Posted in Algebra, Number Sense

Line multiplication and the FOIL method

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Line  multiplication is a nice activity for teaching multiplication especially for more than one-digit numbers. The method is shown in the figure. The horizontal line represents the number 13 where the top line represents the tens digit and the lines below it represents the ones digit. The lines are grouped according to their place value. The same is true for the number 22. To find the final answer, count the number of intersections and add them diagonally. Dr. James Tanton produced a video about line multiplication. Click the link to view.

 

James Tanton related this procedure to rectangle multiplication. For example, the problem 13 x 22 in rectangle multiplication is

#multiplicationIf this is done in class I would suggest that before you show the rectangle multiplication as explanation to the process of line multiplication it would be great to connect it first to counting problem. Instead of counting the points at each cluster by one by one, you can ask the class to find for a more efficient way of counting the points of intersections. It will not take long for students to think of multiplying the array of points in each cluster. Given time I’m sure students could even ‘invent’ the rectangle multiplication themselves. Inventing and generalizing procedures are very important math habits of mind.

Line multiplication or counting intersections of sets of parallel lines is generalizable. You can ask your students to show the product of (a+b)(c+d) using this technique. The answer is shown in the figure below. Note that like rectangle multiplication this can be extended to more than two terms in each factor also. This is much better than the FOIL method which is restricted to binomials. I’m not a fan of FOIL method especially if it is taught and not discovered by the students themselves. Through this line multiplication activity I think they can discover that shortcut.

#multiplication

Posted in Humor

Two Plus Two Apples or Why Indians Flunk

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I found a piece of paper with this little poem inserted in my old notebook. It was written by Beverly Slapin. I realized I was born an Indian and will always be.
two apples

All right, class, let’s see who know what two plus two is. Yes, Doris?

I have a question. Two plus two what?

Two plus two anything.

I don’t understand.

OK, Doris, I’ll explain it to you. You have two apples and you get two more. How many do you have?

Where would I get two more?

From a tree.

Why would I pick two apples if I already have two?

Never mind, you have two apples and someone gives you two more.

Why would someone give me two more, if she could give them to someone who’s hungry?

Doris, it’s just an example.

An example of what?

Let’s try again—you have two apples and you find two more. Now, how many do you have?

Who lost them?

YOU HAVE TWO PLUS TWO APPLES!!!! HOW MANY DO YOU HAVE ALL TOGETHER????

Well, if I ate one, and gave away the other three, I’d have none left, but I could always get some more if I got hungry from that tree you were talking about before.

Doris, this is your last chance—you have two, uh, buffalo, and you get two more. Now how many do you have?

It depends. How many are cows and how many are bulls, and is any of the cows pregnant?

It’s hopeless! You Indians have absolutely no grasp of abstraction!

Huh?

-by Beverly Slapin

Posted in Mathematics education

Bloom’s Taxonomy and iPad Apps

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The original Bloom’s taxonomy include Knowledge, Comprehension, Analysis, Synthesis, and Evaluation. I was introduced to this when I was in college and I must admit it was not of much help to me in planning my math lessons. I just couldn’t fit it. The pyramid image was not of help at all and I think even created the now much ingrained deductive method of teaching. I think teachers must have unconsciously looked at it as a food pyramid so they give a dose of those of knowledge-acquisition activities first before providing activities  that will engage students in higher-level processes

 

Lorin Anderson, a former student of Bloom, revisited the cognitive domain in the learning taxonomy in the mid-nineties and made some important changes: changing the names in the six categories from noun to verb forms and slightly rearranging them. The new taxonomy reflects a more active form of thinking of Creating, Evaluating, Analyzing, Understanding, and Remembering. I also like the inverted pyramid as long as it is not viewed like there is a strict hierarchy of the categories. In fact in my own experience I just make sure that all these are covered in a lesson as much as possible. The way to do this is to teach mathematics through problem solving or engage students in mathematical investigations. Still, the best framework will still be one tailored to mathematics. For me its my list of Mathematical Habits of Mind.

Revised Bloom's Taxonomy

In searching for Bloom’s taxonomy I came across the image below – Bloom’s taxonomy for iPad. It’s a collection of iPad apps classified according to Bloom’s taxonomy.  I found it cute so I’m including it here. This will come in handy once I have my own iPad and start creating math lesson for this device.

There is also such a thing as Learning Pyramid which compares how we learn things and the retention rate in our brain after 2-3 weeks.

Click here for source of image of Bloom’s Taxonomy for iPads.

Posted in Algebra

Guest post: Online Resources to Supplement Math Lessons

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Teachers struggling to improve their students’ math experiences can look online for a plethora of fun and interactive games and programs. All of the online math curriculum help outlined in the following list is free to access and simple enough that a basic computer could run them with no problem. The games are all browser-based, and most will need a version of Adobe Flash Player or equivalent. The first three sites are primarily interactive online games. The last two sites provide printable resources for teachers and students.

1. CSI: Mathematics

This site is the perfect blend of online games and printable supplements. It is organized by grade level and them by subject. Each subject includes a free and printable color resource PDF that gives tips, tricks, and explanations for each subject. There is also an applicable book list included. The PDF would be perfect for in-class reminders and to send home to parents. The site then links to online activity resources for each subject.

2. Sheppard’s Software

This simple and effective site arranges interactive math games by subject. It covers early math through simple algebra. Importantly, it directs students toward games on important concepts like time, measuring, money, and place value.

3. Math Playground

This simple website organizes games by popularity and subject. Students will be attracted to the game through an introductory picture. The games are interesting, although the site is somewhat less organized than the others.

4. Cool Math Games

This is an incredible online resource, and it is first on our list because you just may not have to go any further to find something great for your students. The site is completely free and includes both activities and virtual lessons to supplement teaching. It is visually pleasing and simple enough that elementary-age students could navigate it with some simple directions. It does have an almost overwhelming amount of material.

The best option would be to have a direction and purpose for the class going-in rather than just letting them explore. The website has lessons ranging from basic numeration to pre-calculus work. The one disadvantage to using lessons from this site is the restrictions on printing and downloading. All of the work must be viewed and manipulated through the website. For a class lesson, hook your computer up to the projector.

5. Math.com

This is a site full of resources and games. It is last on the list because it would be the hardest for students to self-navigate. However, it provides an excellent mix of materials including items for homework help and activities for tutors and volunteers.

When it comes to making math interactive and entertaining there is no reason to not try to incorporate as many games as possible.  While technology becomes more and more popular in the classroom, teachers should embrace all forms of online resources in order to supplement their own lessons.

Author: Brandy Olsen