Saturday, March 17, 2012

Math Alive!

The Smithsonian Institute is currently hosting the Math Alive! exhibit.  From the website:

  • MathAlive! is designed to inspire, to spark the imagination, to reveal not only math at work, but the endless possibilities of math. Designed for families and students the exhibition brings to life the real math behind what kids love most - video games, sports, fashion, music, robotics, and more - and creates interactive and immersive experiences that bring to life the math at work in each, whether in design, application or use. 
If you're planning a trip to the Washington, DC area, be sure to include this exhibit in your itinerary.

Visit the Math Alive! website for more information on the exhibit and to sample some of the online games and activities.

Friday, March 16, 2012

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.]

Thursday, March 15, 2012

Dynamic Paper

Be sure to check out the Dynamic Paper app from NCTM's Illuminations site.  This app allows teachers to customize and print out graph paper, grids, number lines, tessellations, spinners, pattern blocks and more.  Users control the number of sections in the spinner and colors.  Likewise, users set the size of the sides of pattern blocks or the range of the number line. 

Dynamic Paper is a great resource for teachers designing activities to support math learning.  It's also an invaluable tool for teachers needing to differentiate classroom learning activities to meet the varied needs of learners in their classrooms.

Check out Dynamic Paper.

Wednesday, March 14, 2012

Friday, March 9, 2012

St. Patrick's Day Activities

  •  Shamrock Paths challenges students to use Pascal's Triangle to figure out how many different ways there are to spell Shamrock. Included in the PDF file are an explanation of how this problem relates to Pascal's Triangle and the numerical solution to the challenge.
  •  Shamrock Patterns requires students to rotate shamrocks to complete the pattern. Download some copies of Extra Shamrocks for students to cut apart and glue on to complete the patterns.
  •  Top of the Morning challenges students to identify the 100th and 1000th letter that will be printed on the electronic signboard.

Wednesday, March 7, 2012

Writing Original Problems

Students use higher-order thinking skills when they write original problems similar to those they have done in class. Have students proof and solve each other's problems before publishing them on the computer, along with the author's name and the date.

These problems may be used as do-nows, homework assignments, or placed in the math center. Teachers who regularly include student writing find that students are highly motivated to solve problems created by their peers AND teachers develop a larger bank of appropriate word problems over time.

An easy way to start is to give students an open-ended problem to solve.  Then challenge students to write their own version of the  problem to share with their classmates.

Tuesday, March 6, 2012

Writing Responses to Open Ended Questions

Use these simple steps to help students master this all-important skill.  Consistent use of these strategies will definitely improve both student confidence and student proficiency.

Review the Rubric with Students

Most teachers use rubrics to score student responses to open-ended questions. Make sure students understand the rubric so that they know exactly what they must provide to get the best score. Teachers use a variety of strategies to provide this training:  [Read more...]

Use the Writing Process Approach

Students are accustomed to editing their first drafts in language arts, reworking their writing to craft a better response. Regularly include these practices in math class. After writing, students should use the rubric to examine their response and annotate, clarify or reorganize, as needed, to craft a tight response. Students should be encouraged to use words, pictures and/or numbers to capture their thinking process in these open-ended questions. They must also understand that there are many different ways to achieve a perfect score as long as they accurately show their thinking toward a correct answer.  [Read more...]

Provide Independent Practice

Students will complete open-ended problems in both classroom and state testing situations, most likely under timed conditions. Teachers should continue to use a combination of the writing process approach (to grow better math writers) and the independent response situation, in which students must apply good practices to producing their best responses under test conditions. A combination of these approaches, used throughout the school year, is the best instruction and preparation for state testing.

Check out the Mathwire Problem Solving collection.

Monday, March 5, 2012

Problem Solving: Stoplight

The Stoplights problem challenges students to figure out how many different stoplights they can create with 3 different color linking cubes if each stoplight must have one cube of each color. 

Use linking cubes so that students can assemble all possible stoplights. You may choose to have students use any three different colors so that each student pair has enough linking cubes to assemble all possible combinations using one of each color to create the stoplight.

In these kinds of organized counting problems, it is important to question students about how they know they have found all of the possible combinations.  A first grader once responded that he knew he had them all because each color gets to be on top two times.  What a great insight!

Download the Stoplights problem student handout.

Friday, March 2, 2012

Seussical Birthday

Check out some Dr. Seuss activities in the Mathwire archives and enjoy this special day.

Tuesday, February 28, 2012

Baseball Multiplication Game

Teachers familiar with the Everyday Math Program are familiar with the Baseball Multiplication Game played to help students master basic multiplication facts.  Players toss two dice, multiply the numbers, then enter the product to score.  According to the table, that product earns a single, double, triple, home run or an out.  Students will enjoy playing against the computer and practice multiplication facts at the same time.

Try Baseball Multiplication online at the Everyday Math Games Site.

Monday, February 27, 2012

Napier's Bones

Once students are comfortable with lattice multiplication, Napier's Bones is a great enrichment activity. Students order the bones as they would write the problem in the lattice. They are then able to read the answer without any writing. It's magic!

Download the Napier's Bones template that students may cut apart to create the bones. The file contains three different versions including blank bones that students fill in to create the bones, developing a richer understanding of the magic of the bones. Copy the handout on card stock and laminate before cutting to create more durable classroom sets.  Teachers may also elect to create a transparency of the bones and cut apart for use on the overhead.

Interactive Napier's Bones:
This site allows students to manipulate the Napier's Bones to multiply large numbers and check the accuracy of their answers.  Simply click on Get Multiplication to start, then click the appropriate bones to add them to the problem on the left.  Students then read the correct answer, post it in the answer box and click Check answer.

Note that the site may also be used to do any problem.  Simply click Reset, then click on the bones needed, and read the answer.  This would be a powerful introduction to Napier's Bones using a whiteboard or projector. 

Friday, February 24, 2012

Coin Antennas

Students draw antennas on coin pictures to represent the value.   Each antenna is worth 5 cents.   This means a dime has two antennas, a nickel has one antenna, a penny has no antennas, etc.   

This strategy capitalizes on students' strength in counting by fives.   They simply point to each antenna as they count by 5s, then count on by ones to include any pennies.   This method is also efficient because students do not need to sort and reaarange coins; they simply draw antennas on coins in the order given.   
This method is especially effective for K-2 regular and special ed. students who will eventually outgrow the need for antennas.   

NOTE: some teachers call the antennas "hairs" and talk about the penny as "bald" because it has no hair.   Whatever works for you and your students is the best strategy. 

Thursday, February 23, 2012

Lattice Multiplication

This algorithm works well for students who are developing multiplication fact fluency. Students may begin using a template to solve multiplication problems, but they quickly learn to draw their own lattice matrix to solve problems. Students love the method and it is very successful with both whole numbers and decimals. Teach students this multiplication algorithm as one of many different algorithms they may elect to use.  Students will very quickly learn to multiply large numbers, using basic multiplication facts and addition/regrouping skills.

  • Student writes the problem in the grid (e.g. 6 x 417).
  • Student then writes the answer to each single digit multiplication in the appropriate square.
  • The tens digit is placed above the diagonal; the ones digit below the diagonal.
  • When all squares have been completed, the student sums the numbers between each set of diagonals, and writes the sum at the bottom of the grid. NOTE: If the sum is greater than 10, regroup the ten to the next diagonal to the left.
  • The student can now simply read the answer from left to right and insert commas, as appropriate.
For a complete overview of this method, see Cool Math's explanation of Lattice Multiplication to view the step-by-step procedure.  Take a cue from the site's explanation and use colored markers when introducing this algorithm to students on the overhead or whiteboard.  

Templates:   Three different templates are included for Lattice Multiplications.   Templates are designed to be used in sheet protectors.   This strategy allows students to use dry-erase markers and re-use the same paper for many multiplication problems.

Wednesday, February 22, 2012

More Seussical Fish

Even the class estimation station can be a part of Dr. Seuss math celebrations.  After reading One Fish, Two Fish, Red Fish, Blue Fish, have students estimate the number of fish in a jar.  Use fish manipulatives, goldfish or gummi fish to fill the estimation jar.  Have students post their estimates on post-it notes, or fill in a class recording grid.

Download the Estimation Station file which includes a class recording grid as well as a form for the actual counting of the contents.  Some teachers keep these records in a class estimation counting folder or binder for future reference.  This is especially helpful when trying to develop how the size of a unit impacts the number of units that can fit into the same jar.  Keeping the documentation allows teachers to call upon the information in future discussions.

Tuesday, February 21, 2012

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.

Monday, February 20, 2012

Seussical Combinations

The book states that not one of them is like another. These original Mathwire problems challenge students to use combinations to figure out how many unique fish could be created.

  • Challenge younger students to find all of the different combinations for  Seussical Fish using attributes from the story. Making an orderly list is an effective strategy for solving this problem. Other students might opt to draw the solution.
  • Older students will enjoy the challenge of the   Fishy Combinations problem which involves many more options. The model solution uses the multiplication principle of counting to easily solve the problem. Students may also use a tree diagram or list all of the possible red fish, then extend this list to the blue and black fish combinations.

Friday, February 17, 2012

Domino Problem Solving

Students will have to really think about dominoes as they solve Garage Sale Dominoes.  A sample solution using an orderly list is included in the pdf file.

After students have solved Garage Sale Dominoes, challenge them to solve More Garage Sale Dominoes in which they must figure out how many different dominoes there are in a complete double-nine set.  A sample solution using an orderly list is included in the pdf file.

Enrichment on the Internet:

Thursday, February 16, 2012

Domino Facts Template

Students insert the template into a sheet protector and use dry erase markers to reproduce the domino, then add in the appropriate numbers.  Finally students, write four number sentences for that fact family.

This activity is easily incorporated into daily lessons as a warm-up or do now! activity.  Teachers may post the domino of the day or use an overhead domino.  Teachers may also elect to distribute dominoes, differentiating the difficulty level to accommodate the needs of varied learners in the class.

Download the Domino Facts Template for student use.

This activity is also an effective math center activity.  Provide tubs of dominoes, domino facts templates and/or have students use the Domino Record Sheet to record the sums of dominoes they select to practice basic facts.

Read more about using the Domino Facts Template on

Wednesday, February 15, 2012

Domino Flash Game

Play Domino Flash to help your students master the domino patterns.  Each student needs a domino mat and counters.  Teachers may use overhead dominoes or Domino Flash Cards, copied on card stock, cut apart and laminated for classroom use.

The teacher shows a domino for a count of 5-10 seconds, depending on the ability level of the students, then covers it.  Students look at the domino as it is shown, then build the domino from memory, using a domino mat.  Students may use round chips to recreate the domino.  Alternately, if the mat is inserted in a sheet protector, students may use a dry erase marker to draw the pips.

The teacher circulates around the room as students work, to observe student performance,   After some time, the teacher asks students to describe the domino they saw and how they remembered the patterns to build.  Finally, the teacher shows the domino again so that students are able to self-correct.

Center Activity:  make the Domino Flash Game materials available for students to play as pairs or triads at center time.  Students love to rotate playing teacher for this game!

Tuesday, February 14, 2012

Domino Parking Lot

Students use a set of regular dominoes and a domino parking mat.  Each student selects a domino, counts the total number of dots (pips) and places the domino in that parking spot.  Dominoes with the same number of dots may be stacked on top of each other in the parking spot, if necessary.

Center Activity:  Teachers may use craft foam and a sharpie marker to create the parking lot mat or download the Domino Parking Lot game mat, copy it on card stock, and laminate the mat for student use.  Provide sets of dominoes for pairs or small groups of students to sort.

Challenge:  As a variation of the game, select target sums and give students a point for each domino they find to park in those spaces.  This motivates students to search for those particular combinations and heightens interest in finding those dominoes to win the most points for the group.  Have one member of the group use a Domino Parking Lot recording sheet, write in the day's winning numbers, then draw in the dots of the dominoes the group finds for those numbers.

Monday, February 13, 2012

Valentine's Day Grab the Candy Game

This game is yet another version of the familiar Battleship game.  Students place candy hearts on the board.  They take turns rolling two dice (one regular die and one marked A-F) and form an ordered pair (e.g. (B,4).  If there is a candy in that square, they take the candy.  If not, the next player tosses the dice.

This game provides practice for young students in naming ordered pairs.  Playing the game prepares them for later years when they will name points at the intersection of the lines.  It is important that students learn to name the spaces using the horizontal (A-F), then the vertical (1-6) to prepare them for the mathematical convention of ordered pairs in (x,y) form.

Download the Grab the Candy game mat which includes both the introductory game mat, shown above, and a traditional game mat using ordered pairs that name the intersections.

Friday, February 10, 2012

Problem Solving: Valentine's Day Treat

Valentine's Day Treat challenges students to list all of the possible sundaes that can be made if students choose one ice cream and one topping from the cafeteria list. Note that students must choose a topping for this problem, otherwise they are simply buying ice cream which doesn't count as a sundae.

This problem may be solved using various methods and is accessible to students of all levels.  Some students may simply list the various combinations.  Other students might elect to use a probability tree.  Others might use a more developed concept of combinations in counting and naming the various combinations.  All are valid methods of solving this problem and reflect students' current understanding of the concept of combinations.

Download  Valentine's Day Treat which includes the student handout and a sample solution.

Thursday, February 9, 2012

Valentine Quilt

This Valentine Quilt is a great culminating activity for a study of the geometry of quilts. Each student designs a 9-square paper quilt patch using only red, pink and white paper. Students may elect to use a traditional quilt pattern or design their own original pattern and name it. Assemble all of the quilt squares into a class quilt.

See Mathwire's Mathematical Quilting menu to view and download resources for teaching a unit on the mathematics of quilts. 

Wednesday, February 8, 2012

Heart Paths

Heart Paths challenges students to find all the different paths that spell HEART if they can only move from a letter to one of the two letters directly below it.  Students use the Heart Path worksheet to record each path, making it easy to check for duplicates.

This problem is best solved using systematic counting, an organized approach to solving the problem.  While many students might not use this approach independently, it is beneficial for the teacher to spotlight students who successfully used this approach.  Or, the teacher might model the approach as students check their own solutions, adding paths they might have missed in a more haphazard approach.

Download the Heart Paths worksheet which includes the problem sheet, worksheet and solution.