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. 

Five Assumptions About Distance Learning - Why It's NOT Working for Every Child!

My husband and I have taught together for over 80 years. We continue to do so because we love what we do; however, we are very disturbed by the terms Virtual Learning or Distance Learning or On-Line Learning. Now don't get the idea that because we are "old", we don't or can't use technology. Technology is part of our lives, especially in our teaching, but when we hear Distance Learning, many people make assumptions that simply aren't true.

Assumption  #1 - Students Have Access to a Computer and to the Internet

Numerous students do not possess a computer nor do they have access to the Internet. For example, my daughter teaches immigrant children, and out of all her families, only two even own a phone. Even though the school may purchase each student a computer, without Internet access, it is worthless. This is a significant issue in rural and lower socioeconomic neighborhoods. Or when the computer goes home, it somehow disappears. A computer can buy much needed food or sadly, drugs for an addict.

I HATE learning at home!

Assumption  #2 - Parents Create a Learning Environment for Their Children

It is obvious that people who believe this have not been in some of the children's homes that I've had the "privilege" of visiting. It's hard to study when chaos reigns or when parents are fighting or the child is expected to babysit younger siblings. Without set class times, children are often interrupted or distracted while studying. Learners with low motivation or unhealthy study habits often fall "through the cracks."

Assumption  #3 - Parents Help Their Children Learn

I believe that most parents want to help their child, but because of work schedules, level of education, not speaking English, etc. many cannot.  And then there are others who simply won't because it takes too much of their time. Participation by the parent is really voluntary. In addition, if students have questions where the parents cannot help, s/he has to wait when the teacher is on-line to assist them.

Assumption  #4 - Students Have an Intrinsic Desire to Learn

Maybe I've just taught too long to believe this one. Most students would rather skip the lessons and play whether it be outside or a game on the computer.  Rare is the child that can hardly wait for that video lesson to come on. After all, there really isn't any accountability since the teacher is just "in the computer." A student must be a self directed learner.

Assumption  #5 - Computer Lessons are Better Because It's Technology

Unfortunately, there are many teachers uncomfortable with teaching on-line. I know because I teach at a community college, and when all in-person classes went to on-line classes in March, many teachers struggled to deliver valuable and worthwhile content. We lost many students because they disliked the on-line classes. I often sat three hours in the virtual math lab with not one student coming for help even though the in-person math lab was always full.

Presently, our college is struggling with enrollment because instead of taking on-line classes, the students have opted to take the semester off. On-line teaching cannot satisfy ALL educational needs and goals. (i.e. hands-on subjects such as public speaking, surgery, dental hygiene, science, sports, etc.) Just because it may be technologically possible to simulate a physical learning experience does not necessarily mean that it is the best way to teach it.

To summarize, I believe we are doing a disservice to most of our students when just on-line education takes place. Many have not been in school since March and have therefore lost three months of learning, not to mention meals, love, encouragement, etc. For countless students, school is their "safe" place. If face-to-face classes don't resume soon, our poorest children are the ones who will be the BIGGEST losers!.

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Ideas on How to Teach Long Division

As presented in an earlier posting, there is another way to approach long division.  However, since many of you are required to present it the long way, here are a couple of ideas to make it easier for your students.

First of all, have the students use graph paper.  The squares help to keep the numbers aligned which seems to be a problem for many students.  If you don't have graph paper, you can download free templates at Donna Young's Free Graph Paper and make your own.  I like the idea of separating the problems with lines to ensure there is no cross over from one problem to another.

Secondly, try using the acronym (a mnemonic device) of Does McDonald's Sell Cheese Burgers.  I have see this acronym many times on Pinterest, but usually the C is omitted.

Check means that after the student has subtracted, they should check to see if the remainder is smaller than the divisor.  If it is equal to or larger than the divisor, then enough was not taken out of the dividend.  This is a step often skipped when long division is taught; yet, if the student doesn't check and make the needed correction, the answer (quotient) will be wrong.

In order to learn division, the student must first have a good understanding of multiplication. Students don’t need to perfectly know all of the times tables, but a majority of the facts or having a reasonably quick strategy to figure out the answer is necessary.

Start by practicing division using the number series the students can easily skip count such as 2 and 5. Then gradually move up to nine. After that, move to division by double digit numbers using 10 since most students know how to skip count by 10. Once the concept is understood, teaching division will become more about guided practice to help your child to become comfortable with the division operation which, in reality, is a different kind of multiplication practice.

The Long and Short of It - Two different ways to do long Division

My remedial college math class is currently working on fractions. (Yes, even many college students don't understand them!) When we discussed how to change an improper fraction to a mixed numeral, long division came up. I showed the class a shortcut I was taught many years ago (approximately when the earth was cooling) and none, no not even one student, had seen it before. I wonder how many of you are unfamiliar with it as well? First let's look at long division and how most students are taught today. We will use 534 divided by 3.

Now if that doesn't make your head swim, I don't know what will. Everything written in the third column is what the student must mentally do to solve this problem. Then we wonder why students have trouble with this process. There is another way, and it is called short division for a reason. This is the way I learned it.......

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I don't know about you, but I would rather have my students doing mental math to solve division problems than writing everything out in the long form. And the paper and frustration you will save will be astounding! So what will it be.....long division or short division?

As a Side Note: Since many students do not know their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math
tool. This resource contains four easy to understand divisibility rules and includes the rules for 1, 5, and 10 as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for use by the student. If you are interested, just click under the resource title page.

Dividing Fractions Using KFC

Ugh - It's time to teach the division of fractions. My experience has been that many students forget which fraction to flip and often, they forget to change the dreaded division sign to a multiplication sign. The other evening,  I was helping my 5th grade granddaughter with her homework. Really, she had completed it by herself, but she wanted me to check it. At the top of her paper were the letters "KFC". I asked her what they meant, and she replied, "Kentucky Fried Chicken." Now I have taught math for years and years, and I had never heard of that one!

She explained that the "K" stood for keep; "F" for flip, and "C" for change. Let's suppose the problem on the left was one of the problems on her homework paper.

First, she would Keep the first fraction. Next, she would Flip the second one, and then Change the division sign to a multiplication sign...like illustrated on the right. She would then cross cancel if possible (In this case it is).  Finally, she would multiply the numerator times the numerator and the denominator by the denominator to get the answer.

She was able to work all the division problems without any trouble by just remembering the letters KFC.

Yesterday, I was working in our college math lab when a student needed help. On the right is the problem he was having difficulty with. (For those of you who don't teach algebra or just plain hate it, I am sure this problem looks daunting and intimidating. Believe me, my student felt the same way!) 

First I had the student rewrite the problem with each fraction side by side with a division sign in between them like this.

Doesn't it look easier already? I then taught him KFC. You read that right! I did! (I figured if it worked for a 5th grader, it should work for him.) Surprisingly it made sense to him because he now had mnemonic device (an acronym) that he could easily recall. He rewrote the problem by Keeping the first fraction, Flipping the second, and Changing the division sign to a multiplication sign.

Now it was just a simple multiplication problem.  Had he been able to, he would have cross canceled, but in this case, he simply multiplied the numerator times the numerator and denominator by the denominator to get the answer.

So the next time you teach the division of fractions, or you come across a problem like the one above, don't panic!  Remember KFC, and try not to get hungry!

Dump and Divide or Better Known As Converting Fractions to Decimals

When working with fractions, my remedial math college students are never quite sure which number to divide by. This same thing often occurred when I taught middle school and high school. So the question I had to answer was, "How can I help my students remember what number goes where?"

First, the student must understand and know the vocabulary for the three parts of a division problem. As seen in the problem above, each part is correctly named and identified.

Side Note: The symbol separating the dividend from the divisor in a long division problem is a straight vertical bar with an attached vinculum (you might have to look this word up) extending to the left, but it seems to have no established name of its own. Therefore, it can simply be called the "long division symbol" or the division bracket. I wish it were named something fancier, but sometimes plain and straight forward is the best!
Now let's look at a fraction that the student is asked to rewrite as a decimal. The fraction on your right is two-fifths and is read from top to bottom as two divided by five. That's easy enough, but when my students enter this into their calculators, many will put in the 5 first, and then press the

division sign, followed by the 2. Of course, they get the wrong answer. Now let's look at the dump and divide method.

First, dump the 2 into the calculator. Then press the division sign; then divide by 5. The answer is 0.4.

I am aware that many of students are not allowed to use calculators; so, let's look at how this method would work using the division bracket. We will use the same fraction of 2/5 and the same phrase, dump and divide.

First, take the numerator and dump it inside the division bracket. (Note: Use N side instead of inside so that numerator and N side both start with "N".) Now place the 5 outside of the long division bracket and divide. The answer is still .4.

Dump and Divide will also work when a division problem is written horizontally as a number sentence such as: 15 ÷ 3. First, reading left to right, dump 15 into the division bracket. Now place the 3 on the outside. Ask, "How many groups of three are in 15?" The answer is 5.

Try using Dump and Divide with your students, and then let me know how it works. You can e-mail by clicking on the page entitled Contact Me or just leave a comment.

Something Else to Think About:  Since many students do not know

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their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math tool. The resource, Using Digital Root to Reduce Fractions, contains four easy to understand divisibility rules as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for the student. Download the preview to view the first divisibility rule plus three samples from the student check off list.

Ten Things That Require Zero Talent

I am an instructor at a community college. Many of my colleagues have unique  posters hanging outside their doors. I found this one by a co-worker's office who teaches English. I believe it is something we can all use during this difficult time when the coronavirus requires that most of us are to remain at home.

Spiders Are Your Friends - Learning about Spiders

Spiders! We see pretend ones in the store as Halloween decorations (some are pretty terrifying) or real ones outside in a web they have created.  For some reason, these creatures are always something that students want to learn about. How are spiders different than insects? What is an orb web? Are all spiders poisonous? How does the spider not get stuck in her own web? These are questions that students will ask because they are curious and inquisitive.

Did you know spiders are really useful animals and serve mankind well? They eat mosquitoes, grasshoppers, locusts and other insects that are harmful to man. A single spider may kill about two thousand insects in its lifetime. Even though you may be afraid of spiders, very few are dangerous. The black widow and the brown house (recluse) spider do have poisonous bites, but there are no other common house spiders known to be dangerous.

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Spiders are not insects, and insects are not spiders. Spiders are arthropods because they have spinning glands used to create silken threads. Sometimes spiders are called arachnids because of their eight legs. Spiders and insects have different attributes. All insects have six legs, but all spiders have eight legs. An insect has a three-part body, but a spider has only two parts to its body. Insects have antennae or feelers, and spiders do not. Spiders can usually be found in basements, barns, garages, or attics. In warm weather, you can find them under rocks or logs, sitting on fences, or in the grass and flowers. There are about forty thousand different species of spiders.

Interested in learning more?  Check out a ten page short mini reading/science unit  about spiders. First, the students read a short passage about spiders. Then they answer several questions about the reading based on Bloom's Taxonomy, or they do an activity related to the reading passage. Activities include dictionary work, spider math problems, labeling the parts of a spider, and completing a spider web. This mini unit is appropriate for grades 3-5 and will take about five days to complete. An answer key is included.

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Want something other than a mini unit? Check out the two word search puzzles and the two crossword puzzles available just in time for Halloween!

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.