Mini-Math Lessons – Absolute Value, Rational Equations

Today’s lessons use a little bit of ‘real-world’ application. You will see images that are part of the graphs, which are used to help illustrate the problems and see how math can be utilized to make a problem easier or more understandable. In the first activity, images are used along with a coordinate grid to help visualize the situation. Students use the slope formula and a tangent formula to help find the angle of a spotlight beam that is positioned to show off art work. The second activity uses images that can then be drawn on to help visualize a real-world situation of a tunnel. This ability helps students the connection of the ‘tunnel’ to circles and the equation of the circle, and they will then use their understanding of solving equations for specific variables to help answer specific questions about the situation. They will also be finding the area and perimeter of shapes within the circle, which requires working with radical equations. There is even an extra portion where the tunnel is now elliptical.

These two activities highlight a couple different things in, the free dynamic math software that is being used. One is the ability to add images to graphs and geometry so that you can use them as part of the problem-solving. This means creating points or objects on top of the images to help visualize and connect the math. There are also some templates that are being used, such as the absolute value, fractions, and you will see how the text stickies allow for math-type when students are writing in their solutions/explanations.

Here are the links to the two activities and the video overview that explains each activity and shows some of the functionality:

  1. Algebra II -Spotlight on Art (absolute value, slope formula, tangent formula for spotlight beam, visual images on coordinate grid)
  2. Algebra II – Do You Have Tunnel Vision? (circle equation, area, perimeter, radical equations).

The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lessons – Dilations & Scale

Today I am going to focus on dilations, a type of transformation that is non-rigid, meaning it changes the size of an object (unlike a rigid transformation, that keeps size and shape the the same).  Dilations do keep the shape of a dilated object the same, but the dilated image is either smaller by a given scale, larger by a given scale. Technically, if the scale is one, then the dilated image is still a rigid transformation, but then, is it really a transformation because it would NOT have changed anything about the shape – it is exactly the same….might be a fun discussion to have with students!

I have only created two activities for today.  The first one is all about exploring dilations and properties of dilation. It starts off just exploring what a dilation is via sliders and moving the dilated image and changing the scale so that there is some sense of what happens in a dilation. The next portion of the activity we really focus on dilation and the proportional relationships between corresponding sides and their relation to the scale. And then we extend into perspective and finding the point of dilation.  The second activity is more of an application of dilation – i.e. scaling objects (so scale models of rooms as an example), and maps and a little bit more on using dilations for perspective drawings/constructs.

Here are the links to the activities and the video overviews that explain and walk through each activity a bit.

  1. Dilation Explorations1 coordinates – Non-rigid Transformations
  2. Dilation Explorations2 – Scale and Perspective


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lessons – Exploring Rigid Transformations (ES/MS/HS)

This week I am taking a themed approach to the mini-lessons and focusing on transformations, both geometric and functions. For today’s mini-lessons, we are going to explore three of the rigid transformations – translations, reflections, and rotations. I’ve decided NOT to do the fourth rigid transformation – glide reflections, though I might come back to that at a later date.

The lessons for today are explorations about what transformations are and specifically what translations, reflections, and rotations do to an object. The goal in these three lessons is really about dragging, moving, noticing and exploring what is the same and what is different between a preimage and it’s transformed image. The goal of all three is that students look at side measures, angle measures, location of the preimage/image, notice orientation, notice distance between corresponding parts, etc. and come up with some conjectures and ‘definitions’ of the transformations on their own through their explorations. We want STUDENTS to figure out that these three rigid transformations keep the transformed image the same shape and size, but move the location and/or the orientation. And that each type of transformation has some special properties about it that can be used to find/construct the transformation of another object (that is a later lesson in the week!!).

As always, the goal of these lessons is really exploration, discovery and making conjectures based on your observations. Students learn and remember more when they are doing math and making observations and testing out their conjectures, and that is the aim here.

Here are the links to the free activities and the overview videos for each lesson:

  1. Translation Exploration – Slide Into Rigid Transformations
  2. Reflection Exploration – Flip Into Rigid Transformations
  3. Rotation Exploration – Turn Into Rigid Transformations

Tomorrow we will expand on today’s discoveries and look at these three rigid transformations in the coordinate plane. Wednesday we will explore dilations (non-rigid transformations) and Thursday we will bring all of this together with some problems that ask students to use the properties they discovered to find the lines of reflection, the points of rotation/dilation, and the transformation rules. Think I might even throw in some challenges – i.e. can you figure out what rigid transformation occurred?

Friday we will spend a bit of time applying the idea of transformations to function transformations and look at quadratic, cubic and absolute value functions.


The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:


Mini-Math Lessons – Measures of Center (Elementary/MS Level)

Today’s lessons are about things kids can do at home, since we are all ‘social-distancing’ and there’s a lot of home-schooling and virtual learning happening all around the country. Getting out of the house and collecting data and/or exploring things around the house or neighborhood are great ways to incorporate math and learn something about your world. The lessons I am sharing today have some specific ‘numbers’ in them, but also offer suggestions to make the data you collect more fun, personal, and relevant/interesting to your own situation. Be creative!! All these lessons can be saved in your own free account and then modified to fit your needs, so make them your own!!

In today’s lessons, the focus is on measures of center and how we might use those measures to make decisions or conclusions about things. And which measures do you use – i.e. does it matter if you use the mean, median or the mode? Also – we will be doing some visualization of the data as well with dot plots, histograms and bar charts and maybe even some scatter plots if we are thinking about relationships.

Below are the links to the free math activities as well as the accompanying videos that walk through the goals of each lesson and offer some quick how-to’s for some of the math involved. Remember, these are really lessons about discovering and exploring with math – it’s about noticing, wondering, trying things out, asking questions, etc.

  1. Measures of Center – Describing Data (lots of reading in this activity – it’s more of a how-to-find each measure w/example)
  2. Simple Statistical Plots – Candy Color Comparison? (this video is about constructing bar graph, histogram, dot plots by hand (vs. using quick-tools)
  3. Data Changes – How Does That Change Measures of Center?

The tool being used in these mini-math lessons is the FREE web-based math software,

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lessons – Slope, Linear Equations & Graphing

Today’s mini-math lessons are focused on linear relationships. I obviously am not going to cover all there is to know about linear relationships and equations, but I thought I would focus on some key areas where students tend to struggle and where visualization and exploration can help them ‘discover’ some of the rules and relationships on their own. It’s a lot more fun to ‘do’ math instead of just hear math and rely on memorizing rules. That is the goal behind all these mini-math lessons – to get students seeing math, moving things, testing out ideas, making mistakes and trying something else. The goal is for them to discover on their own, through context and experience, relationships and rules, instead of relying on memorization. The more they do, the more they see, the more they connect and understand.

With that in mind, the three lessons for today are sort of in the pre-algebra realm of linear relationships. One activity is focused on slope of a line and what that means and how to find the slope of a line. The second activity it about different forms of a linear equation – transformation of a parent function (i.e. y=x or f(x)=x), where students explore different forms of a linear equation to determine what the slope does, what the intercepts do, what the coefficients in an equation tell you about the line. The third activity is more of an application of linear equations, where students can explore lines of fit and writing their own equations, and analyzing data to see if there is a ‘linear relationship’.

Each title below is a live link to the activity on You can play with this activity just by clicking the link – it’s completely free. If you want to save the activity or your work, either to share with others or to modify and create your own version, be sure to create a FREE account on (Anyone can create a free account!!) Below are some helpful how-to’s that will show you how to create an account, how to duplicate and share activities, etc.

Today’s Mini-Math Lessons:

  1. How Steep Is It? (A lesson on what is slope and finding the slope of a line)
  2. Linear Equation Transformations (A lesson on equivalent forms of linear equations)
  3. What’s My Line? (Applications of Linear Relationships)

Note: Tomorrow’s mini-math lessons will focus on measures of central tendency and simple statistical plots.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Halloween – Fun Statistical Exploration Ideas

As I walk down my street and see the spider webs on the bushes, the witches and ghosts flying from the porches, and the glowing pumpkins, it’s hard not to think about Halloween. If you live on a street with lots of younger kids, it’s even harder! On my street, in a small town, Halloween is a big deal – all the neighbors come out, we build a bonfire, adults without kids hang out while those adults with kids walk around trick-or-treating with their children. There are hand-made costumes, adorable family costumes, lots of candy and a few ‘adult’ treats as well. I admit – I tend to dress up just for the fun of it, even though my own children are long grown. All of this has me thinking about the amount of money that people spend on this day of fun – i.e. the decorations, the costumes, the candy. Naturally, this had me out exploring…..

I found several articles, and let’s just say the numbers are pretty astounding:

From one article, it’s estimated that $2.6 billion is spent on candy and $2.7 billion spent on decorations. Wow!  That’s a lot. Though, after looking at the price for one bag of candy, I shouldn’t be too surprised.

Of course, I LOVE when data is displayed visually, since this really helps students in particular see data and make sense of it. Found this chart from Statista about last year’s Halloween and how the spending is divided across categories. The costumes for pets sort of cracked me up!

Infographic: The Shocking Scale Of U.S. Halloween Spending | Statista

As always, I like to think about how this type of information could be used with students. Especially as math teachers, where we are always looking for real-world problems that are in fact real world (not textbook ‘real’….i.e. contrived) and that engage students. Halloween is definitely something that engages students, so there are many ways teachers could incorporate some Halloween real-world data and questions into the math classroom.

There are several articles and places that have data about Halloween, so one way would be to collect data and create graphs of these for comparison. So perhaps comparing the trend of spending over several years and then maybe exploring what might have been happening in the years where spending seemed lower or higher and making some connections there.

Or, having students look at Halloween adds for costumes and/or candy and compare pricing and make some decisions on where they should buy their Halloween supplies, factoring in things such as sale prices, buying in bulk, type of candy, etc.

You could have some fun with candy as well – maybe estimating the number of candy corns in those small bags, and then collecting data (i.e. open several small bags, count the number, record the data, then find statistical measures and graph. Really, same thing could be done with any of those Halloween-size bags of candies, such as M&M’s, Skittles, etc.  A fun exercise in weight vs. quantity.

Students could collect data on costumes (i.e. survey other students in their school on what costume they plan to wear, and see what the trends are for costume type (see chart to the right, where for kids, Princess is #1 and then Superhero).

You could include some health statistics too – i.e. does the number of cavities found in people increase after Halloween?

And let’s not forget about pumpkins!! According to this article, $377.3 million was spent on pumpkins last year (for carving). So – how do farmers plan for the run on pumpkins? How many pumpkins are planted that are not used? Is there a state that grows the most pumpkins? Where are pumpkins grown in this country??There are lots of questions we could ask about pumpkins, so get your students thinking! And, there are lots of statistical plots reflecting the data on pumpkins (and more) that could lead to some really interesting exploration and questions and analysis.

Let your students generate some questions themselves and then help them explore finding information and then presenting the data and their conclusions. There will be a lot of different kinds of math going on, based on the avenue they choose to explore.

Have some fun!!!

Here is a quick video of some data taken from this article that I put into (first as an image, and then I made my own table). Here’s a link to the paper if you want to use it.














You will find more infographics at Statista

Let’s Explore with Geometry and Start the School Year Off Right! (New Features with

I admit it. I am a geometry nut.  It is my favorite subject to teach, which I have been doing for the past 30 years (wow….said that out loud!!). Geometry to me is all about logic and connections and relationships of shapes. It should be hands-on, it should be visual, and with technology, is should be dynamic – meaning you can see and discover relationships through movement and manipulation. There are many good resources out there (for those of you looking for a ‘textbook’, Discovering Geometry has always been my go to – it’s all about learning geometry through hands-on discovery and connections. It’s on it’s 5th edition, and the ebook has dynamic investigation using (formerly used Geogebra), and has made huge strides in advancing it’s geometry functionality, which is what this post is focused on. My goal over the next few posts is to focus on specific geometry explorations using some of’s geometry functionality, but today’s post is an overview of what’s new. has all the tools you would expect a geometry software to have – i.e. points, straight-edge tools, polygon tools, display tools, expressions, equations, etc. It has some others don’t have – i.e. tools for conics for example. Below is a list of some of the added features as we continue to improve the functionality of the software (which is FREE, btw!!)

Quick List of New Functionality:

  1. Compass Tool
  2. Ability to add in images and use them as part of your geometry explorations
  3. Ability to create sliders for transformations (dilations, rotations, translations, reflections)
  4. Trace feature
  5. Multiple Grids, including isometric
  6. Ability to lock constructs
  7. Ability to create a rigid polygon (meaning it won’t change shape once constructed)
  8. Ability to add tick marks to sides and angles
  9. Ability to change the style of points – i.e. dot, square, x
  10. Ability to measure exterior angles explicitly and create angles 0-360
  11. Ability to construct a specific regular polygon (n-gon) by constructing one side and choosing n (number of sides)
  12. Ability to duplicate constructs without have to ‘reconstruct’ them.

I will be creating videos on each of these features and how to use them for future postings, but today, I wanted to show you where you can find the different new features. Be sure to visit and sign up for an account (so you can save any work you do). Both the Free and Basic accounts are completely free and have everything you could need for a classroom (don’t forget there is calculations, graphing, statistics, financial tools, and text as well as geometry!). Below is a quick how-to on finding where all the new features for geometry are – stay tuned for future how-to’s on using the specific features. Meanwhile, why not try and explore things on your own? Have fun!!