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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Football Math Activities for this Week
It is an exciting time of year for our football-loving students! With the big game coming up, football is the focus of many students’ conversations. I feel the most proud and confident about a lesson when it is relevant to my students’ lives. I’ve rounded up some ideas to bring that game-time excitement to your classroom!
Financial Football – Free online game
Kids can play this interactive, video-based online football game to explore financial literacy. There are options for one player, or head-to-head play.
Lesson plan PDFs to accompany the game are available as well, with options for age 11-14 (rookie), age 14-18 (pro), and even an option for 18 years and older (Hall of Fame).
It’s really well done, and students think it is pretty cool. You’ll need an active flash player on the classroom computers to use this.
Creating a Graph
Have students review the history here showing all winners, and determine the best way to represent the data on a graph. They can show all years by creating a graph that best displays the information.
This is a great opportunity to incorporate technology as well. They can do paper or poster graphs to plot it, or use tech tools to create the graph. Try having students work in pairs.
Linear Equations Football Game
Your students are sure to love this fun, football-inspired game for Algebra! With this game, there are so many easy ways to differentiate. You get two sets of “football play” cards. Set A has basic linear equations in slope-intercept form, and Set B has y-intercepts that are not integers (Students must use Point-Slope Form on these cards.)
You also get two different worksheets. Worksheet 1 has students only write equations in Slope-Intercept Form. Worksheet 2 has students finding and recording Slope-Intercept Form and Point-Slope Form.
Students pair up and play the game with their partner. Each turn a new "football play" card is drawn. The offensive player decides which of two points on the coordinate plane to pass the ball to. The defensive player decides which of two defenders will attempt a tackle. A player scores by writing the equation for the line of the pass correctly....as long as the defender did not accurately make a tackle at the same point.
Check out this video to see how the game works!
The Science of NFL football
NBC Learn teamed up with the National Science Foundation to create a 10-part video series centering on the Science of NFL football It covers everything from spheres to health and hydration to the Pythagorean Theorem.
This page is more geared towards a science class, but some videos definitely belong in your math teaching. These videos are perfect for getting your students’ attention!
I especially love this video on Geometric Shapes: Spheres, Ellipses, & Prolate Spheroids!
Mean, Median, Mode, and Range
Use this tweet map to explore the number of times each word was tweeted in each region during the 2009 game.
By hovering over areas and scrolling across the time bar, kids can see the totals. Just give a few question prompts that help them review measures of central tendency. The possibilities are endless!
STEM Lesson with #DeflateGate
This STEM-based football lesson is great for older students. It would work well for high school math, Chemistry, or Physics class. Students explore links between pressure and temperature. They’ll convert units to metric, watch a video, and use the expression P1/ T1 = P2/T2 to determine whether a cold game may have affected the inflation of the ball.
This lesson on the Ideal Gas Law can be just a quick reading of the article in the last few minutes of a class period, or you can go beyond, and try it out with a football and a freezer!
Dollars and Cents at the Concession Stands
Something I’m pretty sure everyone loves at football games are the concession stands! Bring this delicious aspect into your classroom with this creative game from Scholastic.
Scholastic shares a great idea for younger learners. Pass out poster board to small groups and have them build concession stands, drawing and pricing their menu items; using play money and a “register” have students pay for items and make change accurately.
For middle school students, consider printing out the statistics of the two teams in the superbowl, the New England Patriots and Philadelphia Eagles. Using these numbers, have students work in pairs to find the mean, median, and mode, etc. You can also have graph the stats and compare the two teams.
If you teach a subject other than math, Scholastic offers a ton of other ideas to integrate football in your lessons!
For fun math ideas throughout the school year, subscribe here to receive updates straight to your inbox!
Pin these ideas to save for next year!
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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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