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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Is a Square a Rhombus?: The Great Debate & How to Approach it in Geometry Class
Does a square qualify as a rhombus? This big debate has been going on for as long as I can remember! Depending on who you talk to, you will get different opinions; some will say yes, some will say no. It all comes down to your preferred definition of a rhombus.
Inclusive vs. Exclusive
In math, we have two separate definitions of a rhombus:
- Inclusive: a rhombus has four equal sides.
- Exclusive: a rhombus has four equal sides, BUT has no right angle.
Based on these definitions, the inclusive INCLUDES squares, because a rhombus can include right angles. The exclusive definitions of rhombus EXCLUDES squares, because there cannot be a right angle.
Why You Should Consider Using the Inclusive Definition
So, which definition do you use? You may have a strong opinion based on your own belief or how you were taught; this is totally fine! However, if you’re debating on which definition is correct, why not let your students explore this topic and hold their own debate?
It’s also a great opportunity to explicitly teach the above idea. Let students in on the fact that there is indeed a debate, and explore the differences between inclusive and exclusive definitions.
Can your class come up with more examples of inclusive and exclusive definitions in math? This is a great time to investigate a controversy in mathematics!
You may even want to take it a step further and have teams of 2 students find sources on the web that prefer each definition. Let them analyze the argument and decide which they feel is a better, stronger definition and share why.
Don’t miss this chance to integrate debate skills, research skills, and analyzing a source within math class. These opportunities are rare! Students learn best when they cross their curriculum-specific skills over into different subject areas.
Additional Ideas for Teaching Distinctions Between Quadrilaterals
If you are looking for other creative ideas to teach your students about quadrilaterals and their classifications, you have come to the right place!
1) Always, Sometimes, Never: Quadrilaterals
If you head on over to the Math Giraffe Teachers Pay Teacher’s store, you can find a super fun activity for teaching quadrilaterals! This challenge is perfect for getting your students thinking about parallelograms, trapezoids, squares, etc.
The download comes with two versions: a worksheet and a sorting activity. On the worksheet, they determine whether each statement is always, sometimes, or never true and color accordingly. They end up with a design that you can check for accuracy.
They can work in pairs or small groups for the sorting activity. Students sort the cards with statements into the correct category, (“Always true, sometimes true, or never true”).
2) Quadrilateral Fun!
Nichole from The Craft of Teaching provides a great inquiry activity for beginning to learn about quadrilaterals. She cuts out various quadrilaterals, and has her students work in pairs to sort them in a way that makes sense to them on a pre-made graphic organizer.
I love how this activity is student-driven; they have the opportunity to discover the properties of various quadrilaterals themselves! This makes it so much more meaningful for your students, which improves retention.
3) Quadrilaterals Bundle
This discounted bundle on the Math Giraffe Teachers Pay Teachers store has a variety of activities to teach a fun, interactive lesson on quadrilaterals for your High School Geometry Class! It includes a blend of puzzles, activities, and proofs that add the perfect mix of fun and rigor.
These supplemental activities can be spread throughout your quadrilaterals unit!
Here's what's included:
Quadrilaterals: Always, Sometimes, Never
Quadrilaterals Card Sort
Quadrilaterals (Algebra in Geometry) - GridWords
Do you have a passionate stance on whether or not a square is a rhombus? Or any great ideas for teaching Quadrilaterals? Please take a minute to write a comment below! We would love for you to weigh in.
To be extra clear, I sometimes like to write "(non-square)" along with "Rhombus" when I want students to classify by type. This way, their answers are consistent and we can agree to use that as our category while acknowledging that there could be some controversy or confusion if we are not clear on this. (see image below)
You may want to try a similar strategy in your sorting activities or definitions when you introduce the concept of a rhombus vs. a square.
Also, if you haven’t done so yet, enter your email address in the box below to receive Math Giraffe updates straight to your inbox!
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During the day, any student who uses the seat throughout the day can use the Desk Doodle to sketch or doodle along with problems or notes to occupy their hands while engaging their brains.
Each of these 4 versions of Desk Doodles also includes a unique tool for students to self-assess their knowledge. There are three or four symbols that students can color in to indicate how they feel they understand the concept.
You can walk around the room and quickly gauge your students understanding!
In the first option, there are designated areas to jot down notes and thoughts. It also has a large coordinate plane students can use for graphing.
Choose this option for Algebra, Algebra 2, or Pre-Calc classes. They can sketch functions, write key ideas on the clipboard, and main thoughts in the bubble.
Option 2 has a coordinate plane along with a designated x and y-axis. There is a table with an x-column and y-column next to the plane. This option would be great for learning about coordinate planes or graphing lines.
This version is great for middle school Pre-Algebra when learning to plot points, graph basic linear equations, and work with tables.
The third option includes a blank grid, perfect for drawing nets or figures, or finding area or perimeter! Underneath, there is a box to write a formula. Kids can also make marks on the circle when learning diameter and radius.
They can shade the cube to work with faces, vertices, and edges. They can even use their dry-erase markers to write dimensions in any of the figures as they work practice problems. This option is ideal for a high school Geometry class or to swap out for middle schoolers when you get to the geometry unit in 6th, 7th, or 8th grade.
This option features a smaller grid, a number line, and some areas for notes. There is also a nifty area for students to indicate whether they completed their homework or not.
This version is perfect to have out on desks when working with integers and / or fractions. Kids can work with converting between mixed numbers and improper fractions using the fourths in the circles, and they can work with comparing and ordering negative numbers. The number line is so handy to have out on desks when working with integer operations!
Benefits of Doodling
Desk Doodles are a great way to get started with Doodle Notes, or are great to add to your already doodle note-friendly classroom. They work to occupy your students’ hands, while keeping their brains focused.
Doodling in class activates both hemispheres of the brain to increase:
Do you like the sound of Doodle Notes and want to learn more? Check out Doodle Note Club!
Click the image above to download the file.
Learn more at doodlenotes.org
Let us know in the comments below how much your students love Desk Doodles! Also, don’t forget to enter you email to subscribe.
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