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From A Different Angle

Here is a riddle for you.  What did the little acorn say when he grew up?  Give up?  It's Gee-I'm-A-Tree or Ge-om-e-try.  This is what my students are beginning to study.  I absolutely love teaching this part of math, and it is interesting how the students respond.  Those that are visual, love it, but usually, those who do better with the abstract aren't so fond of it. 

I have a beautiful, talented daughter who loves languages.  She is fluent in Spanish and loves to write, write, and write.  To my chagrin, she always struggled in math, especially in high school, until she got to Geometry.  Her math grade changed from a disappointing (let's just say she passed Algebra) to an A.  She thought Geometry was wonderful!!

I enjoy teaching Geometry because there are so many concrete ways to show the students what you mean.  For instance, when introducing angles, (before using protractors) I use my fingers, coffee filters (when ironed, they make a perfect circle), interlocking plastic plates, the clock, etc. to demonstrate what the various angles look like. Here is an example of what I mean. 

To introduce right angle, I have the students fold a coffee filter (which is ironed flat) into fourths, and we use that angle to locate right angles all around the room.  We discuss the importance of a right angle in architecture, and what would happen if a right angle didn’t exist.   
We then use an analog clock to discover what time represents a right angle.  Right away, they respond with 3:00 or 9:00.  Some will say 3:30, but when I display 3:30 on a Judy clock (comes in handy even on the college level), they see that the hour hand is not directly on the three which means it is not a 90 angle.

I also demonstrate a right angle by using my fingers.  What is great about fingers is that they are always with you.  I call the finger position you see on the right, Right on, Right angle.
So are you ready for another geometry riddle?  (I have many!)  What is Orville and Wilbur's favorite angle?  That’s right; it is a right angle. (The Wright Brothers)
Want more geometry riddles?  Check out Geometry Parodies by clicking here.  It's one of my freebies. Also, if you are interested in many different ways to teach angles, take a look at my product entitled: Geometry: Hands-On Activities.

Yes or No? Stay or Go?


Hands-On Equation Balance Beam
www.borenson.com
My Pre-Algebra college students have just started solving equations containing one unknown.  As I tell them, we have now become snoopy detectives looking for the unknown.  Their greatest difficulty is deciding what stays and what goes in an equation. In other words, which term should be cleared by using the inverse operation?  I always start this chapter using Hands-On Equations®.  I have used them for years, and it does give a visual for those concrete learners.  I also refer to the written equation as a teeter-totter or see-saw which must always stay balanced. (Notice that Hands-On Equations® uses a balance beam.)  We also discuss the commandment of "Whatsoever thou doest to one side of the equation, we must doest to the other" and its importance. (I admit that I was with Moses when he received the Ten Commandments, but it "fell upon me" to convey The First Commandment of Solving Equations to future mathematicians.

One Unknown
After much practice with the Hands-On Equations®, we move to actual written equations such as:  x + 9 = 12.  Here's the rub; a few of my students know the answer and do not want to show any of their work.  Maybe some of you have this type of student as well.  Since, after 30+ years, I am still unable to grade what is in their minds, I insist that all steps be written down.  I explain that it's like riding a tricycle to ride a bicycle to ride a unicycle.

First, I instruct the students to look at the equation and determine which terms are out of place.  (Side note: Because my students are easily confused, at the present, we keep all of the unknowns on the left side and all of the numbers on the right side of the equal sign.)  Let's go back to our sample of  x + 9 = 12.  Because the x is already on the left side of the equation, the students write a "Y" over it for "yes".  The 9 is on the wrong side of the equal sign, so the students write a "N" over it for "no". Finally, they write a "Y" over the 12 since it is the correct place.  The students know they must use the inverse operation of addition to clear the 9 because it is a "no".  They therefore subtract 9 from each side of the equation resulting in an answer of 3.

This may seem laborious to some, but what if the equation is:  3 = y - 4?  This always freaks my students out; yet, if they do the yes/no procedure, they will discover that they have one "yes" and two "no's" which means they can rewrite the equation as y - 4 = 3.  The problem can now easily be solved since it is a yes = no, yes problem.


Unknown on both sides
of the equation
The next step is what to do when an unknown appears on both sides of the equal sign.  Usually, my students are sure they are incapable of solving such a difficult problem, but let's use the yes/no method and see what it looks like.  Notice in the sample on the left that we have a yes, no = no, yes.  We start by clearing the no on the left hand side of the equation by using the inverse of -9.  We then go to the right side and clear the no of  y by using the inverse operation of addition.  (Yes, I am aware both can be cleared at the same time, but again simple and methodical is what is best for mathphobics.)  We then divide each side by 4 resulting in the answer of 3.  When the problem is completed, my students are amazed and proud that they could solve such a long equation.  (You might notice in the illustration, a dotted line is drawn vertically where the equal sign is.  This helps my visual students to separate the two sides of the equation.)

If any of you try this approach with your students or have a different method, I would love to hear from you.  Just leave a comment and a short statement of how this process worked for you or what process you use that is even better.  That way, we can learn from each other.

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Hands-On Equations® is Algebra for the visual and kinesthetic learner.  This system, developed by Dr. Henry Borenson, enables students (even those in 4th or 5th grade) to easily and enjoyably learn essential Algebraic concepts and skills. Dr. Borenson received a U.S. patent for his teaching invention.

The Best Laid Plans...

Lesson plans have always been an Achilles heel for me.  I have taught for so long, that how to teach the lesson as well as knowing the content is not an issue.  I always have a Plan B, C, and D ready - just in case.  However, I still write an outline for the day so I know that I have covered the important points. 

My first job, when I retired from our local school system, was teaching math at a private school.  Mind you, I had been teaching math for over twenty years; yet, the administrator wanted me to do detailed plans which must be turned in every Friday.  I grudgingly did them, but would add little comments in the comment section. That space became my way of quietly venting; so, I would write such things as:  "So many lesson plans; so little time. Writing detailed plans is not time well spent.  To plan or to grade, that is the question.  I am aging quickly; so, I need to make succinct plans."

My supervisor finally relented and allowed me to do an outline form of plans.  However, he visited often to observe my teaching, which I didn't mind.  At least he knew what was happening in my classroom.  I learned from teaching and observing student teachers that anyone can come up with dynamite plans, but the question is: "Do the plans match what the teacher is doing in the classroom?"  Remember Madelyn Hunter?  Oh, how my student teachers hated her lesson plan design, but they did learn how to make a good plan.  I still do many of the items such as a focus activity and a lesson reflection at the end.

Science Lesson Plans for a Week in October
 As many of you know, my husband is a middle school science teacher.  He is the "sci" part of my name.  Anyway, he is in his 38th year of teaching, and he still does lesson plans - not the detailed ones we did our first couple of years of teaching, but plans he has.  He has divided his white board into sections using colored electrical tape as seen in the illustration on the left.  He then writes what each class is doing for the week in a designated square.  In this way, the principal, parents, and students know the content that will be covered.  Even the substitute (he is rarely sick) has a general idea of the day's activities.  If plans change, he simply erases and makes the necessary corrections.

So what kind of plans are you required to do?  Maybe there are no requirements for you, but do you still write plans?  Are they in outline form or just brief notes to yourself?  I am interested in knowing what you do; so, please participate in the poll on the left. OR leave a comment to share your thoughts.

Here is a Principal
checking all of
those  lesson plans!
Do you need a lesson plan that is easy to use, and yet is acceptable to turn into your principal or supervisor?  Check out my three lesson plan templates. One is a generic lesson plan; whereas, the other two are specifically designed for mathematics (elementary or secondary) and reading.  Checklists are featured on all three plans; hence, there is little writing for you to do.  These lists include Bloom’s Taxonomy, multiple intelligences, lesson types, objectives, and cooperative learning structures. 

Click on the purple letters if you are interested:  Lesson Plans