Bat for the student recording sheet, directions for whole class play, and data analysis option for extending the game.

Back-to-School Problem Solving

September is a great time for data collection activities as students are naturally curious about their new classmates. Ask questions that require students to analyze data and support their conclusions.

Challenge students to solve some of these problems to practice collecting, organizing and analyzing data in real-life applications. Require students to explain their reasoning and solution to develop mathematical communication skills.

See all Back-to-School Word Problems.

Here's a sample 3rd grade problem that targets student understanding of probability:

Third graders cut apart each of the letters in THIRD GRADERS and place
the squares in a bag. Kevin pulls out one square and looks at the letter.

• What is the probability that the letter he pulls out will be the letter, D?  Explain how you know.
• What is the probability that the letter he pulls out will be S?  Explain.
• On Friday, his teacher says that if anyone can correctly predict which tile he or she will pull out the bag of THIRD GRADERS squares, the class will not have any homework over the weekend.  She asks Kevin's table to predict for the class.  They are nervous because the class is counting on them to say the right letter.
•  Which letter should Kevin’s team predict that they will pull from the bag?
•  Explain how you know that the students should predict this letter.

Note:  It is very important that students explain how they know their answers are correct.  They may use pictures, words and numbers to explain their thinking.

What's Your Favorite Number?

Alex Bellos is fascinated with numbers.  As an author of books on numbers, he's been asked countless times what his favorite number is.  He decided to conduct a survey to find out what numbers people most liked.  He promises to publish the results of this worldwide survey in 2012.


Students can also take part in the survey which simply asks you to input your favorite number and explain why it is your favorite.  It would be fun for students to take part in this easy survey and they'd surely be interested in the results when they are available.  Visit the Alex Bellos Blog to read about the survey idea and to enter your favorite number.


Extension:   Extend this online activity by conducting a classroom survey of favorite numbers.  You might use the same format that Bellos uses online.  Students would collect data, then organize the data presenting it in graphical form before analyzing the results.  Students might elect to organize a schoolwide survey to see if the classroom results match this larger sample.

Data Analysis: TV Survey

TV Survey is an open-ended problem that was designed to introduce students to this real-life application of sampling. Because TV is a part of students' daily life, this scenario is accessible to all students. And, because students often exhibit a keen sense of "fairness," they are sometimes disconcerted by the notion that a small group of people get to decide what all of us get to watch on TV.

TV Survey was also designed to address different Bloom levels so that students have to use higher-order thinking skills to analyze, synthesize and evaluate information about the sampling method.

• Download the TV Survey open-ended assessment.

See more Mathwire Sampling Activities.

Online Resource: Probability Spinners

Math Playground:  Spinner
This online site allows users to customize a spinner by colors, then spin for trials.  There is a graph option to display the results by color as they are generated.  This is an effective online resource for data collection.  Students may easily generate data from larger trials than through the use of a real spinner.  


Many teachers use this kind of online generator after student's initial experience with classroom spinners and discussion of the resulting data from this small trial.  Students generate hypotheses which may be easily investigated using this online resource to capture more data.  A resulting data analysis confirms or disputes the hypotheses and should include a discussion of the use of online generators and randomness.


Note:  This site allows the user to customize the spinner by adding labels to each section (e.g. numbers, names, etc.) and to select how many different outcomes there will be.


Check out Math Playground's Probability Spinner.

Question of the Day

Collecting, organizing and analyzing data are essential skills in today's world.  Some teachers incorporate daily practice by including a question of the day in the morning routines.  Students have markers with their names, and they are accustomed to completing this task when they arrive in the classroom, placing their marker in the correct spot.  If students are non-readers, assign one student to be the interviewer, repeating the question to classmates as they arrive in the classroom.

 

Suggestions:   

• Vary the format of the data organization, providing practice by using bar graphs, clothespin graph, Venn diagram, tally chart, line plot, glyph, etc.
• Use seasonal, weather, holiday prompts to collect data.
• Coordinate math question of the day to current literature, science or social studies units.
• Extend assembly and field trip experiences by graphing data about the particular theme.
• Check out this list of data questions for young learners.

Bar Grapher Tool

NCTM's Illumination site offers a free Bar Grapher Tool students may use to create a bar graph of their own data.  Students simply input the data, select appropriate minimum and maximum values, width of bars, color of bars, then click on Graph Data to create a bar graph.

The graph to the left represents one trial of Mathwire's Cereal Toy Applet.

 This is a useful tool for the math classroom.  Once students know how to create a bar graph with all of the necessary components, the emphasis moves more toward the analysis of results.  Allowing students to use real-world graphing applications provides more time for discussion and analysis of group results.  


Check out the Bar Grapher Tool on the NCTM Illuminations site.

Clothespin Graph

Write each student's name on a clothespin. A piece of foam core board or laminated oak tag makes a great two-choice (Yes/No) graph board. Students simply affix the clothespin to the correct side to indicate their response. Students can easily "see" the results and count to verify the outcome.

• Use this method to record the results of informal classroom surveys:
  • Literature: Which character do you like best? What do you think the character will do next? Which version of the story do you prefer?
  • Daily Routines: Are you buying lunch or did you bring lunch? Present or Absent?
• Ask students to report the results of who won two-player games so that the class can analyze the fairness of the games. 

Suggested Use:  Use a clothespin graph to record the results of students playing the Heads & Tails Game.  

In this game, each student places a penny on the star.  One student is heads and the other student is tails.   Students should place their pennies with the appropriate side up.

The heads student tosses a penny and moves a space toward the head if the outcome is heads.  If the outcome is tails, the student does not move.  Play repeats with the tails student, who moves only if his/her toss is tails, moving toward the tail of the snake.  The first player to read the head or the tail of the snake wins.  The winning player should add a clothespin to the graph to record the win as heads or tails.

Class discussion should focus on analyzing the data to determine if the game is fair or not.   Directions and gameboard are included in the download.   This game was developed by a Monmouth University student for the Probability Fair. 


Download the Heads & Tails Game which includes the game mat, directions and icons for centers work boards.  This game is an excellent addition to a classroom math center, as students may play with a partner and record the results for class discussion at a later time.

Data Collection: Count in a Minute

Count in a Minute: First graders at Flynn School in Perth Amboy, NJ, created a large line plot of how high they could count in a minute. Student pairs worked together to time and count then post the results on the class graph.

• The teacher presented the question to the class: How high do you think most first graders in this class can count in a minute? The teacher led a discussion that included student predictions and the reasons for their predictions.
• The teacher created a large number line on the blackboard to form the basis of the line plot. The teacher used a timer to signal the start and end of the minute.
• Students worked in pairs. First, student A counted aloud while student B listened. When the minute was over, student B wrote the highest number on a post-it and added it to the class results.
• Students repeated the experiment: student B counted aloud while student A listened. These results were then added to the class results.
• Data Analysis:  The teacher led a discussion of the results and whether or not the results were what they predicted. Why or why not?

TV Survey

Sampling is a technique that is used in statistics to gather information about a lot of people or things without testing each one.   For instance, scientists pull a small container of water from the ocean to test to see if the beach is safe for swimming.   Many people follow this same procedure with their backyard pools, testing a small container to check pH levels. 

There are companies that gather this kind of information from a small number of people and make predictions about how a larger group will think or act.   For example, the Nielsen ratings gather information on which TV shows a small sample of Americans watch.   The Nielsen company then publishes which shows Americans watch the most or watch the least based on this small sample.   These are called the Nielsen ratings and they determine how much money advertisers pay for TV commercials.   The more people who watch the show, the more an advertiser will have to pay for a commercial during that show.   The Nielsen Company very carefully selects the people who participate in this sample so that they are representative of the country as a whole.   Unfortunately the end result is that sometimes your favorite TV show will not get a high Nielsen rating, and it may be cancelled by the TV station. 

TV Survey is an open-ended problem that was designed to introduce students to this real-life application of sampling.   Because TV is a part of students' daily life, this scenario is accessible to all students.   And, because students often exhibit a keen sense of "fairness," they are sometimes disconcerted by the notion that a small group of people get to decide what all of us get to watch on TV. 

TV Survey was also designed to address different Bloom levels so that students have to use higher-order thinking skills to analyze, synthesize and evaluate information about the sampling method. 

Download the TV Survey open-ended assessment.

Sampling: Mystery Bag

In the Mystery Bag activity, students take samples from four different bags and use those samples to predict the contents of the whole bag.   Students will tally the results of the samples, compare samples and match the samples to descriptions of the four bags.   Finally, small groups will report sample results to the whole class so that the class is able to review the results of this larger sample and decide if their predictions still hold. 

Sampling is used in polls, advertising, TV ratings, etc.  In this activity students will see firsthand how the size of the sample affects the accuracy.  Individual groups may pull skewed samples, but when the class data is combined, these larger samples should more accurately reflect the actual contents.  NOTE:  Be sure that students understand that they are sampling with replacement, meaning that they pull a tile, record the outcome, then replace the tile in the bag before pulling the next tile.

Download all materials and instructions for the Mystery Bag activity from Mathwire.

Online Cereal Toy Investigation Simulator

This applet, designed by Erin Mulder, allows students to conduct multiple simulations and collect data on a larger sample.   Students may run 20-30 trials more quickly than the dice toss.   The applet is probably best used following the class data collection simulation with 6-sided dice. 

In this simulation, students will need to collect data on the total number of boxes bought to get all 6 toys so they must decide how best to record that data.   The class must also decide on a method of recording the class data (e.g. adding to the class line plot using a different color marker would facilitate comparison of the results).   Discussion should center around how well the small class sampling results matched the larger computer-generated results. 

Directions for Cereal Applet:

• Once the Applet is loaded, click on "Next Box" to buy a box of cereal.
• A box of Cheerios and a toy will be added to the screen.
• Click "Next Box" again to buy another box of cereal and get a free toy.
• Keep clicking "Next Box" until you get all six toys.
• The applet will stop at this point and tell you how many boxes you had to buy to get all six toys.
• Students can then see how many of each toy they got.
• When students have recorded this information for the trial, they click on "Reset" to begin a new trial.
• Students should add these computer simulations to the class data for discussion. 

Run the online Cereal Toy Investigation Simulator.  [Java applet requires Java-enabled browser.]

Cereal Toy Investigation

This activity introduces students to the use of sampling for advertising purposes.   It also generates a discussion about how advertisers use gimmicks to get people to buy more of their product.   Even young students will admit that they have been induced to buy fast food meals in order to collect all of the toys.

The Plan: Many companies are putting toys in their products to try to get customers to buy more.   The company that makes Cheerios thinks this might be a good way to get families to buy more boxes of Cheerios.   They will make six different toys and put one in each box of Cheerios and Multi-Grain Cheerios.   That way kids will want their parents to keep buying Cheerios until they have all six different toys. 

Mrs. Oats, the President in charge of Cheerios, asks you to help figure out if this is a good plan.   She knows that it will cost more to put toys in Cheerios.   She wants to be sure that families will really buy more boxes of Cheerios to get the toys.  Mrs. Oats asks you to find the answers to these two questions:

• What toy should Cheerios put in the boxes to make families want to buy more?
• How many boxes will a family buy to be sure they collect all 6 toys?

"Guess"-timate: 

• Ask students to think about how many boxes a family will have to buy to get all 6 different toys.
• Have students share their "guess"-timates with the class and explain how they decided on that number.
• Write all "guess"-timates on the board or chart paper for later reference. 

Read more about the Cereal Toy Investigation on Mathwire where you can view a step-by-step lesson plan and download materials needed for this data investigation.

The M&M Game

M and M Game: 2-dice Version 

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. 

This game helps students develop an understanding of the probability of a two-dice throw.  As students play the game, they practice addition facts, but they also learn that some sums are tossed more frequently than others.  This hands-on experience is then solidified with a discussion of all of the possible outcomes, creating a chart to visually capture the data.

Next, students play the game again with this mathematical knowledge and discuss how this information impacts the placement of M&Ms.  Couple this game-playing with class data collection and students may then compare theoretical and experimental probability of a larger sample.

This is just one example of how a simple game may be used as a data collection activity to develop student understanding of theoretical and experimental probability concepts.  Some teachers question the use of games in the classroom pressed for time.  In my experience, this rich use of games to practice basic facts AND as a data collection activity is time well-spent and students respond positively to both the practice and the discussion of these hands-on, fun experiences.

NOTE:  This game, created by Susie Siegel as a grad school project, uses a sentence strip as the number line.  The M&M pieces were created using PowerPoint.  Teachers may choose to laminate the pieces for use in a classroom math center.

MATERIALS:

• Download Ms. Siegel's M&M Game directions and recording sheet.
• Download a picture, alternate recording sheet and directions for using that recording sheet.  NOTE:  This recording sheet was designed to fit into a sheet protector so that students may use dry erase markers to play and record data.  This option may be used without the M&M game pieces.
• Download the 2-dice toss recording sheet for a discussion of the theoretical probability of tossing two dice.  Students write one outcome that produces that sum in each box to create a visual display.   NOTE:  use different colored dice for this activity so that students are able to see that tossing 4 on the red die and 3 on the blue die is different from tossing 3 on the red die and 4 on the blue die.  Both must be counted as outcomes.  Students might benefit from actually using two different color markers to record these outcomes as well.
• Download M&M pieces for use in playing the game.

Turkey Glyph

Students in Mrs. Bestle's class in Keansburg, NJ, created turkey glyphs that tell a story about how they celebrate Thanksgiving and what foods they like to eat on that special day.  It's great to have students create the legend(key) for what different parts of the turkey glyph mean (e.g. like white or dark meat, eat home or away, like gravy, feather colors for different vegetables, etc.)

Glyphs are a great visual display of data and may be used to create a seasonal bulletin board of student work.  Be sure to use the display for some data analysis activities. 

• Will more people in this class eat at home or away for Thanksgiving?
• Do more people in this class like corn or potatoes
• How many people in this class like yams?
  
  

Developing the legend(key) with students allows teachers to customize the activity to fit individual classes.  Creating a legend for a glyph is a higher-order thinking skill that develops critical thinking skills. 

 

See turkey glyph for one teacher's legend and lesson plan for this activity.

U.S. Census Activities

Students will implement data analysis skills as they tackle U.S. Census Data. Check out these resources for suggested student activities:

• Scholastic's It's About Us: 2010 Census in School includes options for Teachers, Administrators, Families and Families (Spanish).
• The U.S. Census Bureau's Census in Schools includes activities for kids and teens as well as resources for teachers.