Thursday, September 7, 2017

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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Monday, August 14, 2017

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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Monday, July 24, 2017

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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Monday, May 15, 2017

Testing is Over: Now What?!

Testing is over… now what?  Get inspired with these 14 end of year ideas for your classroom!  Student and teacher tested and approved!

Testing is over now what?  Get inspired with these 14 end of year ideas for your classroom!  Student and teacher tested and approved!  J

Review Project
Put your students in the driver’s seat for their learning.  Before final exams assign students to groups to review topics from throughout the year.  Presentations can include a lecture and an activity to help their classmates practice the skill.  So fun!

Extension Project
For your higher-level classes, give them the opportunity to research a topic you did not discuss in class.  Allow students to set up projects around the room so their peers can review the research and make notes of interesting facts or connections.

Preview Topics for Next Year
I think I enjoy this more than my students but it’s a goodie!  There are definitely times when structure is better at the end of the year.  Have students take notes on a topic related to what you’ve been learning but didn’t have time to completely cover or will be covered in the following year.  As an Algebra I teacher, I love previewing connections to Geometry or Algebra II.

Testing is over… now what?  Get inspired with these 14 end of year ideas for your classroom!  Student and teacher tested and approved!Chalk Practice
Do you catch your students staring out the window when it’s 80 degrees and sunny?  In New England we only get a few of these days before school gets out so I love to bring the practice outside!  Give students a list of problems to complete on the pavement with chalk.  So much more fun and much less complaining!

Outdoor Class
Or hold the entire class outside!  Give students a packet or activity and a clipboard, and let them work independently or in pairs outside in the grass.  This is my favorite thing to do after state testing because it gets students fresh air and allows them to work at a casual pace after they’ve already worked hard all day.

Charades
Played just like the regular version, allow students to group up.  Have a list of academic topics and/or funny memories from class prepared.  Students then select a paper and act it out!

Surveys
Surveys are great way to gather student feedback, especially if you tried something new.  One of my favorite things to ask is the rank of their favorite projects.  I don’t always realize along the way which projects students loved and which were more of a pain.

Organize Classroom
One of the last few days I put my students to work!  They love to help clean and organize shelves and closets.  Best of all, it saves me from having to stay late on the last days of school!

Testing is over… now what?  Get inspired with these 14 end of year ideas for your classroom!  Student and teacher tested and approved!
Clip Art by Sarah Pecorino Illustration
Test Markers and Other Supplies
Students will enjoy doing other monotonous jobs as well!  Have students check markers and pens to see if they’re still working.  Throw away or upcycle whatever is dried up.


Watch a Movie
Setting up a content-related movie is a great plan when collecting textbooks or passing back lots of work at the end of year.  For 8th grade math, Donald Duck in Mathmagic Land is always a favorite!

Advice for Next Year’s Students
Have students write down their best advice for incoming students.  What inside knowledge do they wish they had starting the school year?  Shana of Scaffolded Math and Science has a great resource for this.  You may even be getting your bulletin board done for the fall!

Game Day
When all the learning is done but you still want a semi-structured activity, host a Game Day.  Students bring in their logic-themed games and sign up to play the games of their choice.  Promotes cooperative learning and critical thinking so you can even write some objectives on the board for your administration.  J

Letters to Teachers
I inherited this activity from a retired teacher and it is awesome!  Give students 20 minutes or so to write a letter of appreciation to an adult in the building who had an impact on them throughout their years at the school.  (I teach 8th grade math so my students have been at the school since 6th grade and are heading off to high school.)  Then sort through the letters by teacher and scan for appropriateness.  Deliver this pleasant surprise to teacher mailboxes.

Locker Clean-Out Trick
Have you seen how much stuff gets thrown out on locker clean-out day?  I leave out two boxes and stand near the trash.  Any pencils or other writing utensils being thrown out go in one and any binders, paper, notebooks, supplies go in the other.  Awesome way to start a collection of extra supplies for next year!

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