## Teaching Writing Linear Equations

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).

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.

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).

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!

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.

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.

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!  ;)

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.

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

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!

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.

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.

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.

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

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:

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.

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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## Differentiation Edition 2.2

Differentiation in math education has always been one of my greatest passions, but I always had a difficult time finding differentiation resources that could clearly explain how to make it all work in a math classroom.  Over time I’ve learned from colleagues and developed some of my own strategies, and now I’d like to share with you what has worked for me and my students.

The focus of this edition of my differentiation series is on differentiating homework.  We’ll walk through why, what, and how in this post.

First of all, I do not differentiate every single day.  That’s exhausting, and in many cases, unnecessary.  However, I started differentiating homework based out of the needs of my students.  In my district, students begin tracking into regular versus accelerated classes in fifth grade.  Each year their placement is reevaluated and students can move up or down based on their in-class progress and test scores.  {Whether this is a great method or appropriate for our students is a topic for another day.}  However, the reality is that I end up with a handful of students in my standard eighth grade math classes who were in accelerated math for seventh grade but didn’t quite make the cut this year.  They would be bored stiff if I didn’t offer challenge opportunities for them, and parents really appreciate the extra acknowledgement that their student is a strong math student and needs more than what a standard {undifferentiated} class would provide.  And, on the other hand, I bet you have students with individualized education plans who required reduced or modified homework.  Alas, I began differentiating homework!

Differentiated homework depends on where I am sourcing the main assignment from for that night.  Sometimes I assign math problems out of our textbook.  In that case, I remove a few simple problems and select some above level problems from the same section in the book.  There are many overlapping problems between the two levels, but the students who need a little extra challenge have a couple different problems.  Remember that these are students who wanted to be in accelerated and hope to move back up the following year so they’re willing to do the challenging work.  Buy-in is really important here.  Find a way to sell it to your students if you feel this is something that they need.

Other times I assign a worksheet for homework.  In this case, I usually try to find two different worksheets that cover the same skill but one may be a bit more difficult like containing an extra step.  I have been known to pull out the ole cut-and-paste strategy to replace some problems with something more challenging by covering the problems up with new ones.  I also have recently been writing my own assignments that are differentiated.  Many of the problems overlap but there is a version that is more difficult than the other.

Differentiating for those students who need a shortened assignment can be as easy as circling the problems they need to complete on the worksheet or providing a shortened assignment from the textbook.  In the homework I am creating I provide a one-page version of the regular two-page homework assignment.  For students needing friendlier numbers, I recommend looking for a secondary worksheet option for them, possibly searching the grade level below.

Ok, so there are a few different homework assignments out there.  How do we make this work in the classroom?  Option 1: Post the answers on your website.  However, this isn’t my favorite strategy because a few issues can arise with this.  The main issue is that most of my students don’t bother to log on and check their answers.  They see it as an extra step that adds to the length of their math homework time, instead of understanding that it’s a vital component to make sure they are practicing correctly.  Another issue is that many resources, especially those purchased on Teachers Pay Teachers, can’t be posted online because of the terms of use.

So that brings us to Option 2: Project the answers on the board while students are working on the Problem of the Day or while you are checking the homework.  I typically start with the Problem of the Day projected on the board.  Then about halfway through checking homework, I switch to homework answers.  So by the time I have made it around to everyone (checking for completion only), students have completed the Problem of the Day and have checked their homework and they are ready to rock and roll – quickly going over the POD, asking homework questions, and moving on to the lesson for the day.

When we go over homework, there might be 1-2 questions, several questions, or crazy amounts of questions.  If there are only 1-2 questions, I answer them at the front board.  If there are several questions, I usually call up student volunteers to neatly show their work on the front board so students can look at the ones that they need to.  I elaborate with a brief explanation as needed.  If there are crazy amount of questions, then I might go over one, then have students work together to make corrections, and regroup to see if there are still additional questions.  We may be able to move on after the collaboration or we may need to do some re-teaching that day instead of the planned lesson.  I have not found having different assignments to be an issue here.  After all, students tune out in the same way if they don’t have a question on something that is asked.  The key is not to spend too much time on homework here so it’s not a waste for any student.  Lots of questions from a student or two equals after school help.  Lots of questions from the class means pause where you are and do some reteaching.

Do you teach 8th grade math and want all of this work done for you?  If your response is “yes please!” then check out my 8th grade curriculum that includes differentiated homework.

Catch up on the rest of the series for tons of ideas you can implement in your own classroom!

Comment below if there’s something else related to differentiation that you’d love to read about!

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