​The Vengeance of Vape
It’s 11:47am at Thomas Jefferson High School. The bell rings. 457 high school students slowly gather their backpacks, glance briefly at their phones, and throw away leftover chip wrappers.


Tanya, a high school freshman, begins the trek to her first afternoon class, Algebra II. As she climbs the second floor stairs, she mentally walks through the standard mental checklist:  phone, wallet, keys, vape. She immediately realizes that she left her small, zipped pencil bag on the metal chair against the wall in the lunchroom. Her recently purchased grape flavored USB-shaped vape sits inside.


Turning around with a panicked sharpness, she faces a sea of high school humanity. She dodges and weaves through backpacks and odd stares, desperate for a visual of her recent purchase. As she rounds the curve down the stairs, her peripheral view catches sight of the zipped canvas bag. A momentary sigh of relief. Desperately, she ensures the vape’s presence within the bag. She exhales.


In nearly every middle school, high school, and college campus, scenes like Tanya’s are on repeat. Virtually every demographic is facing a crisis that resembles the tobacco usage peak of the 1960s [1]. At that time, over 40% of U.S. adults engaged in cigarette smoking [2]. Think about that! Nearly half of all American adults smoked cigarettes just a few decades ago. Even for non-smokers, the inhalation of smoke particles was nearly unavoidable.

​Smoking is…Familiar?
Oddly enough, many may hold the scent or image of smoking with a bit of nostalgia. Actors/Actresses often smoked as a sign of class. Teenagers, even young adolescents, used cigarettes to rebel from expectations. Many restaurants maintained a smoking and non-smoking section well in the 1990s. It was ingrained within American culture. Even as data of smoking-related health risks emerged, many Americans were reluctant to accept the statistical reality of their beloved tobacco.

Remember when doctors recommended cigarette brands?!

​Even throughout the 1970s, nearly 30% of American adults still denied that smoking was a cause of lung cancer [3].

Slow Yet Steady Decline
As societal norms and knowledge increased, so did Americans' acceptance of cigarette dangers. Multiple Gallup polls throughout the 1990s repeatedly showed that approximately 95% of Americans finally believed that smoking was harmful to health [4]. Between 1965 and 1991, there was a significant decline in cigarette usage, with a 39 percent decrease from the 1960’s peak. By 2005, 20.9 percent of the American population, approximately 45.1 million people, smoked regularly. The number had dropped by 50 percent [5]. By 2021, only 11.5% (nearly 12 of every 100 adults) reported regular engagement with cigarettes [6]. *If you are seeking a more thorough breakdown of smoking consumption by sex, age, race, education, household income, U.S. Census region, marital status, etc, this Center for Disease Control resource provides such data.

A Flip in the Opposite Direction
Outside of this promising data, however, lies an alarming trend that is quite contrary to the statistical decline of the last 60 years. In 2007, the smoking-prevention movement faced a mountain-sized curveball.

Enter, the e-cigarette.

The e-cigarette, commonly known as the vape, entered the U.S. market in 2007 and shortly after began to dominate the charts as the top tobacco product used by American teens today. Although teens have long engaged with tobacco in some capacity, the alarm today stems from the staggering increase within a short period of time.​

Between 2011-2015, there was a 900% climb in e-cigarette usage. Contributing to this massive spike was the misleading yet engaging marketing strategy of many e-cigarette companies, JUUL being one of the leading forces. Commercials were often filled with bright colors, fashion-forward young adults, and upbeat music. This JUUL commercial highlights these tactics. In retrospect, it is clear that the marketing strategy was, unfortunately, a complete success.


More recently, in the 2022 National Youth Tobacco Survey, an estimated 3.08 million middle and high school students reported using tobacco, with e-cigarettes being the most commonly used product[7]. In some schools, administrative leaders have even installed smoke detectors in the restrooms and intensified consequences as deterrence. For many educators and parents, this reality is not shocking, but rather a lived experience unfolding before their eyes.  
​

What is Really Inside?
What is inside an e-cigarette? For many Americans, the pervasive misunderstanding is that vaping is significantly safer than a traditional cigarette. Besides, weren’t they created as a scaffold for nicotine-dependent cigarette smokers? Although it is true that some e-cigarettes do not contain nicotine, they all contain high-risk chemicals whose health implications have only partially been realized. Consider the following numbers from the National Library of Medicine [10].

• “In 54 samples of e-cigarette liquids, the average nicotine content was 11 mg/ml
• Major compounds of e-cigarette liquids include glycerol (average 37 g/100 g), propylene glycol (average 57 g/100 g) and ethylene glycol (average 10 g/100 g).
• The average exposure for daily users was estimated as 0.38 mg/kg bw/day for nicotine, 8.9 mg/kg bw/day for glycerol, 14.5 mg/kg bw/day for 1,2-propanediol, 2.1 mg/kg bw/day for ethylene glycol, and below 0.2 mg/kg bw/day for the other compounds.”

In another standout publication from the 2016 Surgeon General, there were further explanations of e-cigarette emissions. Secondhand emissions contain "nicotine, ultrafine particles, flavorings such as diacetyl, a chemical linked to serious lung disease, volatile organic compounds such as benzene, which is found in car exhaust, and heavy metals, such as nickel, tin, and lead” [11]. A subsequent publication in 2018 by the Surgeon General noted the immense nicotine that is present in JUUL e-cigarettes, a popular brand on the market.  “All JUUL e-cigarettes have a high level of nicotine.

​A typical JUUL cartridge, or “pod,” contains about as much nicotine as a pack of 20 regular cigarettes [12].

​Are These Really Harmful?
Many of these compounds are known to have significantly negative effects on the body, especially when experienced in high dosages. For example, both glycerol and propylene glycol are capable of altering the gene expression that controls the circadian rhythm of the lungs [13]. Beyond these primary ingredients, the additional compounds are often unknown, mislabeled, or unregulated.


One former vape user, Jay Jenkins, found himself in a coma after casually trying what he thought was a simple CBD-infused vape. Rather than the intended calming effects of CBD, Jay learned [two puffs in] that the vape oil was spiked with a dangerous street drug [14]. Inaccurately-labeled vape devices pose some of the greatest risk, as their components can be a complete mystery. Because federal regulators have been in contentious legal battles with e-cigarette companies since their inception, regulation for these products is ongoing and state-dependent.


Vaping does not only lead to poor organ function. It can cause hospitalization and death. As of February 2020, 2,807 hospitalized EVALI (E-Cigarette, Vaping Product, Use Associated Lung Injury) or death cases were reported within the United States and U.S. territories.

As a Parent, What Can I Do?
Having difficult conversations can be intimidating. Parents often feel pressured to say all the right words. Many are also afraid of the response. However, although vape use is a newer, dangerous trend, it has not been the only dangerous trend or habit teens have tried. Parents of teens in the 1960s can speak to this

In preparation, consider reading this resource from the U.S. Surgeon General. It provides potential questions and reasonable responses a parent can offer.

Most importantly, rather than put off the conversation, simply begin it. Beforehand, you can prepare yourself by maintaining calm body language and a steady voice. This will help open up conversation, rather than shut it down.  

If you are homeschooling, or you'd like printable materials to help support your conversation, check out the doodle notes set below in the teacher tips as well.

As a Teacher, What Can I Do?
Many of the same approaches relevant to a parent are relevant to a teacher, especially regarding calm body language. Keep in mind that there are added boundaries between teacher and student. Begin thinking of meaningful ways you can engage your students on this issue. Yes, even if this is not directly listed within your curriculum. It’s critical! I've created some doodle notes on the dangers of vaping to help your students to analyze the reality of this issue.

​It’s time to step aside and use just a bit of lesson time to address this. It’s worth taking a day to nip it in the bud and save some of our kids from these irreversible damages.

This set of materials is designed to help you approach the topic in your math class, homeroom, science course, or health class.

The goal is twofold: show students who are already vaping what the hidden dangers are, and at the same time prevent other students from trying their first e-cigarette in the future.

​Preparation for the Lesson

• Build up the activity ahead of time. To do this, you can use the bonus slides (which include some of the colored square infographics and stats you've been seeing this article) that come along with the printables to spark curiosity. These can be posted on the classroom door or displayed on a bulletin board.
• If possible, invite a former smoker or vape-user, perhaps a young person, who can speak to the impact vape or nicotine addiction has had on their life.      
• Consider posting one or more of the resources available on the Tobacco Education Resource Library to your class online platform (Canvas, Google Classroom, Blackboard, etc).
• Remind students that the resources are available on more than one occasion. You never know when a student may decide to utilize one of the resources.
• Choose which pages to print. You can use any combination of the pages that are included, depending on your content area and goals. For example, if you teach math and science, you can use Vaping by the Numbers and Vaping Ingredients: What’s Inside during your lesson, and then save the other two pages for if you have extra time or if you notice more student interest or a need for it.  If you teach health, try The Story of Vaping and The Effects of Vaping pages first.  If you are doing this in a homeroom class, or for all of your students because your school is having a problem with vaping, use all four pages across a 2-3 day lesson on vaping.

Different Kids Care About Different Things!  Consider this...
An important thing to remember is that different students will be struck by different aspects of this.  Some vague potential health "danger" may not seem important at all to a particular student, but he may be more taken aback by learning about the financial side of the issue instead.

Or, other teen students who have a hard time visualizing far-off lung problems in the future might not seem to be impacted by most of the health risks, but you might be surprised to realize they care about their teeth and the potential of burns or ulcerations in their mouths in the short term.

Other students will become indignant at the thought of being harmed for someone else’s profit, and motivated to quit or to avoid vaping in the future once they see the profits that are at play.  Other students may be more motivated by learning about the physical harms instead. Different teens have different motivating factors, and find different aspects to be more eye-opening, so certain pages will work best for specific students or groups. 

No matter what page or content seems to hit home for certain teens, these materials cover it all, so they are a ​great tool for helping them become aware.

​And Finally…
Remember that your connection and care are the most powerful tools you have! Your child-parent or student-teacher relationship opens the door to meaningful conversation and a teen feeling truly supported.


One way to empower yourself is to become a ready resource. Review and deeply digest some of the key research points in this article and others. Become confident with the names of available resources so you can share them on demand.


You have more power than you realize! Even one adult can be the difference between an addicted teen and a healthy, breathing, nicotine-free young adult. Be that one.

Details on What's Included in the Lesson Set:

The Vaping by the Numbers worksheet is a look at the stats on teen smoking use and vaping over the years. It’s great for math classes to use when practicing graphing and analyzing data while learning about an important topic that is relevant to teens’ lives. This page goes well alongside the Story of Vaping page so that students can see how the events in the comic-style timeline impacted the rates of youth smoking and why after cigarette use dropped, e-cigarettes came in and took over.

The Story of Vaping page gives insight into the marketing behind smoking, the profits and agendas at play, and how some doctors will even advertise and promote something dangerous if they are paid to do so. This page is great for helping students become aware that these entities do not always have their best interest at heart. Its fun comic format makes a heavy topic more approachable for students. This sheet is helpful for health classes, social studies classes, or homerooms to use and discuss together. Some students will become indignant at the thought of being harmed for someone else’s profit, and motivated to quit or to avoid vaping in the future, so this page can be a great tool for helping them become aware. Other students may be more motivated by learning about the physical harms instead. Different teens have different motivating factors, and find different aspects to be more eye-opening, so certain pages will work best for specific students or groups.

The Vaping Ingredients: What’s Inside? sheet looks at some of the chemicals that are present in e-cigarettes. It also includes a quick peek at the dangers of secondhand smoke. You can use this page in science classes or health classes, or use them with your homeroom group. Since most teens think that only flavoring is inside, this can be very eye opening!

The Effects of Vaping page is an important one to include for all groups of students if possible. It shows students the impact of e-cigarettes on their bodies. This will show them clearly that this is not a harmless form of smoking in any sense. This page will hit home for many students who were not aware of these dangers, or who have a hard time visualizing far-off lung problems in the future, but may care about their teeth and the potential of burns or ulcerations in their mouths in the short term. (Gotta protect those beautiful smiles!)

The PDF downloads are available here: Dangers of Vaping Doodle Notes for Teens

If you have had success in your own classroom (or home) with helping teens quit or avoid vaping, please share any tips in the comments!
​

References:
​1
https://nap.nationalacademies.org/read/11795/chapter/4
2
https://nap.nationalacademies.org/read/11795/chapter/4
3
National Center for Chronic Disease Prevention and Health Promotion (US) Office on Smoking and Health. The Health Consequences of Smoking—50 Years of Progress: A Report of the Surgeon General. Atlanta (GA): Centers for Disease Control and Prevention (US); 2014. 2, Fifty Years of Change 1964–2014. Available from: https://www.ncbi.nlm.nih.gov/books/NBK294310/
4
National Center for Chronic Disease Prevention and Health Promotion (US) Office on Smoking and Health. The Health Consequences of Smoking—50 Years of Progress: A Report of the Surgeon General. Atlanta (GA): Centers for Disease Control and Prevention (US); 2014. 2, Fifty Years of Change 1964–2014. Available from: https://www.ncbi.nlm.nih.gov/books/NBK294310/
5
https://nap.nationalacademies.org/read/11795/chapter/4#45
6
https://www.cdc.gov/tobacco/data_statistics/fact_sheets/adult_data/cig_smoking/index.htm
7
https://www.cdc.gov/tobacco/basic_information/e-cigarettes/surgeon-general-advisory/index.html
8
https://www.cdc.gov/tobacco/data_statistics/fact_sheets/youth_data/tobacco_use/index.htm#:~:text=Tobacco%20product%20use%20is%20started%20and%20established%20primarily%20during%20adolescence.&text=Nearly%209%20out%20of%2010,try%20smoking%20by%20age%2026.&text=Each%20day%20in%20the%20U.S.,youth%20start%20smoking%20every%20day.
9
https://www.fda.gov/tobacco-products/youth-and-tobacco/results-annual-national-youth-tobacco-survey
10
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4304610/
11
https://www.lung.org/quit-smoking/e-cigarettes-vaping/impact-of-e-cigarettes-on-lung
12
https://www.cdc.gov/tobacco/basic_information/e-cigarettes/surgeon-general-advisory/index.html
13
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8787618/
14
https://www.cnbc.com/2019/09/16/investigation-finds-illegal-synthetic-marijuana-in-vape-and-edible-products-sold-as-cbd.html
15
https://www.cdc.gov/tobacco/basic_information/e-cigarettes/severe-lung-disease.html

​

* (Note - Don't miss the free download at the end of this post.)

In high school Geometry, we need students to not only be able to use theorems properly in proof writing and problem solving, we need them to also understand them on a deeper level and be able to reproduce them as needed. We also want them to be able to recall the theorems easily so they know which one they have to use. The way to do this is not just through lecture, then memorization!

There are a lot of circle theorems, and to your students, it probably feels like they all just blur together unless you really strategize the way you teach them.

To introduce new theorems in a way that students can integrate them into their conceptual understanding of how Geometry works, we can use a few strategies. To increase both deep understanding and retention (memory) of theorems, you'll need to:

1.  Have students discover the theorems themselves. They'll need to investigate, explore, and actually see the theorem come to life through an inquiry process. Whenever you can use an investigation to allow students to discover properties themselves, understanding of the deeper concept behind a mathematical property is increased. This way, students are able to reproduce the rules, formulas, and procedures for themselves without memorizing because they understand where the rules came from. (A side bonus of using inquiry is that students get a boost in their problem-solving skills.  They gain resilience to stick with a challenge and learn to think it through for themselves and develop stronger critical thinking abilities.) For more information on how and why we want to use a discovery lesson as an introduction, check out my inquiry learning page.

2.  Engage both brain hemispheres and activate the benefits of dual coding theory. By blending graphic (visual) information with auditory information / words, you'll help students make neural connections that allow them to convert the lesson into long-term memory. You'll see how we do this below, but be sure that you check out my doodle notes hub if you are new to using creative visual note-taking methods in math class.

3.  Use color-coding. Colors and color combinations can have an effect on attention and memory. Neuroscientists have found that color is one of the first things to register in our brains, and has a pretty strong impact on learning. Geometry is especially conducive to color coding. Get students in the habit of always having multiple colored pencils out on their desks, and have them always use different colors to mark up diagrams any chance they get!

*(First, a quick note: All the specific materials shown in the images are here, so you can get them all in one place and then grab the free download at the end of this post to use in your classroom.)

Review definitions if needed before starting.

If your students are not confident with naming the parts of a circle, you'll want to go over the vocabulary before throwing them into working with the theorems. It's difficult to understand new theorems if you don't have a good handle on the terms first. Build familiarity with a quick review of the parts of a circle.

Doodle notes are a great way to do this in a way that will solidify the meanings of the vocabulary terms in long-term memory. For example, if you have students do their notes as shown below, they'll benefit from the hand-to-mind connection. They'll recall how they filled in the sector with the word "sector" in large letters, and how they wrote or doodled the words "arc" and "circumference" in a curved way and added doodles around the circle to extend it around the entire circumference. They'll remember using one color for all the straight line segments, hand lettering for areas within regions, and another color for the curves. The act of coloring, doodling, sketching, color-coding, or hand lettering their vocabulary terms helps them to convert the words to long-term memory.

If your students already are strong in these definitions, you can skip this and move ahead to the investigation.

Kick it all off with an investigation lesson

Before you do any lecture or notes to introduce circle theorems, allow your class to explore them through a discovery lesson. By developing the rules for the properties themselves, they will actually see how and why the theorems work.

They'll understand where these theorems came from when they discover them for themselves.

This type of activity works great in pairs. Have students work with a partner to go through the investigation lesson. They'll discover 4 of the circle theorems themselves, using a protractor and straightedge.

For example, have your students draw 4 or 5 circles of all different sizes using a compass. Then, direct them to place two points on each circle. Have them draw two different inscribed angles that intercept the same arc. They'll take time to do this for all their circles on their own in pairs, record the measures, and start to make their own observations. Think of it like a science lab. They will look for patterns and see if they can draw any conclusions.

Of course, their results may not be perfect. One angle may be 57 degrees and one may be 58 because of variation in the perfection of their lines and their protractor skills and estimations. But by doing it enough times throughout the exploration, they'll start to see some similarities and begin to guess at the property at play.

Once students have started having those "aha" moments as they discover the first theorems and are able to start writing rules for what they've observed, you'll eventually want to bring them together as a whole class for some direct instruction.

Allow the pairs to share what they discovered. Go over the theorems together. Compare how different teams wrote up the rules. Make sure everyone is on the same page and has correctly written up their discoveries.

Then, you can go over the remaining theorems. You can give the rest to them as usual. Students should take notes and ensure they've got the idea. You can provide them with a reference guide as well if you'd like. That can be handy for them as they work through proofs and practice problems later on too.  I have one included in my circle theorems kit if you'd like to use the materials shown here.

Practice Using the Theorems Before Leading into Proofs

Once students are familiar with the theorems, let them try some application problems. Your textbook probably has some basic, boring ones that you can begin with. Do a few in class together, and assign a couple for homework.

Then, let your students level up their practice the next day with some fun challenge puzzles.

This set has a few different circle theorem puzzles at varying difficulty levels. You can let your class try a couple with partners, then let them take on a more challenging one if they are ready for it. Be careful - puzzle #1 is very tricky. Choose which ones to use wisely ;)

Let students see if they really get it

Now it's time to test their knowlege with some critical thinking questions. I love using "always true, sometimes true, or never true" questions to see if students really understand the concept well.

This type of questioning forces them to test different cases and try to find examples and counterexamples. It's great for building logical reasoning skills, and for seeing where there may be gaps in knowledge.

Use always, sometimes, never challenges with partners if you can. They get some really good "math talk" going in the classroom, and then you can go over them as a whole group and check work while addressing the ones that kids got stuck on.

The way I designed mine allows for a quick check as you walk around the room. If they've got it, the colors form a secret pattern that you can use to identify any errors at a glance. (But it's not a predictable enough pattern for kids to guess. Very sneaky...)

Sum up and review before moving on to proofs

Some people like to use their doodle notes as an introduction to a new concept, but a lot of teachers also like them as review. I like blending some of each. You can choose what is best depending on the lesson.  And in this case, since the introduction to the lesson was an inquiry activity, the doodle notes make a great sum-up activity to review all the properties / theorems.

The circle theorems doodle notes make a great reference guide for students. The creativite aspect helps activate all those magical neural pathways, but also, since they can visualize the layout later on in their minds, they can recall the theorems more easily during a test. This is one of the many benefits of doodle notes. Interacting with the visual notes automatically will activate the brain pathways that lead to stronger retention and also take advantage of the powerful brain hemisphere crossover. 

Proof Time!

Finally the best part - write some proofs with the theorems.

The doodle notes you just finished are perfect for pulling back out to have on hand to peek at during proof writing. This is why I like having that step come right before proofs for this unit. They now will have their theorems organized into three categories on that page:

• circle theorems with chords and radii
• circle theorems with inscribed angles
• circle theorems with tangents

Start with a couple of basic proofs using the theorems with chords and radii. These are a nice beginning point. Then, lead into a couple proofs with inscribed angles next.

Then, you can have them try a few on their own before moving on to the theorems with tangents.

My big proofs pack has a lot of proofs you can use for this if you need some, or you can just use whatever you've got in your textbook. It's great to have a wide range of proofs that all follow a nice consistent format. Set your students up for success by having the practice proofs organized the same way as the ones on quizzes like those ones are.

When you get to the point that you are ready to go more in-depth with the theorems involving tangents, you can use my free diagrams here.

This is a nice way to spice it up. You can use the tangent diagrams in so many ways. I like printing them on label paper so they peel off in perfect squares and kids can stick them right into their notebooks. The free sticker templates make that easy. Click on the button below to download them. You'll just print the pages on square label sheets. (Directions and specifics are included in that free download file.)

That way, everyone instantly has all the diagrams right in their notes without having to draw them by hand. It saves a lot of time, and they can still mark them up and use color coding.

Encourage your students to use color coding to match up lines of proof with colors in the diagram. Their markings will not only help them to go through the logic and steps and keep everything straight, but they'll also help you with grading.

As you review the lines of their proof, it's much easier to see where a student went wrong if you can follow their thought process by following their color coding.

If you don't have label paper, don't worry. You can still use these tangent diagrams and have options. You can either...

1. Print these free tangent diagrams on full-page sticker paper, and students just will have to cut across to divide up the squares before peeling the stickers.

​...or...
​
2. Print the free diagrams on regular paper. Kids will just have to use a glue stick. It still is a really efficient way for them to get the diagrams right into their notes, just not quite as quick and easy as the label method.

​You can also use these diagrams just for yourself to easily create worksheets, practice problems, or even quizzes.  Just stick a few on the page, write your "given" information alongside it, and photocopy! It makes it super simple to build a quick handout.

Have fun teaching circle theorems! For an easy to use pack of all the materials shown here, you can grab the Circle Theorems Bundle here.

Hopefully these creative methods and activities will not only help you spice it up and get some variety, but also will build deeper understanding and stronger retention for your Geometry students.

Leading into proof writing is my favorite part of teaching a Geometry course.  I really love developing the logic and process for the students.  However, I have noticed that there are a few key parts of the process that seem to be missing from the Geometry textbooks. 

I started developing a different approach, and it has made a world of difference!  

The Old Sequence for Introducing Geometry Proofs:
Usually, the textbook teaches the beginning definitions and postulates, but before starting geometry proofs, they do some basic algebra proofs. Most curriculum starts with algebra proofs so that students can just practice justifying each step.  They have students prove the solution to the equation (like show that x = 3). 

That's fine. It's good to have kids get the idea of "proving" something by first explaining their steps when they solve a basic algebra equation that they already know how to do. 

But then, the books move on to the first geometry proofs. And I noticed that the real hangup for students comes up when suddenly they have to combine two previous lines in a proof (using substitution or the transitive property).  

They get completely stuck, because that is totally different from what they just had to do in the algebraic "solving an equation" type of proof. It does not seem like the same thing at all, and they get very overwhelmed really quickly.

The standard algebraic proofs they had used from the book to lead into the concept of a two column proof just were not sufficient to prevent the overwhelm once the more difficult proofs showed up. Solving an equation by isolating the variable is not at all the same as the process they will be using to do a Geometry proof.

A New In-Between Step:
So, I added a new and different stage with a completely different type of algebra proof to fill in the gap that my students were really struggling with.  First, just like before, we worked with the typical algebra proofs that are in the book (where students just justify their steps when working with an equation), but then after that, I added a new type of proof I made up myself.

I led them into a set of algebraic proofs that require the transitive property and substitution.  This way, they can get the hang of the part that really trips them up while it is the ONLY new step! We did these for a while until the kids were comfortable with using these properties to combine equations from two previous lines.

My "in-between" proofs for transitioning include multiple given equations (like "Given that g = 2h, g + h = k, and k = m, Prove that m = 3h.") Instead of just solving an equation, they have a different goal that they have to prove. Their result, and the justifications that they have to use are a little more complex.

This way, the students can get accustomed to using those tricky combinations of previous lines BEFORE any geometry diagrams are introduced.  They are eased into the first Geometry proofs more smoothly.  This extra step helped so much.  After seeing the difference after I added these, I'll never start Segment and Angle Addition Postulates again until after we've practiced substitution and the transitive property with these special new algebra proofs.

Here are some examples of what I am talking about. The books do not have these, so I had to write them up myself.
(I am sharing some that you can download and print below too, so you can use them for your own students. There are also even more in my full proof unit.)

Do you see how instead of just showing the steps of solving an equation, they have to figure out how to combine line 1 and line 2 to make a brand new line with the proof statement they create in line 3? 

You can start with ones like this (above), where the statements are already provided and they just have to fill in the second column, and then as usual, after that you will want to lead into some where both columns are blank and they have to come up with the entire thing themselves.

Here is a close-up look at another example of this new type of proof, that works as a bridge between the standard algebra proofs and the first geometry proofs.

The way I designed the original given info and the equation that they have to get to as their final result requires students to use substitution and the transitive property to combine their previous statements in different ways.

Practicing proofs like this and getting the hang of it made the students so much more comfortable when we did get to the geometry proofs. It saved them from all the usual stress of feeling lost at the beginning of proof writing!

Here is another example:

Sequencing the Proof Unit with this New Transitional Proof:
After finishing my logic unit (conditional statements, deductive reasoning, etc.), I start (as most courses do) with the properties of equality and congruence.  I also make sure that everyone is confident with the definitions that we will be using (see the reference list in the download below).  I introduce a few basic postulates that will be used as justifications.  I spend time practicing with some fun worksheets for properties of equality and congruence and the basic postulates.

Then, we start two-column proof writing. The usual Algebra proofs are fine as a beginning point, and then with my new type of algebra proofs, I have students justify basic Algebraic steps using Substitution and the Transitive Property to get the hang of it before ever introducing a diagram-based proof. 

The flowchart (below) that I use to sequence and organize my proof unit is part of the free PDF you can get below. The PDF also includes templates for writing proofs and a list of properties, postulates, etc. that I use as a starting point for the justifications students may use.

The extra level of algebra proofs that incorporate substitutions and the transitive property are the key to this approach.

This addition made such a difference!  By the time the Geometry proofs with diagrams were introduced, the class already knew how to set up a two-column proof, develop new equations from the given statements, and combine two previous equations into a new one. 

Another Piece Not Emphasized in Textbooks:​
Here's the other piece the textbooks did not focus on very well - (This drives me nuts).  Please make sure to emphasize this -- There is a difference between EQUAL and CONGRUENT.  This is a mistake I come across all the time when grading proofs. It may be the #1 most common mistake that students make, and they make it in all different ways in their proof writing.

​I make sure to spend a lot of time emphasizing this before I let my students start writing their own proofs.  I make a big fuss over it.  I require that converting between the statements is an entire step in the proof, and subtract points if I see something like "<2 = <4" or "<1 + <2 = <3".

(The slides shown are from my full proof unit.)

When It's Finally Time for Geometry Diagrams:
In the sequence above, you'll see that I like to do segment and angle addition postulate as the first geometry-based two column proofs. Real-world examples help students to understand these concepts before they try writing proofs using the postulates. 

When you introduce the proof writing process itself, it's nice to have guided notes that students can keep as a reference. Interactive doodle notes are perfect for this because they blend words and imagery to help students retain the information and visualize it as a clear process in their minds.

The doodle notes I created for teaching Geometry Proof Writing are available here if you'd like to use these to accompany your proof lecture.

This way, they can solidify the concept, build mental connections by blending visual and text info, and then have a reference that they can pull out when they are studying or when they are trying a practice proof and get stuck.

After practicing with those transitional proofs I added, my students finally did not have a problem easing into the next level of proofs with Angle Addition Postulate and Segment Addition Postulate.  This made them ready for what used to be such a huge leap with other classes.  We avoided all the struggle that usually comes with introducing proofs.  They did not feel nearly as lost.

​When we finally got into the good stuff, after watching me demonstrate a few proofs, a lot of kids would say things like...

“Ok I kinda get what you are doing, and each step makes sense, but you are just making it look easy.  It seems like you're just making it up."

or

"I understand some of where it is coming from, but there is just NO WAY I could come up with these steps myself and get from the beginning to the end on my own.”

Posters as a Guide for those "I'm Stuck" Moments:
To help them organize the procedure and get "un-stuck" when they were unsure how to progress to the next step, I developed a series of steps for them.  Some kids really depended on this, and some thought that it didn’t help much.  For students who do need that structure, though, this chart can be their friend on their desk at all times for the entire month as we progress through the unit.

​Another group of students seemed to need a reference list of what kinds of things can be used as justifications.  Proofs are so different from anything that has been done before in their math classes.  Each student seems to get stuck on a different part of the process.  I found that having a reference sheet helped them a lot.

They can add to this sheet as they learn more postulates and theorems later on until they no longer need a list.

DOWNLOAD these POSTERS for FREE:   Printable versions of these two pages are included in an email that I send out to subscribers.  If you would like to have the 8.5x11 posters for your students, subscribe to the Math Giraffe email list, and they will be sent straight to your inbox as part of the toolkit for teaching math!

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(An additional free download is available below, so scroll down for the printable download of the special algebra proof samples too!)

Next, we move on to proofs with special angle pairs (supplementary angles, vertical angles, etc.) From there, things tend to smooth out. It gets easier to introduce each new type of proof, because all that is changing is the theorems that we use as we lead into proofs with triangles. 

The format of stepping through a puzzle and getting to the end of the proof stays the same. It's best to keep the structure well-organized and the same throughout. I keep a template handy with my 2 columns in the same format, so I can just fill in the blank space at the top and copy it for any new type of proof. The students can feel the familiarity of the columns and structure all throughout the course.

Try it - Free Download
You can use this sample set to try these algebra proofs in your own classroom.  You'll love the way this additional lesson leads your students into proof writing more smoothly.  This PDF includes a few examples that are half-sheet size.  They work really well as warm-ups.

If you like this format and would like my full Proof Unit, it's available here:
FULL Geometry Proof Unit: Presentation & Printables

Click the images for more information.