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karengreenhaus 8:34 pm on March 27, 2020
Tags: angle of rotation, , dilations, non-rigid transformations, , point of rotation, reflection line, reflections, , rotations, , translations   

Mini-Math Lessons – Transformation Practice

This week I’ve spent most of the time on transformations, so if students have explored the progression of lessons, they would hopefully by now have an understanding of what translations, rotations, reflections and dilations are, including properties. This means they understand the difference between rigid and non-rigid transformations, know the different properties about orientation, corresponding sides and angles, just to mention a few.  Today’s two activities are basically practicing those properties in different ways. The first activity, Transformations – Where Am I Now? focuses on applying the properties of transformation (both rigid and non-rigid) in a coordinate grid and determining coordinates of transformed images, based on a specific transformation and directives. The second activity, Transformations – How Did I Get Here?, is using the properties of transformation to determine WHAT combination of transformations were done to map a preimage onto a given image, and also things like finding the line of reflection or the point of dilation, all of which require an understanding of the properties of transformations.

Here is the link to the two activities and the one video that gives an overview of both.

  1. Transformations – Where Am I Now?
  2. Transformations – How Did I Get Here?

 

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

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

 

karengreenhaus 7:24 pm on March 26, 2020
Tags: , coordinate dilation, corresponding vertices, dilation, , non-rigid transformation, perspective drawings, , scale drawings, scale factor, scale models   

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, ClassPad.net.

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

karengreenhaus 5:48 pm on March 25, 2020
Tags: , Dr. Evil, , , , , , , , Mini-me, , Proportions, Scale, Zeno's Paradox   

Mini-Math Lessons – Multiple Topics (Calculus, Trig, Geometry….)

2020-03-25_13-41-35I am re-sharing some lessons/videos created by Ish Zamora (@seemathrun) because they are so good and if you hadn’t seen them before, I am hoping that you explore them now. (Also, because I have run out of time to create some new ones today, due to some other responsibilities!)  I will be back tomorrow continuing my transformation theme for the week with some new mini-math lessons focused on Dilations.

For today, I am sharing a several activities created by Ish, along with accompanying videos where he talks about the activities. Some of these have been shared in previous blog posts and/or on our FB and Twitter and Youtube accounts, but here they are in one place for you to explore and choose from!

Links to some of Ish’s FREE activity papers and video support:

  1. Exponential Decay – Bouncing Ball  and Calculus Bouncing Ball (Parabolic/derivative)
  2. Proportions & Scale – Dr. Evil   and Trigonometry Mini Me 2 (Trig Intro)
  3. Taxi Cab Geometry – Circles and Distance and Uber/Taxi Cab Geometry
  4. Multiple Representations – A Gutter Problem
  5. The Soggy Grasshopper – Zeno’s Paradox

 

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

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

 

karengreenhaus 6:13 pm on March 24, 2020
Tags: congruent angles, congruent figures, congruent sides, , , , , ,   

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

VA SOL 2015 Released Item Geometry

In yesterday’s post, I explored rigid transformations (translation, reflection, rotation), where the goal was to get an understanding of what these transformations were and their basic properties. Today, assuming that students now have that underlying understanding of what each type of rigid transformation does to a shape, we will explore rigid transformations on a coordinate grid. This is a very typical ‘assessment’ that students might encounter – i.e. if you rotate a given shape with given coordinates about the origin, 90 degrees, what are the coordinates of it’s image? (see image to the right). This requires some experience working with coordinates and the transformations to develop the understanding of what happens as the shapes are transformed within a coordinate grid.

I explore the same three rigid transformations in these three lessons. I would recommend having students first do the initial explorations from yesterday’s post first. Below are the links to the free activities in ClassPad.net as well as the overview videos for each activity.

  1. Translations by Coordinate Rules
  2. Coordinate Reflection Exploration
  3. Coordinate Rotation Exploration

 

 

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

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

 

karengreenhaus 8:26 pm on March 23, 2020
Tags: , , , , , , , properties of rigid transformations, , , , ,   

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, ClassPad.net.

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

 

karengreenhaus 9:30 pm on March 20, 2020
Tags: box-and-whisker, , circle graph, , , , , , , , , , , , , real-world data,   

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 ClassPad.net 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, ClassPad.net.

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

karengreenhaus 8:57 pm on March 19, 2020
Tags: , equivalent linear equations, , linear application, linear equations, linear transformations, parent function,   

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 ClassPad.net. 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 ClassPad.net. (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: