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Fraction Action

In my July 1, 2011 posting entitled Fractions for the Confused and Bewildered, I introduced you to an alternate method for adding fractions called Cross Over, but sometimes students may have to add more than two fractions.  What happens then?  Let’s suppose we have the following problem.

Start out by multiplying the numerator of the first fraction by the denominators of the other two fractions as shown above.  (1 × 5 × 3 = 15)

Do the exact same thing with the second fraction and then add that product to the first.

Now repeat this process using the last numerator of 2 and add that to 33.

The result is the numerator of the answer.  To find the denominator, just multiply all the denominators together just as we do in the Cross Over strategy.

As usual, you may need to reduce to lowest terms or change an improper fraction to a mixed number.  In this example, the improper fraction becomes a mixed number.
For many of us, this may seem like a lengthy and complicated process but for my mathphobic students who have difficulty finding the lowest common denominator, they view this as easy and stress free.  The key is that they have a strategy that works for them.
If you are interested in other alternate ways to teach fractions, check out the resource Fractions for the Confused and Bewildered.

Don't Flip!

My college students in remedial math just finished the chapter on fractions.  Talk about mathphobia.  Dividing fractions was the most confusing for them because it requires finding the reciprocal of the second fraction, changing the division sign to a multiplication sign, and then multiplying the numerator times the numerator and the denominator times the denominator. 

Let me introduce a new method entitled Just Cross.
1)      First and foremost, you must understand what division is.  The statement 8 ÷ 4 means 8 divided into 4 equal sets, OR how many fours are in eight, OR how many times can we subtract 4 from 8?  (Yes, division is repeated subtraction.)
2)      Let me explain this using a hands-on visual.  Let’s assume the fraction problem is:    

The question being asked is, How many ¼’s are in ½?”
 First, fold a piece of paper in half.  The figure on the left represents ½.  Next, fold the same sheet of paper in half again to make fourths as seen in the illustration on the right.  When you unfold the paper, you will notice a total of four sections.  So answering the original question: How many ¼’s are in ½”, you can see that the half sheet of paper contains two parts; therefore:
3)      Using the same example, to work the problem, the fraction 1/4 would have to be flipped to 4/1  nd then 1/2 would have to multiplied by 4/1 to get the correct answer of 2.  That is why the division of fractions requires that the second fraction be inverted and the division sign be changed to a multiplication sign. 
Let’s use the same fraction problem, but let’s utilize a different method entitled Just Cross.
           1)  Cross your arms as a hands-on way of remembering the process.
            2)  Now multiply the denominator of 4 by 1 and the denominator of 2 by1 as seen below.  (We do nothing with the denominators.)  Notice we always start on the left side and then we go to the right side.  I often tell my students it is, "Left, right; left, right"  as if we are marching.  If it is done the opposite way, the answer will be incorrect.
3)      Now simply divide 4 by 2 to get the answer of 2.

No flipping; no reciprocal, no changing the division sign to a multiplication sign.
Just Cross and divide.  Amazingly, it works every time.

I have a resource that features different ways to teach fractions using hands-on strategies similar to the one above.  Just go to Fractions for the Confused and Bewildered.

What You Need to Know to Study Math

Math is hard work, but you can't let that prevent you from being successful.  Anyone who has succeeded in anything has put in "tons" of hard work.  Think about the Olympians and all the practice that is required to even make a local team.  How about anyone who is good at athletics?  Do football players say, "Learning all of those plays is just too hard.  I think I'll quit!"  I don't think so.  The same holds true for learning a subject, any subject whether you like it or not. Below are eleven things to know and think about before you study math or take that next math class.

1)    Remember, an extra step is required to pass math. You must use the information you have learned to correctly solve new math problems.

2)    You must be able to do four things....

a)      Understand the material
b)     Process the material
c)      Apply what you have learned to correctly solve a problem
d)     Remember what you have learned and apply to new material
3)     Math has a sequential learning pattern; material learned one day is used the next day and the next, etc.  All of the building blocks must be included to be successful.

4)     Math classes should be taken each semester with no breaks to enhance the probability of remembering previous material.

5)      Math is similar to a foreign language; practice it or you will forget it.

6)     Math is a skill subject.   You have to actively practice the skills involved to master it – like learning to play a musical instrument, a sport, or using auto mechanic skills.

7)     Math is a fast-paced subject.  You must learn a lot of information in each class so you are ready to move on to the next class.

8)      Society doesn’t help students.   It says it is OK to hate math, to not be able to do it. You will often hear from parents, "I was never any good at math either."

9)    A bad attitude shouldn't prevent you from doing well in math if you decide you are going to do well. You may not like history or English either, but you have to take the required classes and do well in them if you plan on passing/graduating.

10)  Math is objective, and you will receive the grade you earn. There is no talking a teacher into a better grade BECAUSE you must know the material before you can move forward, or you will fail.

11)  Study to make an A on the first test in any math class. It is probably the easiest test, but it counts the same as all of the others. An A shows you know the basics you need to succeed. An A is a good motivator to do well on future tests. An A on the first test improves your confidence that you can do well.


Ten More Tips So You Won't Forget!

Math Study Tips You Won't Forget
I know Moses only gave the Israelites Ten Commandments, but remember, these study tips are not commandments; they are suggestions. Keep in mind, math courses are not like other courses. To pass most other subjects, a student must read, understand, and recall the subject matter. However, to pass math, an extra step is required: a student must use and apply the information they have learned to solve math problems correctly. Special math study skills are needed to help the student learn more and to get better grades. Below are my last ten of twenty Math Study Tips. 

1)      What You Know:  Answer what you know first.  That way, you will be more relaxed when you get to the more difficult questions. 

2)      Read: Read the questions.  Look for words like explain, define, select, give an example, etc.  Look at the points attributed to each question and do what you are asked.
3)      Finished: If you are done early go back over your answers. Make sure that you did what the question asked and check your answers for clarity.

4)      Jot It Down! Write items down such as formulas (or what they are used for) or items you are afraid you will forget somewhere on the test as soon as you receive it. Now you can relax and concentrate just on the test.

5)      Show: Show all of your work; it may be worth points.

6)      Clarity: Reread your work for clarity.  You may know what you mean, but if the teacher cannot make heads or tails of it, you will not earn the points.

7)      Dress Appropriately: Ask yourself, “Is the classroom normally cold? Hot?”  You do not want to be uncomfortable during the assessment. 

8)      Books/Materials: Bring all books and supplemental materials that can be used on the test.

9)      Time and Date: Know the time and date of your test.  Set an alarm, and do what you need to do to be on time for class.

10)  Be Persistent:  After taking 19 steps towards success, you are going to do great! 
The entire list of Math Study Tips is now available on Teachers Pay Teachers. 
It is a free download.  Just click under the above cartoon.


Study Skills - The First Ten

Most of us are familiar with the Ten Commandments.  I am NOT Moses, but I do have ten good study tips for when it comes to studying mathematics.  Read them over carefully as some of them might surprise you.  Feel free to copy these and hand them out to your students before the next big math test.  That's what I did with my remedial math college students, and I was surprised at how positively they responded.

1)      Notes: Organize them and make sure you are not missing anything.
2)      Instructor: What did your teacher tell you to study?
3)      You: Study in a way that works best for you (ex. place and times).

4)      Friends: If you stay on track, studying with friends can help.  Quiz each other, and everyone can explain what they know. 
5)      Tricks: Use methods such as mnemonic to help memorize blocks of information.
6)      Do It Now: No one wants to fail, go to summer school or take the class again.  Work now so you do not have to pay later.

7)      Breaks: Take regular breaks while you study and do not stay up all night.  Lack of sleep will make it hard for you to focus and do your best.

8)      Eat: Eat a good breakfast or lunch before the test.  Not only will a growling stomach interfere with your concentration, but your brain will not function at its best ability when it needs energy.  (Note: Research has shown that eating peppermints will help you to remember what you study!)

9)      Avoid Caffeine: Coffee or coke may give you a quick alert boost, but you will rebound and lose steam.  Drink water; it keeps you hydrated.  (This is a hard one for me.  I am pretty wicked without my morning cup of Java!)

10)  Needs:  Take what you need to the exam or test.  Think ahead.  Do you need a ruler, a calculator, paper, etc.?
Now have your students go back and highlight the study tips they are already doing.  Then ask them to draw a star beside the one they want to work on before the next test.  If you have your students do this activity, I believe you'll find it to be a positive beginning on how to study math.

I Need Math Study Skills?

This semester, I am teaching a new class called Math Study Skills.  We are finding that many of our students who do not qualify for college algebra in reality do not know how to study math.  When you think about it, math is different than other subjects in that it continues to build.  You might do well on the test over Chapter #1, not so hot on Chapter #2, but for sure, you will not succeed on the test over chapter #3.  I am putting together lots of supplemental materials for the class which I hope to share with you on this blog. 

In our next class I am going to ask the students to determine if the following statements about math are true or false.  See what you think.
Math Profile Sheet

  1. In math, there is only one way to get the answer.
  2. To be good at math, you have to be good at calculating.
  3. If you are good at math, you skip steps and do all of the work in your head.
  4. Men are much better at math than women.
  5. There is a "best" way to complete a math problem.
  6. You have to have a mathematical mind to understand math.
Believe it or not, all of these are false statements! I think my students will be surprised as well. But of course the best study skill I can give them is depicted in the cartoon!

Are you interested in a Math Profile Sheet that will help you to measure the mathematical success of your students?  Check it out by clicking under the cartoon.

Anno's Counting Book

Anno’s Counting Book by Mitsumasa Anno is one of the best math picture books for children that I have used with kindergartners and first graders.  This wordless counting book shows a changing countryside through various times of the day and seasons.  It introduces counting and number values from one to twelve. On each page, you can find several groups of items representing the illustrated number, such as 4 fish, 4 trees, and so on. The number is also represented by stacked cubes at the side of the illustration.  The book contains one-to-one correspondence, groups and sets, and many other mathematical relationships.  I purchased the Big Book version so that the entire class could easily see each picture.

Here are a couple of activities that you might try with the book.

1)      “Read” the book to the children and discuss what is happening.  The following questions will help the children to connect what is occurring in the book:

a)      What time of year is it when the story begins?  Ends?  How do you know?

b)     What are the seasons that you see throughout the book?

c)      How is the village changing?

d)     What kinds of transportation do you see?

e)      Compare and contrast what the children are doing in each scene.

2)      Discuss what happens to the trees as the season change in the book.  Are there different kinds of trees in the book?  How do you know? (color of leaves, size, etc.)

a)      Have the students fold a 9” x 12” sheet of paper into fourths.

b)     Have them write the name of a season in each section. (summer, fall winter, spring)

c)      Have them draw the same tree in each section, but show how it looks in summer, fall, winter and spring.

Happy Reading!

Patterns and Problem Solving

In my last two posts, I have presented some patterns for you to look at with the hopes that you would try to dissect them and be able to answer a few questions.  Are you ready for two more?  Here is the first one that I call Consecutive Number Series.

What counting pattern do you see in this sequence?  How would you describe the sequence of numbers that are being added?  What pattern do you see in the answers?  Can you figure out the pattern for 8 × 8 and 9 × 9?  Notice this pattern made a triangle.  Do you know what kind it is?

My next pattern I call The Eights, and you will readily see why. It, too, forms a triangle, but a different kind. Do you recognize the triangle as isosceles?  If you take a ruler, you will find that the base is 4" while the sides are both 3".

What do you think will happen if we take this same pattern and add a 0?  Notice that this pattern does not begin with adding an eight.  Can you figure out why?

I use this type of patterns with my remedial math college students because I consider it important to do some problem solving while recognizing and describing patterns.  After all, problem solving is a part of life.  It doesn't occur in a vacuum. Because students must reason about some specific content, I think patterns are a great place to begin.  Problem solving also helps students to make connections to other parts of mathematics and find some relevance to what they are learning.  And did you know, that problem solvers are typically better test takers?  So take the patterns from my last three postings, and create some of your own questions for your students.  Use them in a journal or as a small group activity.  But whatever you do, have fun learning and discovering patterns in math.

The Beauty of Math Patterns

Some people say mathematics is the science of patterns which I think is a pretty accurate description. Not only do patterns take on many forms, but they occur in every part of mathematics. But then again patterns occur in other disciplines as well. They can be sequential, spatial, temporal, and even linguistic.

Recognizing number patterns is an important problem-solving skill. If you recognize a pattern when looking systematically at specific examples, that pattern can then be used to make things easier when needing a solution to a problem.

Mathematics is especially useful when it helps you to predict or make educated guesses, thus we are able to make many common assumptions based on reoccurring patterns. Let’s look at our first pattern below to see what we can discover.

What can you say about the multiplicand? (the number that is or is to be multiplied by another. In the problem 8 × 32, the multiplicand is 32.)  Did you notice it is multiples of 9?  What number is missing in the multiplier? 
Now look at the product or answer.  That’s an easy pattern to see!  Use a calculator to find out what would happen if you multiplied 12,345,679 by 90, by 99 or by 108? Does another pattern develop or does the pattern end?
Here is a similar pattern that uses the multiples of 9.  How is the multiplier in this pattern different from the ones in the problems above?  Look at the first digit of each answer (it is highlighted).  Notice how it increases by 1 each time.  Now, observe the last digit of each answer. What pattern do you see there?  Using a calculator, determine if the pattern continues or ends.
Recognizing, deciphering and understanding patterns are essential for several reasons. First, it aids in the development of problem solving skills. Secondly, patterns provide a clear understanding of mathematical relationships. Next, the knowledge of patterns is very helpful when transferred into other fields of study such as science or predicting the weather.  But more importantly, understanding patterns provides the basis for comprehending Algebra since a major component of solving algebraic problems is data analysis which, in turn, is related to the understanding of patterns. Without being able to recognize the development of patterns, the ability to be proficient in Algebra will be limited.
So everywhere you go today, look for patterns.  Then think about how that pattern is related to mathematics. Better yet, share the pattern you see by making a comment on this blog posting.

The Eleventh Hour?

The mathematician magician is still here, sharing her tricks.  This week it is the elevens.  Before we demonstrate the trick, I have to get on my soap box for just a moment.  In my humble opinion, all students should know their times tables through 12 even though the Common Core Standard for third grade says through 10 x 10.  Remember, Common Core is the minimum or base line of what is to be learned.  In Algebra, I insist that my students know the doubles through 25 x 25 and the square roots of those answers up to 625.  It saves so much time when we are working with polynomials.

Now to our our amazing mathematical "trick".  Let's look at the problem below which is 231 x 11. 

First we write the problem vertically. Next, we bring down the number in the ones place which in this case is a one. Now we add the digits in the ones and tens place which is 3 + 1 and get the sum of four which is brought down into the answer.

Moving over to the hundreds place, we add that digit with the digit in the tens place 2 + 3 and get an answer of five which we bring down.  Finally, we bring down the digit in the hundreds place which is a two.  The answer to 231 x 11 is 2,541. 

Now try 452 x 11 in your head.  Did you get 4,972?  Let's try one more.  This time multiply 614 by 11.  I'm waiting......  Is your answer 6,754?

Now it is time to make this process a little more difficult.  What happens if we have to regroup or carry in one of these multiplication problems?

We will multiply 784 by 11.  Notice that we start as we did before by just bringing down the number in the ones place.  Next, we add 8 + 4 and get a sum of 12.  We write down the 2 but carry or regroup the one.  We now add 7 + 8 which is 15 and then add in the 1 we are carrying.  That makes 16.  We bring down the 6 but carry the 1 over.  We have a 7 in the hundreds place, but must add in the one we are carrying to get a sum of 8.  Thus our answer is 8,624.

Let's see if you can do these without paper or pencil.  965 x 11   768 x 11    859 x 11   After working the problems in your head, write down your answers and check them with a calculator.  Try making up some four and five digit problems because this is a non-threatening way to have your students practice their multiplication facts.  Have fun!

Never Too Old to Play Games!

I currently teach remedial math students on the college level. These are the students who fail to pass the math placement test to enroll in College Algebra - that dreaded class that everyone must pass to graduate.  The math curriculum at our community college starts with Basic Math, moves to Fractions, Decimals and Percents, and then to Basic Algebra Concepts.  Most of my students are smart and want to learn, but they are deeply afraid of math. I refer to them as mathphobics.
We all have this type of student in our classrooms, whether it is middle school, high school, or college. When working with this type of student, it is important to bear in mind how all students learn.  I always refer back to the Conceptual Development Model which states that a student must first learn at the concrete stage (use manipulatives) prior to moving to the pictorial stage, and well in advance of the abstract level (the book).  This means that lessons must include the use of different manipulatives. 

I use games a great deal because it is an easy way to introduce and use manipulatives without making the students feel like “little kids.” I can also control the level of mathematical difficulty by varying the rules; thus, customizing the game to meet the instructional objectives my students are learning.   However, as with any classroom activity, teachers should monitor and assess the effectiveness of the games. (Refer to the December 13, 2011 posting about Lesson Plans and Research.)   

When using games, other issues to think about are:
1)      Excessive competition.  The game is to be enjoyable, not a “fight to the death”.
2)      Mastery of the mathematical concepts necessary for successful play.  Mastery should be at an above average level unless teacher assistance is readily available when needed.  A game should not be played if a concept has just been introduced.
3)      Difficulty of the rules.  If necessary, the rules should be modified or altered in order that the students will do well.
4)      Physical requirements (students with special needs).  These should be taken into account so that every player has an opportunity to win.

In addition to strengthening content knowledge, math games encourage students to develop such skills as staying on task, cooperating with others, and organization.  Games also allow students to review mathematical concepts without the risk of being called “stupid”.  Furthermore, students benefit from observing others solve and explain math problems using different strategies.
Games can also….
1)      Pique student interest and participation in math practice and review.

Games that Teach
2)      Provide immediate feedback for the teacher.  (i.e. Who is still having difficulty with a concept?  Who needs verbal assurance?  Why is a student continually getting the wrong answer?)
3)      Encourage and engage even the most reluctant student.
4)      Enhance opportunities to respond correctly.
5)      Reinforce or support a positive attitude or viewpoint of mathematics.
6)      Let students test new problem solving strategies without the fear of failing.
7)      Stimulate logical reasoning.
8)      Require critical thinking skills.
9)      Allow the student to use trial and error strategies.
Mathematical games give the learner numerous opportunities to reinforce current knowledge and to try out strategies or techniques without the worry of getting the “wrong” answer. Games provide students of any age with a non-threatening environment for seeing incorrect solutions, not as mistakes, but as steps towards finding the correct mathematical solution.

For a complete listing of the games I have available on Teachers Pay Teachers, click on the page above entitled Games That Teach or click under the quote by George Bernard Shaw.