A New Math Word

My husband's seventh grade class has come up with a new math term! In science they have learned that the prefix "eu" means true. So eubacteria, as opposed to archeabacteria, means true bacteria.  And eukariotic means true nucleus. 

In math, they are learning about variables. Now the definition of variable is "a value that can change". In the the equation y=2x + 5, the variables are x and y. And their values can change.

But sometimes variables or letters are used just to hold a place and the value is pre-determined. An example of this is in the equation 2x +3 = 7. In this case x=2 and the variable is holding the place of the 2.

So the new math word? Here it is: EUVARIABLE!!! It means a true variable and not just a place holder.

Congratulations to my husband and his seventh graders! Good thinking!

Integer Hi-Lo

I came up with a new game and my last period eighth grade group named it! I needed to review multiplying integers with them.

I started by dividing the students into groups of two, three, or four and giving them a deck of cards. I wrote on the board that black cards were positive and red cards were negative. I told them it was that way with the phrases "in the black" and "in the red."

The dealer hands out all the cards to the players. The cards are placed face-down and the players don't look at them. Whoever has been designated to go first puts out one card. If it is black, they are looking for the highest product; if it is red, they are looking for the lowest product. Then the first player lays out another card and announces the product of his two cards. The next player puts out two cards and announces their product until all players have done this. They then decide who has the high or low depending on what they are looking for based on the first card out. The winner takes all the cards and puts them under their deck face-down.

Players are trying to end up with all the cards. If there are more than two people playing, I cut the game off when one person has no cards and the person with the most is declared the winner. If the class ends, the person with the most cards wins.

This is a good game not only for multiplying integers, but also comparing integers. Sometimes students struggle with the concept that -50 is less than -1, and this game helps them review magnitude.

A Kiss For You!

This morning I was going over multiplication with one of my math groups. After doing some basic drill, we were using individual white boards to practice multiplying multi-digit numbers using the standard algorithm. I would give them the problem and wait for them to do it individually. Then I would do the problem on my own white board using think-aloud, so they could compare their work. They were doing a fine job, and only making a few mistakes.

At one point I went to my desk and wrote a note on a piece of scrap paper: "I am going to do the problem wrong. I wonder if anyone will notice. A kiss if they do!" I then proceeded to do the problem incorrectly. It was just one mistake that I made; I said that 7 x 5 =40. At first no one said anything. Then when I paused, one of them said, "What?" I replied that I would be happy to do it again, and proceeded to do that section of the problem again making the same mistake. This time, they spoke up. I showed them my note and praised them for finding my mistake and handed over a Hershey kiss to each one.

They were thrilled! And so was I. A big part of doing math accurately is watching for and catching mistakes. My initial motivation for this was that I had a bag of kisses in my desk that I needed to use up. I had bought the candy for an activity with another group, and being diabetic myself, had no other use for them. But I was reminded that making a mistake and praising students for catching it is a good thing to do!

There's A Lot of Math in a Tree

On the day before school started, I was asked to teach a last period class for remedial students that would help them improve their math skills in addition to the regular grade 8 math class they already were taking. I welcomed the opportunity and it has been a good experience this fall. But there was no room assigned for the class that first week, so students were told to meet me at the office and together we walked outside to the woods behind the school. I handed them meter sticks and clipboards, told them to find a partner and sent them to measure trees in the woods. They were to find the d.b.h. or diameter breast height. This gave us some wonderful data for reviewing measures of central tendency such as mean, median, mode and range. Using all the data, we were able to make some generalizations about the trees in the forest.

Currently, my husband is in a science/math forestry unit with his seventh and eighth graders. The extension forester comes in and they have been doing some incredible explorations that have captured the students' interest and strengthened their math skills and science knowledge at the same time. They are measuring trees, calculating board feet and projecting the dollar amount that various wooded areas are worth.

At this time of year when the autumn colors are so vibrant, it's nice to find so much math in the trees!

Everything Has to Do With Math

I was talking to my friend Catherine. She was saying that she didn't like to put on a wet swimsuit. So I told her the trick I use if I am in a place where the only suit I have is wet: run it under hot water from the sink and wring it out before putting it on. She enthusiastically welcomed my tip and said that it was something worth putting on my blog for everyone to see. When I replied that it didn't have anything to do with math, she replied, "Everything has to do with math!" She proceeded to note that it had to do with the measure of body temperature compared to the temperature of the wet suit. When the temperature of the suit is raised to a temperature equal to or higher than body temperature, there is less discomfort. Not a bad observation for a social worker!

So is it true that everything has to do with math? I have thought about asking students to tell how they used math over their summer vacation. While most will think that they didn't at all, I hope I can eventually get them to realize that math is associated in one way or another with most everything we do. So let's see if it works for what I do.

I just finished washing the dishes. (No, I don't have a dishwasher.) I put the glasses and silverware in the soapy water first and save the dirtier and larger dishes for later. The reason I save the dirtier dishes for later is that I want my dishes to end up clean and the ratio of dirt particles to water needs to be low. When I finally put the really dirty dishes into the water, it increases that ratio of dirt to water. Ratio? Math?

So how else have I used math over this summer vacation? Well, while I was sitting and healing from (yet another) surgery on my foot, I got involved in one of those farm simulation games on Facebook. I plow, plant, and harvest and watch my coins accumulate. I have to buy the seeds from the store, and I try to figure out which will be the best crop for me to grow. It's all based on math!

Thankfully, I haven't spent the whole vacation in a recliner with my foot up. On our trip to Newport, RI, I used a map, roadside mileage markers, and the car's speedometer to try to figure out how much longer it would take. The everyday estimates we make are all based on our knowledge of math.

So is it really true that everything has to do with math? I think it is, whether I can explain the connection or not. But it is a good mental exercise to try to figure out those connections. Maybe I'll contemplate it further when I'm floating in the pool this afternoon!

See Who Laughs!

This is a joke my father used to tell. I tell it to my classes and then watch their reaction. I can tell right away who gets it and who doesn't.

One day a man went into a pizza parlor and ordered a pizza.
The waiter asked him,"Would you like your pizza cut into six slices or eight slices?"
After thinking for a moment, the man replied,"Well, you'd better cut it into six slices; I could never never eat eight!"

This is a good time to point out that fractions that have the same numerator and denominator always equal 1 or 6/6 =8/8

The Book That Might Disappear

My husband came home from an educators’ conference with several new books and one of them caught my eye in addition to hearing him mention it several times. The name of the book is Mathematics: Worksheets Don’t Grow Dendrites. It is written by Marcia L. Tate who is an educational consultant. She has written several books that have the Worksheets Don’t Grow Dendrites mantra in their title and this one is about math. She maintains that learning occurs the best when it centers around activities that involve and engage the student. This is a concept that I have believed for years, but finding or dreaming up enough activities to teach all the math concepts in my curriculum is quite a challenge—something I have spent many years doing.

As I began to flip through the book, it became obvious that it is a valuable resource. Chapter 2 is titled “Drawing and Artwork”. I see a project for “Gallon Man” which is used to help children remember the number of quarts in a gallon, pints in a quart, and cups in a pint. My middle school students have a hard time remembering this and here is an art project that can help! I love it: art projects that teach math!

The other chapters look good too, with a lot of practical ideas that can be used or expanded. You know, my husband loses things pretty easily and this book just might disappear for a while!

My Special Number: 25

For many years I assigned an end-of-unit project to my sixth graders titled My Special Number. This was when students would pick a number on which to perform a number study. Today I offer my own version of this assignment.

My special number is 25. It is a number that is described as odd, composite, and square. It is odd because it is not divisible by two. It is composite because its factor pairs are not only 1 x 25, but also 5 x 5. Having 5 x 5 as a factor pair is also what makes it a square number. In fact it is the smallest square number that is the sum of two other square numbers. ( 9 + 16 = 25) This relationship is found in many right triangles (known as 3-4-5 triangles) used for illustrating the Pythagorean Theorem.

Because 25 is a factor of 100, it is used in a variety of ways. A twenty-five cent coin is a quarter and it takes four to equal a dollar. 25% is equivalent to ¼.

The number 25 has significance in various religions and philosophies. According to Abellio, twenty-five is the symbol for the Universal Word of God. St. Augustin used it to represent the law. In the Bible, King Solomon builds a temple to God that is 25 cubits high, and a Levite could begin practicing his ministry at the age of twenty-five.

But today, the number twenty-five has a special significance. Today marks the day that indicates that FOR TWENTY-FIVE WONDERFUL YEARS I HAVE BEEN MARRIED TO THE BEST HUSBAND I COULD POSSIBLY HAVE!

The Daily Fraction

One ongoing activity that I like to do with my seventh grade classes is the daily fraction. This lends itself well to seventh grade because by then, students have been taught most fraction skills. There are a few that just didn’t get it, while some others have attained mastery, and the daily fraction works with both kinds of students. This activity takes about one minute a day and serves to review and for some re-teach a number of fractions concepts. It is a great point of discussion for students to notice and participate in.

I start at the beginning of the year by making a fraction template on a piece of white cardstock or construction paper. The template has the divide line with 180 as the denominator. I laminate it twice to withstand the daily write-on/wipe-off process. On the first day of school I use a dry erase marker to write a 1 as the numerator and say to the class, “When you go home today, one one-hundred -eightieth of the school year will be completed.”

On the second day, the one has been erased and a 2 is in its place. I ask the students to reduce the fraction and they realize that 1/90 of the school year is completed that day. We compare the two fractions noting that 1/90 is greater that 1/180.

Many days I ask the students to write the reduced form of the fraction on the back of their warm-up sheet, then discuss implications. I might ask them how close to 1/8or 1/4 of the year we are (midterm progress reports or report cards). I might ask for the percent of the year that has gone by or the fraction that is left.
Seventh-graders like to keep track of the year’s progression. Some years, students have felt that the year zoomed by and other years, it seemed very slow to them. But every day they are getting a little reminder of fraction skills through this number study.

So today is the 166th day of our school year. That's 83/90 of the school year completed with 7/90 to go. Or we could say that 92% of the year is finished with 8% left to complete. That translates to fourteen more days of school left!

Sweet Success

About a month ago, I wrote about a student who needed to learn some of the multiplication facts. We picked ten facts she needed to learn and wrote them on flash cards. We practiced and drilled and she got better. She made 90% or nine out of the ten. Now that is a high score that would bring an A grade in any school.

But consistently the one fact that she could not remember was 7 x 6 = 42. Whenever that fact would come up, the look on her face clearly communicated to me not only that she didn’t know it, but that she knew this was the one that she didn’t know and was convinced she would never be able to remember.

So one day, I let her use some art supplies to make a poster that now hangs in my room. And sure enough, even with her back to the poster, she can remember that 6 x 7 is 42. The visual and tactile experience of creating this poster focused on what she needed to learn has brought her to 100% success.

It's All About Pizza

Even though I have started my fractions groups by using giant cookies and a recipe for mushroom sandwiches, the example I use the most when talking about fractions is pizza. I spent some time this morning cutting out large red paper circles. Some of them I cut into halves and some I cut into fourths. I will use them as visuals when I talk about pizza slices and improper fractions and mixed numbers.

Improper fractions simply tell how many pieces there are and what kind of pieces they are. They are usually not the best way to express a number, but sometimes they are necessary. So if I have nine fourths, I have nine pieces that have been cut into fourths (which means four for every whole pizza). Using my visuals, students can see that every group of four makes a whole pizza. They find that two whole pizzas can be made with one piece left over. After some practice with this concept, I help them to verbalize the analog for how to change an improper fraction to a mixed number.

Then we move on to starting with a mixed number and changing it to an improper fraction. The pizza slices come in handy once again as students can beak down the wholes to see how many parts there are. Together, we come up with the steps for doing this process without the pizza slices.

The worst part of all of this is that the kids complain that these exercises make them hungry. It has an effect on me too—“Hello, Pizza Hut? I’d like to order a large pizza to be delivered to the school!”

From Cookies to Mushrooms

Today I will see how much my RTI students remember about the fractions skills I went over on Monday. I made a handout with one of my favorite snack recipes (that I got from my mother-in-law). Then I ask students to answer some questions. Here it is:

If you like mushrooms, try this recipe sometime.


Hot Mushroom Sandwiches

2 4-ounce cans mushrooms
1 T onion, finely chopped
2 T mayonnaise
5 slices rye or wheat bread
grated Parmesan cheese


Drain and chop the mushrooms. Add the onions and mayonnaise and stir. Spread evenly on the bread slices. Sprinkle with Parmesan cheese. Broil until lightly browned. Serve hot. Makes 5 sandwiches.


If two people share this recipe, how many sandwiches will each one get?

Express your answer as an improper fraction:

Express you answer as a mixed number:

Write a complete sentence that best answers this question:



Because five sandwiches are being shared or divided by two people, the improper fraction is 5/2. As a mixed number, that would be 2 1/2 sandwiches.

After doing the drill and this activity, we will review reducing fractions by playing NASCO's Fraction Simplification Bingo.

Fractions With Cookies

Fractions, which are usually taught in fifth or sixth grade, are a challenge for many students. So in middle school, we find many students that have difficulty with fractions. My school's RTI (Response to Intervention) time was recently designated as a time to work on fractions with both seventh and eighth graders. So today I started with two new groups to focus on fractions. My sixth period group was with seven eighth graders and the seventh period group was with eight (very energetic) seventh grade boys.

I was keenly aware that I was starting an instructional unit with students I hadn't worked with before and that I needed some way to get their attention, yet establish some limits. I continue to look for hand-on visual models to explain mathematical concepts that I am trying to get accross.

When the seven students came into the room, I had two large cookies on the side table. (I made these chocolate chip cookies from one mix, pressing half the dough into the bottom of an eight-inch round cake pan.) Of course their attention was mine right away! I told the students the problem that we would work on later. There were two large cookies and they needed to figure out a way to share them evenly. I quoted what my mother used to say,"If you can't share them, share and share alike, then nobody gets any!" I told them we had some drills to do first, but they should be thinking about this problem.

After the drills, which then went quite smoothly, we came back to the cookie problem. Two large cookies split among seven people seemed to be quite a problem for them. The first idea was that they would share a part with me and cut each cookie into eight parts. I declined and they worried out loud about a cook that wouldn't eat her own cooking! After explaining diabetes limitations, I told them if they could come up with the fraction that each person would get, I could help them know how to cut the cookies. I gave them a hint: when I was a kid and Grandma gave us a bag of candy, we divied it out by saying "One for you, one for you..." until they were all given out. I asked them how they would have to cut the cookies so they could divy them out evenly. They knew it would have to be in sevenths. And they realized they would each get a part from both cookies. So they would be eating 2/7 of a cookie. I then got out my protractor and a piece of paper. They knew that there are 360 degrees in a circle, so we figured that 1/7 of a circle was about 51 degrees. I made a 51 degree paper wedge to use as a cutting guide. After sanitizing their hands they cut the cookies perfectly!

While they were munching on pie-shaped cookie pieces I explained that we had divided two cookies by seven people and each got to have 2/7. The line in a fraction means "divided by" in this kind of problem. I had another similar word problem printed out that I put on a clip easel.

"Camp Winnetaka has 300 campers. The chef made 90 extra-large pizzas. If the pizzas are shared equally among the campers, how much will each camper get?" By referring back to the cookie problem, students were able to see that 90 pizzas were being divided by 300 people. So each one would get 90/300. I told them that it wasn't likely that the chef would cut each of those pizzas into 300 tiny pieces and then let each camper go and get one piece from each pizza. They realized the need to reduce and we were able to spend the rest of the period working with individual white boards on reducing fractions, improper fractions and mixed numbers. (Each camper got 3/10 of a pizza.)

The seventh grade group had to split three cookies among eight people. They came up with two solutions which caused a bit of an argument until I stepped in. The first proposal was to cut each cookie into eight pieces and everyone would get three. The second solution was to cut two cookies into four pieces and give everyone one piece, then to cut the last cookie into eight pieces and give everyone one of those. It was a great time to point out that both would be the same amount.

At the end of the period, I asked them to individually write a few sentences about how they thought they were doing with the skills we had worked on today. I listed them on the board: word problems dividing food among people, reducing fractions, changing improper fractions to mixed numbers and back. The students were honest and most expressed that there was something in the class that had helped them today, especially with that type of fraction word problem. I thoroughly enjoyed teaching these classes and am looking forward to Wednesday when I see them again!

Happy Birthday to Me!!!!!!!!!!

Today is my birthday and I am staying late after school to do the best thing a teacher could be doing on her birthday! You see, I have spent the whole year teaching with little to no supplies. I have scrounged around and made my own games. I have brought things like markers and teaching resource books from home all year long. But a few weeks ago, some grant money came through and I got to place my order. And today I am unpacking all these boxes. Such richness: a stapler, pencil sharpener, and three-holed punch for my room. Dozens of math games, manipulatives, and demonstration materials. Plenty of paper, glue, pencils, markers and more. And neat little storage bins to organize it all. It's just like Christmas---actually it's like my birthday! Happy birthday to me!

Survivor Blew It!

OK I have to admit it: I am a fan of CBS's Survivor. It is the only reality show that I watch, much to the amusement of my friends who think it is out of my character. But Tuesday(NCIS and The Mentalist) and Thursday are my regular TV nights. And tonight's episode of Survivor really blew it mathematically speaking.

The immunity challenge involved a math problem, and the first player to solve the problem won the immunity idol, and could not be voted off during tribal council. Of course, contestants had to go through an obstacle course and memorize pieces of the math problem before they could even begin to solve the problem. But Survivor blew it when they accepted a wrong answer as correct because they themselves didn't follow standard mathematical rules. The problem was this: (believe me we replayed and paused several times to get a good look at it!)

6 + 2 / 4 x 3 / 3 - 7 +6 + 5 /3 - 1 x 1

Now one player, JT, had written the letters PEMDAS in the bottom left corner of his slate. PEMDAS is the mnemonic to remember the standard order of operations which governs the order in which problems are solved. Order of operations is commonly taught in middle school grades (usually seventh). If you follow the order of operations to solve this problem, you should get 6 1/6 for an answer. But if you merely solve the problem from left to right, then the answer would be 1, which is the answer given by Stephen, who won immunity. But solving from left to right is not correct mathematical practice.

This is where I am suppressing the urge to sermonize, preach, and ask questions that are sarcastic in tone.

If you're still reading this--Thanks! I needed to vent!

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!

Math Vocabulary: The Need for Intentional Instruction

Another valuable insight that came from my classroom observation was that my students were not strong enough readers to take what they read and translate it into mathematical operations. They needed more work on vocabulary. The administrator directed me to a website which has a table of math vocabulary words that translate to mathematical operations. I made a chart with this information, then made a sorting activity by putting the words on separate cards along with cards for the operations of addition, subtraction, multiplication, division, and equals. I used this activity to focus on teaching the mathematical association each word had. Instead of just pointing it out as we did the word problems, I made math vocabulary the focus of a teaching mini-unit. We did this activity regularly for several days and then sporadically in the weeks that followed.

Today, as we approached word problems and translating them into algebraic expressions, I had the students use a highlighter to mark vocabulary words that signaled specific mathematical operations. They were then able to determine the algebraic expression that was needed because they knew what operation to use based on the vocabulary words. I realized this had indeed been a successful undertaking when I saw the students answering the assigned tasks independently with correct answers.

The valuable insight that I gained here is that I needed to stop and do some intentional instruction focused specifically on what I wanted them to learn. The overall task of solving word problems needed to be set aside for a time to work on the individual skills needed to complete the overall task. Sure, as we had done word problems, I had pointed out what various words meant and how they would translate into math. But these kids needed to have intense training along with the message that they needed to know how to do this. And it paid off in increased learning and skills for my students.

Book Review: What Successful Math Teachers Do

This year when I made out my annual goals, one of them was to read the book What Successful Math Teachers Do, Grades 6-12.It is written by Alfred S. Posamentier and Daniel Jaye and was published by Corwin Press in 2006. (ISBN 1-4129-1619-4) I had seen an advertisement for it and was interested. In the meeting where I discussed my goals with the principal, he directed me to Barbara in the library and she ordered it for the school. Two weeks later, the book arrived and I have kept it on my night stand, looking at it frequently. I wanted to buy my own copy and ended up doing just that, because it is a valuable reference for math teachers, and because my lunch that contained beets dripped on it a little bit.

I understand that there is a counterpart to this book for math teachers of grades K-5 also.

After observing countless mathematics classrooms, the authors compiled a list of 79 strategies that are present in math classrooms where there is a high rate of student achievement. These are listed and explained in the book. After a strategy is given, it is followed by sections titled, "What the Research Says," "Teaching to the NCTM Standards," "Classroom Applications," and "Precautions and Possible Pitfalls."

The strategies are further organized into chapters titled, "Managing Your Classroom," "Enhancing Teaching Techniques," "Facillitating Student Learning," "Assessing Student Progress," "Teaching Problem Solving,"and "Considering Social Aspects in Teaching Mathematics."

I will admit that this is not the kind of book I can sit down and read cover to cover. For me, it is easiest read by reading one or two strategies at a time. Because I have been teaching for quite a few years now, I spent less time in the classroom management and assessment sections and focused a bit more in the problem solving and teaching techniques sections.

This is a book that I would love to discuss with other math teachers in a group setting. I will admit that in my mind, I was able to dismiss a few of the the strategies as less important than some of the others and would enjoy some discussion on them. Just three of the strategies that caught my eye were:

#37 Teachers can help students learn to ask better questions.

#59 Have students study written model solutions to problems while learning and practicing problem solving.

#66 Help students learn without relying on teacher-centered approaches. Give them carefully chosen sequences of worked-out examples and problems to share.


I am hoping that someone will want to be in a group to discuss this book--I think the strategies in it are interesting and valuable, not only to new teachers, but to seasoned veterans as well.

Supporting the Spiral Even in Middle and High School

I was recently observed by one of the building administrators. Over the years, I have been observed quite a bit, but I can't say that the countless hours put into observing my teaching and then the written summary have ever resulted in something that actually helped me with my teaching and subsequently student learning. Most years, I got the impression that administrators sat in my classroom to reassure themselves that I was really teaching the students, and then they could go back to their offices happy and content. Many years, I have not been formally observed, but when I am, they note that I am creative and capable etc. Years ago, as a new teacher, when I was yearning for some valuable and helpful insights, the best anyone did was, "The letters on your bulletin board are a bit crooked." Needless to say, for me observations have just been something that happen from time to time with little value or effect.

But this year is a refreshing breath of fresh air. On this observation in addition to the numerous commendation comments, there were two observations that I could sink my teeth into and put into practice. I was actually excited to get this feedback because it gave me a better sense of direction with what I needed to do and also clarified something I had been thinking about already. One suggestion, I will be dealing with in a subsequent post, but the other reaffirmed my concept of how students learn and how instruction helps them learn.

The observation was done while I was guiding a small group of seventh grade students through some math word problems. (In light of my last post, I hesitate to use the p-word here, but for the sake of clarity,I am.) There was no common theme in the problems, so it would be classified under mixed practice rather than just addition or subtraction etc. My students especially need practice reading problems and figuring out how to solve them, so I try to do exercises like this frequently. I did not write the problems myself and one of the problems required a lot of "guess and check" work. Now "guess and check" is a viable method that we teach for solving math problems, but it can also be downright tedious. This particular problem could have been solved by using algebra, but my students have not had algebra yet and the answer key said students should use the "guess and check" method. So after we went through all the hassel of trying every possible solution and eventually arrived at the correct one, I mentioned to my group that when they got a little older and learned algebra, there would be a simpler and easier way to solve this question. Then we moved on.

The administrator's observation was that at that point after we had the correct answer, I could have modeled how to solve the problem using algebra. The students would have been able to follow some, if not all of the process, and it would have served as an introductory/pre-teaching moment. It also would have piqued their curiosity and helped to lay a foundation for the math instruction that lies ahead. DUH! I know this and it makes sense, but it took an outside observer to point it out.

For a number of years now, elementary math education (as well as other subject areas)has been based on the spiral concept. I won't say that I'm always a big fan of how some curriculums implement the spiral, but I am a big believer in the concept. Whenever possible, you introduce a concept or skill before students are expected to master it. They are given several opportunities to practice the skill so that when mastery is expected, it will be there. Future learning is built upon the prior knowledge.

Perhaps the realization/reaffirmation that the spiral is important for middle and high school students as well as elementary was my philosophical highlight of the week. I am now even more conscious and deliberate in trying to work in a few more moments that support the spiral.

Just a Thought about Problems

Last week, during vacation from school, I had surgery on my foot and my dog died. Earlier this winter, we lost electricity for three days which caused our basement to flood because the sump pump wasn't working, resulting in significant damage. Last summer, I was in a car accident that gave me whiplash that I am just getting over now. In the past few years I have been in a position where I have been responsible for making health care decisions for several family members who were incapacitated. And then I sat with them while they died and subsequently made the funeral arrangements. THESE ARE PROBLEMS!

When students come through my door and I have math tasks for them to perform, why in the world do I refer to them as problems? They are tasks and challenges, but really not problems. Believe me, a number of my students have real problems of their own and don't need any more. I am trying to teach my students sound practices for approaching math and I want them to feel empowered rather than burdened by it. So every now and then I choose words other than "problems" in math. I hope students can view math with a welcoming attitude rather than misery.

What Good Mathematicians Do: Check Your Work

A few weeks ago, I hung a large poster in my room titled,"What Good Mathematicians Do". The poster, by McDonald Publishing Co., has eleven things listed on it. This week I asked students to sit facing me with their back to the poster. I asked them to tell me any of the things they had noticed or could guess were on the poster.

The most common answer in all of the groups was, "Check your work!" In fact, for most groups, this was the only answer they could initially come up with. I facillitated a few activities and discussion around the poster and was able to drive home a few more points, but I found it interesting that the "Check your work" answer stood out so prominently.

I guess I shouldn't have been surprised. At every grade level, teachers remind children to check their work. It has been drilled into them, even though it seems that many choose to ignore this advice. "Check your work!" is a common admonition for just about every math task from kindergarten on up.

What if math teachers were just as diligent and consistent at teaching the other aspects of being a good mathematician? What if "Know how to explain your work," and "Ask questions," were called out as frequently as "Check your work"? I think we would have higher achieving students. If they were just as familliar with the other success strategies, students would probably be more capable of checking their work, and ultimately more successful.

The Significance of Valentine's Day to Math Teachers

For a math teacher, Valentine's Day has another meaning. After we finish with the party, candies, cards and romantic dinners, the calendar reminds us of another upcoming event. Valentine's Day, February 14, means that we have just one month until the mathematician's holiday: Pi Day, March 14.

Teachers begin getting ready for this event early. There are a variety of activities and teaching opportunities leading up to this special day. Granted this year's Pi Day is on a weekend, but it can be celebrated a day early just as we are having Valentine's parties today. Of course the significance of Pi Day being on the 14th day of the 3rd month is to help children remember that the value of pi is 3.14.

I like to read the book Sir Cumference and the Dragon of Pi by Cindy Neuschwander. (ISBN-13:978-1-57091-164-4) It is a fairy tale that establishes in a creative way the definition of pi. Other titles in this series include:
Sir Cumference and the First Round Table
Sir Cumference and the Isle of Immeter
Sir Cumference and the Great Knight of Angleland
Sir Cumference and the Sword Cone

Main characters in these books include Sir Cumference, his wife Lady Di of Ameter, and their son Radius who is half as tall as his mother. Children in fifth and sixth grades enjoy these books, but may not fully understand all the concepts. I have read them to seventh and eighth grade classes who enjoyed them as well.

So after all of the mushy lovey stuff of Valentine's Day, it's time to haul out the Pi Day materials and let the real fun begin!

The Queen of Mean

The mean is one of those measures of central tendency (along with median, mode and range) that students must learn and use to analyze data. I use a simulation presented as a small group demonstration, to illustrate what the mean is and how it is found.

I start with 5 or six sheets of different colored construction paper spread out on a table. I tell the students that these represent houses. I then use unifix cubes or something similar to represent people/families that live in each house. The cubes that live in a house are all the same color (ie the brown cubes are referred to as the Brown family.) Each house has a different number of people living there.

Then the Queen (sometimes it is me and sometimes I just refer to her) makes a decree that there must be the same number of people living in each house. So people must be moved to fulfill this mean-spirited decree which breaks families apart, and the Queen is dubbed "the Queen of Mean" (I then embellish and repeat this part of the story about how mean she is.)Students can work to figure out how many must be in each house. If they have no clue how to do this, I suggest taking all the people and spreading them around one at a time. (This is the same as adding all of the people together, then dividing by how many houses there are--the steps for finding the mean.) I emphasize several times to put all the people together, then divide them out.

After several simulations with different numbers of people, students get the idea of how to find the mean. I then take the cubes away and ask if they can do it without using cubes. I pose another problem and let them go through the steps of solving it. From this point on, whenever a student doesn't rmember how to find the mean, I say, "Do you remember the Queen of Mean?"

"Oh, yeah!" they respond and I don't have to say anything else.

Pulling Teeth and Getting Blood from Turnips

The task before me is like pulling teeth and getting blood from turnips! OK--well maybe it's not quite that bad. I have been working today with students on writing answers to math problems; not just the kind of math problem where you give the answer in one or two words, but those that say "Show your work." and "Explain your thinking." These are the kind of problems we frequently find on standardized tests.

We start by making an effort to understand the problem. This usually means reading it over a couple of times at least. We then define the specific question(s) that we must have an answer to. We look at what information is already given to us in the problem and decide what operations/methods will be used to solve the problem. All this is mapped out on scratch paper, not yet ready for the final draft.

If we get to the point where we get a correct answer, we feel pretty good about ourselves and are tempted to communicate that particular answer with just two or three words. But there I am with my turnip juicer and tooth pullers insisting that everyone must now write in great detail how they arrived at their answer. Then we look back to make sure it isn't just details and explanation, but that a clear statement answers the direct questions that were asked.

Many students find this to be a hard process. Writing and language tasks may not come easily to them. In their minds, they consider tasks like this to be appropriate for language arts classes, but not math. They want to give up. And there I am with my dental tools and juicer as well as any incentives I can come up with.

Slowly but surely, I start to see some improvement. I have modeled how to answer these questions what seems like a bazillion times. And I see my students beginning to write just one more detail or begin to show their work. The process of organizing their thoughts into words to explain how a problem is solved is starting to show through.

Model, think aloud, model some more. Practice, practice, practice. I hope and pray that it works!

Ready-made Powerpoint Presentations

I have been using Pete's Power Point Sation to find powerpoint presentations that I can use with my students. It is found at: http://www.pppst.com/. There are powerpoints for just about all subjects. I can save these powerpoints in my own "instructional presentations" folder and modify (if desired) and use them in my classes. Most of the presentations last just a few minutes. There are some jeopardy games which I downloaded and use as a template by changing the categaories, questions and answers.

This site is a tremendous timesaver that enhances the quality of my educational program. My own introduction to a class one day, can be followed up the next day with a short presentation that reviews what I went over, but in a little diffferent format. Students respond well to this medium, and math concepts are reinorced.

The Vocabulary of Word Problems: When Math Teachers Must Teach Reading

Some students struggle with math because they struggle with reading. And the trend in mathematics education lately, is to include a lot of applications which involve more reading than ever. Gone are the days where students would do math problems in isolation, so those who didn't read well could still do well in math. Math books contain more written words and language-based tasks than ever. So math teachers find themselves teaching reading skills, especially as they pertain to understanding word problems. This is nothing new; good math teachers have always taught math vocabulary and applications.

For word problems, the basis seems to be vocabulary. Most of us have lists of catch phrases that signify certain operations. For example, "how many more?" indicates subtraction. But we cannot rely totally on vocabulary and catch phrases. Sometimes the situation described in word problems describes a mathematical concept without any signal words or catch phrases.

I am currently trying to get my students to recognize when to use a proportion to solve a problem: when there are three pieces of data and two labels. They all can solve proportions quite well, but sometimes have a hard time knowing when to use one. Having a general guideline has helped them tremendously.

With students in Title I classes, I have found it necessary to slow down and require the students to read (and reread) problems. I have noticed that sometimes math teachers skip having students read instructions, but paraphrase them instead. This can sometimes cause confusion which leads to incorrect work. When a student asks for help with a problem, a common response I have is "Read the problem to me."

Every now and then, it is helpful to remind myself that I am not only a math teacher; I must teach reading as well.

Another Way to Remember Multiplication Rules for Integers

Back when Mike Cirre was my principal, he told me this way to help kids remember the rules for multiplying integers. (The same rules are used for division--just substitue the sign.)

I start by making a chart on the board with the headings: person, enter/exit, and we feel.
I explain that there are two kinds of people: positive and negative. Then I explain that those people can use the classroom door to come into our class (thus adding to the number of people in the room)or they can leave (which subtracts from the number in the room. And whenever people come or go, we feel either good or bad deending on what kind of person they are and whether they came or went. All of these situations can be represented by + or - signs.

So we start to fill in the chart on the board. When a positive person enters the room, we feel good because they cheer us up. (+ + + ) When a positive person leaves the room, we feel bad because we are sorry to see them go. ( + - - )When a negative person enters the room, we feel bad because they bring their negativity with them. ( - + - ) But when a negative person leaves the room, we feel good because we are glad to get rid of them. ( - - + )

By this time I have filled in the chart so it looks like this:



+ + +
+ - -
- + -
- - +

The chart is then interpreted, a positive times a positve is a positive, a positive times a negative is a negative etc.

This is an amusing way to remember the rules for multiplying and late dividing integers. Invariably, while I am doing this with the class, someone will walk in the classroom door which will send the students into a burst of laughter, but that ultimately contributes to the effectivveness of learning.

Mathematics and Art: Symmetry

Symmetry is a mathematical principle that is used a lot in art as well. Students learn about symmetry and lines of symmetry in geometric shapes during the elementary grades. Their understanding of symmetry helps them in higher level geometry, but can also help them appreciate and analyze a number of works of art as well. Imagine that--mathematics is a basis for aesthetic experiences.

Symmetry is all around us in nature, and many works of art are judged to be more attractive when there are clear lines of symmetry which come from a central point.

Last year for a skills lab series, I did a two-week exploratory titled "Mathematics and Art". Symmetry was the first topic studied. I made a powerpoint using pictures of nature, buildings, and works of art that had clear lines of symmetry. I found the pictures by searching www.images.google.com. I just put in the word "symmetry".

This year, I briefly revived that powerpoint and students once again enjoyed viewing the pictures and identifying the lines of symmetry.

Adding and Subtractting Integers: A Flow Chart

The seventh grade is in an integers unit. Some are still having a hard time keeping the procedures straight. On Friday, we created a flow chart which served to further illustrate the process. This exercise and the resulting visual, served as the catalyst for the AHA! I GET IT! experience for a number of the students.

Computation vs. Calculator

There are those who feel that the use of calculators make students dependent on them so they never learn the basic facts. There are others who maintain that calculators are necessary as a time-saver and for those with disabilities who cannot learn math facts and computation skills. But having a successful middle school math program actually depends on a healthy balance between the two: computation and calculator.

As students start to solve higher level math problems, using a calculator allows them to do it faster. They can effectively perform and learn the steps required in the math problem without getting bogged down in the sometimes laborious task of computation.

On the other hand, students who don't regularly use their computation skills, sometimes lose some or all of them. Most middle school math curriculums don't come with computation exercises. Teachers who choose to include this piece usually do so with a daily drill or problems of the day at the beginning of the class. My favorite resource for this is the A.D.D. (Arithemetic Daily Developed)series by Cuisenaire. These small practice sheets give three mental math questions, a word problem, and several review problems. The drill sheets are based on NCTM standards. I have seens students'abilities and test scores improve as a direct result of using this resource.

I recently posted a poll where I surveyed about the importance of learning basic facts. Even though there were few responders, they both answered that it is very important for students to learn basic facts. I agree wholeheartedly--students who know the basic facts and can perform computation problems acurately will have more options in life and will do better in math overall.

But when you teach math to middle schoolers, the stark reality is that some of them have not learned the basic facts. This could be for a number of reasons: poor memory, gaps due to moving or truancy, immaturity, poor educational practices, lack of support for learning...and the list goes on.

Middle school math teachers and especially title I and special ed remedial math teachers must decide how much time to spend on basic computation and how often to allow calculators. Some students who don't seem to have the capability of remembering math facts are able to reason and problem solve because they understand math concepts. These students should be encouraged to use the "tools" they need. The danger of focusing only on math facts computation is that students view the exercises as drudgery and may not achieve success, whereas they may enjoy some other aspects of the math curriculum.

So like all aspects of life, we must seek a balance. Computation and calculator useage are both necessary to help students to learn and improve.

A Math Dictionary: Terrific Website

My husband told me about a fabulous website. It is www.amathsdictionaryforkids.com (Sorry, I haven't figured out how to post it as a link here yet.) I immediately went to the library and have scheduled a day for my students to use the computers. I made a worksheet that instructs them to go to this site and then use the site to answer the questions. This site is an online interactive math dictionary. My description sounds boring, but please check it out--it's great. My purpose in wanting my students to be familliar with it, is to provide them with a tool for finding math definitions and information.

The questions I am asking them to answer are:

1. In the number sentence 6 / 3 =2, which number is the dividend?
2. If you have two gross of pencils, how many do you have?
3. How many square meters are in a hectare?
4. What is a vinculum?
5. What is an algorithm?
6. Why are 31 and 36 relative primes?