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Slope for Vertical and Horizontal Lines

I work in the Math Lab at the community college where I also teach. Last week, I had two College Algebra students who were having difficulty with slope.  They knew the equation y = mx + b, but were unsure when it came to horizontal or vertical lines. By the way, they were using their graphing calculators which I made them put away. (The book said no calculators.) I feel that if they construct the lines themselves, it puts a visual image into their brain much better than if the calculator does it for them. Sure enough, one of the sections in their math books gave the picture of the line from which they had to write the equation. They were amazed that I could just look at a graph and know the slope, give the equation, etc. When I taught high school math, my students couldn't use a graphing calculator until the middle of this particular chapter as I wanted them to physically draw the lines.

First, for those who have no idea what I am talking about, slope is rise over run.  Rise is how far a line goes up, and run is how far a line goes along.  At the right, the line goes up 3 and has a run 5; therefore, the slope is 3/5.  Rise/Run (Rise divided by Run) gives us the slope of the line.

When a line is horizontal, it has no rise, only a run. So the numerator would be zero (for no rise) and the denominator would be a number such as 5 for the run.  0 ÷ 5 = 0  This is true for any horizontal line.

A vertical line is different.  It has rise, but no run; therefore there would always be a number in the numerator, but always a zero in the denominator.  Since we cannot divide by zero, the slope is considered undefined. (I do use rise over run stating that a horizontal line might have 0/5 which is equal to 0 and that a vertical line might have 3/0 is undefined because we can't divide by zero. Our college algebra book uses O/K for okay and K/O for knock out which I like, but I still think the students need to know why.)

I wanted these two students to have a picture that would help them remember the difference.  I thought of a table for the horizontal line and asked them what would happen if the legs of the table were uneven.  They agreed that the table would have slope.  Therefore, the table would have a slope of zero if the legs were even.

I then went blank.  In other words, by creative juices stopped working, and I could not think of a picture that would help them visualize undefined. Since Teachers Pay Teachers has a forum,, I asked my fellow math teachers if they had any ideas.  Here is what some of them came up with.

The Enlightened Elephant suggested using a ski slope. She talks about skiing down a "cliff", which would not be possible (although some students try to argue that they could ski down a vertical cliff) and so the slope is "undefined" because it doesn't make sense to ski down a cliff.  Skiing on a horizontal line is possible so it's slope is zero,  She also talks about uphill (positive slope) and downhill (negative slope). 

Math by Lesley Elisabeth tells her students to use "HOY VUX" (rhymes with 'toy bucks')

             Horizontal - Zero (0) slope - y = ?   
             Vertical - Undefined slope - x = ?

All horizontal lines are =7 or = -3 etc., and all vertical lines are =1 or = 6, etc. Students forget this so the acronym HOY VUX helps them to remember. Once they've mastered the slope concept in Algebra I, for the rest of the school year, for Algebra II (especially equations of asymptotes - a line that continually approaches a given curve but does not meet it at any finite distance) and even in calculus classes for tangent lines, HOY VUX is just faster and more practical. 

Animated Algebra created a video lesson on the Slope Intercept  ($5 on TPT).  She has a boy skateboard down a negative slope, literally right on the graph line. Karen then shows the same boy taking an escalator up on a line that has a positive slope. Later in the lesson, she rotates the line clockwise, each movement with a click, to show the corresponding slope number to link the line to the slope.  She includes lots of other visual cues to help students focus on and pay attention to the concepts.

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Teaching this year has been more challenging than ever, and you deserve something special. Teachers Pay Teachers is celebrating YOU and the end of the school year by offering resources, ideas, worksheets, power points, assessments, etc. at discounted prices by having a two day sale.

We invite you to take part in this two day sale that will run from May 4th from 12:01 AM (ET) through May 5th 11:59 PM (ET). Don't miss out as most sellers will have everything in their stores discounted from 5% to 20%.  PLUS, when you check out, if you enter the special code of THANKYOU21, you will receive another 5% off of your total purchase. Here's a great chance to stock up and save money at the same time.

Of course, my store, Scipi Products, is no exception; so head on over and purchase those items you might have on your wish list while they are on sale. Let's make the end of the school year special by saving you money!

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The Mysterious Case of Zero, the Exponent - Why any Number to the Zero Power Equals One


Sometimes my college students like to ask me what seems to be a difficult question. (In reality, they want to play Stump the Teacher.)  I decided to find out what sort of answers other mathematicians give; so, I went to the Internet and typed in the infamous question, "Why is any number to the zero power one?"  It was no surprise to find numerous mathematically correct answers, most written in what I call "Mathteese" - the language of intelligent, often gifted math people, who have no idea how to explain their thinking to others.  I thought, "Wow!  Why is math always presented in such complicated ways?"  I don't have a response to that, but I do know how I introduce this topic to my students. 

Since all math, and I mean all math, is based on patterns and not opinions or random findings, let's start with the pattern you see on the right.  Notice in this sequence, the base number is always 3.  The exponent is the small number to the right and written above the base number, and it shows how many times the base number, in this case 3, is to be multiplied by itself. 

(Side note: Sometimes I refer to the exponent as the one giving the marching orders similar to a military commander. It tells the base number how many times it must multiply itself by itself. For those students who still seem to be in a math fog and are in danger of making the grave error of multiplying the base number by the exponent, have them write down the base number as many times as the exponent says, and insert the multiplication sign (×) between the numbers. Since this is pretty straight forward, it usually works!) 

Notice our sequence starts with 31 which means 3 used one time; so, this equals three; 32 means 3 × 3 = 9, 33 = 3 × 3 × 3 = 27, and so forth. As we move down the column, notice the base number of 3 remains constant, but the exponent increases by one. Therefore, we are multiplying the base number of three by three one additional time.

Now let's reverse this pattern and move up the column. How do we get from 243 to 81? That's right! We divide by three because division is the inverse operation (the opposite) of multiplication. How do we get from 243 to 81? That's right! We divide by three because division is the inverse operation (the opposite) of multiplication. Notice as we divide each time, the exponent by the base number of 3 is reduced by one. Let's continue to divide by three as we move up the column. 27÷ 3 = 9; 9 ÷ 3 = 3. Now we are at 31 = 3 which means we must divide 3 by 3 which gives us the quotient of 1. Notice, to fit our pattern, the 3 in the left hand column would have to be 30 ; so, 30 must equal one!

This works for any number you wish to put in the left hand column. Try substituting the base number of three with two. Work your way up the sequence dividing by two each time. You will discover that two divided by two equals one (20 = 1). Therefore, we can conclude that any natural number with an exponent of 0 is equal to one. (Zero is not included; it's another mystery to solve.)

What happens if we continue to divide up the column past 30 ?   (Refer back to the sequence on the left hand side.)  Based on the pattern, the exponent of zero will be one less than 0 which gives us the base number of 3 with a negative exponent of one or 3-1 .    To maintain the pattern on the right hand side, we must divide 1 by 3 which looks like what you see on the left. Continuing up the column and keeping with our pattern, 3 must now have a negative exponent of 2 or 3-2 and we must divide 1/3 by 3 which looks like what is written on the right.

Each time, the exponent by the base number of 3 is reduced by one. Let's continue to divide by three as we move up the column. 27÷ 3 = 9; 9 ÷ 3 = 3. Now we are at 31 = 3 which means we must divide 3 by 3 which gives us the quotient of 1. Notice, to fit our pattern, the 3 in the left hand column would have to be 30 ; so, 30 must equal one! This works for any number you wish to put in the left hand column. Try substituting the base number of three with two. Work your way up the sequence dividing by two each time. You will discover that two divided by two equals one (20 = 1). Therefore, we can conclude that any natural number with an exponent of 0 is equal to one. (Zero is not included; it's another mystery to solve.)

As a result, the next two numbers in our pattern are..............

Isn't it amazing how a pattern not only answers the question: "Why is any number to the zero power one?" But it also demonstrates why a negative exponent gives you a fraction as the answer. (By the way math detectives, do you see a pattern with the denominators?)
 
         Mystery Solved!   Case Closed!

This lesson is available on a video entitled:  Why Does "X" to the Power of 0 Equal 1?

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Want simple, visual answers to other difficult math questions? Try this resource entitled Six Challengig Math Questions with Illusratrated Answers. Many of the answers feature a supplementary video for a more detailed explanation.

A Go Figure Debut for a Digital Teacher Who is New!

 

Ian has been a classroom teacher 12 years in two states (Virginia and Massachusetts) and has worked with students in a general education and advanced academic setting in grades 3-5. What he loves most about teaching is inspiring awe and wonderment.  Ian pushes his students to think beyond the standards, to question the world around them, and to seek out answers to their wonderings. He encourages his students to utilize technology, to demonstrate their understanding and to showcase their findings. His classroom has a certain buzz of excitement while students work on their latest problem-based learning activity, such as delivering the news for the southeast region, or researching a signature dish for their Midwest Cooking Star episode. His students enjoy many healthy, competitive digital games to review and reinforce concepts.

Ian is a soon-to-be father of four, who loves to spend the summers on the beaches of Massachusetts, the falls hiking and exploring the outdoors, and the winter months building snowmen and sledding. His family is a running family, with a combined 15 marathons between him and his wife.

Ian's Teachers Pay Teachers store, Mr. Kidders Keys To Online Learning, contains 112 resources with 12 of them offered as free. The purpose of his store is to help facilitate a digital friendly classroom and is intended to make the hectic life of a teacher more manageable. It features a variety of digital morning meeting, virtual, or brain break games, some of which are also content based. His three main lines of games are Zoomed In! - Reveal! - and Dash and Discover! In addition to the digital games, he has a growing library of digital math resources, mainly for grades 3-5: digital escape rooms, self-graded Google Forms with embedded video tutorials, interactive skip counting activities.

FREE Resource
One of those digital math resources is entitled Grade 4 Multiplication: One Digit by Multi-Digit Multiplication Word Problems Google Form. Enjoy this FREE grade 4 Google Form multiplication quiz that is a self-graded word problem assessment involving one digit by multi-digit questions. It gives instant feedback for teachers! Just download it by clicking the link under the resource cover on your right.

Only $3.00
Ian's featured paid resource is a mystery image game that will boost your students' social-emotional health and get them engaged with online learning. Animal Adaptations Edition of Reveal! The Mystery Picture Game is ideal as a review or pre-assessment of animal adaptations. The Google Slideshow can be used during distance learning video calls or in the classroom. This involves NO PREP and will leave your students smiling.

Finally, Ian has a few interactive, narrated virtual field trips that explore American history through famous works of art. All the digital activities in his store are ready to use, no preparation required, no need to print or copy anything (some even come with optional printouts). Take some time to check them out!