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## How to introduce negative exponents to improve understanding

Students often misinterpret negative exponent notation and if this is cleared up, in the beginning, a lot of frustration and time can be saved.

## Common Mistakes with Negative Exponents

There are often a lot of misconceptions about understanding and teaching negative exponents. Before we begin looking at the solutions, let's take a look at the common problems.

#### Student Mistakes

Students love to create or rework rules for negative exponents. Be aware that you may find negative signs in the strangest of places.

For example, you may find mistakes like this:

$2^{-3}=(-2)(-2)(-2)=-8$

Teacher Mistakes

Just take the reciprocal can be a real answer later, but first students must understand the why. Why does this work and how can I know this rule? We have all seen students get done with the homework super quick, but then bomb the test. Simple may make the lesson go faster, but an easy gain is an easy loss.

Students, to retain these rules, must have an understanding of why and how they work in a way that works for them.

## Using Common Language and Vocabulary

Most students will want to see several different ways of looking at this concept. But one thing is for sure, if we can connect this learning for them between the examples, they will develop a solid understanding of the concepts with negative exponents.

Some of the most common vocabulary: Exponents, exponent rules, repeating functions, reciprocals,

## Activities to Help Understanding

#### Patterns

I like to give the students this table and ask some questions. What do exponents mean? What is happening? Can this pattern continue? How do you know it can/cannot? Are we still multiplying?

I like to do this a few times, to allow all of the students to have a chance to see the patterns.

#### Tangible

I like the idea of giving the students something they can picture to work within their minds. For example, thinking of cake, as we go into the negative exponents we get a fractional part of the cake, we do not get a negative cake.

And while seeing the pattern does help, they also need to understand the rules. At this time we are ready to put it all together.

#### Exponent Rules

Depending on your class and their level of understanding, you may need to review the exponential rules. Specifically, they need a good understanding of the product rule and the quotient rule.

I find it best to plug in positive numbers and explore the possibilities. The students will begin to come up with "shortcuts." This discovery is useful! We can then introduce the rule of, $x^{-a}=\frac{1}{x^{a}}$ But now they understand why this works and we can move forward.

#### Challenges

I found the following challenge on the website, http://math.stackexchange.com/questions/629740/how-would-you-explain-to-a-9th-grader-the-negative-exponent-rule and I loved the potential of teaching it this way. It builds on the exponent rules. I would stop talking at the arrow and see where the students take the conversation. I bet it would give you a lot of insight into their understanding.

#### Video - How I Feel About Logarithms - by Vi Hart

This video moves very fast, but for your students that are intrigued, this video will motivate and challenge them. The negative exponents begin around minute 7, but you need the whole video to understand the vocabulary. I can see this working for some and not so much for others, but I enjoyed it, and I hope you get some additional ideas from it as I did. I suggest watching this to be sure it is a good fit for your class.

"Sometimes to make the harder things simple, first you have to make the simple things harder"

-Vi Hart

## Rewards of a Job Well Done

While it may take more time up front to ensure that students understand this concept, in the long run, it will save you time. The ability to see patterns and complete the calculations without help will be priceless. When you give students an in-depth understanding and allow time for processing the concepts go into the long-term memory, students gain quite an advantage as they continue their math education.

## 3 Super Bowl Lesson Plans that Keep Kids Engaged

I used to hate the Monday after the Super Bowl. The kids were tired, I was tired, and it felt like a horrible situation of force-feeding information to the unwilling.

So I began looking for a lesson that would keep them engaged. And I found it! Now, many of you know I am a huge fan of the lessons at Yummy Math.

## Scientific Notation

The lesson I found, called The cost of a Super Bowl XLVII ad was perfect. Incorporating scientific notation (which conveniently the biology teacher asked me to review), and prediction, the cost of the ads was a great way to get the kids interested in the math. I showed a few of their favorite commercials, and they were hooked.

For more lessons on feel free to check out the Unit 8 Livebinder.

## Graphing Exponentials

Yummy Math one-upped them with this year's lesson. Again using the cost of an ad, this lesson incorporates graphing, creating a line of best fit, making predictions, comparing linear and exponential functions and other skills. Super Bowl Ads 2015 is copied and ready to go for tomorrow morning.

Unit 8 has more lessons on all aspects of exponentials.

## Measures of Central Tendencies

For the middle school teachers out there, you are not forgotten! Yummy Math also has a way to predict (or check your prediction) of the score using statistics. Typical Super Bowl scores? This activity could also be used as a review if you are heading into statistics soon.

Are you looking for more statistics? Unit 4 has you covered!

## Thank you, Yummy Math!

To ensure the kids love the lesson, I find a few commercials (yes, I make sure they are appropriate) and show those during the class. I love it! This day has gone from one I used to dread to a one I treasure. The kids and I enjoy it, and I hope you will too! Thank you, Yummy Math!

# Modeling Exponential Functions

#### Unit 8 - Week 22

• Graph exponential functions (F.IF.7.e)
• Identify intercepts and end behavior (F.IF.7.e)
• Translate exponential functions (F.BF.3)
• Compare the key components of a linear function graph to an exponential function graph. (F.IF.4)
• Fit a function to the data; use functions fitted to data to solve problems in the context of the data. (S.ID.6.a)

### Day 109 - Fit a function to data

• PPT includes creating the equation from two points and a situation

# Radical Functions and Rational Exponents

Do your students hate radical functions? If they do, follow these three tips to not only make learning them easier for your students but teaching them easier for you. Let’s jump right in!

## Tip #1 - Radicals Are Simply Variables

I once had a student working very hard to understand Algebra though it wasn’t her strong suit. When we got to radicals, she admitted to me that she hadn’t paid much attention the year before and had no idea what radicals were.

I went through the process of teaching her how we use symbols for real numbers the way we use the pi symbol for 3.14. Once I explained it that way, she was able to grasp the fact that radicals stand for real numbers that are irrational and, therefore, difficult to write out.

Make sure your students have a firm grasp on the fact that these symbols stand for irrational numbers. If they miss this basic step, it will be difficult for them to move on to the next thing, something we must remember as teachers.

## Tip #2 - Connecting to Previous Learning

I have said it before and I will say it again: make sure you always remind your students of something they already know when you are teaching them something new.

Your students should already understand variables, so remind them of variables as you begin to teach radical functions. When I do this, I stop for a moment and ask my students easy questions like, “What do you get if you multiply x by x?” They can tell me that it equals x squared. I ask, “What happens if you add x to x?” That’s an easy one; it’s 2x.

Now they can see that if they know how to work with variables, they can work with radicals.

## Tip #3 - Review, Review, Review

Begin or end each math class by reviewing basic facts. Give a speed drill of the square root facts, progressing to the ones they inevitably forget.

My kids always forget the square root of 1 because we spend so much time reviewing the larger numbers. The truth is that you can’t expect your students to have success with more complicated topics if they can’t rattle off their square and cube roots without thought.

I use charts of exponent equations with my students at the beginning of class, and for my struggling learners or my ESL/ELL students, I have them keep the charts on their desk as I teach. Just reviewing these basic facts goes a long way toward the students retaining the information and being ready to tackle more difficult concepts.

I also do this is to get students off of the calculator, which can be very frustrating for a teacher. Beyond that, once we get to more complicated lessons, the calculators won't help anyway. This is why I recommend doing these types of drill charts in every class session.