See You Later Alligator, Teaching Greater Than and Less Than

I originally posted this article back on May of 2011, but as I view products on Pinterest or on Teachers Pay Teachers, I feel a need to revisit it. I have seen alligators, fish, movable Popsicle sticks, etc. as ways to teach greater than or less than to children. Even though these may be good visual tools, to be honest, there are no alligators or even fish in mathematics.  Because many students still fail to understand which way the symbol is placed, (once in awhile I have a college student who is confused) here is a different method which you might wish to try. First of all, every child knows how to connect dots; so, let’s use that approach. 

Suppose we have two numbers 8 and 3. Ask the students, “Which number is greater?" Yes, 8 is greater. Let’s put two dots beside that number. 8 : Now ask, “Which number is smaller or represents the least amount?" You are right again. Three is smaller. Let’s put one dot beside (in front of) that number. Now have the students connect the dots.....

    

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It will work every time! When two numbers are equal, put two dots beside each number and connect the dots to make an equal sign. What makes this method a little different is that the students can visually see which number is greater because it has the most dots beside it; so when reading the number sentence, most of the time it is read correctly.

In a free handout entitled Number Tiles - Math Activities for the Primary Grades a greater than and less than activity is included which can be used over and over again. It's yours for free. Just click on the title to download your free copy.

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Using Lattice Multiplication in College - A Good or Bad Idea?

I work in the math lab at the college where we teach. All of our students have free access to this tutoring. This is one of my favorite things to do since I am working mostly one-on-one with students. Last week, I had a student who is taking Fundamentals of Algebra. It is a remedial class for those who do not pass the math test to take College Algebra. In this particular class, students are not allowed to use calculators. In other words, it helps if you know your math facts (and sadly, many don’t).

She had to multiply a three-digit number by a two-digit number in one of her word problems. Pretty easy, right? She kept telling me that she could only multiply using the “lettuce” form of multiplication. Many tutors in the lab had no idea what she was talking about; so, I asked her, “Do you mean lattice multiplication?” Once she demonstrated it to the group, I knew that she was using lattice multiplication.

For those of you unfamiliar with it, here is an illustration of what it looks like. 

Step #1

Draw a grid so that each digit has its own box. If it is a 3-digit by 3-digit problem, you will need a grid that is three by three. If it is a 3-digit problem by a 2-digit problem, you will need a grid that is three by two.

Step #2

Write the digits for the first factor going across the top of the lattice (grid), one digit per box. Write the second digits on the right going down the lattice, one number per box.

Step #3

Now, divide each box in half by drawing a diagonal line, starting in the top right corner and moving to the bottom left corner. Use a ruler if you like nice straight lines. You can have the diagonal lines continue so that they are outside of the grid boxes. This will help you with your answer!

Step #4

Work through the lattice and multiply each number together. Write the answers in the box. The number in the tens place goes in the upper part of the box, the number in the ones places goes in the lower part of the box. If there is not a number in the tens place, put a zero.

Step #5

To finish, you just add down the diagonal lines. Remember to regroup if necessary

Step #6

Finally, to figure out your product, read the numbers from the left of the grid around to the bottom of the grid.  In the example above, 789 x 461 = 363,729.

This form of multiplication dates back to the 1200s or before in Europe. It gets its name from the fact that to do the multiplication you fill in a grid which resembles a garden lattice, something you might find ivy growing on. Although it works, it is a pretty lengthy process for a college student, especially when she only has so much time to complete a test. 

The “vertical” method of multiplication is more efficient because most can write down and solve a 3-digit by 3-digit multiplication problem faster than many students can even draw the lattice. Unfortunately, I showed this student several other ways to multiply, but she failed to grasp their significance. She said that is how she learned in fifth grade, and she was too old to change! With all of that said, lattice multiplication will make it easier for her to transition from lattice multiplication of whole numbers to multiplication of polynomials. 

Writing Papers - Using a Graphic Organizer

I am currently teaching a Personal Development college class which is required for all new in-coming freshmen. In this class, we learn about learning styles, AVID strategies, how to take notes, how to read a textbook, etc. Their final project is a poster with an accompanying paper.  Here are the guidelines I give my students when it comes to writing the paper.

1) This paper should link and connect your ideas with any aspect of self, identity and personality concepts, mindset or learning styles we have discussed in class. In other words, use the class readings and discussions as a “lens” through which you view this person. Do this by using specific vocabulary used in class (e.g. conscious identity claims, growth or fixed mindset, grit, introvert or extrovert, learning style, soft and hard skills, etc.). 

2) Be sure to discuss how and what made this person successful. You might discuss their background, how and where they were raised, what challenges they overcame to succeed, how they reacted to failures and mistakes, what gave them the desire to succeed. 

3) This is not a facts paper about the person. This is about the character traits and attributes of the individual. Although facts can be included, most facts should be on the poster part of this project.

The first semester, the papers were just awful. I could use other words, but needless to say, they were painful to read. The next semester, I created A Graphic Organizer for Writing Papers. My students were amazed at how much easier writing a paper was. Many had never used a graphic organizer like this in English; so, this whole concept was new to them. (This was hard for me to believe, but I guess on the college level, such visuals are rarely used.) 

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This graphic organizer not only helped my students to arrange ideas thus communicating more effectively, but it also facilitated understanding of key concepts by allowing the students to visually identify key points and ideas more efficiently.

The blank graphic organizer found on Teachers Pay Teachers is divided into 11 sections, one for each paragraph. The students write the main idea followed by five details for each paragraph, not in sentence form but in a few words. Separate grids for the introduction and conclusion paragraphs are included. Even though there are 11 paragraphs, the organizer can be reduced to include as many paragraphs as you desire. My students were required to write a paper that was about two pages in length (500 words) when typed; so, this worked well in getting them to that point. Why not take a peek at the preview to see what you think? And if you choose to purchase the item, I would love your feedback.

I trust your students will find this graphic organizer easy to use as well as being a helpful aid in writing papers.

Algebraic Terms - Finding the Greatest Common Factor and Least Common Multiple Using a Venn Diagram

I tutor math at the college where I teach. Many of those students have been confused on how to find the greatest common factor for a set of algebraic terms. Having an elementary background, I introduce them to a factor tree which, believe it or not, many have never seen.

When just a rule is given by an instructor, often times, students get lost in the mathematical process. I have found that utilizing a visual can achieve an understanding of a concept better than just a rule. A Venn Diagram is such a visual and helps students to follow the process and understand the connection and relationship between each step of finding the GCF and LCM.

It's important to always begin with the definitions for the
words factor, greatest common factor and least common multiple. If a student doesn't know the vocabulary, they can't do the work! I continue by explaining and illustrating what a factor tree is (on your left) and how to construct and use a Venn Diagram as a graphic organizer.

Let's suppose we have the algebraic terms of 75xy and 45xyz. I have the students construct factor trees for each of the numbers as illustrated on the left.

Then all the common factors are placed in the intersection of the two circles. In this case, it would be the 5 and the xy. 

The students then put the remaining factors and variables in the correct big circle. Five and three would go in the left hand circle and the three 2’s and the z would be placed in the right hand circle.

The intersection is the GCF; so, the GCF for 75xy and 40xyz is 5xy.   To find the LCM, multiply the number(s) in the first big circle by the GCF (numbers in the intersection) times the number (s) in the second big circle.

5 × 3 × GCF × 2 × 2 × 2 × z = 15 × 5xy × 8z = 240. The LCM is 600xyz. 

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This method is applicable and helpful in algebra when students are asked to find the LCM or GCF of a set of algebraic terms such as: 25xy, 40xyz. (LCM = 200xyz; GCF = 5xy) or when they must factor out the GCF from a polynomial such as 6x²y³ + 9xy². Using a Venn Diagram is also an effective and valuable tool when teaching how to reduce fractions. 

Are you interested in finding out more about this method?  Then download my newest free resource entitled: Algebraic Terms and Fractions - Finding the Greatest Common Factor and the Lowest Common Multiple Using a Venn Diagram.