April 07, 2014

Gizmo of the Week: Target Sum Card Game

1020DETApril is National Math Awareness month, and this year’s theme is Mathematics, Magic, and Mystery. Card games are a great way to make the connection between mathematics and magic. The Target Sum Card Game Gizmo is a fun and engaging game that utilizes basic arithmetic skills, estimation, and logic.

In the Gizmo, students are dealt a hand of cards with the digits 0-9. The goal is to arrange the cards in different place values so that the sum is as close as possible to a given target. This is relatively easy to do when the number of cards is small, but the challenge increases rapidly as the number of cards increases.

Students can modify the game in a variety of ways and even play each other head-to-head. After using the Gizmo, try playing the game with real cards!

Posted by Heather Jones at 07:36 AM in Math (Real World), Using Gizmos | Permalink

March 31, 2014

Gizmo of the Week: Binomial Probabilities

Before the start of the NCAA Men’s Basketball Tournament, which ends next weekend, Warren Buffett offered a prize of one billion dollars to anyone who could create a perfect bracket. It only took 25 games before the last possible winning entry was eliminated, and Buffett’s money was safe.

126DETHow many brackets are possible in the tournament, and what are the chances of creating a perfect bracket? Students can explore these questions with the Binomial Probabilities Gizmo.

In this Gizmo, students use tree diagrams to illustrate the possible outcomes of a series of binomial experiments. Students then derive the formula for finding the probability of r successful outcomes in n experiments.

By the way, there are over 9.2 quintillion (9,200,000,000,000,000) possible brackets in the NCAA tournament. Warren Buffett’s money might be safe for a while!

Posted by Heather Jones at 07:54 AM in Current Affairs, Math (Real World), Using Gizmos | Permalink

March 19, 2014

Celebrate International Forest Day with Gizmos

639DETDid you know that March 21st is International Forest Day? This global celebration of forests provides a platform to raise awareness of the importance of all types of forests and of trees outside forests. Forests are the most biologically-diverse ecosystems on land, home to more than half of the terrestrial species of animals, plants and insects. Forests also provide shelter, jobs and security for forest-dependent populations.

In celebration of National Forest Day, use the Forest Ecosystem Gizmo to teach students about the delicate balance within a forest ecosystem.  Or, have students try out the Measuring Trees Gizmo to investigate how precipitation affects tree growth, while reinforcing mathematic concepts like circumference and diameter.

If you're planning activities for International Forest Day, share them with the community on our Facebook page.

Posted by Heather Jones at 12:09 PM in Current Affairs, Math (Real World), Science, Science (Real World), Using Gizmos | Permalink

March 10, 2014

Gizmo of the Week: Circumference and Area of Circles

Friday March 14 (3.14) is Pi Day! Get ready with the Circumference and Area of Circles PiGizmo, which has been recently updated with new lesson materials.

In this Gizmo, students can set the radius of a circle, and then use the provided geometry tools to measure the circle’s circumference. The area of the circle can also be displayed. Students can quickly discover that the ratio of the circumference to the diameter is always equal to the same number, pi.

Don’t forget to do the real-world experiment too. Have your students use rulers and string to measure the diameters and circumferences of as many circular objects as possible—pot lids, Frisbees, plates, and of course pies!

Posted by Heather Jones at 11:08 AM in Math (Real World), Using Gizmos | Permalink

February 17, 2014

Gizmo of the Week: Critter Count (Modeling Multiplication)

Designed for younger learners, the Critter Count Gizmo provides students a fun and engaging introduction to multiplication as repeated addition. For example, 2 × 3 means “two threes,” or 3 + 3. This example in the Gizmo would be modeled with 2 leaves and 3 critters on eac1013DETh leaf.

Once students are comfortable with using repeated groups to model multiplication, they can click the Array button to rearrange the critters in a rectangular array. Using the example above, 2 × 3 would be modeled with 3 critters in 2 rows, arranged neatly as a rectangle.

Students can choose the type of critter, the number of leaves, and the number of critters on each leaf. It’s simple and fun, and drives home the concept of “multiplication as repeated addition.”

In addition, the Gizmo begins to form a connection that will serve students well as they progress in math – namely, the connection between multiplication and area of a rectangle.

Posted by Heather Jones at 09:07 AM in Math (Real World), Using Gizmos | Permalink

December 16, 2013

Gizmo of the Week: Road Trip (Problem Solving)

While holiday road trips have changed tremendously over the past 20 years with the addition of GPS, cell phones, and computer car systems, there’s still a lot of planning that goes into these family journeys. Where will we stop? When will we need gas? Whose car should we take?

1036DETThe popular Road Trip (Problem Solving) Gizmo allows students to map their winter travels plans or create their own imaginary holiday journey using mathematics and critical thinking skills. First, students choose a vehicle to drive, and then fill up the tank with gas and go! Students solve real world problems like comparing gas mileages of different vehicles, discovering the shortest path between two cities, and planning a budget for their trip. When students are complete, they can share their holiday trips with their friends.

Happy holidays and safe travels!

Posted by Heather Jones at 11:12 AM in Math (Real World), Quick Tips, Travel, Using Gizmos | Permalink

October 30, 2013

Celebrate Fall with New Lesson Materials

Et gizmoCelebrate the fall season with new Lesson Materials! Six of our math Gizmos have updated materials just in time for a little fall fun. 

Use the Elapsed Time Gizmo, along with its newly updated Lesson Materials, to give students an application of “borrowing” when subtracting, in a real context. (Perhaps you and your students want to calculate elapsed times after a class apple-bobbing contest?)

If Geometry is more your speed, try the Chords and Arcs Gizmo. The updated materials guide students as they explore how a central angle in a circle is related to the arc and its intercepts (and if the visual reminds you a bit of a piece of pumpkin pie, well, so be it). Students can also explore the relationship between chords and their distance from the circle's center.

Other Gizmos with updated Lesson Materials include:  Arc

Adding Real Numbers

Multiplying with Decimals

Adding Vectors

Geometric Sequences

Wishing you all a happy and safe fall season!

Posted by Heather Jones at 02:49 PM in Math (Real World), Site Announcements | Permalink

October 21, 2013

Expert Corner: Changes to Roller Coaster Physics

KurtKurt Rosenkrantz is a science curriculum writer and Gizmo designer for ExploreLearning. Kurt holds a Master of Science in Geology from the University of Cincinnati, and a bachelor's degree in Earth Science from Harvard. He taught high school and middle school science for eight years before joining ExploreLearning in 2005.

A while ago, a teacher named Joshua Buchman suggested a way to improve our popular Roller Coaster Physics Gizmo. In the Gizmo, a toy car rolls down a track, over several hills, and into an egg. The egg will either crack or not.

Rc physics gizmoIn the original Gizmo, the egg would crack if the momentum of the car was over a certain threshold value. Mr. Buchman pointed out that it was more likely that the kinetic energy of the car, rather than its momentum, would be the critical factor. He argued that the car would need to travel a certain distance into the egg, overcoming the resisting force of the eggshell, for the egg to crack. In other words, the car would have to do a certain amount of work to crack the egg, and the work it could do depended on its kinetic energy.

This argument made sense to us, but we wanted to check that it was true in practice before changing the Gizmo, which was designed with the help of real-world experiments that took place in the EL offices a decade ago. To investigate, I bought toy cars, a track, and several dozen eggs. I set up the track at a steep angle and went to work.

Rollercoaster2Right away I realized that I needed to establish a consistent definition of “egg breaking.” It turns out that a very tiny impact can cause a small fracture in the egg, and that the fracture grows bigger and bigger as the force of the impact increases. Eventually I decided that the most consistent criteria I could use was “eggshell breaks completely into two halves.” So, any fracture that did not go all the way around the egg was considered a negative result.

Rollercoaster3After several very messy sets of experiments using cars of different masses, I plugged the data into a spreadsheet. Sure enough, the minimum kinetic energy required to break the egg was much more consistent than the minimum momentum required to break the egg. With experimental results supporting the scientific argument, we decided to make the change. In the updated Gizmo, the car now needs to have a minimum kinetic energy of 0.25 J to break the egg. We have adjusted all of our lesson materials and assessment questions to reflect this new result.

We hope you enjoy the new-and-improved Roller Coaster Physics Gizmo, and thanks again to Mr. Buchman for bringing this to our attention!



Posted by Heather Jones at 08:47 AM in Math (Real World), Site Announcements, Using Gizmos | Permalink

October 07, 2013

Algebra Lesson Materials Updated

The ExploreLearning curriculum team has been hard at work serving all your algebra-related needs!  We’re happy to report that we’ve updated the Lesson Materials for these Gizmos:

Direct and Inverse Variation

Standard Form of a Line

Absolute Value with Linear Functions108DET

Point-Slope Form of a Line

Solving Equations by Graphing Each Side

Cat and Mouse (Modeling Linear Systems)

Cat and Mouse (Modeling Linear Systems)- Metric

Linear Inequalities in Two Variables

Linear Programming

Arithmetic Sequences

We’re always trying to support real, conceptual understanding of these topics. For example, using a fun and playful setting, the Cat and Mouse Gizmo helps students connect real-world meaning to slope, y-intercept, and the intersection of lines.


Posted by Heather Jones at 10:49 AM in Math (Real World), Site Announcements, Using Gizmos | Permalink

October 04, 2013

New! Number Sense Lesson Material Updates

Calling all elementary and middle school math teachers! Some of your favorite Gizmos just got an update. Check out our new Lesson Materials for each of these Gizmos:

Beam to Moon (Ratios and Proportions)

Beam to Moon (Ratios and Proportions) – metric98DET

Adding and Subtracting Integers

Adding and Subtracting Integers with Chips

Modeling One-step Equations

Modeling and Solving Two-step Equations

Square Roots

Mean, Median, and Mode

The Beam to Moon Gizmo uses the concept of weight differences between the earth and the moon to teach proporations. You can even use this Gizmo as part of an interisciplinary lesson to integrate related math and science concepts.

Posted by Heather Jones at 08:40 AM in Math (Real World), Site Announcements, Using Gizmos | Permalink

September 30, 2013

Growing Geometry Gizmo Lesson Materials

CaptureGeometry is the branch of mathematics concerned with points, lines, angles, shapes, and solids.  Without a true understanding of the properties and relationships of these figures, students are likely to struggle as they progress toward higher math courses. Gizmos are such a great fit when building students' understanding of geometry because they provide an interactive, inquiry-based experience that reinforces the conceptual understanding students need.

The ExploreLearning team is constantly working to keep your Gizmo resources updated, and we are pleased to share these math Gizmos have all new Lesson Materials.

Rotations, Reflections, and Translations


Surface and Lateral Area of Prisms and Cylinders

Surface and Lateral Area of Pyramids and Cones

Parallel, Intersecting, and Skew Lines

Constructing Parallel and Perpendicular Lines

Circumference and Area of Circles

Constructing Congruent Segments and Angles

269DETAs always, our goal is real student understanding, beyond just formulas or memorization. For example, in the new Student Exploration Sheets with the Rotations, Reflections, and Translations Gizmo, we work to help students see that each type of transformation is really just a specific rule – a function, essentially – which can be applied to any figure, point by point, to create the transformed figure.  In the Mathematical Background, found in the Teacher Guide, we expand on this “transformations as functions” idea, to help support teachers in how they might want to present this topic.

Reminder: Our updated Lesson Materials have 4 documents (Student Exploration Sheet, Answer Key, Teacher Guide, and Vocabulary Sheet), each available as a .doc or .pdf. (You'll need to be logged in to see all four documents.)

Posted by Heather Jones at 11:31 AM in Math (Real World), Site Announcements, Using Gizmos | Permalink

June 07, 2013

Expert Corner: Get a Jump on Fall with Reflex

Kathleen Kaplan PicKathleen Kaplan is a Regional Manager for Professional Development at ExploreLearning. With a M.Ed. in Secondary Math Education, she taught all levels of high school math before joining ExploreLearning in 2012. Kathy combines her passion for math and science education with her technology background to partner with teachers in increasing student success.

As a teacher, can you imagine what math class would be like if all of your students were fluent in their math facts? If you are already a Reflex customer, you can help make this dream come true for your students and fellow teachers by encouraging your students to use Reflex over the summer. You can find great implementation ideas on our blog that will inspire both students and teachers like this one from Virginia:

An elementary school in Loudon County Public Schools, Virginia, is gearing up for the summer months with a great idea inspired by the summer vacation tradition. They call it a "Reflex Road Trip!"

Reflex Road TripStudents need to collect three "Green Lights" a week on Reflex on their way to their destination. Each traffic light on the log they created symbolizes getting the Green Light, and each light will need to be initialed to prove they did it (and they have five a week on their log in case they go the "extra mile").

At the end of the summer, when students have completed the required Green Light days, they earn a special Reflex Driver's license!

Reflex Driver LicenseWe think this is a terrific idea for encouraging summer Reflex usage, and we look forward to hearing about how the students will be starting next school year with a strong math facts fluency foundation!

If your school or district does not have Reflex yet, talk with our sales staff about ways to bring the program to summer school.

Instead of losing ground this summer, help your students come back more ready to succeed than ever. And, students will enjoy Reflex so much, they may even thank you for it!

Take a free Reflex trial and see why it's the most effective (and the most fun!) math fact fluency solution.

Posted by ExploreLearning at 09:35 AM in Math (Real World) | Permalink

March 14, 2013

Happy Pi Day, from ExploreLearning!

How many diameters does it take to exactly cover the circumference of a circle? And what does this have to do with March 14? If you’re a math person, or just a fan of numerical oddities, I have a feeling you can see where I’m going with this.

It’s Pi Day, boys and girls!

So, first and foremost, please celebrate responsibly. But secondly, celebrate with Gizmos!

Some math Gizmos related to circles, cylinders, and pi:

Circles: Circumference and Area

Prisms and Cylinders - Activity A

Or, if you’d prefer a couple selections from our science Gizmo library:

Measuring Volume

Measuring Trees

So, from the math nerds at ExploreLearning, happy Pi Day to you and your students! We hope you have at least 3.14 times your normal amount of fun in class today.

Posted by Dan at 01:16 PM in Fun/Humor, Math (Real World), Science (Real World), Using Gizmos | Permalink

March 06, 2012

Expert Corner: Pi Day

kurtDan Moriarty is a curriculum writer and editor for ExploreLearning, and our chief "demo movie" maker for Gizmos and Reflex. He holds a Master's degree from the University of Virginia in secondary math education, and he taught high school math before joining ExploreLearning.

Well, March 14 is nearly upon us again. You can refer to March 14 as “3-14” if you’re into shorthand. And of course that makes us mathy types think about π (“pi”), which equals about 3.1415926535…, or if you are okay with rounding, just 3.14.

Pi PieOf course Pi Day isn't just for the math crowd, since this irrational number plays a big role in many science lessons as well. If you’d like to celebrate Pi Day with your students, Gizmos can help.

Basically, the number π shows up whenever you want to measure something circular. Or more generally, whenever you measure anything involving or derived from circles; such as cylinders or sine waves. So if your students are not familiar with π yet, just start by showing them a circle. It could be a plate or a jar-lid or anything else that is basically flat and circular. Review these questions with them:

- What is the diameter of a circle? (Distance across the circle.)

- What is the circumference? (Distance around the circle.)

- And then the kicker... how many diameters would it take to exactly cover the circumference?

The answer to that last question is of course π — thus the formula C = πd — but if students have never seen this before, it could be pretty surprising. The Circles: Circumference and Area Gizmo allows students to explore this relationship quite nicely. For further extension, the National Council of Teachers of Mathematics offers a nice lesson plan around this sort of exploration with real objects.

Measuring TreesIf your students already are familiar with π but you want to apply it to some real-world use cases, we have some Gizmos for that as well.

The Measuring Trees Gizmo gives students a chance to learn about tree rings, as well as delving into an ecology lesson. The Gizmo also allows students to measure a tree's diameter (which is pretty tough in real-life without cutting the tree down!) and its circumference. This is another place that students can discover and use the C = πd formula in context.

Measuring VolumeFinally, one of our newest Gizmos is well-suited for Pi Day. The Measuring Volume Gizmo allows students to find the volume of liquids and solids. They can determine the volume of some solid 3-dimensional figures using formulas. They will discover that the formulas for the volume of a sphere and of a cylinder, since they are circular, involve π. In addition, they'll use the "submerge it in water" technique for finding the volume of irregularly shaped objects.

So, we hope that Gizmos can find a place in your Pi Day celebrations on March 14! And as always, if you celebrate with Gizmos, please do so responsibly. : )

Posted by ExploreLearning at 02:43 PM in Math (Real World), Using Gizmos | Permalink

February 10, 2011

Expert's Corner: Fractions

ThomThom O'Brien has been with ExploreLearning for eight years in a variety of roles, including working with teachers to integrate Gizmos into more effective teaching in math and science. Thom has a Master's degree in Instructional Mathematics and he taught 7th grade math before joining EL.

Fractions are one of the most frequent lessons in a young student's education. Many schools introduce them in grade 2 and continue to teach fraction concepts through grade 7. In fact, the NCTM Focal Points document (2006) and the National Math Advisory Panel (2008) have both recommended that teachers spend larger portions of their time teaching this valuable topic.

Conceptual understanding of fractions is important because they play a pivotal role in higher-level mathematics. Teachers' toolkits for explaining fractions include such diverse resources as pattern blocks, egg cartons, Cuisenaire rods and candy bars. In order to build conceptual foundations, students need to "see" fractions through a variety of different models.

Toy FactoryGizmos are particularly well suited to helping teachers move through fraction models effectively and efficiently. ExploreLearning has many Gizmos devoted to fractions that help teachers provide multiple representations of the concept.

Here are some great Gizmos to try with your students. The Toy Factory Gizmo can be used to demonstrate fractions as a part of a whole or part of a set. The Comparing and Ordering Fractions and Fraction Garden Gizmos can be used to help students compare fractions and set the building blocks in place for adding and subtracting fractions. Also, Gizmos such as Multiplying Fractions and Multiplying Mixed Numbers help students learn to multiply fractions.Fraction Garden

Using Gizmos when studying fractions allows teachers to concentrate on building students’ conceptual understanding. Gizmos allow students to evaluate pictorial representations of sets, manipulate numerators and denominators, and bridge the symbolic fractional representation with the abstract understanding of fractional numeric value.

Posted by ExploreLearning at 04:03 PM in Math (Real World), Using Gizmos | Permalink

January 12, 2011

Expert Corner: Piecewise Functions

Betty KorteBetty Korte is a Regional Professional Development Manager for ExploreLearning. Her credentials include 17 years teaching mathematics, with 14 years as the department chairperson, and a M.S. in Education with an emphasis in teaching mathematics.

Rainfall and Bird BeaksOne of the most exciting aspects of Gizmos is their versatility. I recently watched a colleague demonstrate the Rainfall and Bird Beaks Gizmo and participated in an excellent discussion on natural selection. I thought that a statistics teacher could use the very same Gizmo to study distribution and variance. I worked with the Fraction Artist Gizmo at the elementary math level recently as well, visualizing a high school teacher using the simulation to introduce infinite geometric series (with |r| < 1) in Algebra II.

The Distance-Time Graphs Gizmo has a seemingly endless array of pre-Algebra and Algebra applications, from graph sense to linear theory. Its strength lies in its simplicity. Students discuss (or model) the actions of the runner relative to the graph. Through these discussions, they construct meaningful definitions for such abstract concepts as rate of change, y-intercept, and parallel lines.

Distance-Time Graphs GizmoBecause the runner can change speeds and direction during the simulation, higher-level concepts can also be introduced. For instance, an Algebra II topic that challenges many students is piecewise functions. A piecewise function is simply a function whose definition changes depending on the input value. In theory, this is not difficult for students, but the notation can be overwhelming, especially if it is presented too early in the learning process. A better way to structure the learning is to allow the students to develop a concrete understanding of the function and then move to the abstract formulation.

Students first create a scenario where the runner changes speed or direction during the simulation. They describe what they see in words then translate these descriptions into algebraic sentences with increasing precision. Once this step is complete, they are ready to use the complex notation that defines the function. Because they construct the notation themselves, it no longer seems difficult. Students should also be able to come full circle and create a graph or scenario from the notation.

Watch the video "Using Distance Time Graphs to Study Piecewise Functions" for further details.

Apart from the stated lesson objectives and the curriculum correlations, there are many more "outside the box" uses for Gizmos.

Posted by ExploreLearning at 03:59 PM in Math (Real World), Using Gizmos, Video | Permalink

October 05, 2010

Expert's Corner: Conceptual Foundations in Math

Bridget MulveyThom O'Brien has been with ExploreLearning for eight years in a variety of roles, including working with teachers to integrate Gizmos into more effective teaching in Math and Science. Thom has a Master's degree in Instructional Mathematics and he taught 7th grade math before joining EL.

Have your students worked through math problems, performing the mechanics of each step, but not having the foggiest idea why that procedure works? Some students have become masters at solving problems just by mimicking steps, rather than by really understanding what they're doing, and why. This disconnect can be the result of a lack of a deep conceptual understanding of the topic. Providing students opportunities to visualize the concepts, discuss their thinking, and work in small groups can help students build these conceptual foundations.

Today's mathematics teachers can infuse lessons with practice that supports conceptual learning. A great way to do this is with visual models of mathematical concepts and problems. Obviously, Gizmos are a great support for visual learning. Try just about any math Gizmo — for example Comparing and Ordering Fractions. This Gizmo helps students develop a visual representation of least common denominator and gives them a basis for understanding how to add and subtract fractions with unlike denominators.

Quilting BeeIn addition, teachers can move math classrooms towards conceptual problem solving with the language used in the room. Mathematical communication is saturated with "doer" verbs; write, draw, build, graph, multiply, for example. Simply adding in some "thinker" verbs such as think about, decide, explain, reflect on, and consider, help teachers take students down the road toward more complex mathematical thinking. As an example, try the Quilting Bee Gizmo. As a warm up activity, ask students to reflect on symmetry by having them find it in the world around them or in magazine pictures. Then with the Gizmo, ask them to extend their thinking by considering additional lines of symmetry in the quilts they have been working with.

Read the research behind Gizmos for more information on how simulations can be powerful tools for improving student learning.

Go go GIZMOS!!!

Posted by ExploreLearning at 03:38 PM in Math (Real World), Using Gizmos | Permalink

January 05, 2010

Expert's Corner: Function Machines

Dan Moriarty

Dan Moriarty is a curriculum writer and editor for ExploreLearning, and is also our chief Gizmo video producer. He holds a Master's degree from the University of Virginia in secondary math education, and he taught high school math before joining ExploreLearning.

Functions are a topic that math teachers at many levels teach. Linear, quadratic, cubic, absolute value, trigonometric… these are all different types of functions that students encounter as they advance through their studies.

But what is a function? All too often, the definition sounds something like this: "A function is a relation between a set of elements called the domain and a set of elements called the range (or co-domain), that maps each element in the domain with exactly one element in the range." This definition is technically true, of course, but to most kids, it doesn't make much sense.

So, math teachers search for a simpler way to present this concept, often characterizing them as "input-output machines." An input value goes in, the function machine does something to it, and it comes out as a single output. This works well, but how do you SHOW kids this?

Three related Gizmos - Function Machine 1, Function Machine 2, and Function Machine 3 - provide a nice introduction to functions, using the "input-output machine" theme. For starters, students can select a pre-set machine and send input numbers through it as a guessing game. What is that machine’s function? What does it do to each input number?

Gizmo screenshot

Students can then program their own machines - but not display the function - and challenge their classmates to figure out their function. They can get more advanced as well. The machines are stackable, so they can experiment sending input numbers though multiple machines. This illustrates the concept of composite functions.

Gizmo screenshot

In addition, these input-output pairs can be displayed as points on a graph. This is a perfect way to begin making the connection between a data table and a graph, which is the first step toward graphing functions.

For more ideas on teaching with the Function Machine Gizmos, take a look at the Teacher Guide and the Student Exploration Guide, found in each Gizmo's Lesson Materials. In addition, we have just published a new Teaching with Gizmos: Function Machines movie on our Videos page. All of these short videos help demonstrate how to easily use Gizmos in your classroom.

Posted by ExploreLearning at 03:18 PM in Math (Real World), Using Gizmos | Permalink

May 05, 2009

Expert's Corner: Carnival Probability

Lisa Bickel is the National Training Consultant (Mathematics) for ExploreLearning with a background in educational publishing. Lisa holds a B.S. degree in Applied Mathematics from Pennsylvania State University, and has led the development of math textbooks and software for middle school and high school students.

When I lead Gizmo training workshops, I like to suggest ways for teachers to bring in current events and make learning with Gizmos even more fun. What better way to celebrate spring than to have a Gizmo Spring Carnival! You can challenge your students to use different Gizmos and award tickets, or prizes, for wins.

Spring carnivals are popular at many schools. Games and prizes - a great way to celebrate the end of the school year - and the end of state testing! School carnivals also offer an opportunity to talk about games from a mathematical perspective.

What makes a game fair? In a fair game, a player is equally likely to win or lose. Consider the game shown:

Double your fun!

Only 1 ticket to play!

Spin the spinner twice and multiply the numbers.

Win 2 tickets if the product is odd.

It sounds fair, doesn't it? It's actually not. Because the player wins when the product is odd, the player has a huge disadvantage. Of the 36 possible products, only 9 are odd, so the probability of winning 2 tickets is one-fourth, or 0.25.

To make this game fair, the expected value must be zero. In this case, 3 tickets should be paid for winning. You can see this by finding the expected value of this game. Multiply the probability of winning and the number of tickets won. Multiply the probability of losing and the number of tickets lost. Then add.

Expected value = (P(winning) × tickets won) + (P(losing) × tickets lost)
= (0.25 × 3) + (0.75 × −1)
= 0

What better way to celebrate spring (and the end of state testing) than to have a Gizmos Spring Carnival! You can challenge your students to use different Gizmos and award prizes. Here are a few Gizmos that will work. Have fun!

Try these Gizmos, and others, at a spring carnival in your classroom:

Spin the Big Wheel (Probability) - Have students create a fair game and run the game with 1000 spins. Award prizes for the game with the closest experimental and theoretical probabilities.

Target Sum Card Game (Multi-digit Addition) - Have students play several times and award prizes for landing closest to the target.

Cannonball Clowns (Number Line Estimation) - Have students estimate a distance (such as from New York City to Paris). Award prizes for the closest clown launch to that distance.

Go Gizmos!

Posted by ExploreLearning at 03:13 PM in Math (Real World), Using Gizmos | Permalink

March 03, 2009

Expert's Corner: Pi Day

Dan Moriarty is a curriculum writer and editor for ExploreLearning, and our chief Gizmo "demo movie" maker. He holds a Master's degree from the University of Virginia in secondary math education, and he taught high school math before joining ExploreLearning.

The number pi (π) shows up a lot in formulas involving circles. The area of a circle with radius r is A = πr2. The circumference of a circle is C = 2πr (or C = πd).

The value of π as we have all learned, is about 3.14. (In reality, it's a never-ending decimal that starts out 3.14159265358979…) But, how do we know this? How do you come up with the value of π? (And no, I don’t mean "hit the π button on your calculator"!)

I remember doing a classic activity when I was a student in 5th or 6th grade. We all brought in different-sized cans from home. Using string and rulers, we measured the circumference and the diameter of each can and then divided to get the ratio C/d. After putting our results together, we could see that all the ratios C/d were pretty similar (though not exactly the same). I remember thinking that the "right answer" must be 3. In reality, if we could’ve measured perfectly, we should’ve been getting π. But, measuring and eye-balling aren't very precise – particularly in the hands of 11-year-olds!

In the 2nd century B.C.E., Archimedes came up with a clever way to make a very good estimate of π, using areas. The more sides a regular polygon has, the closer it gets to being a circle. Archimedes used two regular 96-gons – one inscribed inside a unit circle (radius = 1), and the other circumscribed around it – to figure out that π (the area of the unit circle) had to be between 223/71 (3.140845…) and 22/7 (3.142857…). That's a gap of only about 0.002 – not bad!

In 1897, a bill introduced in the Indiana General Assembly suggested in a roundabout way that π equals 3.2. That estimate is off by about 0.06, making it about 30 times worse than the estimate made by Archimedes 2000 years earlier! Luckily, this bill was never passed (and – who knows – may still be stuck in committee today).

Today, computers have computed π to over one trillion decimal places, believe it or not. Fortunately for us, it's normally good enough to remember that π is about 3.14. It is from this value that we have Pi Day, March 14 (3/14).

So, have some fun with it and celebrate one of the most important numbers in mathematics! Happy Pi Day, everyone!

Posted by ExploreLearning at 03:07 PM in Current Affairs, Math (Real World) | Permalink

February 05, 2009

Math and Careers

Interesting blog post on EdWeek.org about how math is used in various jobs.

"Taking math seriously, and learning to enjoy it, will probably make your life easier in high school. It will almost certainly help you get into college and increase your odds of succeeding once you get there.

But what kinds of career options are out there for students with talent in math and a love for that subject..."

We certainly think that using Gizmos will help students enjoy math. Read the full posting here.

Posted by Raman at 09:38 AM in Math (Real World) | Permalink

September 30, 2008

New Prime Number Discovered

Big news in the math world!

Mathematicians at UCLA have discovered a new prime number, and it has over 13 million digits! It took a network of 75 computers to crank this number out. Prime numbers include numbers such as 2, 3, 7 and 29 that are divisible by only two whole positive numbers: themselves and the number one.

The newly discovered prime number is a special type known as a Mersenne prime number, named after Marin Mersenne, a 17th-century French mathematician. Mersenne prime numbers can be expressed as 2P-1, or two to the power of "P" minus one. P is itself a prime number. For the new prime, P is 43,112,609.

Read all about it here:

One of our resident math experts, Dan Moriarty, reminds us of this related numerical fact: Every whole number can be factored into a product of prime numbers. To study factoring numbers into primes, check out this Gizmo: Finding Factors with Area Models

Posted by Julia Given at 04:53 PM in Math (Real World) | Permalink

March 14, 2007

3.14 = pi = March 14, 2007

For those that enjoy math don't forget that today is Pi Day. I never realized that there is also a day called Pi Approximation Day (July 22nd).

Here in the office we were wondering when e Day would be. Using the approach they did for Pi Day, we guess that would be on April 12th. We are looking forward to it!

Posted by Raman at 10:44 AM in Math (Real World) | Permalink

January 05, 2006

Math Goes to Hollywood

If you've ever watched the TV show NUMB3RS then you'll know that the show revolves around a gifted mathematician who helps the FBI solve crimes using math. But how accurate is the math? Well, check this out:

On NUMB3RS, [CalTech math professor] Gary Lorden's job is to help the scripts credibly utilize bona fide mathematical techniques such as cryptography, combinatorics, number theory, and epidemiology statistics in solving crimes. Besides reviewing scripts for mathematical authenticity, he has also been asked to come up with math or physics concepts and equations to provide the mathematical background to what some of the characters are doing, saying, or thinking. The show actually uses a whole team of mathematicians from the California Institute of Technology, including Lorden, Nathan Dunfield, Dinakar Ramakrishnan, and Richard Wilson. Even students can get a share of the glory. David Grynkiewicz served as a hand double, writing the problems on a blackboard and on notepaper.

Cool. Now I like the show even more knowing it's rooted in reality. (Well, at least the math parts. It is a TV show after all).

The blurb above comes from an entertaining article called "The Academic Path to Hollywood" in the most recent Chronicle of Higher Education.

Posted by ExploreLearning at 01:35 PM in Math (Real World) | Permalink | Comments (1)

1 is the loneliest number...but not the largest

Researchers at a Missouri university have identified the largest known prime number. ... The number that the team found is 9.1 million digits long. It is a Mersenne prime known as M30402457 -- that's 2 to the 30,402,457th power minus 1. ... "We're super excited," said Boone, a chemistry professor. "We've been looking for such a number for a long time."

Read more about this exciting number!

Posted by Raman at 09:00 AM in Math (Real World) | Permalink

July 07, 2005

Going to Hollywood? Learn Math!

Do you plan to head to Hollywood to become a screenwriter for shows such as the Simpsons or MadTV? If so, you had better learn a bit of math, since they actually have chic math clubs where writers get together to discuss math on television.

This NPR story had a humorous and humiliating take on the Hollywood Math Club.

Link to NPR story
Simpson's (and Futurama) Math Home Page

Posted by Raman at 10:10 AM in Fun/Humor, Math (Real World) | Permalink | Comments (0)

May 24, 2005

Math Question No Touchdown for Football Fans

From the Boston Globe, now here's a case of "math in the real world" not working out quite as expected:

On an end-of-grade test this month, seventh-graders had to calculate the average gain for a team on the game's first six plays. But the team did not gain 10 yards on the first four plays and would have lost possession before a fifth and sixth play.

The team opened with a 6-yard loss, a 3-yard gain and a 2-yard loss, which would have made it fourth down with 15 yards to go for a first down. The team's fourth play was just a 7-yard gain, yet it maintained possession for a 12-yard gain and a 4-yard gain on two additional plays.

So do the creators of math problems have an expectation to be grounded in reality or is the football example above just a variation on the old "imagine a spherical cow" or "the frictionless ice" or "the chicken and a half can lay an egg and a half every day and a half" types of fiction for the sake of example that comes with the territory in math and science problems?

Posted by ExploreLearning at 09:22 AM in Math (Real World) | Permalink | Comments (2)

May 19, 2005

Should you wear red when taking an exam?

Scientists from the University of Durham in England found that, for the competitions in the Athens Olympic Games, the athlete wearing red won 55 percent of the time.

Does red really make a difference in competition? Some mathemticians feel that since this is a small sample, it is nothing unusual from a statistical point of view.

I wonder if students wearing red perform better on exams. If anyone has any data, feel free to send it in!

I know that my favorite hockey team does pretty well wearning red, and even have red in their name: the Detroit Redwings. It is also well known that Tiger Woods always wears red on the final day of a golf tournament, and he has done rather well. Hmm...

Here are several links to news sites with the story:

Posted by Raman at 10:24 AM in Math (Real World) | Permalink

March 11, 2005

Mmm...Monday is Pi Day

Picture of pi Be sure to celebrate Pi Day on Monday, March 14th. Pi has been studied for thousands of years, and has an interesting history.

Did you notice that March 14th could be written as 3/14, and pi is approximately 3.14. As Mr. Spock would say, "Fascinating."

Posted by Raman at 01:58 PM in Math (Real World) | Permalink

Math Challenges Students on SAT

test form for multiple choice questions As I was listening to NPR this morning I heard that the new SAT exam has increased the difficulty of the math questions. I hope you have all been learning a lot of math from our Gizmos!

Good luck to all on the SAT tomorrow morning. Don't forget to set the alarm clock.

Posted by Raman at 01:30 PM in Math (Real World) | Permalink

January 24, 2005

Can math predict the worst day?

January 24. Monday. Can this really be the worst day of the year? A British psychologist has developed a mathematical formula to predict the worst day for people.

y = [(W + (d-d)) * TQ]/(M * NA)

What do you think of the formula? What are all those variables? Do you think the feelings of people can be predicted from equations?

Now I'm heading off to the dentist. Can you predict how I feel?

Posted by Raman at 09:00 AM in Math (Real World) | Permalink | Comments (0)

October 11, 2004

Can Math Help in Terror War

Can Math Help in Terror War? 

Wired News reports on how order theory and data mining are being used in the global war on terrorism:

Theoretically, [mathematician] Jonathan Farley [of MIT] said, abstract math could help intelligence officers figure out the most efficient way to disable a terrorist network.

Say it's cheaper or more practical to go after a terrorist cell's "middle management" rather than its leadership. How many of those lieutenants would you have to remove in order to disrupt communication between the top dogs and the field operatives? Are there one or two key individuals whose capture would completely cut off the chain of command?

Order theory is all about such questions.

Fascinating stuff. And a cool tonic to any who are inclined to label the study of mathematics as "boring" or as having no practical application to "real life."

Posted by ExploreLearning at 04:21 PM in Math (Real World) | Permalink | Comments (0)

May 07, 2004

Dip Trip for Rover?

The NASA Mars Rovers have provided a wealth of information over the past months, but will this be the final destination for Opportunity?

BBC News Story

Posted by Raman at 09:42 AM in Math (Real World) | Permalink | Comments (0)

February 24, 2004

Random Coin Toss?

National Public Radio ran a very entertaining story this evening. A professor has done research to see whether a coin toss is truly a random event. The conclusion - NO! If you have seven minutes to listen to a fun story...

Listen to The Not So Random Coin Toss on the NPR Site.

Posted by Raman at 06:03 PM in Fun/Humor, Math (Real World) | Permalink | Comments (0)

December 03, 2003

"Math Is Hard"

A recent Washington Post article discusses why learning math can be so difficult for students. More and more research suggests how important gender differences can be:

JoAnn Deak, a psychologist and author of "Girls Will Be Girls: Raising Confident and Courageous Daughters," said most schools approach math in the early grades "as if there is one kind of brain" -- though neuroimaging suggests that most girls develop language skills faster and most boys develop spatial and visual abilities faster. This helps explain why boys traditionally have been seen as "better at math," and why some girls have steered away from it.

Different teaching approaches early in a child's life can make up for these gender differences, Deak said, but most teachers don't try.

Have any of you who are math teachers had any success with trying different approaches to teaching math based on gender?

It'd be interesting, too, to do a study with Gizmos to see if they benefit one gender more than another in learning new math skills.

Posted by ExploreLearning at 09:35 AM in Edu/Tech, Math (Real World) | Permalink | Comments (2)

October 23, 2003

Principia Woochuckia

Yesterday the sky took on its winter look for the first time this autumn. You know what I mean? Emily Dickinson describes it perfectly in her Poem No. 258:

There's a certain Slant of light,
  Winter Afternoons--
    That oppresses, like the Heft
      Of Cathedral Tunes--

Exactly. And fittingly yesterday the arborist (also known as the guy with a pickup and chain saw) dropped off my first load of wood for the coming winter.

Which brings me to today's math topic: How much wood would a woodchuck chuck if a woodchuck could chuck wood?

While we don't yet have a Gizmo to answer this question of the ages, this doesn't mean mathematicians haven't deeply considered the problem. If you're wondering, according to Stephen Lavelle "roughly 1*10 18 kg of wood could potentially be chucked by a woodchuck operating at maximum efficiency (this is only an approximate maximum limit)." There you have it. Heh heh …

Posted by ExploreLearning at 08:41 AM in Fun/Humor, Math (Real World) | Permalink | Comments (5)