Weigh the Wangdoodles

Weigh the Wangdoodles is an algebra-based game on www.mathplayground.com. It is an excellent way of introducing middle school students to basic algebra by using visual representations. Before playing this game, I usually have my students try all three levels of Algebra Scales, another game found on the same website.

I have been asked to explain the steps and reasoning for the game Weigh the Wangdoodles.

The game is based on solving systems of equations and shows three colors of Wangdoodles on scales, two at a time. Pictures of the wangdoodles are shown below the equations given. The variables stand for the colors blue, reed, and green. For example, one problem showed this setup:

G + R =9 B + G = 12 R + B = 17

The steps are simple: Add (combine )two equations , then subtract the third to get a simple equation that can be solved.

So

G + R + B + G = 9 + 12 or 2G + R + B =21

Then subtract

R + B = 17

to get

2G = 4

So G = 2.

You can then use that value to find the other two using the original equations. And the Wangdoodles are weighed! Check out this game!

A Checkbook Unit

For a number of years now, I have carried out a checkbook unit in my seventh grade math classroom. This year, I am in a different position at a different school and I took my unit to a high school special education classroom. This unit takes a week or two to introduce and runs for several months concurrently with other instructional units. I always enjoy doing this unit, even though it is a bit more work for me.

I designed the unit myself, trying to make it a simulation of a real-life experience. I made all the forms (checks, deposit slips, etc.) to look like the forms I get at the bank. For this reason, students and their parents sign an agreement saying they understand the checks that are issued must not be used in the real world, thus constituting fraud. I start with a unit overview telling each student how much they have in their account, how much their paycheck will be each week,and how much they will have to pay in bills each month. I then do a lesson on how to write checks and keep record of the transaction in a ledger (generously donated by a local bank.)
Then students are ready to start the business of writing and depositing checks to a box labeled with the bank name in the back of the classroom. Some of the features of my unit include:
vocabulary introduction, practice, and quiz
explanation of debit cards--then kids come to school to find out they went shopping the night before and are given the receipts and told to record it in their ledger
an explanation of income taxes--followed by a game to determine how much they pay or get back (This is usually done around April 15.)
That's Life Situations--stories that explain extra payments they must make (ie. Your cat Fluffy coughed up an abnormal sized hairball, so you took her to the vet. Make your check for $50.oo payable to County Veterinarian. )


I keep the account ledger for the bank and am able to give a total for the students to compare to their total. This can be laborious at times, but it is very helpful for the students.

This year, the checkbook system has also served as a reward/accountability system in the classroom with bonus checks given for hard work and appropriate behavior, and fines imposed for poor behavior. Food and other items can also be purchased in the classroom with this system. It was very rewarding to find out that after practicing these skills, one of my student's mother opened a checking account for him and is helping prepare for being in the real world soon. So all in all, students are getting math and life skills from this one simulation and I continue to enjoy doing it.

Take a Study Walk

Today I did something I have never done before. My student that needs to learn some multiplication facts said she was feeling tired. So rather than sitting down and going over the flash cards, I invited her to take a walk with me through the unoccupied halls of the school. As we walked, I held the flash cards so she could see them and she practiced giving the correct answer.
Roaming around the halls of the school when we were supposed to be in a classroom made it feel like we were getting away with something. But she was learning and we were both getting a bit of exercise. When we returned to our sedentary spot, she did very well at remembering the correct answers. Go figure--you don't always have to be sitting down to learn!

When Middle Schoolers Still Don't Know the Multiplication Tables

Sad to say, a number of middle school students admit they don't know the multiplication tables. What I have discovered is that they actually know most of them, but a significant few of the facts are pulling them down. Yesterday, I gave one of my students a blank multiplication chart and asked her to fill in the ones she knew for sure and to leave blank (and later highlight) the ones she didn't know. Filling in the 0's, 1's and 2's was a breeze and she was also able to fill in the 5's and 10's pretty easilly as well. Then came the 3's and 4's only leaving a few of the higher ones blank. Because of the commutative property, many were able to then be completed in another column. I told her that 11's followed a pattern which she readily saw and showed her how she could fiure out the 12's by multiplying the digits. That left just ten multiplication facts among the higher numbers which were selected for special attention. The student then made her own flash cards while I made a list of the facts to focus on in the next few weeks. We chose five for her to study and try to remember this week. And today when I saw her, she consistently answered four out of the five correctly.

I'm hoping that this organized approach will help this student to master the basic multiplication facts, and in turn improve her math scores. I think this was a good approach for her, because first of all, the chart showed her that she knew more than she had previously thought she did, and I was able to convince her that learning what she didn't know was not really that big of a job. We've been playing a game that requires multiplying, and every time one of her facts comes up, I clear my throat and she smiles.

Usually success comes when the huge task is defined, and organized into more manageable portions. Consistently working on these little things can have a big payoff later on.

Connecting Math to the Real World : A Great Website

Every now and then, kids will ask "Why do I have to know this? When will I ever use it?"

Even though we are tempted to say "You'll need it to pass the test and ultimately graduate!" there is a better answer. My daughter-in-law, who is a college math teacher,sent me a website that I have found very intersesting and useful for answering such questions. It is:

http://www.xpmath.com/careers/topics.php

At this website, you can click on math concepts such as fractions, algebra and geometry concepts, and many more to get a list of occupations that need and use that concept. Or click on the career, and along with a brief summary and salary level of the job, find out which math skills are used in that field. The possibilities for student research are numerous here!

If there is one message that I want kids to get from using this website, it is that math gives you career options. People who have mastered math concepts have a variety of fields open to them. And in our changing world, that is a good thing!

Overcoming Student Temperaments and Moods in Math Class

Today I observed once again that student moods, outlooks, and temperaments directly influence how much learning takes place in my math groups and also how I teach. For the first three periods today, I introduced the concept of slope to my seventh grade groups. Some were easeier to teach than others because of student responses based on their moods and temperaments, and my role and responses had to be adjusted to meet the needs of each group. Let me explain what I mean.

After I had introduced the concept with some practice examples and discussion of slope in real life, I announced that I would like to know the slope of the back stairs located not far from my classroom door. So we grabbed our measuring sticks and went off to measure the rise and run. We came back to the classroom and used calculators to divide and come to a value for the slope of the stairs. Then I addressed my group again. I said, "We measured the rise and run in centimeters, so what would happen to the slope if we measured it in inches instead?"

Notice that the response to my question was different each period.
The first period response was,(with a moan) "Oh, do we have to do that?"
The second period response was, "Let's do it!" Whereupon they jumped up and grabbed up the measuring sticks and headed for the stairs.
The third period response was, "Well, centimeters are smaller than inches, so I think the slope in centimeters will be smaller than the slope in inches."


Now to my way of thinking, the responses during the second and third periods were much more encouraging than the first period response. But each response was indicative of the moods and temperaments of my students. And if there's one thing I have noticed over the years, it is that I cannot control other people's moods and responses; I can only control my own. However, I do acknowledge that my own outlook and responses can influence those of my students--at least most days.

For example, the mood in my first period group was one of stress and apprehension. One of the students had shared that she had done something the previous day that she would probably get in trouble for today. She said that it was just a matter of time until she was called to the office with another student to face the music. (Shortly after we measured the stairs in inches, her prediction came true and she later returned in tears.) The first period students struggle with sluggishness and low motivation as well as considerable disorganization at home. I'm trying not to exaggerate here, but whenever I see them interested in anything academic, it is a momentous occasion. Today they needed some excitement and positive outlook for this project, so I shared mine. I remained positive, but firm in my desire for the group to perform this task, and they did just fine, discovering that the slope does not change just because we used different units to measure.

My second period group can be slightly A.D.D.-ish. (By that adjective I mean that they might have a touch of Attention Defecit Disorder.) Quite frankly, after the first period response, I was thrilled with this one! We went with enthusiasm to re-measure the stairs. What I wish I had done was to slow them down a bit and ask them again to make a prediction or at least think about and discuss the process more. Anyway, this group too, was able to realize that the slope was the same either way and seemed to understand what I said about slope being a ratio.

Then came the wondrous third period group. They analyzed the situation and used prior knowledge to make a prediction. They took the risk and said what they thought. Then they went to perform the project, thoughtfully and enthusiastically. My role became to guide them through the educational experience. When they returned, they didn't mind that their original premise had been wrong. They saw the relationship between the size of the units of measures and the number it took to measure the distance given. They also were able to analyze why the slope remained the same--because the lengths of the rise and run of the stairs had not changed, only the size of the unit of measure. The analysis and discussion in this group was very rewarding for me.

All three groups learned the concepts I was trying to teach them today, but they certainly did it with diferent outlooks and approaches. Each group needed me to be in a different role. The first group needed me to be positive, yet calm. They needed some curiosity and enthusiasm, so had to borrow some of mine. The second group also needed me to be calm, but also needed me to push them to more thoughtfulness. The third group needed me to facillitate the activity by asking questions, but then step back and let them wrestle with the answers. And all three groups indicated they felt they understood the concept of slope and beamed when I said, " You've done some good work today. Good job!"

Remembering the Nines

"I know most of the multiplication tables. I'm just not good at the nines," a new student said to me recently.

"Then you are in luck!" I replied. "I know a couple of things that can help you."

The first method is pretty widely known among elementary teachers. It is the finger method. A student holds out their hands in front of them, imagining that each finger represents a number from 1 to 10; left to right. Whatever number is given to be multiplied by nine, that finger is bent down. Then count the number of fingers before the bent finger for the tens digit, and the number of fingers after it for the ones digit and you have the answer.

For example, if the student is trying to figure out 9 x 4 they bend down finger number four which is the index finger on the left hand. There are three fingers before it and six fingers after it so the answer they are looking for is 36.

The second method starts with a story that I tell. I learned this over twenty-five years ago when I did my student teaching in Milwaukee. My supervising teacher, Emma Beck told this story to her class, and I have told this story many times since.

Once upon a time there was a man who was looking for a job. He went from company to company and eventually found one that had an opening and wanted to hire him. The supervisor said, "There is just one thing. We have a test that we want you to take. It will only take a few minutes." The man was handed the test and it looked like this: (I have this written on the board ahead of time.)

1 x 9 =
2 x 9 =
3 x 9 =
4 x 9 =
5 x 9 =
6 x 9 =
7 x 9 =
8 x 9 =
9 x 9 =
10 x 9 =

The man took one look at the test and groaned. Now he was really wishing he had paid attention in math class and done the homework assignments he was supposed to do. He had no idea, but he wanted this job and he refused to give up. "Well," he said to himself, "I don't know how to do this, but I'm sure the answers will be numbers and I know the numbers, so I'll just count the problems." And going from top to bottom he started to write the numbers.

1 x 9 = 0
2 x 9 = 1
3 x 9 = 2
4 x 9 = 3
5 x 9 = 4
6 x 9 = 5
7 x 9 = 6
8 x 9 = 7
9 x 9 = 8
10 x 9 = 9


The man stopped and looked at what he had done so far. Surely that couldn't be right, so maybe he should do it again, but this time he would start at the bottom and go up.

1 x 9 = 09
2 x 9 = 18
3 x 9 = 27
4 x 9 = 36
5 x 9 = 45
6 x 9 = 54
7 x 9 = 63
8 x 9 = 72
9 x 9 = 81
10 x 9 = 90

So I then conclude my story by asking: "Do you think the man got the job?"

This is a good time to ask students if they see a pattern. The sum of the digits in each product is nine. The number in the tens digit is one less than the number being multiplied by nine. So the number in the ones digit is what is added to the number in the tens digit to make a sum of nine. Middle school students can write this algebraically.

Number to be multiplied by nine: n
Tens digit of the product: n-1
Ones digit of the product: 9 - (n-1)

These methods really help students remember the nines table!