If you’re getting ready to teach exponents, you might be looking for a simple way to introduce the concept and help your students avoid common mistakes. Exponents can seem intimidating at first. It can be daunting when your students aren’t sure how to write them, evaluate them, or understand what they actually represent. With the right strategy, your students can build a strong foundation and confidently use exponents in their math work. Today, I’ll walk you through how I approach teaching exponents using a combo of vocabulary, visuals, and guided practice that makes the learning stick.

See Teaching Exponents in Action

Watch the two videos below for a full breakdown of how I teach exponents. We start with the core concepts and then move into guided practice. These are great for modeling in your classroom or brushing up on your own before teaching the lesson.

In this first video, I walk through essential exponent vocabulary like base and exponent. I then explain what repeated multiplication looks like and cover common mistakes.

This second video gives your students a chance to practice with you and check their understanding using real examples.

Key Takeaways From Videos

• Teach your students that the base is the number being multiplied. The exponent tells how many times to multiply it.
• Emphasize that exponential form is not the same as base × exponent. Example: 2³ = 2 × 2 × 2 = 8, not 6!
• Highlight special rules: Any number to the 0 power equals 1, and anything to the 1st power equals itself.
• Show comparison problems like 3⁵ vs. 5³ so your students can explain which is greater and why.

This kind of explicit modeling and discussion helps your students avoid common pitfalls. It also builds real understanding instead of just memorizing rules.

Extra Tips to Help You Teach Exponents

I found that having students move back and forth between expanded form and exponential form made a huge difference. For example, give your students something like 6 × 6 × 6 and ask them to rewrite it as 6³. You can also give them 4² and have them write it out and find the value.

Quick classroom tip: I always kept a visual reminder on the board that showed common errors. One of the most frequent ones is students thinking that 2⁴ means 2 × 4. Having a chart with ✘ 2⁴ = 8 ✔ 2⁴ = 16 helped catch that mistake early.

Also, powers of ten are a great opportunity to explore patterns. Once your students realize that 10² has two zeros and 10⁴ has four, they start recognizing those powers more quickly, and estimating large numbers becomes easier, too!

Additional Resources to Help You Teach Exponents

If you’re looking for a way to reinforce everything covered in these lessons, check out my Exponent Notes and Practice for 6th Grade Math. I include fold-it-up vocabulary pages, guided notes, practice activities, and exit tickets. It’s everything you need to help your students master exponents from start to finish.

You’ll get:

• A vocabulary foldable to help students define base, exponent, squared, and cubed
• Printable and digital practice pages with problems in both exponential and expanded form
• Comparison questions to deepen understanding
• Exit tickets to check for mastery
• A digital version perfect for Google Classroom

It’s a complete, scaffolded approach to teaching exponents that’s easy to prep and even easier to reuse year after year.

Save for Later

Remember to save this post to your favorite math Pinterest board for quick access to these resources to help with teaching exponents!

The post How to Teach Exponents With Step-by-Step Examples appeared first on Cognitive Cardio Math.

Today, I’ve got five math lessons to help you ace your upper elementary or middle school teacher observation!

Before we get into these math lessons, let’s think about how you normally plan for your teacher observations. 

• Do you try to find a math lesson with all the ‘bells and whistles’ to attempt to knock your administrator’s socks off?

• Or, do you stick to your normal math class routines so your observer gets a good look at what your math class typically looks like (and knock their socks off with a well-run classroom and academically engaging lesson)?
• Or do you aim for something in between?

Teacher Observation Approach

Looking back at my early teaching days, I probably leaned more toward the ‘bells and whistles.’ This was likely because I felt I needed something ‘special’ for those days to show that I could really design a fun, engaging math lesson.

But as I became more confident in my abilities (teaching, facilitating, and managing the classroom), I tended more toward sticking with our routines, PLUS an engaging activity.

This way, classroom visitors could see that my math students knew how to come in to class, follow our warm-up routines, discuss math with each other, and be self-directed….no matter when the visitors dropped in.

And then visitors could also see how students interacted with the math lesson itself.

So, in this post, I’ll summarize and link to 5 math lessons I’ve used in both my ‘every day’ teaching and in my teacher observation lessons. 

Five Awesome Math Lessons for Teacher Observations

These five math lessons are some of my very favorites, over the course of my 24 years of teaching math, at several different grade levels (2nd, 4th, 5th, 6th).

1) Remove One Game ~ for Math Teacher Observation

This lesson uses a probability game to help students discover the chances of rolling certain numbers when rolling two dice and finding their sum.

It’s a great lesson to introduce or reinforce probability concepts. 

Remove One is also an excellent lesson for leading to a discussion of the probabilities as fractions, decimals, and percentages.

This game only requires paper, markers of some kind, and one pair of dice. 

All students participate with their own markers, keeping them all very much engaged and awaiting the next roll of the dice. 

Get all the details about this lesson in the Remove One blog post.

2) Ratios, Proportions, and Goldfish Math Lesson

This ratios and proportions lesson is a real-life application of tagging part of a population to then determine the approximate number of fish in an area.

Materials needed for this lesson include goldfish, paper bowls, large spoons and a recording sheet.

Students work in groups in this lesson, so if you want to show off some cooperative work, this is a great one!

Get all the lesson details in the Ratios and Proportions and Goldfish blog post.

3) The Factor Game ~ for Math Teacher Observation

An all-time favorite (mine AND students’), The Factor Game is an excellent lesson to help students practice with factors, while discovering how to use strategy in their game choices.

Materials for this lesson are simply paper and pencil/markers.

The Factor Game is a great lesson to show how you can lead students in a game that helps them think about strategy and how to think ahead, WHILE reinforcing their knowledge of factors and multiples. 

We know students struggle with factors when they’re trying to simplify fractions, factor expressions, and just divide numbers! 

So, this lesson is a fun, highly engaging way to practice with factors.

Learn how to play the game in the blog post, The Factor Game.

4) Problem Solving and Footloose for Teacher Observation

This combination of problem solving plus Footloose task cards can be used with ANY math topic. 

Have students solve problems collaboratively (after practicing during previous class periods) and then have them move on to using the Footloose task cards.

Although it’s a seemingly simple lesson, this combinations shows an observer how students can function in a group setting to solve word problems and how they can then move to a self-directed activity.

Read about how this went during one of my teaching observations in the Using Collaborative Problem Solving blog post.

5) Truth or Dare Math Game ~ for Teaching Observation

Truth or Dare is SUCH an engaging math game that you can use for just about any concept!

Students play in groups, so you can show off your group work routines.

Students can self-differentiate in this game, so your observer can see one way you incorporate differentiation.

Click here to See Truth or Dare Games on TPT

The game also offers a chance for discussion, so an observer can see how you’ve helped students develop their discussion skills.

Materials needed for this lesson are the game cards, recording sheets and a pencil!

If you do choose to use Truth or Dare for an observation, I’d be sure to play it a time or two with other concepts so students are familiar with the game and its expectations.

• Reviewing some standard Math Talk guidelines before playing would also be helpful!

Get all the math Truth or Dare details in this post.

Good luck with your teacher observations in math class!

The post Five Math Lessons to Help You Ace Your Teacher Observation appeared first on Cognitive Cardio Math.

Note-Taking in Upper Elementary and Middle School

How much do your math and ELA students love taking notes?! I’m guessing it is not their favorite thing. What if you could change that? Now you can with engaging graphic organizers for math and ELA.  These magical note-taking graphic organizers are interactive and guide students to practical and useful notes.  Keep reading to find out more.

See Math Wheels on TPT

I used to use fold it ups (or foldables) quite often in both math and language arts. However, my middle school classes were always 40ish minutes long and often, no matter how prepared I was, creating the fold it up just ate up too much of the class time. 

Using Fold it Ups for Taking Math and ELA Notes

When we used fold it ups like the free Ratios Fold It Up pictured, sometimes there wasn’t enough time to add the notes before math or language arts classes was over.

 

And the fold it ups were kind of hard to keep organized. 

• We tried using folders and envelopes, and in my last years of using them, I tried using a bound book of fold it ups.
• I took all my math fold it ups and organized them in the order we’d use them during the school year, added some blank pages in between them, and had them bound as a book for each math student. Then all the fold it ups stayed in the book.
• Some remained attached on their original page because they were never totally cut out; some were glued or taped onto the blank pages.
• This method of organizing the notes was the one that worked best for my math students.

I never did that with ELA…because I stopped teaching that class before I had this idea, lol.

But then, I started making math wheels….

Math Wheels for Taking Notes

My new favorite note-taking method for middle school became the doodle wheel graphic organizers for math and ELA. I love these wheels for note-taking, for so many reasons!

Benefits of Using Doodle Wheels as Note Taking Graphic Organizers for Math and ELA

1) All the notes on these doodle wheels are on one surface/one side of a page:

• No folding or unfolding to add info…and then again to find the info

See Math Wheels on TPT

2) The graphic organizers for math and language arts are engaging! 

• Students add their notes to the labeled sections, using various colors, which is great for color-coding their math or language arts concepts.
• Students can add their own doodles or images to connect the learning to their own background knowledge; this is a fantastic way for students to get a little creative in math (or ELA) class.
• Students can color the headings, doodle arrows, and images that are included in the wheel sections.

3) The wheels have a patterned background that typically includes some practice problems, so students have their notes and practice examples all on one page.

• This is super helpful when they need a refresher later in the year…..they can look at the notes and the problems they solved.

4) It’s SO easy to store these graphic organizers!

• Math and ELA wheels can easily be stored in a folder or envelope.
• Or, they can be hole-punched and kept conveniently in a binder.
• If you use a type of interactive notebook, they can be added to that.
• And then students can reference these notes ALL YEAR!

 5) Students get to color the background, so they get that little added coloring benefit as part of their day:-) 

• They may choose to color in a way that helps them remember or focus on a particular aspect of the content. 
• This is another opportunity to be creative in math class, while using color and coloring to help them learn.

More Graphic Organizers for Math and ELA

Have you tried math or ELA doodle wheels for taking notes? If not, I hope you will! There are several free ones here on the blog:
Rounding Decimals Wheel
Fraction, Decimal, Percent Wheel
Percent of Number Wheel

You can access all my doodle wheel graphic organizers for Math and ELA on TPT:
Math Wheels
ELA Wheels

Happy note-taking!

The post Using Doodle Wheel Graphic Organizers for Math and ELA appeared first on Cognitive Cardio Math.

I’m sure you use a variety of task card activities and games in your middle school or upper elementary math classroom, but have you ever used Footloose activities?

 

I’ve mentioned the activity in my blog posts before, but have never really explained it on this blog (I did on my old one, over 5 years ago), so unless you’ve used one of my Footloose activities in your classroom, you might not know how it works.

• Remind students to walk away from the board when they’re working on a card – sometimes they like to hang out and do their work at the board, and that blocks other students from finding the cards they need.
• Remind students they can only have ONE card at a time – some students like to grab a couple so they don’t have to get up for a new one as often, and then other students can’t find the cards they need.

The post Elementary and Middle School Task Card Review Activity for Any Subject: Footloose appeared first on Cognitive Cardio Math.

I recently asked this question on Instagram: “What’s the most challenging math topic to teach/most difficult for your students to ‘get’?” I got a variety of responses, but the one that came up most often was fractions – remembering the ‘rules;’ students finding common denominators when they were multiplying; students (older students) not being able to find a common denominator; and so on. So if your answer to this question is “least common denominator” just know that you are not alone.

The Ladder Method for Finding the Least Common Denominator

Today, I’m going to share how to use the ladder method to find the lowest (least) common denominator, and hopefully, if your students have struggled with this, it will help them (and you!). Before I explain how it works, I want to share that I’ve used the ladder method for several years, after many years of teaching GCF and LCM the ‘traditional’ way  – the way I’d been taught! And during those years, I’d often get frustrated by the fact that students would miss the GCF because they missed factors, or they couldn’t find the LCD because the numbers got too big so they just multiplied the denominators…..or they listed out the multiples, but made a mistake in one list, and so they never found an LCM/LCD. I’m sure you know what I mean!

The ladder method took these issues away, and it also added something I didn’t initially expect – it appeared to improve number sense for many students who struggled with their multiplication facts or with the idea of finding factors and multiples. It helped them understand HOW numbers were related to each other by making the breakdown of the #s more visual (using prime factorization does this as well, but the ladder method provides a little more organization to the process, and I think that’s helpful).

Finding the LCD

So, here’s how to find LCD with the ladder method.
Let’s use the problem 9/28 + 13/42

Step 1

Put the denominators in the ladder.

Step 2: “Remove” common factors.

a) Choose a common factor of the denominators and place it outside the ladder. The factor can be any factor. I tend to use prime numbers, because we also use the ladder method for prime factorization, and that just keeps things consistent. Students often like to start with small numbers – they are faster and maybe a little ‘safer’.  In this case, I’ll start with 2.

 b) Divide the numbers in the ladder by the number outside (2) and write the quotients below the original numbers.

c) Look at the quotients and determine whether they have a common factor. They can both be divided by 7, so add a 7 outside the ladder and divide again.

Check the new quotients for common factors. 2 and 3 have no common factors, so the ladder is complete.

Step 3: Multiply Factors

Now, to find the LCD (LCM), multiply all the numbers (factors) on the outside of the ladder (2 x 7 x 2 x 3 = 84). The numbers on the left
of the ladder are the factors that the denominators share. The numbers on the bottom are the factors they do not have in common. Multiplying all of them gives us the LCM.

Step 4: Find Equivalent Fractions

So now the common denominator is 84.
Finding equivalent fractions should be a similar process to what students already do. They will find what 9 and 13 were multiplied by to get 84. Once they go through the process several times, they may notice that the number the numerator needs to be multiplied by is always the factor at the bottom of the ladder, under the OPPOSITE denominator. So, in this case, 9 is multiplied by 3 and 13 is multiplied by 2, to get 27/84 + 26/84.

Once students get the hang of using the ladder method for least common denominator, they can find the LCD very quickly!

I hope this is helpful for your teaching on this topic.  And. . . I hope you get the chance to use it and see how your students respond!

I created a ‘least common denominator’ notes sheet (free) and a least common denominator math wheel in case you want to share either of them with your students. If you have any questions, please let me know!

Click HERE to head to YouTube for a ‘live’ explanation of using the ladder method for LCD/LCM.

The post Finding the Least Common Denominator with the Ladder Method appeared first on Cognitive Cardio Math.

Teaching Fraction Division in Middle School

Have you ever spent time teaching fraction division to your 5th or 6th grade math students (or even to older middle school students), only to find them “flipping” the wrong number? You, or a previous teacher, may have taught students to “skip, flip, flip,” “invert and multiply,” or “multiply by the reciprocal.” Students may have learned the steps to divide fractions or learned a fun fraction division song to help them remember.

But somehow they still flip the wrong fraction. OR students change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred…and then they don’t flip anything.

Why does this happen? I’m going to say that it happens because students don’t understand why they’re flipping anything – it doesn’t mean anything to them.

Teaching Fraction Division Using Common Denominators

Because of these fraction division issues, I started using another way to teach fraction division. Perhaps you’ve heard of it, or you already use it.

• I never learned this method when I was a student (and didn’t see it for many years of my teaching career), but I like it, and it makes more sense to some students.
• I was fortunate to learn this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course I said no.

The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators.

These are the steps:

• Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to finding common denominators).
• Step 2: Create a new fraction with the numerator of the first fraction over the numerator of the second fraction…this is your answer.
• Done (unless you need to simplify)!

I was shocked – it seemed SO simple!

Fraction Division Example

Check out this example – it’s a simple one, for starters:
5/6 divided by 2/3.

1) Find the common denominator of 6 and 3, which is 6.

• This gives you 5/6 divided by 4/6.

2) The first numerator (5) becomes the numerator in the answer.

• The second numerator (4) becomes the denominator.

3) Simplify

Fraction Division Example 2

Let’s look at another one, with mixed numbers:
1_4/7 divided by 1_3/4.

1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4.

2) Find the common denominator of 28 and make equivalent fractions.

• This gives you 44/28 divided by 49/28.

3) The first numerator (44) becomes the numerator in the answer.

• The second numerator (49) becomes the denominator.

4) No simplifying, in this case.

Why Does Using Common Denominators Work?

Once the denominators are the same, we’re dealing with fraction pieces that are the same size.

When we look at 5/6 ÷ 4/6, there are 5 pieces and 4 pieces that are the same size (our numerators).

• So, we’re really looking at 5 ÷ 4, or how many times 4 fits into 5, which is 1 and 1/4 times.

Student Response to Dividing Fractions Using Common Denominators

I teach both methods to my sixth-graders. Some really like it…it makes sense to them.
Others stick to the flipping method, but I don’t know if this is because they like it better or because it was the first way they learned it.

Most of them have been taught something about fraction division in 5th grade so they have the reciprocal background.

Fraction Division Using the Reciprocal

Multiplying by the reciprocal: if students are going to use it, I think it’s important that they understand WHY it works.
It may be tough for them to understand in 5th or 6th grade, but if they learn the common denominator method first, the proof may then make more sense to them. I found a great article on the NCTM website that uses the common denominator method to prove why multiplying by the reciprocal works – check it out, if you have time:-)

Fraction Division Math Wheels for Taking Notes

Not too long ago, I made two math wheels, to use to teach both methods of dividing fractions – taking notes with the wheel will be more fun!
Students can save their wheel to use as a reference throughout the year.

Resources to Teach and Practice Fraction Division

Select the images to see the resources on TPT.

If you need some extra tools to help teach or review fraction concepts, check out the Fraction Review Toolkit!

Save to Pinterest!

Not teaching fraction division yet? Save this post to your favorite fraction Pinterest board so you can come back to it when you need it!

The post How to Teach Fraction Division….Another Way appeared first on Cognitive Cardio Math.

Finding The Factor Game Again!

I was looking through my middle school math folders on my computer, and came across a document called “The Factor Game.” 

When I saw the document, it occurred to me that in trying to think of some new things to do with my classes, I forgot to play The Factor Game this year when we started talking about factors!

I was so disappointed. 

Of course, we can play The Factor Game next week, or any other time, since practice with factors is always helpful, but I just couldn’t believe that I had forgotten about it.

I’m sure many people know about The Factor Game, or use a version of it, but for those who don’t, I’ll explain how to use this math game. It’s great for 6th grade, but also upper elementary grades ~ and even 7th or 8th grades if students need factor practice!

How to Play The Factor Game in Middle School Math

1) Players start with a simple game board, with the numbers 1-30 

• In the past I’ve alternated between giving students a pre-printed sheet and having them write their own numbers on their own paper
• Sometimes I have them play at the whiteboard or with mini whiteboards, which they enjoy

2) Choosing Numbers

A. Player 1 chooses a number, and marks it on the game board

• They might circle it, square it, triangle it, color it etc
• Player 1 receives points equal to the number they chose

B. Player 2 then marks all of the remaining factors of Player 1’s choice

• Player 2’s points are all of those remaining factors

3) Next, Player 2 chooses a number and gets the value of that number as their points

• Player 1 identifies all of the remaining factors of that number, receiving those numbers as points

4) Play continues this way until all possible numbers have been used.
*Special rule – players can’t choose a number that has no factors available for the other player. If they do, they lose their turn, receiving no points.

The Factor Game ~ Example

Let’s look at this with actual choices and numbers.

When I introduce this game to my math classes, we play it on the whiteboard, and I always allow them to go first (the first time we play). 

Often, they will choose 30 as their number, since it’s the biggest and gives them lots of points.

When I go ahead and mark all the remaining factors of 30 (my points), they are shocked! 

• In the images in this post, their points are circled in red and mine are squared in blue.

After the first turn, they have 30 points and I have 42! 

I normally have 2 students record our scores on the board, while 2 others keep running totals on a calculator, but points can all be tallied at the end if you want.

For my first turn in this example, my choice would be 27. 

This leaves students with only 9 as a remaining factor, since 1 and 3 are already gone. 

Needless to say, they’re usually a bit disappointed when they get so few points!

After their experience with 30 and its factors, students start thinking a bit more for their next turn.

They will often consider 25 or 23, but I remind them that they must leave a factor for me, and 1 and 5 are already taken. 

So, let’s pretend they choose 21. That leaves 7 for me.

The game continues, alternating turns, until all possible numbers are used, as shown in the “finished game” picture.

You can see, because of the way numbers were chosen, some numbers are not used.

Playing the Second Time

The second time we play as a group, I go first.

I choose 29 as my first number, and they are disappointed to only get 1 point!

We continue to play the second game, and they get a better idea of what makes a good choice. They start to look ahead and think about and discuss what factors I might get if they make a certain choice.

Students Play With a Partner

After our games as a class, I have the students work in partners, and they normally get one or two games played before we stop to discuss at the end of the period. 

• We always talk about what the best and worst first choices are, and bring the idea of prime and composite into the discussion.

I’ve been playing this game with students for at least 15 years (I think), and every group of students has loved it! 

I found the game when I was involved with Mathline, so long ago. If you’d like to see the original lesson plan, with extension ideas, click here.

You can also watch the video of me explaining the Factor Game. I did this on a Facebook Live and then uploaded to YouTube:-)

Have you played the Factor Game in middle school math or upper elementary math? If not, I hope you’ll introduce it to your students!

The post Playing The Factor Game in Middle School Math appeared first on Cognitive Cardio Math.