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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.

A Little Math Humor

I just love to laugh in math class.  When I do, my students are surprised because they believe math is no laughing matter.  I like to have a small, humorous snippet on my tests or a cartoon hanging in my room.  Here are a couple I found on Pinterest which I absolutely love!

Apparently, x is lost!

Thank goodness, x has been found!
P.S.  Do you know the value of x in the above problem?