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.