Writing Two-Column Proofs: A Better Way to Sequence Your Proof Unit in High School Geometry
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."
"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!
Enter your email address here to get the pieces shown above sent to your inbox:
(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.
Algebra Tiles to Print on Label Paper for Interactive Notebooks
If you are planning to use Algebra Tiles for factoring and want students to get familiar with them, you will likely have a set on hand for them to use as a manipulative. They'll practice using them in class as you model it. But the challenge is that they don't have a permanent representation of what they arrange on their desktop.
They'll lay out the algebra tiles on the table, start to understand the concept, and learn how to factor using algebra tiles. But they won't have anything to take away as a model to review from. It can be nice to have another version that they keep in their notebooks. Just a few examples will do. This way, they can have a copy to take home.
These sticker templates are an easy way for kids to have a representation of the algebra tiles in their own notebooks.
To use the templates, just pick a color (I've included standard coloring that matches the algebra tiles you probably have in your classroom, as well as another fun color option), then print a page for each student. They'll quickly cut down the lines for the smaller ones, and then they can peel and stick them right into their notebooks as you do notes and examples.
I'd recommend only printing one or two sets of stickers per student. That way you don't need to get too many sheets of label paper. This way, you can do a couple of examples with the real algebra tiles, then do one or two different example problems that they can model in their notes with these stickers.
Of course, they can also then just draw the tiles themselves later on if they want to keep modeling problems this way on worksheets or homework. They will not need to use stickers every single time. It just offers a nice in-between option for them where the hands-on effect of a manipulative and the permanent drawing version to keep in a notebook can meet halfway.
If you use these too long, it will likely become too cumbersome and not be worth it anymore. Try just using a sheet of these when you introduce each new way to model with algebra tiles.
The template (free download below) has a link to the right size & shape of label paper as well so you can get the correct one to fit these templates. This limits the cutting. However, if you have plain full-page sticker paper available that you use for my other templates, you can just use that too. The students would just have to do a bit more cutting, since the squares won't be pre-cut for them. The Avery label paper I've linked to just makes it easier.
Try these other creative math teacher hacks too:
8/26/2022 6 Comments
Misdiagnosed ADHD, Depression, and other Issues Can Often Actually be Attributed to Lack of Sleep
This Math Giraffe article was originally published in SnowDay Magazine for Creative Teachers, Issue 4. This is purely informational, and is not health advice. Consult a medical professional with any questions.
Did you know that children in China continue midday napping through elementary school, middle school, and even into adulthood?! More sleep in children has been found to significantly improve happiness, control, and grit. Adversely, sleep deficiency can cause a number of problems, like chronic disease and behavioral problems.
Some researchers are surprised to find that many of the diagnoses our children and teens are receiving can instead be attributed to extreme sleep deprivation based on their age and needs.
We can’t change our school day schedules overnight, but we can educate our students’ parents about the widespread impact of a good night’s sleep, as well as the negative impact of sleep deficiency.
Impact of Healthy Sleeping Habits
Every single person needs sleep every night to support their circadian rhythm, or sleep-wake cycle. Studies have found that children’s mental and physical health are directly affected by sleep. Healthy and consistent sleep habits are associated with better language development, academic achievement, and socio-emotional and behavioral functioning in young children. Other studies have shown that kids of all ages who regularly get an adequate amount of sleep have improved attention, behavior, learning, memory, and overall mental and physical health.
So, how do we help children develop healthy sleep habits?
First, students need the recommended number of hours of sleep per night. Depending on their age, the numbers vary:
Infants under 1 year: 12-16 hours
Children 1-2 years old: 11-14 hours
Children 3-5 years old: 10-13 hours
Children 6-12 years old: 9-12 hours
Teenagers 13-18 years old: 8-10 hours
For teens who wake up at 6am to get on the bus, for example, this would mean getting to bed between 8 and 10pm. Is that the reality for most of your students? Not likely. But are parents and teens aware that it could be the root of their problems with weight, hormone imbalances, trouble focusing, or depression?
Beyond just the time spent in bed, it needs to be quality sleep. It’s important for parents to consider bedtime routines. Consistent bedtime routines are critical for younger children, (daily, positive interactions that end with the child sleeping). Healthy bedtime routines should include things like brushing teeth and bathing, reading books together, and physical contact for younger children. They may look different for older children and adolescents as they become more and more independent, but should still be happening.
Parents of teens often drastically miscalculate the hours their child actually spends sleeping. They imagine them resting from the time they “clock out” for the day until the alarm goes off, but adolescents are frequently spending hours scrolling, reading, texting, or even just thinking while their caretakers assume they are asleep.
These reasons combine to cause a lot of confusion when a student starts to have trouble. Experts in pediatrics and in education both tend to look toward certain common explanations or diagnoses, when in fact, for many students, sleep should be the very first thing that is questioned.
Diagnoses that Could be Explained by Sleep Deficiency
There are many diagnosed behaviors and conditions in classrooms that could be a result of sleep deficiency. Not always, but oftentimes, students suffer from obesity or are misdiagnosed with ADHD or depression. Professionals will follow along with treatments accordingly when really, a lack of sleep might be the root cause.
The wonderful upside is that in cases where sleep is successfully identified as the root of the issue, the problem is easily solved. Working to get more rest is free of any negative side effects, and will have additional benefits for the whole child.
Attention-Deficit/Hyperactive Disorder is prevalent in today’s schools. Millions of children have been diagnosed with ADHD in the US, and the number has only risen over the years. But, did you know that sleep deprivation and ADHD exhibit some of the exact same symptoms?
The behaviors associated with ADHD interfere with a child’s social and intellectual development, which can lead to problems with relationships with other children and adults, at school and at home. But, ADHD might not always be the underlying cause.
That’s right, many cases of ADHD are misdiagnosed, and forming healthy sleep habits could solve the problem. Some symptoms ADHD and sleep deprivation have in common are inattention, hyperactivity, anxiety, agitation, nervousness, insomnia, and weight loss.
Often, when a child is acting out and we don’t see an obvious cause, doctors jump to an ADHD diagnosis. Sometimes, though, the child is suffering from sleep-disordered breathing, causing disruption to sleep patterns. At the NIH, they studied children with ADHD who were on medication, and fixed their sleep-disordered breathing. Within just 6 months, 70% no longer showed ADHD symptoms and could be taken off medication.
The Child Mind Institute shares, although sleep disorders are rare in children, a lack of sleep (even if it’s not considered a sleep disorder) can cause or worsen ADHD symptoms. So, if a student is diagnosed with ADHD, you may want to advise parents to pay attention to any sleep disturbances, and follow up with their physician.
It’s also worth noting that ADHD medication can play a role in sleep deficiency. Children who take stimulant medication might experience symptoms, like deficient sleep. If the child takes the stimulant too late in the day or if it keeps working too long, they might have trouble falling asleep. It’s worth bringing up any of these concerns with your doctor.
As you probably know, obesity is prevalent in the US. It affects about 17% of children and 35% of adults. Obesity often leads to many other medical problems, like cardiovascular disease, diabetes, and hypertension. Studies have also found it might cause sleep apnea in children.
Most people automatically jump to poor diets and exercise habits as the cause of this epidemic. Although these definitely need to be considered, poor sleep patterns are often overlooked. Most studies have shown a convincing association between a lack of quality sleep and increased weight gain in children. Different factors determine quality sleep, such as the duration of sleep at night (Is the child getting the recommended number of hours of sleep per night?), or sleep patterns (Is the child following a strict bedtime?).
As mentioned earlier, depending on their age, children and adolescents need a certain number of hours of sleep, and multiple reviews have determined there is a relationship between sleep duration and childhood obesity. The underlying explanation(s) are still unknown though. Some theorize it is because poor sleep can disrupt hormone production, which can lead to an increase in appetite.
Children and adolescents who suffer from obesity are at a higher risk for various sleep disorders. Although researchers are still continuing to study this relationship, many believe this association involves metabolic and neuroendocrine/hormonal physiology and other factors. These sleep disorders can be categorized into four functional categories, including insufficient sleep quantity, poor sleep quality, inappropriate timing of the sleep period, and primary disorders of excessive daytime sleepiness.
It’s totally normal for a child or teen to feel down or irritable from time to time, but if it’s happening consistently there could be a deeper problem. Depression can occur early in life and cause serious consequences later in life. There are, of course, many different factors that can lead to depression or depressive symptoms.
Research has indicated that depression (in children and adults) is often linked to insomnia and other sleep problems. Depression and lack of quality sleep also share common symptoms, like fatigue, loss of interest, and depressed mood. Researchers have found that depression in children can be affected by the duration and quality of sleep they get each night.
It is rare to see depressive symptoms in youth before puberty, but the rates increase drastically in adolescence. Some argue that before puberty children’s sleep is “protected” against disruptions. Young children are able to reach the deepest level of sleep, whereas adolescents often cannot. Many suffering from depression, including children and adults, report “sleep complaints” like having trouble falling asleep, having difficulty waking up, etc. These are considered subjective findings; there are also objective measures of research done through laboratory-based sleep studies that examine some of the physiological characteristics of sleep. Some studies have found that children with subjective sleep complaints do not always correlate with objective measures. This suggests a patient’s perception of their sleep may differ from objective measures.
Clearly, quality sleep and these chronic diseases have a complex relationship. Many research reviews use the term bidirectional, functioning in two directions. Children and adolescents diagnosed with ADHD, Obesity, or Depression might experience sleep deficiency because of their diagnosis. It could also be the other way around, with sleep deficiency causing a misdiagnosis.
Please consult with a doctor if you have any concerns.
What you Can Do as a Teacher
Teachers can only do so much. If you suspect any students are struggling due to sleep deficiency you certainly can’t go home with them and make them go to bed at a reasonable hour. But, you can do what you do best... educate!
Educate parents and students (depending on their age) about good sleep hygiene. Grab the parent handout below, fill in the blank (using the info on the sheet), and make copies to send home with students.
More Ways to Help Your Teens
Read these next if you are a teacher (or parent) of teen students:
>> CDC: Sleep in Middle and High School Students
>> PubMed.NCBI: Attention-deficit/hyperactivity Disorder With Obstructive Sleep Apnea: A Treatment Outcome Study
>> NCBI study: Sleep Disturbances in Pediatric Depression
>> Harvard.edu: Sleep Deprivation and Obesity
>> ScienceDaily: Children's Mental Health is Affected by Sleep Duration
>> CDC: Data and Statistics about ADHD
>> PubMed.gov: Obstructive Sleep-Disordered Breathing in Children
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