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.
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.
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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).
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.
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 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 the 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.
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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.
In today’s post, I want to give you some tips and tricks to help you when teaching geometric constructions.
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.
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.
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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.
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.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!
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.
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.
High school geometry students are expected to know far more than they used to. Curriculum standards that used to be taught at the high school level have been moved to advanced 6th and 7th grade.
Your students are “expected” to know many geometry terms when they enter your class, which is why I recommend starting the year with the Visual Dictionary project.
Creating the Visual Dictionary
The goal of this activity is to understand what the students retained and what would I have to re-address.
Your students can use tablets or computers for this project, but technology isn’t required.
The students are placed in groups of two, so they can learn to start working in groups at the beginning of the year.
A real-world example of an "acute" angle.
They have a list of about 25 words. For each word, they have to find a mathematics definition and a real world picture of the word. One teacher I know had one student sign out a camera and take the pictures, which turned into a cool final result.
Give the students about a day and a half in class and five days overall. That amount of time gives the students the opportunity to produce quality work and to really be creative.
Once the projects are complete, they upload the pictures to a tool like Glogster, Prezi, or Google presentation. Then you can refer to the projects the rest of the year when you are working on specific standards and vocabulary.
The following week, you can share all of the pictures with the class so all of the students can see what other groups found.
Vocabulary is so important for students to know, use, and understand, the two in-class days for this project are definitely worth it.
This past Christmas I received a π cake mold from my oldest daughter. She knew I would love it and she was right! (I added a link to it below. It works great!)
I celebrate the pi day at school and home every March 14, so imagine the fun I have put this day together.
The following video begs the question, how many dominoes is that? Spoiler alert: the second (needed information) and third act (the answer) are at the end. This would be a great way to begin class.
Pi Day In Math Class
Is there any better day in the life of a math geek? Ok, that might be a little bit of an exaggeration, but it is a very fun day to be a math teacher.
In my class, we always eat pie on pi day. The kids bring it in, and I supply the cups, juice and paper goods. We always have a great time discovering the ratio of pi and enjoying the wonder of the randomness. The students always seem to gather interest in the wonder of numbers. Isn't that what it is all about?
The class is set up as stations with different activities depending on the mathematical level of my students. When they walk in, I am playing Pi Songs. They circulate through stations ranging from creating the music pie to measuring and finding the common ratio of pi, to pinning the tongue on Einstein. Don't forget that he shares his birthday with pi! And of course, the end of class has a memorization of the pi contest.
I always want to try something new with pi day and therefore have made necklaces by assigning each digit a different color bead, found our birthdays within pi and we have sung pi day songs. The next day, when the sugar buzz has worn off we discuss what we discovered and observed, the kids always amaze me at how much they take away from a day that would look like nothing more than high school kids at recess. I hope they have great memories of Pi day and a deeper understanding of pi.
There are some great resources online. Some of my favorites are below
Youtube.com there are some great projects done by students, perhaps have your students do the same.
Do you have a favorite π day activity that you do with your students? Please add it to the comments below! Thanks!
Interesting find while searching online
To type the π symbol, you can use your number pad on your keyboard. Hold down the alt key and type 227 on your number pad. Let go, and you have a pi symbol. Amazing. For more helpful tips on inserting math and science symbols into your documents, you can see my list here.