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## How to use four 4s to build math fluency

I love seeing students problem solve and brainstorm. There is something about seeing the concepts connect and the light in their eyes as they say, "don't tell me!"

When reviewing the order of operations (or as a holiday filler, or when 1/2 the class is gone on a field trip, or.) I love to give my students the four 4s challenge.

The idea is to reach the given number using the order of operations using only four 4s.

I like to give lots of examples of different combinations to get their minds thinking. Here a few examples to help your students get started:

$1=\frac{44}{44}$

$9=(\frac{4}{4})+4+4$

$47=(4!+4!)-\frac{4}{4}$

$60=4*4*4-4$

$81=(4-(\frac{4}{4}))^{4}$

$100=4*4!+\sqrt{4}+\sqrt{4}$

I like to give them time to play with them. Some of my classes will need more examples, and some will not want one more answer. I allow them to work together. I love hearing their conversations as they work. I like to give them a leveled challenge, and I don't allow this challenge to go home or let anything that connects to the internet as the answers are too easy to find.

If they can get at least ten additional answers, they get ___ points

If they can get at least 20 additional answers, they get ___ points

Etc.

This strategy allows me to personalize it for each person's needs if necessary.

I love seeing the students think outside of the box to get these answers. It truly is amazing! The students that are not usually my "star" performers will often be excellent at this because they don't have to follow an algorithm. Each problem is different.

Before we finish, I like to ask some questions to help them see the patterns.

• Which numbers are the hardest to find?
• Is there anything special about prime numbers?
• Can there be more than one answer for each number? Prove it.
• What other patterns can you find?
• Where there strategies that helped you move quickly through this sheet?

I love the development of language during these discussions. It took a fun puzzle and increased it's value even more.

Here is a great answer key with a lot of variations: http://mathforum.org/ruth/four4s.puzzle.html

## 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.

## Motivation - The One Thing Students Must Have!

How often do we hear teachers say that the kids are not motivated? We teach, but they don't want to learn.

Pin this post for a friendly reminder.

Many teachers believe that motivation is intrinsic and that there is nothing we can do about that. But, I believe there are some things we can avoid that will help our kids stay motivated.

### Perfection

Expecting yourself or your students to be perfect kills learning. Learn to forgive yourself and be patient with your students. This expectation alone can bring back enthusiasm for not only your students but for your career!

### Everyday Monotony

Having everything the same every day; same seats, same bellringer, same lecture, same problems (different page).
I noticed my second-hour class was bored. It was two days before spring break, and they were restless. I simply gave the instructions in the classroom and allowed them to sit in the hallway (I have minuscule windows in my room) near the sunshine. It rejuvenated them and me!

### Make them Guess

On the other hand having no structure and forcing kids to figure out what they are expected to do is exhausting. Finding balance is key. Excellent communication and an agenda on the board is very effective.

### Sarcasm

Honestly, this is the hardest one for me, but I had to remove this from my room to be effective. It killed kids motivation to try.

### Negativity

Don't allow it from yourself and don't allow it from your students. Done.

Give tasks that mean something to students. Balancing application and fluency can be hard at first. Before each lesson check to see that you have both. You will be amazed at how this can turn a kid around.

## Inverse Functions Lesson

I like teaching inverse functions. This inverse functions lesson plan will help you find connections with your students. With the real world context, students understand this concept well by the end of class. I love working on the ideas of computers, maps, and math puzzles. Be sure to stay to the end for your free copy of the inverse functions worksheet set.

### Standard: CCSS.MATH.CONTENT.HSF.BF.B.4.A

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

### I Can Statement for the Inverse Functions Lesson Plan

I can find the inverse of a linear function.

This introduction is our students first experience with inverse functions, and while the full standard also incorporates the inverse of exponential and rational functions, we are beginning with linear functions during this lesson.

### Essential Questions About Inverse Functions

Why is the idea of "inverse" so important in mathematics?

I like this question. It forces the students to think critically and evaluate their understanding of the concept.

### Bellringer Worksheet

The bellringer worksheet consists of 4 questions asking students to create a linear equation from a table. This bellringer should reinforce their understanding of linear relationships and functions before beginning the day's lesson.

### Understanding Inverse Functions Activity

The activity for today helps our students understand the essential question. We take a look at some direction to our friend Tracy's home, and we ask the student to give the directions back. This activity is not only ideal for understanding inverses but also this is an excellent technical writing exercise as well.

We can then connect that activity to a math question. Students may have seen this before, and they are stumped that a teacher could know what number they started with after doing some silly calculations. But now they will know the magic trick.

### Practice - Inverse Functions Worksheet

The practice sheets are designed to give students a contextual, algebraic, and graphical understanding of inverse functions. The inverse function worksheets build upon themselves and will allow students the opportunity to ask questions and work through the work using math vocabulary.

### Exit Slip - Formative assessment for Inverse Functions

The exit slip is two relatively simple inverse questions that will allow you to quickly assess your students to determine if anyone is falling behind. The first question will give you a glimpse into their algebraic ability to manipulate the equation, while they second problem will give you a look at their understanding of inverse functions.

## The Correlation vs. Causation Activity

Students love this correlation vs. causation lesson, because -- once they've grasped it -- they begin to see this phenomenon all around them.

The first time I taught this lesson was eight years ago during an election year when political commercials were all over the place. After this lesson, the history teacher (yes, this is also a history standard in most states) pulled up some of those commercials and we had great discussions about the difference between causation and correlation and how advertisers assume we don't know the difference.

Students that did not love math were coming into my classroom to have high-level discussions around this subject, and healthy and educated debates began to take place. Real learning was happening.

### The Statistics CCSS Standards

S.ID.9 - This is the main standard that we discuss during this lesson.

High School: Statistics & Probability » Interpreting Categorical & Quantitative Data » Interpret linear models » 9

Distinguish between correlation and causation.

This standard is a great example that will prepare our students to decipher and interpret information in their lives.

### The Lesson Opener

I like to begin this lesson with some great graphs from co.design, but make sure your students' maturity can handle the content. These graphs make it evident that -- even though the graphs match -- the information is not a causation.

The conversation then can move into the definitions of causation and correlation with context and meaning. The discussion will help all of your learners connect the learning to previous understanding so they retain the information. The power in this is incredible and will immediately increase confidence. You will know this is true when you hear them saying the lesson was "easy."

### Strong Examples of Causation vs. Correlation

Both of these websites are perfect for showing students the ridiculousness of the assumption that correlation always means causation.

### Now, for the Activity!

The fun part of this activity is that the students will find data with a common independent variable (usually time) and show a correlation, trying to trick people into believing causation. The process of thinking through and considering the data can be a bit rough, but remind your students that simply recognizing a positive or a negative trend may be enough.

To download the complete activity, enter your email below, and I will send it to you today.

## "What does a typical day in your classroom look like?"

It's a question I'm asked a lot, so I'm taking a minute to run you through a few key things that happen in my classroom every day.

### The Greeting

Connecting with every student on the way into my classroom is one of the most important aspects of each day. I shake their hands, ask them a question, and straighten out any misconceptions or misbehavior from the day before in a private and non-threatening way. Personal details may come up, such as soccer games, track meets, Orchestra concerts, or how their family is doing.

I take the time to connect with my students so that they know I value them, regardless of whether they believe they're good at math.

### The Bellringer

Every class begins with a bellringer. A bellringer is a short assignment, just four or five questions and can accomplish two things:

1. Reviewing a concept from the day before
2. Assisting in remembering a concept previously learned that we'll need later in the lesson

I typically make bellringers straightforward and easy to complete with a partner. They allows me time to take attendance, do the bookkeeping, deal with last-minute issues, and give the kids that always seem to be at my desk at the beginning of class the attention they need to thrive during class.

### The Activity

Next we move into the meat of our lesson. This is what I usually call an activity, which is time given to students to comprehend, process, evaluate, and problem-solve, in a safe environment where there are no penalties for mistakes.

These activities are included in the PowerPoint to spark discussion and make communication easier. It helps me to find misunderstandings before they develop into problems later in the unit.

#### I believe this is the most important time in my class.

Activities leverage different learning styles. Examples include:

• Using Algebra Blocks
• Creating flowcharts
• Engaging in interactive or investigations on Desmos
• Using Gapminder or other websites not created by teachers.

Regardless of the method used, all of these things are developed with a sense for questions to allow students time to examine and connect this current learning to what is coming and what has happened before. The connection of ideas will allow for retention, confidence, and increased engagement.

### The Presentation

After the activity, I have a good idea of where their misconceptions are, so I walk them through a PowerPoint. My PointPoints always include the activity, but they also include concrete examples. I work very hard to connect those examples to the activity they just completed.

### The Practice

If there's time remaining, I always have a practice sheet ready, either for in-class work or homework. These sheets are developed to give students practice and confidence.

### The Exit Slip

With about five minutes remaining in class, I always try to sum up what we've learned. I then ask a few students what they've learned so we can make those connections one last time, and then I hand out an exit slip. This exit slip usually has one or two questions for the student to complete that will be collected by me at the door as they leave. I love these exit slips because I can quickly look through them for any misunderstandings and make a plan to fix it the next day.

### You can get a free Algebra Unit Sample. Just fill out the form below.

If you're ready to implement this type of lesson plan in your classroom, then the A1T Membership is for you.

• Bellringers
• PowerPoints
• Activities
• Practice pages
• Exit slips
• 15 hours of training ready for you!

So whether you are looking for the CEU's or those never-ending artifacts for your evaluations, the certificates are ready for you to print as soon as you finish the training. Remember that you can cancel at any time.

And if you're looking for purchase orders to spend that last-minute money before it disappears from your school year, feel free to drop me a line or click the purchase order button at the bottom of the annual pricing section to become a member.