Help Students Understand Reasoning & Proof by Explicitly Teaching the Distinction in a Way that Textbooks Don't
This concept (accompanied by free downloads below for both middle and high schoolers) gives a boost to students in grades 6-10. This will help set them up for higher level Algebra as well as Geometry and proofs. These skills strengthen reasoning and really have helped my students understand math on a deeper level.
When getting ready to introduce geometry proofs, I have learned that it’s essential to teach transitive property vs. substitution before jumping into proofs with geometry diagrams. This structure is missing from the curriculum I have seen. So, I’ve built my own resources to slowly build these skills that so many students are missing. It has made such a difference for my own classes. Be sure to take time to include these resources (free below) to give your own kids this leg up in Algebraic reasoning!
I've put together materials and videos showing the perfect way for you to clearly explain the difference between the transitive property and substitution. This is a tricky distinction for students; so it’s important to be clear right off the bat!
Keep reading for the ideal explanation of the difference between transitive and substitution property, and how to effectively prepare your students for Geometry (and higher level Algebra) proofs!
Transitive Property vs. Substitution
This is tricky for our students, so read carefully to make sure you are able to give an effective and clear explanation. I have found it helps to teach this with individual cards for each variable, (See my video, below).
For high schoolers, explicitly teach this difference.
For middle school, scroll down to the picture puzzles for skill building instead.
So, let’s say we have 2 given equations:
a + b = c and a = g
Since a is equal to g, we can replace a with g and make a new equation→ g + b = c
When two things are equal, we can replace one with the other, and we know that the equation will still be true. This is the Substitution Property. Substitution is the replacement of one piece.
On the other hand, the Transitive Property is when two numbers, variables, or quantities are equal to the same thing (not necessarily each other right away as the given).
Let’s say we have two different equations:
x + y = g and x + y = z
The key for Transitive Property is that one entire side of the equation has to match. So, it’s not just replacing one piece. In these given equations, because z and g equal the same expression, they must equal each other.
z = g
z and g must be equal, because they are equal to the same QUANTITY. This is the Transitive Property.
It doesn’t work unless the entire side of the equation matches. For example, if the equations were:
x + y + m = g and x + y + p = z
x would not equal z because the equations do not match.
I’ve created a video to show you step-by-step. Consider showing this video to your students as well!
Watch the Video:
Introducing Geometry Proofs
Once your students understand transitive property vs. substitution, I like to get them practicing using JUST these properties with a new, transitional style of algebraic proofs, before we jump into geometry proofs. FYI: textbooks don’t seem to teach this; this is just a trick I’ve learned to improve understanding of working with proofs!
Most curriculum jumps right from the type of algebra proof that is just solving an equation (justifying each step to get to the solution x = a number) into the first geometry-based proofs. If you find a book that includes this style of proof as a bridge between the standard algebra proof and geometry proofs, let me know! So far, I've had to develop my own in-between practice.
These proofs teach students how to COMBINE two previous lines in the proof using the transitive property and/or substitution as the justification.
Taking a couple of days to develop JUST this thought process helped my students so much.
After practicing these proofs, they had no problem easing into the next level of proofs with Angle Addition Postulate and Segment Addition Postulate. (Click here for a fun worksheet for practicing with these postulates.) This made them ready for what used to be such a huge leap. We avoided all the struggle that usually comes with introducing proofs. They did not feel nearly as lost.
(For these additional free Geometry proof resources, enter your email to subscribe at the bottom of this post. They'll be sent right to your inbox.)
FREE MATERIALS to build these skills
Here are two resources to help you build this skill:
1. My special breed of proofs
Include these algebra proofs as a bridge between algebraic and geometry proofs with justifying reasoning just for combining lines using transitive property and substitution. Click here for FREE samples of algebra proofs!
my special breed of transitional proofs
The key is that these are DIFFERENT from the typical “solving” style algebra proof. Look closely!
2. SKILL BUILDING Picture equation puzzles
This free set of fun challenge cards can be used in middle school OR high school to help your students build up the knowledge they need for geometry proofs (and algebra reasoning)! This free download comes with 4 basic cards, 4 medium cards, and 4 difficult cards for those students who want a challenge!
free skill-building puzzle sets
The set is differentiated to meet the needs of each of your students at any level.
For more tips on teaching these skills before having students prove their logic, go to Introducing Geometry Proofs!
This makes a huge impact on student comprehension, so be sure to download the sample proofs if you just did a quick skim or don’t quite know what I am talking about ?
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Engaging Bulletin Board Ideas
Spruce up your room and set the mood for Math Madness this March! I’ve rounded up a few of my favorites and linked the creators to spark your creativity.
Keep in mind, you can always add a basketball themed problem of the week (or day!) to the board to make sure there is an aspect of learning; you want to make the best use of your time decorating. Twist a word problem or challenge problem from your current lesson in the textbook and just add a basketball theme to it! You won't even have to switch the numbers or answers from the book if you use a little creativity to re-write it.
The games last all throughout March; the final four games are on March 31st and the National Championship is on April 2nd. Consider using this bracket above or printing out this gameday schedule for your board and assign students to fill in the teams that are yet to be announced!
You may want to set up a bracket like the second bulletin board that actually tracks student progress instead of the basketball games. Have students participate in a math tournament like my End of Course review tournaments, and let the teams compete in this way instead of on the graph!
Tom DeRosa shares a great Probability Activity using the NCAA’s data since 1979. Students find the probability of a given seed winning the tournament, as a fraction and percent.
Then, they use these numbers to answer a few questions about probability. They discuss the difference between experimental and theoretical probability. Using real-life data about basketball makes the work relevant to their lives!
March Madness Trashketball
This FREE download brings real-life fractions and decimals practice and lets your students get in on the basketball action! They get to “shoot hoops” by shooting crumpled up paper into the trash bin.
They keep track of the shots they make versus the shots they take, and they use this data to practice fractions, statistics, and decimals!
STEM Idea: Basketball Tower Challenge
If you need a STEM lesson idea, look no further! Try this fun idea by Sarah Wiggins at More than a Worksheet. All you need is newspaper, masking tape, and a few basketballs. Have students work in pairs or small groups to build a newspaper tower that can support a basketball. This challenge sounds simple, but is actually very tricky! Students will definitely be challenged.
Before students create their tower, have them create a hypothesis and introduce the STEM engineering design process with the free graphic organizers she provides.
Algebraic Expressions Basketball Game
This game allows middle schoolers (Pre-Algebra) to play a basketball game practicing with expressions!
Or, this similar game helps students practice solving equations. Track their points using a bracket to run your own tournament throughout the month!
Geometry in Basketball
This presentation offers a lesson for your Geometry students, showing how geometric principles play into every aspect of the sport.
Let kids explore it, then write up a few questions using the information they discovered. They can swap with a partner and use the presentation to try to answer the other set of questions!
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Consider kicking off Pi Day class with a fun, instructional video on Pi! I love this Math Bites with Danica McKellar video all about pi; It’s funny, light-hearted, relatable, and most of all informative and really helps students understand the concept of pi; it really gets the point across that pi is an irrational number. Plus, it’d be the perfect time to serve up and eat some Pi Day treats!
Pi Doodle Notes
If you’ve tried doodle notes, you already know kids love them! They are the perfect blend of creativity and rigor; doodle notes activate the left and right sides of the brain, which improves memory and retention. To read more about the many benefits of doodle notes, go here!
These Pi Doodle Notes are perfect for Pi Day or anytime! Here’s what it includes:
- approximations of Pi
- definition of "irrational"
- meaning / utility of the number Pi
- room for a history fact about Pi
- room for a formula that includes Pi
Pi Day Stations
My favorite way to set up class on Pi Day is implementing Pi Day Stations. I have a post about how I plan and manage the perfect Pi Day using stations, here.
I wanted to structure the class period so that the learning activities could run themselves, while I monitored and served pie to the students. Here are the stations I came up with:
I settled on the idea of stations, or centers, that would cover all the activities I wanted to do. This filled the time so nicely. No time was wasted. (As always, I am all about KEEPING the rigor, even on a fun day!) Plenty of learning was accomplished, but students still felt the joy of a fun celebration day.
For station 1, I laid out a bunch of different round objects, and had kids measure with string. They recorded their observations for a few and then derived an estimate for pi.
In Station 2, students receive a Pi Fact Sheet with fun tidbits about pi and its history. Using the sheet, they are asked to develop something creative to share a few of the facts. It can be a poem, song, graphic, etc....
I leave out special Pi-themed paper for drawing or writing. I also set out some blank white paper, so each student can choose.
Stations 3, 4, and 6 are fun worksheet-style stations. Students do a word search in one. In another, they apply pi to determine what size pizza is the best deal by finding price per square inch of pizza. The 3d one focuses on a spherical ball.
For Station 5 students read books about Pi; my favorite one is Sir Cumference!
You can purchase it (affiliate link):here.
Most students, if asked their favorite part of Pi Day will be quick to answer with, “The food!” After all, they probably don’t normally get the chance to eat anything in class, let alone sweet treats.
I discovered the coolest item at Kroger a few days ago- miniature individual pies! Avoid the mess and hassle of serving pies or other foods, and give students the choice to eat their favorite pie! I found mine at Kroger, for only a dollar, but I also saw these pies on Walmart’s website for only 50 cents! Create a signup sheet for students to choose which flavor of pie they want, and then ask students to chip in $$$!
Get Students Laughing with Pi Jokes
It’s always a good idea to begin class with some smiles and laughter! Break out a few of these hilarious and dorky Pi jokes throughout the class period, and set the mood for the class period. Here are some favorites:
Reading this too Late? Pin It for Next Year:
I hope you try one or more of these ideas! Don’t forget to plan your Pi Day the right way- Check out Perfecting Pi Day for my tips and tricks for planning the perfect day and avoiding my previous mistakes!
Do you have anything awesome planned for Pi Day? We’d love to hear in the comments below!
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