## Friday, November 16, 2012

### Stay Sharp Math Arcade

For the past ten years, our math learning center has offered a program called Stay Sharp! to help students maintain fluency with basic math facts during the summer months. We convert one of our larger classrooms into a math arcade and provide a variety of learning games and challenges. Students may opt to work in teams or attempt the challenges on their own. This is a favorite among children who attend the math center and they feel quite proud when they return to school with instant recall of all their math facts.

Many of the activities on Math Playground mirror the approaches and teaching philosophy we support at the learning center. Math Playground partnered with a developer of educational games to bring the excitement and benefit of our Stay Sharp! summer program to students who visit the website year round. The Stay Sharp Math Arcade contains practice games that help students with both basic and advanced math skills. Students may compete against other players or they may choose to play against the computer. The games address very specific math concepts and provide engaging learning environments that children enjoy. We currently offer games that reinforce math concepts related to addition, subtraction, time, money, multiplication, division, fractions, ratios, proportions, decimals, integers, and pre-algebra.

## Monday, October 29, 2012

### Geometry and More with Geoboards

The geoboard just might be my all-time favorite math manipulative. There are so many interesting questions that can be explored with this easy to use math tool. When I first introduce students to their geoboards, I encourage open-ended exploration. At this phase, students usually create various shapes without consideration of each shape's properties. Once they're comfortable with this, I then engage my students with specific questions:

- Can you make a rectangle whose perimeter is 10?
- How many shapes can you make whose area is 12?
- Can you make a shape whose perimeter is larger than its area?
- How about a shape whose area is larger than its perimeter?

After my students understand the difference between perimeter and area, I then focus on patterns and relationships and provide even greater challenges:

- Build a rectangle and a square with equal areas. What do you notice about the perimeters? Is this always true? Can you find a counterexample?
- How can you find the area of a right triangle? What about other types of triangles?

Geoboards can be used throughout our students' study of math. Preschoolers can simply design various shapes while  middle school students can explore advanced topics like Pick's Theorem . In addition to geometry, the geoboard can also serve as a tool for exploring fractions and algebraic thinking.

Are geoboards part of your math program? How do use this manipulative to promote mathematical reasoning in your students?

## Sunday, October 21, 2012

### Versatile Pattern Blocks

Pattern blocks have many uses and ours are as quiet as a mouse! Use them to explore transformations, discover symmetry, compose and decompose shapes, investigate fractions, introduce algebraic thinking, create patterns, and engage students in authentic problem solving. These colorful shapes can provide learning opportunities for students throughout elementary and middle school.

One of my favorite ways to use pattern blocks is to enhance my students' conceptual understanding of fractions. The various shapes enable my students to move beyond traditional models for fractions, pizza pie circles and candy bar rectangles, to more elaborate structures. This leads to greater flexibility in my students' ability to visualize fractions which improves their problem solving skills. We begin by defining a particular combination of shapes as one whole. From there, we challenge students to build various fractions of the whole such as 1/3 or 1/4. When they are comfortable and familiar with these tasks, we proceed to the next level. What does it mean to build a shape that is 4/3 or 5/4 the size of the original whole? How about 5/3 or 7/4? Ultimately, we explore fraction algorithms visually. What does 4 ÷ 3/4 look like with pattern blocks? What does such an expression mean? When might we need to divide by fractional numbers in the real world? Modeling algorithms in this way creates a deeper understanding of division with non-whole numbers.

Do you use pattern blocks with your students? Please share your favorite activity in the comments.

## Tuesday, October 16, 2012

### Escape from Fraction Manor

Would your students be able to create and order fractions if doing so meant they could help Cleo the Cat escape from the spooky and dangerous Fraction Manor?

In this fun problem solving game, students collect cards as they journey through three levels of Dr. Fractionstein's castle. Watch out for the monsters! They will try to prevent students from finding all of the cards. When each level is completed they are presented with a series of math puzzles. The cards that the students have collected contain digits that must be arranged into a series of fractions in a given order. The puzzles increase in difficulty at each level.

Escape from Fraction Manor addresses both the Common Core State Standards for Mathematics and the Principles and Standards for School Mathematics.

Common Core State Standards for Mathematics

4.NF.2: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Principles and Standards for School Mathematics

Math - grades Grades 3-5: recognize and generate equivalent forms of commonly used fractions, decimals, and percents

Math - grades Grades 6-8: compare and order fractions, decimals and percents efficiently and find their approximate locations on a number line

## Friday, October 12, 2012

### Real World Math

Year after year, students make the steady ascent along the rocky trails of Math Mountain. Arithmetic gives way to algebra. Polygons lead to polyhedra. Functions progress from linear to quadratic to exponential. But what's at the summit? What will students do with all this knowledge?

When will we ever have to use this stuff?

Math Apprentice hopes to answer that question. Designed for students in grades 4+, Math Apprentice invites students to play the role of an intern at one of eight businesses that use math. Students are given an overview of the math by an animated, virtual employee. They may then choose to freely explore math concepts or solve a specific problem.

The math in the activities is a mix of grade appropriate concepts and advanced mathematics. I think it's important for students to interact with math beyond the standards. This is often where the real joy of math can be found. Even young students can access difficult concepts if they are presented in a meaningful and engaging way.

There are eight careers to explore:
• At the Sweet Treat Cafe, students analyze graphs, scale up recipes, and find the best buy.
• Students learn about ratios and conversion factors at the Wheel Works Bike Shop.
• At Game Pro, students use the Pythagorean Theorem to find the distance between the villain and the hero.
• Students become computer animators at Trigon Studios. They use sine and cosine function to manipulate characters and props in a movie scene.
• While interning at Doodles, students use various functions to create works of art.
• At Space Logic, students match robot speeds to distance vs time graphs and program a space rover to reach its destination.
• At Builders Inc, students must create room shapes whose dimensions meet the customer's specifications.
• While working at Adventure Rides, students determine the height of a roller coaster hill that will give the speed that is needed.

Laura Rose has written a comprehensive summary of Math Apprentice for Connexions in which she describes how Math Apprentice can be used in a middle school classroom. She suggests the site could be the cornerstone of a semester long project about math in the real world. It's my hope that students will spend time with Math Apprentice and internalize its underlying message: math is the path to anything you want to be.

## Monday, October 8, 2012

### Spirograph Math

For our first real world project of the year, I introduced my precalculus students to a Spirograph* toy. I passed out a variety of gears and asked half of the group to rotate a gear around the outside of another fixed gear while the others rotated gears around the inside. Stunning images appeared from both groups.

We compared the two processes and looked for patterns. The images depended on three variables: the radius of the fixed circle, the radius of the moving circle, and the placement of the pen. Would it be possible to derive equations for the position of the pen? If so, could we then write our own Spirograph* program? Much to our delight, we could. You may try our version here

Even young students can appreciate the intricate beauty of the curves generated by mathematical equations, if not the math itself. Give your students time to explore this app. Have them vary the radius of each circle and the distance from the pen to the center. What connections can they make? They may not have the mathematical language or concepts to accurately describe what's happening but they will gain an appreciation for the power and complexity of math - something that could spark a life-long interest.

*Spirograph is a registered trademark of Hasbro, Inc.

## Thursday, October 4, 2012

### Problem Solving with Fractions

This fraction game introduces students to a character with an interesting dilemma. Walker wants desperately to get home but the road has gaps that prevent him from reaching his destination. At each gap, Walker is presented with increasingly challenging tasks involving fraction pieces. Students must help Walker solve these problems in order to move him closer to home. In the process, students apply their knowledge of fractions to an engaging problem within a meaningful context. Students work primarily with unit fractions at first but are later shown how to build any fraction needed.

Fraction concepts covered in this game include:
• unit fractions
• equivalent fractions
• comparing fractions
• adding and subtracting fractions
• multiplying and dividing fractions

This game is best presented as a whole group activity before students explore on their own. An interesting component of the game is a tool called the combinator. After solving simple challenges with unit fractions, students are introduced to higher level fractions such as 2/3 and 3/4. Students build these fractions by physically dragging unit fractions to the combinator. This tool provides a powerful visual for students who are beginning to develop a deeper understanding of fractions and how they are composed.

## Tuesday, September 25, 2012

### Can Your Third Graders Do This?

Solve for A, B, and C using the following equations.

2A + B = 18
B + C = 12
3A = 15

My third graders can! However it looks more like this:

My students think this is great fun. They have no idea they are exploring linear functions or algebraic relationships. All they know is that these problems make them think and they seem to like that.

I usually introduce algebraic thinking problems to third grade students during our unit on multiplication and division. As you know, this topic does go on for quite some time and it can get a little, dare I say, dull. Algebraic reasoning problems give young students a chance to apply their knowledge of basic math facts to fairly complex problems. Problems like this inspire young minds and satisfy their need for a greater challenge. My students are incredibly proud when they are able to solve one of these math problems successfully.

To make things even more interesting, I ask my students to create their own scale problems. We begin with two scales which I improvise with pieces of plain copy paper. I then give the students a variety of objects such as base ten blocks, colored cubes, and geometric tiles. They choose two types of objects to work with and begin creating their scale problems. They have to decide upon a value for each scale and then check it to make sure it works. After that, the students switch places and try to solve the problem. It's one of their favorite activities and it gives me great joy to see them so actively engaged in problem solving.

Give it a try. You won't be disappointed!

## Tuesday, September 18, 2012

### Teaching Transformations

I designed the math game, Shape Mods, and the accompanying Transformation Workshop to provide students with opportunities to practice geometric transformations. The object of the game is to transform the starting green figure into the final red figure using anywhere from one to four transformation blocks. The blocks include translation, rotation about the origin, and reflection across horizontal and vertical lines as well as y = x and y = -x. Once the blocks are in place, students can watch the transformations play out in the order they chose.

We've been projecting Shape Mods on the whiteboard and having our 5th and 6th grade classes engage in spirited competitions. Each team is given a set of pink, white, blue, and yellow discs to represent vertices. This helps students visualize the steps before locking in an answer. Students have also enjoyed practicing independently.

How will you use Shape Mods in your classroom?

## Saturday, September 15, 2012

### Stories for Em

I generally work with students who would be considered above average in school. But every so often a student comes into my life for whom each new math concept is an exhausting struggle. Math is an endless menu of incomprehensible and unrelated steps to be memorized and catalogued. That there could ever be any purpose to, let alone any joy in, this cryptic jumble of numbers, formulas, and procedures is unimaginable.

These are the students that inspire creativity, awaken passion, and elicit reflection. They are the reason I teach and they are the students who make me want to be a better teacher.

Em came to be my student last year at the start of grade 7. A portfolio of sixth grade work revealed a math program that was largely focused on computation. Em had countless examples of multiplication and division of whole numbers, fractions, and decimals. While it was clear that Em attempted to dutifully follow the algorithm of the day, there were signs that something was seriously amiss.

Em needed to prepare for a private school entrance exam. The test primarily consisted of problem solving, pattern recognition, and general mathematical concepts. Em could only confidently answer 2 of the 50 questions on the diagnostic test. As we worked our way through the problem set, numerous content holes, flawed reasoning, and misconceptions were exposed. Em had managed to mimic the computational steps necessary to pass classroom tests and quizzes but had escaped any real mathematical learning. Em did not understand place value, could not order simple unit fractions, saw no relationship among equivalent fractions, did not understand the purpose of a decimal point, and lacked number sense.

Em and I worked together regularly for 6 months without any significant progress. I thought I had tried everything - visuals, manipulatives, real world examples, common language, even acting out problems. Just as I was about to give up, the connection I so desperately sought finally made an appearance.

We were exploring decimals when Em called the decimal point a period. While privately lamenting this student's misunderstanding, I wondered if perhaps there might be something to it. I asked Em to explain further. Em went on to tell me that the decimal point marks the end of the whole numbers and the start of the "smaller pieces", the pieces that weren't quite whole yet. In Em's mind, that was very similar to the way a period ends one thought but can also signal the start of a new one. From there, Em told an elaborate tale of a fantastical world of whole numbers and pieces, how they are kept apart by the will of the decimal point, how the wholes and pieces organize themselves into groups by size, and how these groups are either 10 times bigger or 10 times smaller than groups on either side. Em also described how the decimal can make numbers grow or shrink by moving its location and that it is always present even when there are no "pieces".

Em understood decimals better than any 7th grade student I had ever met. No standardize test in the world would ever ask Em to tell the story of the wholes and the pieces. Yet, that was the only way Em could confidently share her knowledge.

Since that time, classes with Em have been rather magical journeys into far-off lands where numbers and symbols come to life and tell their stories. Em is in 8th grade today and is struggling with the rigidity of her pre-algebra course. Her class has been studying the distributive property and combining like terms. Em confided that she just didn't get it. I mentioned something about helping the expression escape from its parentheses prison. Before long, Em had crafted a story about the number guard that stood watch outside the prison, the banning of subtraction, and the look-alike law.

And every homework problem was solved perfectly.

## Wednesday, September 12, 2012

### Algebra for All

Solve for A, B, and C using the following equations:

A + B = 26
B + C = 45
C + A = 33

How would you go about it? Would you begin by combining equations? Or would you start by making substitutions? Is this a problem a 5th grader could solve? Without guessing and checking?

What if the problem looked like this?

Is it easier to solve? It's exactly the same problem, isn't it? The visual representation of the problem makes a huge difference, though. Now it's obvious that the two green blobs are the same size even though they exist on different scales (or different equations). And the two blue blobs are the same. Aha, the two red ones as well.

I have prealgebra and algebra students who ask if the A in one equation has the same value as the A in another, whether the two equations are part of a system or not. Imagine if those students had been solving math puzzles like this one throughout elementary school. Would their concept of variable be more clear?

Elementary age students at the math center think problems like this are great fun. They have no idea they are exploring linear functions or algebraic relationships. All they know is that these problems make them think. Algebraic reasoning problems give young students a chance to apply their knowledge of basic math facts within fairly complex scenarios. Problems like this one inspire young minds and satisfy their need for a greater challenge. Our students are incredibly proud when they are able to solve one of these math problems successfully.

How would a young student solve such a problem? We ask our students to compare any two scales and find what they have in common. Let's take the first two scales. Students will point out that the blue blob is common to both. Then we have them look for differences. Students notice that the scale weights differ and the partner blobs are different. We ask them what they think might be causing the weight on the second scale to be greater. It's obvious to students that the bigger red blob on scale two is causing an increase in weight.

Once they understand the effect of changing the partner from a small green blob on the first scale to a bigger red blob on the second, we can look at the quantitative aspects of the problem. We then ask what is the difference in weight between the two scales. Students will do the computation and find the difference is 19. That's the difference between scale one and scale two. What else does this number mean? What other difference does it describe? Students will relate 19 to the difference in weight of the green blob and the red blob. The red blob weighs 19 more weight units than the green blob.

We write this as:

R = G + 19

Now we have a relationship between the red blob and the green blob. This relationship tells us that if we replace a red blob with a green blob plus 19 weight units the scale reading will stay the same. So let's do it.

We head over to third scale. The equation there is:

G + R = 33

We replace the red blob. We get:

G + G + 19 = 33

What if we take 19 weight units off the scale?
What will the scale read then?

33-19 = 14

Our new equation is:

G + G = 14

At this point, students recognize this as a doubles problem and easily find the value of G to be 7. They then use this value to find the weights of the other blobs.

Modeling this problem with young students as a whole group activity is a very powerful. They excitedly share their insights and answers. We'll do several of these together before they work independently to solve similar problems. Eventually we make our way toward the original abstract problem. We replace the green blob with the letter G, then the blue blob disappears and gives way to the letter B, and finally we part ways with the red blob and bring out the letter R. The letters remain on the scale however so the context is reserved. Once the students are comfortable working with letters, we then remove the scale. Students solve systems of three equations by the end of 5th grade. More importantly, students learned how to think through abstract problems, a skill that will forever be of value.

## Monday, September 10, 2012

### Tactical Rescue Missions for Intergalactic Good

While foraging for markers, a student in one of my math and programming classes stumbled upon some old science equipment I keep in the closet. The air-propelled rocket launcher was promptly brought out of retirement and set up in the long rectangular space at the rear of the math center. It wasn't long before a rousing game of "hit the target" was underway. Based on the number of times the soft foam rocket came perilously close to my head, it would seem I was the target although everyone agreed they were aiming for the algebra poster.

Sensing an opportunity among the chaos, I grabbed a hula hoop out of the closet of science and placed in on the floor on the other side of the room. The hula hoop proved a more interesting target and it wasn't long before the discussion headed in the direction of angles and velocity. An impromptu lesson on projectile motion ensued.

We measured launch angles and landing distances and recorded flight times. We refined our understanding of velocity and used horizontal motion data to find starting velocities. Through our experiments, we hit upon combinations of velocities and launch angles that would land our rocket inside the hula hoop. One student, who grew frustrated with the trial and error process, asked,

"Can we calculate the velocity and angle if we know where we want the rocket to land?"

Since this was a math and programming course, I suggested we write a program that models the experiment and perhaps make a game based on this student's question. After several weeks of brainstorming, coding, revising, and experimenting, we came up with this:

There are both elements of game design (points awarded, incentives for calculations and good guesses, increasingly difficult levels, and interesting sound effects) and elements of instructional design (timed flights, recorded data, and explanation of math equations). We also replaced the theme of destruction typically seen in these games with a more positive rescue mission plot. I am incredibly proud of this group and all that they have accomplished this year. I know some will be moving on but I hope we can continue our work next year.

## Friday, September 7, 2012

### Stumbling Upon Misconceptions

A seventh grade student came to the math center to prepare for a test on fractions. She brought in a review sheet with various practice problems which she completed with time to spare. The student, somewhat anxious about the test, asked if I could make up problems on the whiteboard. I complied and wrote out the following problem:

My student proceeded to simplify by canceling common factors.

And then declared this was all she could do.

Me: I think you can simplify this further.
The student tested adjacent numerator-denominator pairs: 24 and 7, 5 and 8, 8 and 27
Student: No, that's it. I can't simplify this.
Me: Have you tried 24 and 27?
Student: (spins around) I wanted to but aren't they too far apart?
Me: Nah, I think the limit is around 3 feet.
(I couldn't resist.)

After 45 minutes of rather predictable practice problems, we finally had a teachable moment.
It makes me wonder what other misconceptions our students have that we never uncover. And it reminds me to never stop pushing past the surface.

## Tuesday, September 4, 2012

### Euclid Comes To Programming Class

A student in my middle level programming course brought in a word problem from school.

"The Billy Bonkers candy factory is having a contest. The candy makers placed a silver ticket in every 600th chocolate bar and a golden ticket in every 720th chocolate bar. Anyone who purchases a chocolate bar containing both tickets wins the grand prize. If 10,000 chocolate bars are sold, how many grand prize winners will there be?"

Once we determined this was a Least Common Multiple problem, we talked about various ways to solve it. One student suggested writing out the first few multiples of each number and looking for common numbers. Another student had learned about factor trees and knew how to use prime factors to find the LCM. Yet another student showed us how to use a Venn diagram to organize the prime factors.

While discussing the relative efficiency of each method, one student declared, "It would take forever to do this if we had a lot of numbers."

"It sure would seem that way," I thought to myself, relishing the near perfect segue this statement introduced. And before I could get the words out, another student asked if we could write a program.

"We certainly could try," I said aloud.

We began by attempting a simpler problem: finding a common multiple of two numbers; not necessarily the smallest one. Everyone agreed that the easiest method was to multiply the numbers together. We then compiled a list of products and LCMs for various pairs of numbers. I wanted the students to find a connection between the product and the LCM. How do the divisors relate to the original number pairs?

2 and 5: Product = 10; LCM = 10 (divide the LCM by 1)
3 and 6: Product = 18; LCM = 6 (divide the LCM by 3)
4 and 6: Product = 24; LCM = 12 (divide the LCM by 2)
8 and 12: Product = 96; LCM is 24 (divide by LCM by 4)

The connection eluded the students and it took quite a few hints to help them see that the product of two numbers is related to their LCM by the greatest common factor. We had the beginnings of an efficient algorithm.

LCM = (number1 x number2)/GCF

Or did we? Had we just shifted the difficulty in finding the LCM to the similarly difficult task of finding the GCF? Is there a quick way to find the GCF of two numbers?

We talked through some known methods (listing the factors of each number, using prime factorization, applying the venn diagram) and opted to look for a better way.

We uncovered some interesting facts about the GCF. It's never larger than the smaller number or the difference between the two numbers. Consecutive numbers always have a GCF of 1 making them relatively prime. One student suggests an algorithm that tests if each number, from the smaller of the pair down to 1, evenly divides both numbers. The first number that worked would be the GCF. We look at the numbers 6 and 20. The program would test 6, 5, 4, 3, 2, 1. The number 2 evenly divides 6 and 20 and would be the GCF. I asked what would happen if the two numbers were as large as the numbers in the word problem - 600 and 720. Is it really necessary to test all the numbers from 600 to the GCF? Someone suggests testing only the factors of 600. Nice! But how do we find all the factors of 600? And even if we had an algorithm for this, we'd have to test several factors before hitting upon the GCF.

Time to introduce Euclid's algorithm. We divide the larger number by the smaller number and find the remainder. If the remainder is zero, the smaller number is the GCF. If not, we continue the process. During each iteration, the original smaller number becomes the dividend and the remainder becomes the divisor. When we finally get a remainder of zero, the last divisor is the GCF.

Finding the GCF of 600 and 720 became an astonishingly easy task.
720/600 = 1 r 120
600/120 = 5 r 0
GCF = 120

The students were amazed! Two steps. That was all. After trying a few more number pairs, the class was satisfied that the procedure was foolproof. Back to the original problem; finding the LCM of a group of numbers. The equation we had discovered earlier was:

LCM = (number1 x number2)/GCF

The plan was to take the first two numbers in the set, apply Euclid's algorithm to find the GCF, then use the GCF to divide the product of the two numbers. We would repeat the procedure with the current LCM and the next number in the set until we reached the final number and, therefore, the final LCM.

Naturally, this begs the question, "What can we do with this?", which I will ask the class at our next meeting.

## Thursday, August 30, 2012

### Busy Learning

Many of my students attend a private K-8 school that offers a very traditional math curriculum. Students in grades 5, 6, and 7 spend many months studying shopkeepers' math. The main focus is percentages - discounts, sales tax, tip, simple and compound interest, commissions, annuities, etc. Each of these variations is taught as a series of formulas to be memorized. The vocabulary is beyond the comprehension of most 12 year olds. Even the numbers themselves are quite tedius. Students spend hours calculating 17.25% of \$12,650.85 by hand.

A group of 7th grade boys came in with homework from their new math unit - an introduction to linear relationships. They each had a worksheet with about 15 tiny grids upon which lines were drawn. The students were asked to calculate the slope of each line. That was it. No context and no explanation. That was their starting point.

"We don't really know what we're supposed to do," said one student.
"Yeah, it's just a bunch of lines," said another.
"We're lost," said the third member of the group.

The other two solemnly nodded in agreement.

Maybe it was the fact that I had just watched Stand and Deliver over the weekend and wanted to be like Jaime Escalante but I grabbed the worksheet, pretended to scrutinize it, and delivered the following line with exaggerated astonishment and admiration.

"Whoa!!! You guys are doing this? Wow! I can't believe it."

"What do you mean? What is this stuff?," they asked, almost in unison.

I walked to the door, checked to see if anyone was around, then closed it.

"Alright, I'll show you," I said.

One of the advantages of working in a defunct science center is access to a closet full of physics toys. I grabbed a few matchbox cars, ramps, and books and set up an impromptu math lab. What ensued over the next 45 minutes was a spirited look at independent and dependent variables, data plotting, best fit lines, and, of course, the meaning of slope.

At the end of the class, one of the fathers came in to pick up the students. I was putting away supplies but I heard him ask his son if he got his homework done. The boy said no. The father, clearly upset, asked his son for an explanation. The boy said....

"Dad, I can do my homework later. I didn't do it here because I was busy learning."

Yeah!

## Tuesday, August 28, 2012

### A Math Problem Revisited

One of the advantages of teaching in a learning center rather than a classroom is that I often get to see the same student at various stages of his or her academic life. A student who is in one of my precalc classes today was the very first student I ever taught. Back then it was number facts and place value. Today it's trig identities and parametric equations. While I haven't seen this student every year since first grade, we've reunited for key math events. We were together for whole number operations, met again for fractions, and stayed close through algebra and geometry. The opportunity to observe students over the entire length of their K-12 careers has been an incredible learning experience. It has given me a unique perspective regarding how children learn math and how various math programs succeed or fail in preparing students for each new stage of concept development.

One of the math classes I'm teaching combines problem solving with computer programming. The group is made up of 8th graders I had taught a few years back. They had participated in a math olympiad course. One type of problem that appeared frequently dealt with simultaneous equations. For example,

At the movie theatre, tickets cost \$6 for adults and \$5 for children.
A total of 24 people went to the movies and paid \$128 for tickets.
How many children went to the movies?

At the elementary level, the students were encouraged to create a chart to keep track of the number of adults and children, and total cost. The process is simply an organized version of guess and check. The efficiency of this method is dependent on the extent of number sense developed by the students and the number of potential solutions.

I thought it would be interesting to revisit this type of problem four years later from a different angle. If we removed the constraint of the total number of people, would we be able to find the exact number of reasonable solutions to the problem? In this case, only positive integer values would be sensible since the problem deals with whole people.

We had done some hand graphing of linear equations, so I led the group in that direction. The students identified the variables and came up with an equation to graph, 6x + 5y = 128. They determined that there were 4 combinations of adults and children that would result in a total cost of \$128. The graph showed that while there were an infinite number of solutions to the equation, only 4 solutions made sense in the context of the problem.

We then talked about other situations that would result in an equation of this type and developed the general equation, ax + by = c. Next, I challenged them to create a computer program that would find all positive, integer solutions of this generalized equation. From the graph, the students knew that we needed to test all x values from zero to some equation dependent maximum value. The students were familiar with programming loops and agreed that this would be the best approach. Here's the pseudo-code version of what the group invented.

var a:Number;
var b:Number;
var c:Number;
var x:Number;
var y:Number;
var nMax:Number = c/a rounded to the next highest integer;

for(all values of x from 0 to nMax)
{
y = (c-ax)/b; //rearranged equation in terms of y
if(we divide y by 1 and get a remainder of zero)
{
then y is an integer
this xy pair is a solution
store these values
}
else
{
y is not an integer
this xy pair is not a solution
}
}

So far so good. My students were very proud that they were able to work out the logic in this program. They had to figure out a way to calculate the maximum value of x, determine if a number was an integer, and decide what to do when y was an acceptable value. Each of these areas led to rich discussions and deeper connections. Our next step is to find a way to display our results in both graph and chart form.
If I'm fortunate enough to teach these students a couple of years from now, perhaps I could challenge them to develop an algorithm based on parametric equations.

## Monday, August 27, 2012

### Fun For Whom?

I drive my students crazy.
I know I do.
And this is why.

Student: Is this right?
Me: What do you think?
Student: I don’t know.
Me: Well, how did you get your answer?
Student: I just did what you showed me before.
Me: Ok, but what does that look like for this problem?
Student: (utterly exasperated) Why can’t you ever just say yes or no?

Me: (to myself) Because saying yes or no is the easy route and you deserve better than that from me. Because doing so discourages analytical thought and makes you dependent on the approval of others. Because this is the one thing I can do to ensure you develop confidence in your own problem solving abilities. Because this tiresome process is my gift to you; you just don’t know it yet.

Me: Oh, it’s much more fun this way!

## Friday, August 24, 2012

### Save the Zogs

"Four frightened Zogs have left the safety of their planet and are floating around in space. The Duplicators, a band of space travelers with the ability to imitate others, have infiltrated the floating Zogs. This is making the rescue mission very difficult.

Fortunately, the Zogs are very clever. They can assemble themselves along a straight line path. The Duplicators cannot exist on this path. If the rescue team can determine the equation of this line, then the Zogs will be saved. The Duplicators will be left behind.

To rescue the Zogs, you need to learn as much as possible about linear equations and the lines they create. What happens when the slope is zero? What effect does the y intercept have on the position of the line? The more you know, the more Zogs you can save."

The game begins with simple horizontal and vertical lines, builds to equations with non-zero slopes passing through the origin, and ends with equations that have both non-zero slopes and y-intercepts. A tracking tool enables students to rotate and move a line into position to help them visualize whether or not the line passes through the correct Zogs. Once the visual is in place, students must then identify the equation the describes the line.

What middle school student can resist the plight of the Zogs?

## Wednesday, August 22, 2012

### Middle Zeros

I had an opportunity to spend some time with a very enthusiastic group of third grade students at a local school. They were subtracting three digit numbers by regrouping and were doing quite a spectacular job. That is, until we reached the dreaded Middle Zero.

The problem was 601 - 347. The following is an actual transcript of the dialogue that ensued.

Me: Does anyone know how to do this problem?
Student 1: You can't take away 7 from 1 so you have to make the zero a 9.
Me: You mean I can just turn a 0 into a 9? How can I do that?
Student1: Well, you make the 6 into a 5 and then the zero becomes a 9.
Me: I still don't understand how the zero becomes a 9. And why does the 6 become a 5?
Student2: There aren't any tens so you have to borrow a ten from the hundreds column.
Me: Ok, that makes sense. How many tens are in the hundreds column?
Student2: 6

and so on...

It was fairly obvious that they had been introduced to the mechanics of regrouping but they had not achieved a true understanding of the process. The beauty of mathematics and the amazing "aha!" moments are revealed only when we investigate the math beneath the surface.

Third grade students are generally very competent when it comes to regrouping tens to ones in the case of 85 - 29 or 371 - 158. Problems arise when there are no tens in the tens column. This requires a visit to the hundreds column whereupon one of the hundreds is regrouped as 10 tens.

For example, in the problem:

402 - 138

the number 402 would have 3 hundreds, 10 tens and 2 ones. Now it's possible to regroup one of the tens into 10 ones and add those to the ones column. The final version of the number 402 is 3 hundreds, 9 tens, and 12 ones.

While this is no doubt trivial and rather obvious to you as a teacher or parent, third grade students find this incredibly confusing. In fact, I'm convinced that even with the benefit of base ten blocks and dozens of practice problems, students are still not really sure what's going on here.

What if we approach problems with middle zeros a little differently. First, help your students see numbers in terms of tens. If you use the number 80, ask how many tens that is. If the number is 527, help them see not only that there are 5 hundreds, 2 tens, and 7 ones but that there are 52 tens. Once that is understood, you can explain regrouping numbers with middle zeros more easily.

Let's revisit the problem, 402 - 138. A ten is needed to make 12 ones. Turn this into a single step process by having the students think "40 tens". Now they can use one of those tens to change the number of ones from 2 to 12 and they'll have 39 tens left. It's much simpler, there's less crossing out, and students really get it.

This method works with numbers containing more than three digits as well.

How do you teach regrouping with middle zeros?

## Monday, August 20, 2012

### Comparing Fractions

My fourth grade students are learning how to compare fractions. They've mastered this concept for very specific types of comparison problems, for example, when the denominators are the same and when the numerator is 1. Now I'm trying to teach them how to decide if a fraction is less than or greater than 1/2. They seemed to really grasp the idea that there are many ways to express the quantity 1/2 using fractions - 2/4, 3/6, 4/8, etc. The students recognized that the denominator is always double the numerator. In fact, one particularly clever student suggested the fraction 1/8 / 1/4, remembering the relative sizes of the fraction pieces we had used during the beginning of the unit. Despite this, my students are having difficulty determining whether a fraction such as 3/7 is more or less than 1/2.

To help students visualize more difficult fractions, I designed a simple mathlet that draws two partitioned rectangles whose shaded sections represent the fractions typed in by the students. My students set one fraction to 1/2 and then alter the other fraction for comparison. I think they are starting to develop a better sense of relative sizes among fractions. Feel free to try it with your own students.

## Friday, August 17, 2012

### Invert and Multiply

When it comes to the dividing fractions, I have to admit that I graduated from the school of "Don't Ask Why, Just Invert and Multiply". Were you taught something similar?

There are so many things wrong with that approach. It reinforces misconceptions that students may have about the mysterious and magical nature of math. Is dividing 1/4 by 1/3 really so incomprehensible that we shouldn't bother to explore its mind-boggling complexities? Fortunately for the current generation of students, that sort of teaching disappeared along with transistor radios and hoola hoops.

Or did it?

I had the opportunity to teach a 5th grade class recently. They had been immersed in a unit on fractions for several weeks and were just wrapping up a section on division. To find out what the students had learned so far, I placed the following problem on the board:

A lively conversation ensued.

Me: Does anyone know how to solve this problem?
Student1: Make it a multiplication problem.
Me: Like this?

Student1: No, you have to flip over the fractions.
Me: Oh, you mean like this?

Student1: No, they don't both get flipped. Only one of them.
Me: Which one?
Student1: I'm not sure. I forget which one.
Me(to myself): Uh huh. Where have I heard that before?
Me: Does anyone have a different approach?
Student2: Well, half of 2/5 would be 1/5 so that's the answer.
Me(to myself): Ok, there's some number sense here but we're not quite there yet.
Me: Do you think the problem is asking us to divide 2/5 into 2 equal parts?
Nearly the entire class nods a self assured yes.
Me: If that was true, wouldn't the problem read: 2/5 divided by 2?
The class expresses a collective look of bewilderment.
Me: Actually, the problem asks you to figure out how many halves there are in the fraction 2/5. It works just like whole numbers. When you are given a problem such as 40 divided by 8, you are being asked to figure out how many groups of 8 are in the number 40. So how many halves are in the fraction 2/5?

Lots of blank stares. I better try another approach.

Me: Who thinks there's at least one 1/2 in 2/5?

Way too many hands go up. Ok, we've got a number sense problem here. The division lesson is over for now. Where's that giant number line? We've got some work to do!

The old invert and multiply mentality is alive and well today and continues to ensure that students have absolutely no idea what is going on with division by fractions. While invert and multiply is a tried and true method for dividing fractions (provided one remembers which fraction to invert), it is not the only way to divide fractions.

Suppose you were given the following problem:

You don't even need to invert and multiply in this case. You can actually just divide the numerators and the denominators independently and get a more direct answer.

Students should be taught to assess each fraction problem to determine whether or not it would be easier to just simply divide. Perhaps then, students wouldn't automatically apply a poorly understood algorithm to every problem. Such a skill would serve them well when faced with more advanced math down the road.

How do you teach division of fractions to your students?

## Friday, August 10, 2012

### Animated Word Problems

Engaging students has always been a considerable challenge for teachers. Fortunately, there are a vast amount of resources and technologies available to help meet that challenge. One resource for upper elementary and middle school students is a catalog of animated word problems that guide students through the process of solving multi-step problems. The narrator and animated host is a young woman named Infinity Quick, or IQ for short. The introductory animation shows IQ receiving a new word problem on her computer. The movie is paused, giving students a few minutes to tackle the problem on their own. They may then proceed to watch the problem solving unfold.

The solution to each problem is presented with diagrams, charts, and calculations. The video solutions may be paused or replayed as many times as needed. Some of the video solutions contain interactive components, allowing students to further connect with the math lesson. After the video presentation, students are invited to try a similar problem on their own. They are then taken to the practice room where a new interactive problem is presented. If help is needed, there are a series of hints available to the student at the press of a button. The hints make use of illustrations that are familiar to students who have previously viewed the video. To further assist students, problem solving tools such as a calculator and a sketch pad are also available.

There are currently 19 problems available covering topics such as fractions, ratios, percentages, probablility, geometry averages and algebra.

## Monday, August 6, 2012

### Learning in Mathland

The book that influenced my approach to teaching mathematics to children is Mindstorms by Seymour Papert. Subtitled Children, Computers, and Powerful Ideas, this book provided a glimpse into a world of which I knew little but eagerly wished to learn more. As promised, powerful ideas emerged quickly and continued to be revealed at every turn. The most compelling idea for me, the one that figuratively jumped off the page, was the author's concept of Mathland. Papert envisioned a learning environment designed to make the development of mathematical concepts a more natural process. This novel vision captured my imagination and laid the groundwork for what would later become our after school math center.

So many questions arose. Could math achievement be improved by weekly trips to an authentic Mathland? What would such an environment look like? What did we want students to learn and how would they learn it? After a few false starts, we eventually succeeded in making our version of Mathland a reality. Each week students were transported to a world of numbers, mathematical ideas, and meaningful challenges. Gear Math was a hands-on, problem solving program designed to teach multiplication, division, ratio, and proportion to upper elementary students. Students programmed robots to traverse number lines and coordinate systems and designed containers based on volume and surface area restrictions. Most importantly, the residents did very well in school math despite the fact that curriculum objectives rarely overlapped.

How was that possible? Students learned to think and act and approach problem solving much like a mathematician would. This was accomplished by immersing students in credible learning environments and by providing challenges that were not only engaging but also demanded intellectual investment. Getting the right answer was important but it wasn’t the only goal. Making sense of those answers was critical. Explanations were reworked and refined. Multiple representations were encouraged. Our students developed a familiarity with math that all the practice sets and timed drills in the world could never accomplish. And, most importantly, it laid the foundation for future success in math. As our students moved from elementary to middle school, high school, and beyond, they always carried a piece of Mathland with them.