ExploreLearning News : Math (Real World)

Expert's Corner: Fractions

Thom 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.

Gizmos 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.

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.

Expert Corner: Piecewise Functions

Betty 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.

One 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.

Because 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.

Expert's Corner: Conceptual Foundations in Math

Thom 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.

In 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!!!

Expert's Corner: Function Machines

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?

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.

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.

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!

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!

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.

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:
http://www.msnbc.msn.com/id/26914730/?GT1=43001

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

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!

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.

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!

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

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?

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:

• NPR: Study: Red Is the Color of Olympic Victory
• CNN: Researchers: To win in sports, wear red
• BBC: Reds have a sporting advantage

Mmm...Monday is Pi Day

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."

Math Challenges Students on SAT

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.

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?

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."

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

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.

"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.

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 …