Mastering Area in Geometry Using Strategic Acceleration.

Geometry is often where math starts to feel tangible again. After months of abstract algebra, students finally get to work with shapes, angles, and visual spaces. But when teachers introduce the unit on area, a familiar hurdle appears: many students stumble, confusing perimeter with area, or freezing up when faced with composite shapes.

In the past, catching students up on area meant falling back on old-school remediation. A student struggling with the area of a triangle or circle might be pulled out of class to do elementary school grid worksheets, drilling formulas they memorized and forgot yesterday. This approach alienates older students, making them feel like they are permanently falling behind.

So, how do we modernize math recovery for the geometry classroom, specifically when tackling area? The answer lies in strategic acceleration.

Traditionally, geometry instruction relies heavily on rote formula memorization: Base times height divided by two. Pi r squared. But when students only memorize formulas without understanding why they work or what area actually represents, they struggle the moment a shape is turned sideways or transformed into a composite figure.

If a student lacks foundational number sense or fraction fluency, forcing them to memorize formulas doesn’t fix the gap; it just masks it. Traditional remediation treats this by slowing down the entire class or pulling kids out of grade-level content. Strategic acceleration, however, keeps students anchored in high-rigor geometry while providing just-in-time support right when they need it.

Imagine your geometry class is diving into a real-world project: calculating the cost of materials to lay sod and build a patio for a backyard landscape design (involving composite shapes made of rectangles, triangles, and semicircles).

Instead of spending two weeks re-teaching basic multiplication or decimals to the whole class beforehand, a modernized recovery approach uses a quick, targeted diagnostic:

• If a student struggles with multiplying decimals while calculating square footage, the teacher provides a quick, five-minute targeted scaffold during the design phase.

• If another student forgets how to decompose a complex shape into simpler triangles, the teacher uses visual manipulatives or a digital applet right then and there to rebuild that spatial intuition.

Students stay engaged because they are working on a compelling, grade-level project, but they receive the precise building blocks required to succeed.

To make area recovery stick, classrooms are shifting away from flat worksheets toward tactile learning. Students struggling with area formulas benefit immensely from physically cutting paper shapes apart to prove that a parallelogram is just a rearranged rectangle, or using grid overlays to see why a triangle's area is half of a rectangle's.

By blending grade-level geometric challenges with just-in-time arithmetic support, we turn an area unit from a source of frustration into an opportunity for visual discovery.  Let me know what you think, I'd love to hear.

Is Interactive Geometry Software Really Good.

Technology has transformed the way geometry is taught in classrooms. Interactive geometry software allows students to rotate shapes, manipulate angles, create perfect constructions, and instantly visualize mathematical relationships. These tools are powerful and engaging, but many educators are beginning to ask an important question: Do students actually learn geometry better when they draw it themselves?

While digital geometry programs offer convenience and precision, physically drawing shapes by hand may provide deeper learning experiences that strengthen spatial reasoning and conceptual understanding. As schools continue integrating technology into math instruction, many teachers are rediscovering the value of traditional geometric drawing.

Geometry is unique among math subjects because it is highly visual and spatial. Students are not only solving equations — they are learning to understand shapes, relationships, measurements, and movement in space. When students physically draw triangles, circles, angles, and polygons themselves, they engage more actively with the concepts.

Drawing geometry by hand requires students to slow down and think carefully about what they are creating. Using rulers, protractors, and compasses forces students to pay attention to measurements, angle sizes, symmetry, and proportion. Instead of simply clicking and dragging points on a screen, students must make decisions throughout the construction process.

This hands-on work helps strengthen spatial reasoning, which is the ability to mentally visualize and manipulate objects. Spatial reasoning is important not only in mathematics but also in science, engineering, architecture, art, and many everyday tasks. Research and classroom observations suggest that physically creating geometric figures can improve students’ ability to understand how shapes relate to one another in space.

Another benefit of drawing geometry manually is that it often reveals misconceptions more clearly. When students construct figures themselves, mistakes become learning opportunities. A poorly measured angle or uneven triangle encourages students to analyze what went wrong and make corrections. This process builds deeper conceptual understanding and problem-solving skills.

Digital geometry tools, while highly useful, can sometimes make the process feel too automatic. Software can generate perfectly accurate shapes instantly, which may prevent students from fully understanding how those shapes are formed. Students may learn how to operate the program without fully grasping the geometry behind it.

For example, a student using software can easily create parallel lines or bisect an angle with a few clicks. However, drawing those constructions manually requires understanding why the steps work. The physical process reinforces the mathematical reasoning behind the construction.

That said, geometry technology still offers tremendous advantages. Interactive software allows students to explore transformations, rotations, reflections, and dynamic relationships in ways that are difficult to replicate on paper. Students can test ideas quickly and visualize concepts that might otherwise remain abstract.

The real solution may not be choosing one method over the other but finding a balance between both approaches. Traditional drawing methods help build foundational spatial reasoning and deeper conceptual understanding, while digital tools enhance exploration and visualization.

Many effective geometry classrooms now combine the strengths of both. Students may first draw constructions by hand to understand the process and then use technology to experiment further, test patterns, and explore more advanced concepts. This blended approach allows students to develop both precision and conceptual flexibility.

As education becomes increasingly digital, there is growing recognition that some traditional methods still provide unique benefits. Physically drawing geometry encourages patience, focus, reasoning, and spatial thinking in ways that technology alone may not fully replace.

Geometry is not just about producing correct figures — it is about understanding relationships in space. Sometimes, the simple act of drawing shapes by hand may help students see mathematics more clearly than any screen can provide. Let me know what you think, I'd love to hear.  Have a great weekend.

Construction lite

Today in two different classes we used apps to help the students learn.
In the college prep math class we used free graphing calculator to learn what the graphs of sin and cos looked like and what happens if you add this or that to the equation.  We discussed it, played with equations and the kids really had a blast.
We started with y = sinx.  I took time to have them read the unit circle so that we could connect the unit circle to the sin graph.   They graphed y = 3sinx on a different line so they could have both equations on the screen at once.  This way they were able to tell me what happened to the graph by adding the 3.  I had them graph y = sinx + 2,  y = sin(x+pi/2) and y = 3sin(x + pi/2)+2.  After comparing and contrasting the graphs, I was able to lead them to seeing how the transformations from earlier math classes applied to trig graphs.  It was awesome.
I downloaded and had my geometry students use construction lite which simulates using the compass and straightedge.  The first two sections are unlocked in this free version of the app.  The sections are lines and angles.  We looked at copying a segment and constructing a perpendicular bisector.  The app shows the student how to do each activity and then allows them to do it themselves using the digital compass and straightedge. 
I just let my students loose to try the sections themselves and as one figured out how to do something, I would have him or her go help others so those who finished early were working as peer tutors with the others.  Although only two topics are available, there is a sketch function that will allow me to have students follow the construction directions in the books and I don't have to keep track of all the compasses, pencils etc.  The kids enjoyed themselves.

Next step in process

I've got the iPads and I've got the students trained to grab an iPad and scan the QR code for the warm-up.  They do this before they get their folders.  After yesterday's origami lesson, I was going to have students write a short summary of the relation of origami to points, lines, planes and intersections but the e-mail was not set up.  So I spoke to the tech dept and they assured me that with the new OS7 I would not need e-mail because it has airdrop which will allow students to turn in their work.  There are assignments I'd like to have them do in a paperless way so that I do not have to correct the paper. This is also going to cut down on the "I turned it in" but its found in their folder/locker/home syndrome.  I can hardly wait to try it out.
Once that is up and working, we can use a spread sheet to program rate of change and use that for an activity in a couple of my classes.  I can have them make presentations on showme or Haiku deck and have them turn it in.  I am just excited.
Ohhh in regard to the origami exercise yesterday, first thing my students said was "When do we get to do that again?".  I am thrilled because they don't usually get excited over math in general.  I know we can use it for angles, types of triangles, etc.  This is going to be a fun year.

Foldables, origami and geometry.

Today in Geometry we had a blast doing origami.  We just started learning about points, lines, planes and intersections yesterday.  Rather than bore them with standard learning activities using points, lines, planes and intersections, I decided to incorporate origami.  After the warm-up I showed a video on the science of origami which was quite fascinating.  Dr Robert Lang, a person who can fold just about anything from a single sheet of paper, shared how origami has helped in science.  I didn't know that the way a scorpion is made in origami is the same type of folding used to put the airbag in the steering column.  The video was only 1:46 long but the kids were so interested in the material, they asked me to show the video again.
I had loaded two origami apps onto the iPads earlier this month.  One is Origami instructables while the other is "How to make origami" .  Personally, I prefer the latter because each step allows you to have the app show the actual fold if you want more than just the directions.  My students had so much fun, they didn't want to put away the iPads when the bell went.  Tomorrow, I will have the students create a summary of how they used point, lines, planes and intersections in this activity.
A couple weeks ago, we had a short after school class on Foldables created by Dinah Zike.  I wasn't sure my high school students would buy into them but they did.  They were focused and on task.  For Algebra II, we will be filling in the information over the next couple of weeks while in College Prep math, we created a foldable for the 6 basic trig ratios including as they relate to the unit circle.  It was awesome. I think that high school kids still like to make things and don't often get the chance unless they are enrolled in an art class.  I have to pull out my book on using foldables in math.

Origami

I sometimes include Origami folding in my geometry class for two reasons.  First, it makes students pay attention to the directions and second it uses angles, bisectors, triangles, and other such geometric shapes and terms.  I found a nice app called animal origami and downloaded it.  It has several different animals that  you can make following the step by step directions.  I picked it up the day it was being offered for free.  I did a quick check and there are several free to very low cost origami apps which show how to fold the paper step by step.
The ways I work origami folding into Geometry is to have students:
a.  follow directions and then write a paper  on all the geometric shapes and figures they used in the process.
b.  unfold their creation to use a protractor to measure the angles of all the folds off of certain lines.
c.  label and identify geometric shapes and figures on the paper itself.
d.  Write a set of directions on making the figure.

After they've had some practice with fun shapes, I pull out the origami book I have for geometry and have them follow directions for specific items such as their own specialized angle reader. 

Project based vs menu driven learning?

I was looking at a link to project based learning ideas and realized for as long as I've been teaching and as long as I've heard of project based learning, I still am rather unclear about it.  This is even though I worked in a district with levels, end of level tests and the idea of moving towards project based learning.  From my understanding, you assign a project, have the students work through each part of the project and at the end, they turn in the final project and in the process, they have learned several to apply several math standards.  In menu driven learning, students are taught the material, given the menu to create several small projects based on learning styles to turn in for points. 
Where my thoughts are going with this is are they mutually exclusive.  Can you create the over all project and break the parts down into a menu so each student or groups of students will be able to create the final project using different smaller parts that works best for them.  I realize each project will have to broken down into steps so students are not overwhelmed but could each step have three different choices on how to accomplish it?  Since so many of my students are ESL, I have to do a break down so each student knows the steps they have to follow but not necessarily how to do each step.  For instance, when I assign the "Design your dream bedroom" I have them draw 6 scale drawings to show the 4 walls, floor and ceiling so I know where all the windows, doors, lights, and how the room itself looks.  I then have them create an estimate for the finishing cost of the room including flooring, paint, wall paper, ceiling tiles, etc.  I need to look at being able to combine both projects and menus to give students a better choice and increase their learning.

More on centers

I have been giving some serious thought to using iPads with centers.  I have a book with printed instructions for using centers in Algebra I and ll, geometry and pre-algebra.  I can scan the various activities into PDF form.  If I have an app on the iPad that allows the students to read and write the answers on the file, they can then send it off to me so that I can grade it.  By using centers, I can increase their group activities and hopefully decrease their dependence on me. I look forward to trying this.  If I store the pdf files in the school wiki, I can use a QR code to send them to the proper page, provide instructions and any additional information.  I am trying to change the main focus from teacher oriented to student oriented.

Thoughts on the volume app

I have given some thought to the best way to use the volume app I mentioned a few blogs ago.  The videos are well done but it has a mechanical voice which may make my students dislike the videos  I am going to create a set of questions to answer as they watch the videos  This way they have to actually pay attention to what is being watched.  I am also going to include some problems based on the material in the videos so they have to attempt to work some problems to see if they understand the process. When they are done, they can check a QR code to see if they get the answers correct.  I am going to use those QR codes more often beginning in the fall. 
I am now on holidays but will spend the summer actually writing lesson plans and units with more integration of the iPad.  As I come up with ideas and plans, I"ll share the information with my readers.  I will be starting Monday and hope to post something from then.

idea for teaching 30/60/90 triangles.

I am going on memory right now and will do some additional research this summer but I seem to recall something on iTunes university about math in nature. This series shows real life examples of math in nature.  Then they go on to show the mathematics behind it.  I recall watching the episode on nautilus and if I remember correctly, it used 30/60/90 triangles.  I am thinking I can show the episode on the nautilus and the math behind it.  I can then have them create their own nautalis on the geoboard app on my iPad.  This would happen toward the end of the triangle section.
I am adding to this as I was grading papers after having read a bit on fractions and realized I can use the Geoboard app to have students create representations for fractions such as what is 1/3 or 1/3 x 1/2.  I  can also use it to show the distributive property and multiplication of binomials.  This opens more uses of the app for me.

I should be letting you know on Monday when I am offering my e-book for free.

2nd website and mindmapping

The second website I use is www.aaamath.com.  I often use it for my Algebra I or Pre-Algebra level students who need to practice various topics.  Today I'll be sending my students here to read and practice prime factorization.  This website has a topic, information on doing it, practice and often offers games.  My 9th graders tend to be much more focused in class when they have an iPad assigment.
I gave some thought to mindmapping last night and came up with a couple ideas of how to use it in my math classes.  So far the ideas are for geometry but its a start.
Mindmapping ideas
1.  Use it for classification of triangles.  The center topic of triangles has three arms.  One for angles, one for sides and one for both.  There are more arms off of each of those topics to list the type such as right, acute, scalene or right scalene.  Add more arms for the characteristics such as acute would have three acute angles, or right scalene might list must have a 90 degree angle with three different length sides.  A 30-60-90 degree triangle would be an example.
2.  Classification of quadrilaterals.  This would work the same way as the triangles one.
3.  Altitudes, bisectors, and medians.
This could easily be done on a mindmapping app on the iPad.  I use Simple Mind.  It is free and it allows you to use a different color for each branch of the map.  Students often remember color better than something in black and white.
I'll give some more ideas for mindmapping in a day or two.

Infographics

I've been reading about infographics and I think it is something worth trying.  Since my college prep is in the middle of a project, I think on Monday I will give them a choice of actually writing the short report or preparing an infographic on logs and natural logs and their use.  I think they might have more fun doing the infographic rather than a report since it is a bit more creative.  From what I've read about infographics, these would be great for the probability unit as students can see the real life applications of probability.  Bingo is a huge business out here and parents will often go play to 3 or 4 in the morning.  Maybe an infographic on the probability of Bingo would be a good place to start the unit. 
I am hoping to discuss a couple math apps I discovered on iTunes that will help students with the distributive property and with the special binomials such as (x^2 - 4).  I even have ideas of how to do a graphical representation on Geoboard but I want to do more research on other apps that might help me with the graphical representations.
If anyone knows of one, let me know.

Virtual manipulatives in the classroom.

Currently I use two different ones in my math classes.  One is Geoboard for the iPad.  It is just like the real life geoboards we have in our classrooms except this app comes with two different boards and lots of colored rubber bands but no chance of loosing either the board or the rubber bands.  I've used it for students to create various types of triangles such as those classified by sides, or angles, finding the number of triangles in a polynomial as a starting point to find the equation telling the interior measurements of the angles, tessellations, transformations, etc.  If you use a geoboard in real life, you can use this app to do the same thing.
The second app I've used in geometry is Geometry pad.  It allows students to use a coordinate plane for geometry.  It allows students to make all sorts of quadrilaterals and polygons, triangles, circles with radius, arcs, chords, etc.  There is an option for points, angles, lines, rays, segments, perpendicular bisectors, medians, altitudes and angle bisectors.  you can draw, type, use a ruler, and do certain functions.  I've used this one when the book has a coordinate plane problem, transformations, midpoints, diagonals, etc.  If I have to resort to worksheets for certain activities, I will have my students use this app to help them complete the worksheet. 
Due to iPads not having java, I am unable to use many of the virtual manipulative websites on the internet.  I found another app that has great ratings and I hope to down load it tonight to play with.  Its virtual manipulatives and focuses on decimals, percents and fractions.  Although it is rated for middle school, many of my lower performing high school students are weak in that area.  I think this could be used in that class to help them learn how decimals, percents and fractions relate to each other.  I hope to give more information on this app on Monday or so.