“I can’t wait to grow up!” 

“Having independence from my parents is all I want!”

“Being on my own is going to be the absolute best.” 

Have you ever had flashbacks to these types of thoughts or conversations? …And subsequently cringed? For most, meals magically appearing on the table and bills never having your name was a portion of childhood that ended before you knew it mattered. 

It goes without saying that there are endless benefits of living a fully adult life: freedom, growth, and independence to name a few. Yet those benefits can easily be tainted by a complicated financial life. As a classroom teacher, you can play a pivotal role in positively shaping a teenager’s financial future. 

​If you teach middle or high school students, introduce them to the basics of budgeting so that they don't end up in a sticky situation in life. So many graduates are not prepared for a healthy financial future. Don't let your students be in this group.

The Unfortunate Reality
Think about the following statement from page 31 of the Federal Reserve’s recent publication, “Economic Well-Being of U.S. Households in 2022.”

”When faced with a hypothetical expense of $400, 63 percent of all adults in 2022 said they would have covered it exclusively using cash, savings, or a credit card paid off at the next statement (referred to, altogether, as “cash or its equivalent”). The remainder said they would have paid by borrowing or selling something, or said they would not have been able to cover the expense.” [1]

This means that 37% of Americans, not a negligible amount of people, could not pay for an unexpected $400 expense. With many expenses far exceeding this amount (a child’s ER visit, a plumbing issue, a veterinarian visit, etc.), it is logical to assume that the number of Americans unable to cover an even larger expense is dramatically higher. 

Imagine one, or many, of your students fall within that 37%. The Federal Reserve states below the reported ways American say they would cover that expense. [2]

​

​Notice the three highest categories: Put it on a credit card, would not be able to pay, or borrow. All three options lead to either interest, increased stress, unease within relationships, or a combination of the three. 

Further complicating the issue, many high school students are not being taught any money management skills in the classroom. As of March 2023, only 8 states have fully implemented personal finance course requirements [3]. Eight! More states are in the process of making this change. As a whole, it is common for personal finance classes to be available, but most are not a requirement. 

​

3) Explanation of the Details
There are a multitude of ways to budget! Today we’ll outline two possible approaches. However, before beginning this instruction, ensure you front load with this key message: A. Budget. Is. A. Work. In. Progress. It is a plan that shifts and changes as life circumstances (expectations, children’s costs, medical bills, income, etc.) change. 

Frame this idea with lots of hope. Phrases such as “Do not get discouraged if your numbers don’t add up right away!” or “It is normal to be shocked at all the costs!” or “Remember your why!” can lessen the frustration and disregard for this critical practice. Repeat this idea throughout your unit. 

First, familiarize students with the idea of withholdings.  Students will need to know that taxes, healthcare, etc. are taken away before the paycheck even touches your bank account. Secondly, give students a sample salary. This will then be divided into the needs, wants, and savings sections. Now, you are ready to teach two standard budgeting plans. 

​

​This explanation is based on the budgeting approach within Math Giraffe’s Finance Doodle Notebook. You can get the workbook here if you'd like to have resources to bring this critical skill to life! 

Using the pages in the book, or just a blank sheet of paper, walk students through the process:

1. Choose the 50/30/20 or 70/20/10 plan. Which one do you think will work better for you right now?

2. Divide appropriately and choose one color for each category. We will be color coding!
 
​Calculate what 50%, 30%, and 20% are for your total budget. (Or if you chose the 70/20/10 approach, calculate based on those percentages.) 

3. Using a piece of scratch paper, make a complete list of what subcategories will fit inside each category. No numbers needed yet! Repeat this process for both the needs and wants categories. For example:
 - Needs: Food, transportation costs, housing, utilities, etc. 
 - Wants: Streaming services, shopping, gifts for friends, restaurants, etc. 
 - Savings:  Emergency fund, funds for big purchases, retirement, etc

4. Play with your numbers. Begin plugging in the percentage costs. Remember those numbers must fit within the allotted portion: 50%, 30%, etc. 

5. Come back to your pie chart. Now it’s time to break down each colored category with all the subcategories you identified. This will help you visualize where the money is really going! 

Decide a pattern for each sub category (transportation, restaurants, etc). Be sure to note that pattern on the key to your left. 
​
7. It is perfectly okay to re-work this if it needs to change! Reiterate to students that budgeting is a learned skill that takes patience and practice. Model this patience. ​

​In the sample above, the students will select patterns to use for the more specific categories within each color-coded section. This visual budgeting process helps them break down the costs and really understand how the plan they chose (either the 70-20-10 plan or the 50-30-20 plan) really works. The workbook pages shown are available here if you'd like your students to have the structured guidance.

On a Personal Note - For the Teacher
​
It is worth recognizing that finances are incredibly personal. What did you see modeled to you as a child? Did anyone have conversations about spending, saving, or the risks of debt? Was money a stressful or taboo topic? Our history and current financial landscape can have a strong effect on how willing we are to engage with these concepts. You will likely feel more confident in your instruction if you first familiarize yourself with the concepts and do a bit of self-reflection. 

Reflection Questions for You, the Adult:

• Do I maintain a monthly budget? If not, would I be willing to start? (Going through the process yourself, the struggles included, will lead to far greater authenticity in your classroom instruction.) 
• With what tone will I talk about budgeting? As a tiresome, limiting to-do item or an empowering practice that can lead to freedom?

​

​
Contentment is a powerful practice but is incredibly countercultural: Acrylic nails, DoorDash orders, Starbucks, and Amazon Prime are now considered normal parts of many young adult lives. For this reason and countless others, including contentment in your instruction is both healthy and critical for lasting success. 
​

Communicate with Parents
In the high school setting, this may or may not be part of your current classroom procedures. Consider sending a message or brightly colored handout that shares a few key points about what you’ve taught. It could open up good conversation within a few families.

Talk about Generosity by Planting Seeds
By first discussing students’ WHY, you are already deeping the impact of your instruction. However, there is another layer that adds meaning: Generosity. 

Throw out a few ideas about organizations that you can give to: A children’s hospital, an animal shelter, your church, even a family member or stranger who needs a boost! Invite students to share aloud who or where they may want to help. This could plant seeds that will bear fruit for years to come. 

​

Looking for More?
Alright, so you’re sold. Money matters, and teenagers need to have the building blocks for how to manage it. With a topic as big as money, let Math Giraffe lighten your load. Net worth, taxes, mortgages, and more are taught in Math Giraffe’s Finance Doodle Notebook here.

The more of these lessons you include in your classroom, the more empowered your students (future adults, remember!) will be. 
1-Federal Reserve publication: ”Economic Well-Being of U.S. Households in 2022”
2- Federal Reserve publication: ”Economic Well-Being of U.S. Households in 2022”
3- NextGen Personal Finance “NGPF’s 2023 State of Financial Education Report”

​

​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...
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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.