Mr. L's Math» Teacher to Teacher Press

Meet Mr. L

Bill Lombard — Wed, 17 Jun 2009 22:58:24 +0000

Bill is the 2006 California League of High Schools northeastern California Educator of the Year

Bill Lombard currently teaches at Foothill High School, ten miles east of Redding. He has taught at the high school level for twenty years, at the middle school level for six years, two years at Shasta Community College, and three years at the adult/teacher level as the Manager of the Teacher Education and Computer Center based in Redding. For the last fifteen years Bill has presented mathematics curriculum to teachers at conferences and workshops.

He and his co-author, Brad Fulton, have written more than a dozen books for math teachers as well as over twenty Teacher Resource Manuals for teacher training seminars. Brad and Bill host a companion website, Teacher to Teacher Press. On this site you may find more materials to help math teachers. Known throughout the country for motivating and engaging teachers and students, Brad and Bill create activities that help teachers who believe mathematics must be both meaningful and powerful.

Bill’s goal is to make mathematics both understandable and meaningful for his students, and on this website he shares some of his materials, lessons, and other information to help both teachers and students to be successful in the math classroom.

Seminar leader and trainer of mathematics teachers

California Math Council and NCTM/NCSM presenter
Lead trainer for summer teacher training institutes
Trainer/consultant for district, county, regional, and national workshops

Co-author of mathematics curriculum

Simply Great Math Activities series: five books covering all major strands
Math Discoveries series: four books bringing math alive for students in middle schools
Teacher Training Resource Handbooks for elementary, middle, and secondary teachers

Workshops, conference sessions, and keynote addresses

All workshops provide participants with complete and ready-to-use activities. These activities require minimal preparation, use materials commonly found in classrooms, and give clear and specific directions and format. Participants will also receive journal prompts, homework suggestions, and ideas for extensions and assessment.

“Excellent…relevant…useful…practical…engaging.”: - Tizoc Tirado, kindergarten teacher

“I have learned more about math in these five days than in all my career.”

- Annie Jackson, 5th grade teacher

“Brad and Bill’s math activities are the best I’ve seen in 30 years of teaching!”

- Wayne Dequer, 7th grade math teacher

“The best and most practical and relevant workshop I have ever attended. Thank You!”

- Leslie Lowman, high school teacher

Math Made Fun – a Video Series for Math Mavens and Future Math Mavens

Fourteen Factors of a Successful Algebra Program – by Brad Fulton

Bill Lombard — Mon, 05 Jan 2009 01:11:05 +0000

NOTE: This post is written by Brad Fulton, Mr. L’s coauthor and partner from Teacher to Teacher Press.

Many factors go into creating a successful program for preparing students for 8th grade algebra. Here are ones we feel have contributed to the success of our program. While you may decide that some of these strategies may not work at your site, the more of them you implement, the more results you should see. You may be able to put into practice your own ideas which are not on this list.
1. Expose as many of your 8th graders as possible to an algebra program in the 8th grade. Include ELD and resource students. When in doubt as to whether a student is ready to take algebra, let them try the course.
2. For students who require more support, avoid pull-out programs. All remediation should be in addition to not in place of regular instruction. Dovetail intervention classes to support the instruction of the regular algebra program.
3. Use collaborative time to develop continuity along grade levels for algebraic instruction. Use common vocabulary, strategies, and teaching techniques to avoid students having to memorize extraneous information. For example, how do you teach students to approach solving equations?
4. Begin early with preteaching of concepts. Encourage teachers to look at the standards their students will face in subsequent years. How can we lay the foundation for these future skills?
5. Teach algebra concepts to all 8th graders. Teach the rigor of Algebra 1 to the students who are ready for the traditional course. For many struggling students, it is not the algebra itself that creates problems, it’s the mathematical rigor of negative numbers and fractions.
6. For this reason, teach initial concepts with positive whole numbers. As students gain fluency and familiarity with concepts, introduce negative numbers, decimals, and fractions.
7. Many errors are not the result of problems with algebra, but with negative numbers. For this reason, students should have ongoing instruction and practice with integers.
8. Research, not routine, should guide instruction. Read what valid research has said about mathematics instruction. Experiment. Keep what works. Don’t be afraid to toss a long-standing or accepted instructional practice if you find a better way.
9. For all concepts, try to teach multiple strategies for problem solving. Demonstrate algebraic approaches, graphical models, and any other tactics that are appropriate. For example, when teaching students to solve systems of equations, show them how to solve them by graphing, elimination, and substitution.
10. Teach algebra as a natural extension of arithmetic. For example, teach multiplication of binomials as an extension of two-digit by two-digit multiplication. This will not only provide critical opportunities for students to improve their number skills, it will provide a seamless transition from the familiar world of numbers into the more abstract world of algebra.
11. It is critical to make math enjoyable. Math can be beautiful and engaging or it can be dull and coma-inducing. We hold the keys for this. Engage students in math projects that allow students to experience the beauty of math as they delve extensively into the skills they have studied. This is the opposite of the “mile wide and inch deep” approach as it lets students dig deeply into specific skills and concepts. See examples of this on the “Student Work Site” section of our website: www.tttpress.com.
12. Don’t be afraid to step outside the textbook. Texts are a valid and powerful tool, but they are not a cure-all or a golden bullet. They are reference works just as dictionaries are. They are also the most difficult reading material in schools. Students learn from teachers, not from textbooks.
13. This brings us to a very important point. Probably the most critical factor in helping students learn algebra has nothing to do with numbers and everything to do with the teacher. A personable teacher who communicates a caring attitude working in the supporting environment of a nurturing school will have much greater success in teaching students than an expert mathematician who fails to connect with students. I used to teach math; now I teach students.
14. Above all else, set high standards for all students. Expect everyone to achieve them. No one ever jumped high by aiming low, and even if we miss, we jump higher than before.

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Algebra Man, and His Trek Through the Hundreds Chart, by Brad Fulton, guest blogger

Bill Lombard — Wed, 23 Apr 2008 03:12:38 +0000

[N.B. - This post is written by my coauthor from Teacher to Teacher Press, Brad Fulton, with some slight tweaking from me. The activity will work well for students from grades 6 - Algebra 2. This is an extension of the activity Hundreds Magic, which can be found at the end of this activity under Related Posts.]
My 8th grade students enjoy discovering the algebraic patterns hidden within the hundreds chart. They are all familiar with this chart that they have seen since Kindergarten. One of our activities involves “Algebra Man.” I display a transparency of the hundreds chart and a second transparency of Algebra Man on top of it as shown.

I move the upper transparency around until a student yells, “Stop.” Then the students race me to find the total of the numbers inside Algebra Man. The total for the placement shown is 580. I always win this contest and boast that it is because I am so talented at math. The students insist there is a trick and demand a rematch. After a few times, some of the students see a pattern and discover the trick. (Turns out I’m not a rocket scientist after all!) Once a few discover my secret, we explore the problem using algebra.
The students notice that for any number on the chart, the number to the right is one more and the number to the left is one less. Also, the number beneath the given number is ten more and the number above it is ten less. This is always true no matter where Algebra Man runs. If we think of the number in his waist as n, then the following diagram shows the values of all ten numbers inside Algebra Man:

Many of my students say that they got the answer a different way. Often they simply look two spaces to the right of his waist and put a zero after the number. I express this algebraically on the board. The number two spaces to the right of his waist would be called n+2. Putting a zero after it is the same as multiplying by ten. Thus their method is 10(n+2). I show them that these are equivalent being the distributed and factored forms of the same expression:

10n+20 = 10(n+2)

Algebra Man works on any hundreds chart, even one that begins with -49 and ends with 50. He also works on a calendar, but the formula for his sum is slightly different. Challenge your students to find the formula when Algebra Man runs around on a calendar. This is just one of many algebraic explorations that can be made on the hundreds chart. If your students are like mine, they will want to explore all the “what if’s” they suggest. One time my students wanted to find out what happens to Algebra Man’s formula when he is standing on his head. Now I was curious.
Related Posts:

Hundreds Magic – a Simple Diagram Produces Patterns, Number Sense, and Algebra Proofs

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Algebra Problem Solving with the new Common Core Standards

Hundreds Magic – a Simple Diagram Produces Patterns, Number Sense, and Algebra Proofs

Bill Lombard — Sat, 16 Feb 2008 13:36:14 +0000

Students will be so amazed at the patterns they find in the hundreds chart they will think you are a “mathemagician“. Once hooked, they will use algebra to understand the magic.

A favorite activity of my partner, Brad Fulton, and mine whenever asked to do a workshop for teachers, this chart can be found in primary grades to teach skip counting, multiples, factors, prime numbers, and pattern recognition.

We show you how to incorporate simple arrangements of numbers and transparent chips to arouse the curiosity and imagination of students with number sense that leads naturally to algebraic proofs that all students can see and understand.

Numerically, in the second graphic the sum of the blue chips equals the sum of the violet chips. This arrangement of chips leads to the same sums no matter where this arrangement is placed. This leads algebraically, to a simple proof. If x is allowed to equal the number 2, then x+1 equals 3, x+10 equals 12, and x+11 equals 13. Adding the blue chips and the violet chips results in the same sum of 2x+11. Since this works every time, you have a simple algebra proof. Other proofs are outlined in the activity. You can probably find more after playing with the chart.

The six-page downloadable pdf activity will print well at your site, so you can make Activity Masters and Overhead Transparency Masters for all your classes. Pick up a few transparent colored chips and you have materials for lesson plans any day of the year. Send a comment and let me know how you do with your classes!

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Menu Math – a Surefire Way to Introduce Algebra to Any Student

Bill Lombard — Wed, 13 Feb 2008 13:47:06 +0000

This activity is so easy to teach you might feel guilty for taking a
paycheck! Students will quickly pick up on the meaning and use of variables and will be able to substitute values into formulas to solve them. As a bonus, students can learn the Distributive, Associative, and Commutative properties as well!

My writing partner, Brad Fulton, and I have presented this activity to thousands of teachers over the years. The response has always been one of amazement that so much math can be found in so simple a format. Since the activity was written, we have found so many extensions to the basic format that Menu Math could be a 100-page book by itself. As you become comfortable with the activity (complete instructions provided) you will find new ways to use Menu Math with your students. We have heard stories from teachers of students from primary grades through high school of kids who “finally get it” after using this simple approach.

This four-page downloadable activity is is pdf format so it will print well at your site and allow you to make an overhead transparency of the Menu Math graphic shown in this post.

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Four In a Row – A Number Sense Game with a Chess-Like Strategy

Bill Lombard — Sat, 09 Feb 2008 13:54:53 +0000

Practicing multiplication of integers, variables, and binomials becomes a game in this highly motivational activity. As students use strategy to find four products in a row, they will solve and resolve numerous problems.

Chess-like strategy characterizes this highly engaging game that helps students practice multiplication. Activity masters include game boards for integers, algebra, and a bonus Blank Game board for you to make up your own game.

When Brad Fulton and I conduct many introductory teacher workshops, this is the activity we finish with and teachers love it – they get just as competitive as our students do.

The six-page activity Four In a Row is in pdf format so it will print well at your site and you can make Overhead Transparency masters of the 3 game boards.

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Neighbors – a Math Game that Develops Number Sense and Strengthens Order of Operations

Bill Lombard — Sat, 02 Feb 2008 14:05:27 +0000

Students will enjoy competing against the teacher in this activity that richly develops number sense and operations. They will learn to apply order of operations and put square roots, exponents, and factorials to use. Once they learn the game, they will enjoy playing it in pairs or teams.

To play the game, use the digits 1 – 6, or the numbers 1 – 10. Select three of these at random by using a die or spinner. Then have the students use them to create a problem that has one of the game board numbers as its answer. They should mark it out with an “x”. Their goal is to cover the largest possible answer.

Each time a team creates a new answer, it must be a “neighbor” of a previous one. The new answer may be a neighbor vertically, horizontally, or diagonally.

Some possible Journal Prompts that can be used when playing this game: If the numbers 3, 5, and 6 are selected, what is the largest score possible? What is the smallest score? Show how you get these scores.

This game is quite extensible; the teacher can create new game boards by using different numbers, or simply rearrange the numbers on the existing game board. The teacher can also modify the rules; if one team gets ahead by a large number of points, they may only be permitted to use two numbers. Let your imagination be your guide as your students build number sense, gain practical experience with Order of Operations, and have fun!

This 4-page activity is in pdf form so it will print well at your site. After downloading Neighbors and using it in the classroom, let me know your experiences.
- Mr. L

Backwards Math – An Activity for All Operations and All Levels of Students

Bill Lombard — Tue, 29 Jan 2008 15:10:20 +0000

If you’re looking for an activity that can replace some boring sections in your math book, here’s a great one! Students will not only work intently for long periods of time, they will want to learn even more advanced operations such as exponents, roots, and factorials. Classes and groups of students will be challenged to work together to explore and complete this task. Students are given the answers to 100 math problems and then challenged to use four 4’s to make expressions that equal these numbers. Along the way they will work hundreds of math problems and learn a lot about order of operations, an important concept in beginning algebra and algebra readiness.

BackWards Math is a pdf file (62 k) so it will print well at your site. It will open in a new window. Try this out with your students and you will be extremely pleased with the results. Let me know how it goes. I will post extensions of this activity for those who wish to explore this method further.

- Mr. L

The Tax Collector – Number Sense and Problem Solving in a Game Format

Bill Lombard — Mon, 28 Jan 2008 15:11:41 +0000

This is a fun activity that reinforces factor concepts as it pits the student taxpayers against you, the tax collector! A game lasts about ten minutes. It can be used as a beginning or end-of-class filler. Be forewarned that if you begin your class with it, the taxpayers may revolt and demand that you keep playing until they beat you!

This activity is one from Simply Great Math Games by Brad Fulton and me, at Teacher to Teacher Press.

Click the link: The Tax Collector Activity for the 4-page activity. It’s in pdf format (91k), so it will print well at your site – it will open in a new window.

Feel free to make up new factor banks and change some of the rules to fit you and your class as you collect taxes!

- Mr. L

Tangrams – A Way to Visualize Fractions, Geometry, and Algebra

Bill Lombard — Thu, 24 Jan 2008 15:19:26 +0000

How would you like an activity that can be used at a wide variety of grade/ability levels, and covers fractions, decimals, percents, geometry, algebra, and language? Well, if this floats your mathematical boat, then read on.

Many tangram activities ask kids to make different shapes from the seven tans, but don’t really ask them to stretch their mathematical muscles. This activity encourages students to “see” the geometry and the fractional relationships inherent in the tangram.

Spatial reasoning is a critical skill that must be developed and nurtured in order for students to be excellent problem solvers. There are few activities that surpass this one in creating this skill. As teachers gain experience with tangrams, they see how to incorporate algebra as well. My partner, Brad Fulton, and I have presented this activity to thousands of teachers in workshops and at conferences. It never fails that many teachers have an “Aha moment!” while doing the activity and reflecting on its power to cover so many mathematical bases for students.

Tangrams activity is a 4 page pdf file, so it will print well at your site.

Give it a try, you’ll be energized, and your students will thank you!

- Mr. L

Algebra Problem Solving with the new Common Core Standards