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Yes or No? Stay or Go? Solving for "x".


My basic algebra classes have just begun solving equations containing one unknown. As I tell them, we are inquisitive detectives looking for the unknown.

My students' 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 and which term should stay where it is?

Hands-On Equation
 Balance Beam
www.borenson.com
I always start this chapter using Hands-On Equations®. I have used them for years because it provides a visual for those concrete learners. I also refer to the written equation as a teeter-totter or a see-saw which must always stay balanced. In other words, the equal sign is the pivotal point and both sides of that = sign must be the same.  (Notice that Hands-On Equations® uses a balance beam.) We also discuss the importance of the"Whatsoever thou doest to one side of the equation, we must doest to the other". (Out of necessity, 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 are 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 the word, "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. They now have exactly what they want, a Y and N on the right side and a Y on the left side. They now must clear anything that has a "N" over it.  The students recognize they if they use the inverse operation of addition, they can clear the 9. They therefore subtract 9 from each side of the equation resulting in an answer of 3.

Many algebra teachers will have the students write the step x + 0 = 9.  You may wish to include this step in the process, but since my college students readily see that +9 and -9 make zero, they put an X over the two opposites to show that they cancel each other out or when added together, they equal zero.

What if the equation is: 3 = y - 4? This always freaks my students out; yet, if they do the yes/no process, they will discover that they have two "no's" and one "yes", not a yes, no = yes.  This means they can rewrite the equation as y - 4 = 3 to get a yes, no = yes. The problem can now easily be solved like the one above.

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 "N" on the left hand side of the equation by using the inverse of -9. We then go to the right side and clear the 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 my 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 learn essential algebraic concepts and skills. Dr. Borenson received a U.S. patent for his teaching invention.

Metrics - Not Going the Whole Nine Yards!

Did you know that there are only three nations which do not use the metric system: Myanmar, Liberia and the United States? The U.S. uses two systems of measurement, the customary and the metric. Yes, since our country does use the metric system, we have given more than an inch, but we haven't gone the whole nine yards.

Today, when we shop for groceries, soda is sold in liters. Medicine is sold in milligrams, food nutrition labels are metric, and what about a 100-meter sprint or a 5K race? Still, we are the only industrialized nation in the world that does not conduct business in metric weights and measures. To be or not to be a metric nation has been a question of great consternation for our country for many years.

Here are some reasons why I think our nation should go to the metric system.
  1. It's the measurement system 96% of the world uses. 
  2. It is much easier to do conversions since it is based on units of ten. Water freezes at zero, not 32°, and it boils at 100, not 212°. 
  3. Teaching two measurement systems to children is time consuming and confusing. 
  4. It is the "official" language of science and medicine. 
  5. Its use is necessary when you travel outside of the United States. 
  6. Conversion from customary to metric is often fraught with errors. Because the metric system is a decimal system of weights and measures, it is easy to convert between units. 
  7. There are fewer measures to learn. Once you learn the meaning of the prefixes, you can easily convert mass, volume and distance measurements. No further conversion factors need to be memorized except the specific power of 10. For the Customary System you have to remember 5280 feet = 1 mile, 4 quarts = 1 gallon, 3 feet = 1 yard, 16 oz. = 1 pound, etc. 
  8. And just think, I would have less clutter in my kitchen since I wouldn’t need liquid and dry measuring cups or teaspoons and tablespoons! All I would need is a scale and liquid measuring cups!
So, while most nations use the metric system, the United States still clings to pounds, inches, and feet. Why do you think Americans refuse to convert? I’d be interested in your perspective and ideas.

A Go Figure Debut for Amy Lynn who is new!

Amy Lynn's Store
Amy Lynn has been teaching for 12 years. For her first three years, she was a K-5 science teacher at a magnet school in Tampa, with a side role of being the drama teacher. From there, she and her husband moved to NYC so she could pursue acting, and he could pursue music. While in New York, she taught second grade for seven years at a Christian school while starting up a drama club on the side. (She studied and performed improv at night with the upright Citizens Brigade Theater).

Amy then discovered she was pregnant! A son was born, and the three of them lived in a tiny (280 sq. ft.) studio apartment. (Talk about tiny house living!) At that point, they were a bit tired, and having accomplished a great deal, they packed up and moved back to Tampa, where she is now an elementary special needs teacher. As you can see, her teaching experience is broad...from inner city science...to a private school...to now. Knowing that no day or year is the same is what she loves about teaching.

Amy claims to be eclectic, as is her classroom. It is colorful, with a side of the arts and a side of Zen. She incorporates skits into many lessons (Her drama background seems to keep rearing its head). She has a corner filled with tactile stress balls/tubes/instruments that students can visit when they just need a mental break. Other teachers stop in and use them too, which is always fun. She LOVES fun. She loves it when her students laugh and feel safe to be themselves.

Only $7.00
Amy married the boy next door whom she met as a seven year old and has loved ever since. (They are those kind of people.) They both work hard during the week; so, on the weekends, when not at church, they can be found on the beach, on a bike trail, or just at home, reading books together in a hammock.

Amy is a HUGE fan of creating materials that any teacher can use with as little prep as possible. One such resource is entitled Word Wall - 100 Printables for Any Word Wall. This product includes 100+ printables/activities/skits to use with any class word wall.  It can be used as centers, seatwork, early-finishers or more! Skit writing and performance pages, creative writing, seatwork for individual and for partner work, dictionary and Thesaurus skills, poetry, games, and much more are included in this activity pack of 106 different printables and games for ANY WORD WALL!

Amy currently has over 134 products in her store. Of these, 24 are free, many of which are science items.

One such freebie is on food chains. It contains four fun and simple activities to help engage students as they learn! Included is:
Free Resource
  • A graphic organizer for students to color and fill in their prior knowledge
  • Two "Create a Food Chain" pages
  • A full color page that can be used as seat-work, and then as a colorful wall display of what your students have learned!
Her store is as varied as her 12 years of teaching have been. She has many desk/center labeling tags to help teachers change the looks of their rooms, packets of 100 standards aligned anytime reading/math/writing/language printables per grade level, yearlong Dolch activity packs, and Christian resources as well. There seems to be something for everyone!

Her current goal, (You can see a sample by clicking on her "superhero theme" category at her storefront, as this category is almost complete) is to create grade level standards aligned themed printables that will assist teachers as they endeavor to incorporate their school’s yearly themes. These will include themed 100th day packs, 100 standards aligned printables, reward coupons, free "getting to know me" pages, as well as end of the year memory books/ writing sets. So if anyone knows their school’s upcoming themes and would love these types of resources, she would love all your suggestions! Just go to her store and click on “Ask A Question.”

Pi Day is on March 14!



Mathematicians love to celebrate anything related to math; consequently, Pi Day was invented by some anonymous mathematician.  It is observed every year on March 14th because if this date is written in a month - day format, we get 3-14 which is similar to 3.14, the estimate we use when working with Pi.

The symbol for Pi comes from the 16th letter of the Greek alphabet which is written as “π”.  When used in mathematics, this symbol stands for a constant which is the ratio of a circle’s circumference to its diameter.  This is approximately 3.14159...  Remember using the formula C = π d in high school ? This formula is used because the circumference of a circle is about three times the diameter.

We classify Pi as an irrational number because it cannot be written as a simple fraction. The decimal goes on forever without ending or repeating any numbers. In other words, it is infinite.  While only a few digits are needed for typical math calculations (we usually use 3.14) mathematicians (not me) have calculated Pi to over one trillion digits beyond its decimal point. Pi’s unlimited nature makes it a fun challenge to memorize, but I doubt if anyone has memorized a trillion digits!  Now there is a great bonus assignment for some math teacher to try! 


Since Pi Day is a reason to celebrate, let's have some fun with Pi.  On the left you will see a pumpkin "Pi".  (Was it baked by a mathematician?) On the right, is some "pi" in the sky!



Have a great and glorious "Pi" Day which, in my opinion, 
is just another way to celebrate math.


St. Patrick's Day Myths and Fun Facts

March 17th is St. Patrick’s Day; so, for fun, let’s explore some of the
myths surrounding this Irish holiday as well as a few fun facts.

Myths

1) St. Patrick was born in Ireland. Here is a surprise; St. Patrick isn’t Irish at all! He was really born in Britain, where as a teen, he was captured, sold into slavery, and shipped to Ireland.

2) St. Patrick drove all of the snakes out of Ireland. It’s
true there are none living in Ireland today, but according to scientists, none every did. You can’t chase something away that isn't there in the first place!

3) Since the leaves of a shamrock form a triad (a group of three), St. Patrick used it to describe the Trinity, the Father, the Son, and the Holy Spirit so that people could understand the Three in One. However, there is nothing in any literature or history to support this idea although it does make a great object lesson.

4) Legend says each of the four leaves of the clover means something. The first leaf is for hope; the second for faith; the third for love and the fourth leaf is for luck. Someone came up with this, but since a clover is just a plant, the leaves mean absolutely nothing.

5) Kissing the Blarney Stone will give you the eloquent power of winning or convincing talk. Once upon a time, visitors to this stone had to be held by the ankles and lowered head first over the wall surrounding the Blarney Stone to kiss it. Those attempting this were lucky not to receive the kiss of death.

Fun Facts

1) The tradition of wearing green originally was to promote Ireland otherwise known as "The Green Isle." After the British invasion of Ireland, few people wore green because it meant death. It would be like wearing red, white, and blue in the Middle East today. When the Irish immigrated to the U.S. because of the potato famine, few were accepted and most were scorned because of their Catholic beliefs. For fear of being ridiculed and mocked only a small number would wear green on St. Patrick’s Day. Those who didn't adorn green were pinched for their lack of Irish pride. This “pinching” tradition continues today.

2) Did you know that in 1962, Chicago, Illinois began dying the Chicago River green, using a vegetable dye? An environmentally safe dye is used in amounts that keep the river festively green for about four to five hours.

3) The Irish flag is green, white, and orange. The green 
represents the people of southern Ireland, and orange
signifies the people of the north. White is the symbol of peace that brings the two groups together as a nation.

4) Boston was the first city to celebrate Saint Patrick’s Day way back in 1737, but they did it in August, not March!

5) A famous Irish dish is cabbage and corned beef which I love to eat!

It is estimated that there are about 10,000 regular three-leaf clovers for every one lucky four-leaf clover you might find. Those aren’t very good mathematical odds whether you are Irish or not!

Want some activities for your classroom? Check out my St. Patrick's Day resources.

March Fraction Word Puzzles

Pot of Gold Glyph

St. Patrick's Day Crossword