3 Teaching Methods That Teach Properties Of Bisected Line Segments

Want to teach properties of bisected line Segments? Here are three quick methods every geometry teacher needs to know.

Method #1: Allow Students To Make Conclusions And Connections

Here's a simple method that so many geometry teachers miss out on when teaching midpoint and bisectors.

Here's what I mean: as the teacher, we must connect new information to prior learning for every lesson. Research shows that this will help our students retain the new knowledge that they are learning.

The connection is important to you because it means by allowing our students to do the constructions themselves and see the bisection and make the measurements, we can get the most reluctant learners engaged.

This method is important because  when you ask questions like

Will this always be true?
Can we do this will a different line?
Are the steps always the same?
Is there another way to answer this question?

Helping the students to make the relationships and connections, will place this new information into their long term memory.

So the thing for you to do here is to plan time into your schedule where students can do the constructions. Prepare with compasses and sharpened pencils (to save precious time) and cardboard or thick paper to protect your desks.

Method #2: Creating A Proof

Here's the crucial thing you need to understand with this method: when students conclude that a bisector is a midpoint, we need to progress to the next level of ensuring that the students can prove that conclusion.

The most important thing to understand with this is that if students can successfully prove geometric conclusions, they have a deep understanding of the methods of geometry.

This method is important because as the students find the pattern, they are more engaged than traditional instruction that tells them the definition of bisector or midpoint. We need students that will own their learning and when they find patterns and make conclusions that is precisely what they are doing.

My advice at this point is to find at least three questions that will help your students find the patterns if they are unable to do it on their own. Examples include:

What do you notice about the measurements?
How are these two examples the same?
What do you notice?.

Method #3: Use An Exit Slip To Ensure Understanding

The main idea for every geometry teacher with this method is that ensure understanding by all students. While to us math teachers, this can seem like a more straightforward concept, this is where students can fly under the radar.

The most important thing with this method is that you can quickly identify the students that did not understand or for some other reason did not master the content.

This method will help geometry teachers because by planning a lesson and daily work where each student feels successful on the first day, your students will have much more success with the whole unit.

To put this into action, you should create a simple exit slip (or download mine) and use them to browse for misunderstandings quickly.

So there you have it! The 3 methods just waiting to let you wave bye-bye to all the traditional publishers and attain the dream for every geometry teacher to teach properties of bisected line segments!

Still not convinced? Well, if you want to teach properties of bisected line segments, this fantastic free lesson plan spells it all out for you! Check it out here https://highschoolmathteachers.com/optin-geometry-lessons/ 

How To Teach Trigonometric Ratios for Retention

If you want to teach trigonometric ratios, here are 3 quick methods every geometry teacher needs to know.


Method #1: Connect Trig Ratios To What The Students Already Understand

Here's the big thing geometry teachers must understand about this method: as the teacher, we must connect new information to prior learning for every lesson. Research shows that this will help our students retain the new knowledge that they are learning.

Image result for knowledge

 Geometry teachers need to understand this method means that for trig ratios to make sense and be remembered, we need our students not to see only a fraction, but for that fraction to tell them something about the angles. We need to ask questions that get students to connect the size of the angle in a right triangle with the fraction created.

This is important because when you ask questions like
*What happens to the sin ANGLE as that ANGLE gets larger? Smaller?
Helping the students to make the relationships and connections, will place this new information into their long term memory.

You should plan out at least ten questions you will ask your students so that they can connect this new learning with something they already understand and feel confident about from previous learning. Add these to your lesson plans.

Method #2: Show The Problem Done Three Different Ways

The key to this method is when students make connections on their own; they not only own their learning, they can retain it longer and recall it easier.

So, here's what this means: when you put the problems on the board. I like to use a minimum of three. You can, as the teacher and guide, start asking questions to lead them in the right direction of understanding.

For example, I would draw one large right triangle and label all the sides lengths and angle measurements. Next, I would begin writing true statements about the triangle while encouraging the students to find a pattern.

Image result for image of right triangle with measurements

 Geometry teachers need to know this method because as the students find the pattern, they are more engaged than traditional instruction that tells them the definitions of trig functions. And we all understand that engaged students are going to learn more and understand more.

To put this into action, you should find three more examples of right triangles where students can look at patterns and relationships.

Method #3: Start With Success

The bottom line for you with this is that this is the first lesson in a unit of lessons. Give them success today. Help them to feel confident. This is the most important lesson of the whole unit. Make sure every student walks out today feeling confident.

Image result for success

 The most important thing to get with this is students will be more engaged, more willing to complete work, and are much less likely to feel overwhelmed during the rest of the unit.

This method is important because by planning a lesson and daily work where each student feels successful on the first day, your students will have much more success with the whole unit.

So the thing for you to do here is to take a look at your students and make a realist plan for the first day working with trig functions.

It will amaze you how much faster you can get things done as a geometry teacher just by understanding these three teaching methods and how to use them to your advantage.

Hey, one more thing before I forget, if you're a geometry teacher serious about teaching success, this free lesson plan I just released "the free Introduction to Trigonometric Ratios Lesson Plan and Resources along with nine other lessons" helps you teach trigonometric ratios and more! Check it out: https://highschoolmathteachers.com/optin-geometry-lessons/ 

How to Teach Transformations with These Ready To Use Free Resources

I love teaching transformations. It's so fun to get visual with the students and allow them to see the world around them with new vocabulary and new appreciation.

Take a look around for examples

Transformations are everywhere. I like to ask the students to brainstorm as many reflections and then translation and finally, rotations that they can think of in 5 minutes or less.

Once they do this, we can share on the board all their different answers. As the other students start to answer they can add to their lists. After a bit, as their brains are still processing the new information they just learned from their classmates, I like to add have them add to their list again.

With the challenge of not listing something on somebody else's paper, their answers can get very creative. Usually, too creative, and someone will oversimplify the definition. This is a great time to emphasize common misconceptions and mistakes around transformations.

And if they need more ideas, this video shows so many. While I don't enjoy the music during this video, I really like that it is simply a slideshow of images we can use during class as discussion pieces.

Use technology to encourage curiosity


Let them play

I like to put a shape on the board that has undergone a translation to see if the students can recreate it within the program. We always start with easy ones, making sure students are understanding the formulas and patterns they are seeing.

If the class is doing well we can then start combining transformations to challenge their understanding.

And don't forget to ask...

"How do you make it look like this?"

"Is there another way to do that?"

Check for understanding

And in all my lessons, I like to have practice pages where the students can show me independent mastery of the skills we've learned in class. This along with exit slips can give me a clear picture who understands the materials.

If you'd like my complete lesson plan for free and transformation,

please enter your name and email below and will be sure you get it right away.

Just for fun!

Common Core Standards:

CCSS.MATH.CONTENT.HSG.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Mathematical Practices


Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Algebra Connections:

Translating Graphs Lesson Plan

The fastest path to teaching math vocabulary [angle definitions lesson plan]

The Fastest Path to Teaching Math Vocabulary

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Do your students often forget the definitions that they need to understand to be successful in geometry? If so, this blog post is for you. We will use angle definitions specifically to address this topic.

We will talk about ways to help your students remember the essential definitions that come up in Geometry. By using some simple strategies that utilize precisely how the brain works we can get this done quickly.

Assessing angle definitions prerequisites using familiar items

The first thing that we want to do is evaluate student understanding of the definitions. This informal assessment should be done early in the year when they come into our classrooms. 

This Geometry lesson plan is often done the first week in my class. It gives me a full understanding of where my students are when they come in so that I can help them in the best way possible.

One of the strategies that I love utilizing is connecting items that are common for the students to see on a daily basis outside of my classroom. 

In the connecting activity for angle definitions that I'm showing here, we have simple questions that ask them to understand the analog clock. Depending on your school and the clocks they use, they may even need to draw out the clock themselves. And while time telling is often learned in second grade, with modern technology many of our students have forgotten some of these basic concepts.

Using something like a clock to help our students understand angles will give us a trigger memory to help students recall this information in moments of forgetfulness. Now when they forget, we can ask them to remember the clock activity to bring the memory forward.

I also like to have my students engaged in the material in more than one way. And so for my kinetic learners, using a protractor or other tools of measurement is very helpful to cement the learning into their memories. 

So in this lesson, we use protractors to measure the angles and make relationships and connections that will help our students remember the information.

Why Repetition Matters

Repetition is a great way to build fluency and confidence with your students. 

During this practice page on angle definitions, I utilize this strategy right at the beginning of the year to help my students feel successful. Therefore I can build on success later in the unit.

The repetition in the practice page builds on itself and allows students to try new ways of solving similar problems. The brain is now making connections and solving problems. 

Even if they didn't feel like they remembered anything from middle school when students started this lesson today, they now have built confidence and use the vocabulary in their geometry work.

Incorporating student ownership of the angle definitions

By allowing time for students to make connections between their world and the content that we are trying to teach them, they gain more ownership of the material. 

This new information is no longer the material the teacher wants them to learn, but now is material that they own within their context.

This shift in ownership is the big payoff of making connections, and this is how we can help our students to retain more of the information each day.

If you want your students to learn and retain the geometry definitions above about angle definitions, then the fastest and easiest way to do that is by making connections. 

Use these strategies to help them with the prerequisites, give them repetition to build confidence, and allow them time to own the material themselves.

If you would like the complete lesson plan in this blog post, fill in your name and email below, and we will send it to you right away.

CCSS.MATH.CONTENT.HSG.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.


Easy and Engaging Activity to Introduce Constructions

In today’s post, I want to give you some tips and tricks to help you when teaching geometric constructions.

Safety First

When working with geometric constructions, you have a lot of students handling sharp instruments. The first thing I like to do is go over the safety instructions. I start by assessing the students’ fine motor skills. There are a couple of ways you can do that.

Drawing Circles

First, have the students draw circles on blank paper. I like to have plenty of paper available for this. I use recycled paper that I pull from Xeroxes that weren’t able to be used. As I walk around the room, I am often surprised at which students have trouble drawing a simple circle using the compass.

I want them to be able to move their compass and continue to draw intersecting circles. If they can do this well, they will end up with 6 circles that all intersect and that meet back up where they began. If they are able to do this easily, I know their fine motor skills are developed.

Assign Partners

I know it is exhausting to continue to make the rounds of the classroom trying to give constant feedback to each student. Assigning partners allows the students to give feedback to each other as they work through the assignments.

I used to go home so tired when I first began introducing constructions from trying to answer all the questions. Once I began incorporating partners, or even group work, I started noticing a couple of things.

First, one student might be great at drawing the circles while the other had trouble. The student who was great at it could help the one who wasn’t which saved my voice and my energy.

Secondly, the students learned the vocabulary more quickly when I partnered them up. Instead of saying “ruler”, they were using “straight edge.” Words like circumference, diameter, radius, bisector became more familiar to them as they used them to help each other.

Prepare for Accidents

It is important for you to look around your classroom and spot things that could be problematic. Are the desks too close together for safety? Are the kids pressing too hard and scratching the desks?

I once taught constructions in an old science lab classroom, so the kids were standing. It was an ideal set-up because the kids had more room to work. I wasn’t worried about them scratching the desks with the sharp instruments and they had the ability to lean forward or to the side as they were working.

If you don’t have a space with standing desks, there are a couple of other ways to accomplish this. First, you could spread the desks further apart to give the students more room to work.

Secondly, you could ask your custodian for cardboard to cover the desks to avoid scratching them. The backs of old notebooks work too.

I always remind you to tell your students things that build their confidence, so be sure to tell them that mistakes will be made as they learn this concept and it is okay.

Online Tools

There are many tools online to help with the teaching of geometric constructions. I have gone through a ton of them; some are really good and some are not. But there are a few that I think are outstanding and I want to share those with you.

  • https://www.geogebra.org/m/Xfayrrj8  You can have your students go through this tutorial to learn how to do constructions online. This is my favorite one for teaching that.
  • https://www.mathopenref.com/tocs/constructionstoc.html This is the link I give to students when I give them constructions to do on paper as homework. This website has the constructions directions written out in a step-by-step manner so they can move along one step at a time. It can also run on a loot for your classroom.
  • https://www.khanacademy.org/math/geometry-home/geometric-constructions If your students just need to be talked through the construction process again, there are plenty of online helps for that on Youtube. However, I know that Youtube is blocked in many schools, so this one from Khan Academy is very organized and helpful. This link is to the page that contains all of their videos on this topic.

I hope all of this helps you to teach constructions this year and greatly lowers your and your students’ frustrations. As always, please tell me in the comments how I can be of help to you!



Unit 1 - Geometric Transformations

During this unit, students will begin to develop detailed definitions. The work they will do will help them to explain the geometry in the world around them, communicating to solve problems. Geometric Transformations can be found in many careers, and I often take the time to point them out as often as possible.

  • Week 1 – Definitions (G.CO.A.1 & 2)
  • Week 2 – Rotations, Reflections, and Translations (G.CO.A.3, 4, & 5)
  • Week 3 – Congruence (G.CO.B.6)

Common Core State Standards for Mathematical Content

Congruence — G-CO
A. Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on
The undefined notions of point, line, the distance along a line, and distance around a circular arc.
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not (e.g., translation
versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections
that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using,
e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that
will carry a given figure onto another.
B. Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given
rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.

Common Core State Standards for Mathematical Practice
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.


Unit 1 - Geometric Transformations