Teaching Writing Linear Equations

Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!

My students would tell you that I say this about every lesson, but I really think that teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!

Explore the Ideas
The unit that precedes writing equations is typically centered on slope and rate of change.  I get so excited when we start looking at equations and graphs in terms of slope that I usually start pointing out patterns or leading students to make connections and draw inferences about the y-intercept themselves.  At this point, we typically spend a little time looking at arithmetic sequences.  We use the arithmetic sequence formula, and then discover patterns to formulate our own formula (which – ahem – is basically slope-intercept form).
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


CCSS 8.EE.B.6 mandates that we “use similar triangles to explain why the slope m is the same between any two distinct points [and] derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.”  I’ve created a discovery worksheet that facilitates the exploration of these big ideas.  You can scoop it up here.
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Get the Basics
Organized notes and repetitive reminders of the steps are super important here.  I will typically create an anchor chart that outlines the steps given each potential set of information.  Notice all of the exact language that I use (also great for your ELLs).
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Demonstrate a variety of examples and allow students ample time to practice their new skills.  I teach in a 55-60 minute class period, and I spend two days on each topic: writing equations given a graph (or slope and y-intercept), writing equations given the slope and a point, writing equations given two points.  The first day we take notes and begin to practice the new skill.  Then we continue hands-on practice on the second day.  Check out my differentiated notes and practice here.  Opportunities for differentiation here!
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Make it Real
I’m not talking about textbook word problems here snore.  Cow population, birth defects, what??  There are so many examples of linear relationships in our students’ lives.  Use examples about social media, sports, music, etc.  I have a set of rate of change and initial value task cards that uses real examples. 
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Students also love the analysis and discussion required for this discovery worksheet about planning a road trip.  Ask students to draw inferences, make predictions using their equation, discuss limitations, and – most importantly – allow time for debate.  You will turn those adolescent brains right on when they get to argue, and your colleagues in the humanities will thank you for practicing supporting arguments with evidence.
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Practice Makes Permanent
Here’s a fun way to practice a new skill or review at the end of the unit:
~ NOTE ~ Depending on your comfort level with thinking on the fly this lesson could potentially have ZERO prep.
1)   Everyone sits at their desk with a piece of notebook paper and a pencil.
2)   4-5 students at a time go to your front whiteboard.  (I rotate through the class to make sure everyone gets up once before repeating volunteers.  This means I often end up with a few “non-volunteers” at the end.  I typically motivate students to volunteer early because they know the rule is that the practice problems generally get harder as we go.)
3)   Announce a slope and a point (for example).  Everyone at their desk writes down the information on their paper at the same time that the volunteers on the front board use the whiteboard markers.  Everyone in the class writes the equation given that information.  I focus on the students at the front and provide subtle redirection as needed, scan the class to make sure everyone is working, and provide assistance to a student or two at their seats if needed.
4)   We discuss things that are great or potential pitfalls to watch for then repeat with a new group at the front board.
5)   Everyone loves it!  And seriously I very rarely prep for this lesson.

And of course I’d be remiss if I didn’t mention that I have a super popular OMG game for this topic.  Haven’t played yet? 
O. M. G you don’t know what you’re missing!  ;)
Teaching students how to write linear equations is my favorite unit.  There’s ample opportunity for exploration and discovery, applications, and extensions!


Extend the Unit
So many opportunities for extension here!  You could introduce relationships with parallel or perpendicular lines, scatter plots (and really collect data), standard form of linear equations, graphing calculator extensions and more!

Here’s a fun one for your high flyers:
Write the equation of the line perpendicular to 2x-y=8 that shares the same x-intercept as the given line.
(Answer: y=-0.5x+2)

Join in the conversation!  What are your favorite ways to teach students how to write linear equations?

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4 Common Misconceptions and Solutions for Solving Linear Equations

4 common misconceptions and solutions for solving linear equations

Solving linear equations is one of my favorite topics to teach.  So fun!  However, there’s always challenge when addressing misconceptions.  Read on to learn about some of the errors you can expect and how to address them.

Undo the Operation v. Combine Like Terms
So it’s time to teach your students how to solve equations that involve combining like terms!  Be prepared that some students will struggle with knowing when to simply add and subtract numbers and when they need to balance the equation and undo the operation they see.  I tell my students that if the terms are on the SAME side, they SLIDE them together.  If the terms are on OPPOSITE sides, they use the OPPOSITE operation.  Helpful hint when teaching students how to combine like terms: Use shapes to identify like terms.  Circle the x-terms and put boxes around the constants and be sure to keep the sign in front of the term with it.  This visual approach really helps my students see which sign belongs where.  Lots of practice helps solidify this concept for students.  Practice makes permanent!

4 common misconceptions and solutions for solving linear equations


Zero Solution v. No Solution
Special solutions bring their own set of misconceptions.  The biggest issue I have seen is mixing up no solution and a solution of zero.  This mistake commonly occurs when students try balancing the equation using the constants first.  Then they end up with something like 3x=x and think this is no solution because they are not sure where to go from there.  In an effort to avoid this mistake I encourage my students to balance the equation using the terms involving the variable first.  This discovery worksheet has really helped my students understand this case.

4 common misconceptions and solutions for solving linear equations

Rational Coefficient v. Rational Expression
Inevitably every year at least one student consistently makes the mistake shown below.  One way that I explain this is using order of operations.  To assemble the expression, x was multiplied by 3 then decreased by 2 and finally divided by 4.  Therefore when solving we must undo the division by 4 first because we use reverse order of operations.  In a less “mathy” way I tell students that the numerator is trapped until they unlock it by multiplying by 4.  If your students are struggling, it may help to show a comparison.  Show the case where you would add 2 first and compare and contrast the two equations. 

4 common misconceptions and solutions for solving linear equations

Clearing Fractions v. the Distributive Property
By the time we get to equations with the variable on each side involving rational coefficients some students give me the “really??” look.  I focus a lot on clearing the fractions in the first step so that students who struggle with operations with fractions don’t need to deal with them in every single step.  When I teach students to solve equations involving distributing a fraction, I start with friendly numbers but then move to cases where distributing does not eliminate the fraction. 

We know it’s actually super easy to eliminate the fraction before distributing, but conceptually this can be a challenge for students.  Students need to clearly understand that they only need to clear the fractions outside the parenthesis because they will be distributing to the other terms and, therefore, those will be affected by the change too.  When students continue to question why I didn’t multiply everything by 12, I explain that each side of the equation has three factors.  It’s the same as multiplying 2x3x4.  Once I’ve multiplied 2 and 3, there is no need to also multiply the 4 by 2.  This discovery worksheet facilitates a great discussion about this case.

4 common misconceptions and solutions for solving linear equations

Join in the conversation!  What other misconceptions do you see your students demonstrating?

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How I Learn 100 New Student Names in 2 Days

Each year I have about 100 or so new student names to learn.  It takes me about two days to get them all down.  I think it is so important to learn them quickly – including correct pronunciations!  Here are five strategies that help me to learn 100 new student names within the first two days of school:

Yup, you read that right.  Each year I have about 100 or so new student names to learn.  It takes me about two days to get them all down.  I think it is so important to learn them quickly – including correct pronunciations!  Here are five strategies that help me to learn 100 new student names within the first two days of school:

Each year I have about 100 or so new student names to learn.  It takes me about two days to get them all down.  I think it is so important to learn them quickly – including correct pronunciations!  Here are five strategies that help me to learn 100 new student names within the first two days of school:

1)  I seat students alphabetically by last name. I always learn first and last names at the same time. I think this is actually easier because I can think about where in the alphabet their name should be when I'm trying to remember. (There are some exceptions for preferential seating modifications.)

2)  Students find their seats on day one by reading the seating chart and finding their own seat.  I do not attempt to say their name before hearing them say it themselves.  For attendance I ask them to say their name, then I repeat it.  I make it very clear that I want to be corrected if I pronounce any part incorrectly.  {My first year a student didn’t tell me until the end of the year that I had been mispronouncing her name I never wanted that to happen again!}

3)  I give students at least 15 minutes of quiet work time each of the first two days. I spend that time studying my seating chart and matching faces to names.
Each year I have about 100 or so new student names to learn.  It takes me about two days to get them all down.  I think it is so important to learn them quickly – including correct pronunciations!  Here are five strategies that help me to learn 100 new student names within the first two days of school:

4)  I demonstrate my bravery by attempting all names without a seating chart at the end of day one. This practice is a great way for me to check-in with myself and see which names I really need to focus on. Also, students get a kick out of it and it's a great way to develop relationships right off the bat!

5)  I make connections to people I already know. Did I have their sibling in class? I will probably learn their name right away. Do they remind me of a former student? I try to link the two names together in my mind.

6)  A temporary solution I use is to somehow connect what they're wearing to their name. (Example: Amy is also my cousin's name. Cousin Amy loves the color orange and today Student Amy is wearing orange.)

By the end of day 2, I can typically recite all first and last names when they are in their ASSIGNED seats. (It takes an extra week or so for recognition outside the classroom.) When I feel really brave, I let them switch seats randomly so I can try again. I love it and so do my students!  My eighth graders love to try to stump me!


What strategies do you use to learn your students' names?

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