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Solving and graphing inequalities notes with pencil

After teaching solving and graphing inequalities for a few years, I noticed my middle school math students kept making the same mistakes. I immediately started teaching inequalities different, and deliberately had some conversations with my students which made all the difference.

Now, I call them conversations, because the key is to not just tell them how to do it. The key is to lead the conversations with thoughtful questions. Here are three conversation topics that are essential for teaching inequalities in math.

Essential Conversations while Teaching Inequalities

1) There is a difference between x > 7 and x > 8. Graph both solutions on the same graph. Then have the conversation why they are not the same. Have students give examples of numbers that are included in x > 7, but not x > 8.

2) Start a conversation about x < 7 and 7 > x. Some ideas to direct the conversation: think of three numbers that are less than 7 and think of three numbers that 7 is greater than. Do your solutions work for both inequalities? Why? Have the students graph both, and notice that they are the same. Have students practice changing from the variable on the right, to the variable on the left. Though both are correct and mean the same thing, it is standard to have the variable written on the left-side of the inequality.

3) Understanding inequality graphs on a number line. There is a more effective way to teaching graphing than saying that the inequality points to the way the arrow should be drawn. When teaching this way, students often do not understand why. Also, if the students do not happen to flip a solution such as 7 > x so that the variable is on the left, their arrow will be incorrect. You could teach students to test a point such as 0. If 0 works, then point the arrow towards 0. If 0 does not work, then point the arrow away from. However, make sure that you are explaining that the arrow represents every single solution to the inequality. I often say that since it is impossible to write out every solution, the arrow visually shows every solution.

Having these conversations with your students definitely takes more time teaching. However, my experience is that they will retain solving and graphing inequalities better and you will spend less time reviewing.

Guided notes are a great way to keep your thinking and conversations on track while teaching. Here are some solving inequalities guided notes that may help you while teaching. Click on the image below to take a look at the notes.

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Solving and graphing inequalities guided notes with pencil

Christmas math worksheet activities for middle school math

I've got some great activities to keep your middle school math students engaged right before Christmas and the Holidays.  Burnt out students + winter blues + holidays around the corner often means less engaged students.  I'm here to tell you that it doesn't have to be that way.  Keeping your students working on quality holiday-themed resources is a win-win for both teachers and students. 

Christmas Activities for Middle School Math

These Christmas math activity worksheets are a must-have for middle school. These 5 different activities cover a variety of math tasks.  Also includes ideas and ways to easily differentiate between grade levels or different ability levels with in your classroom. Take an in-depth look at each activity.

  • Snow Day: This task requires students to reason through the size of a snowman and calculate the volume of the snowman. Easier level: Students can calculate the area of the 2d snowman. This task also includes a challenge as an extension.

  • Santa vs. The Grinch: Students calculate who wins in a sleigh race between Santa and the Grinch. Higher level: Students use systems of equations and solve both algebraically and graphically. Lower level: Students reason through the task using problem-solving skills. This task also includes a challenge as an extension.

  • Geometric Snowflake: Students learn how the Koch snowflake is created and find the area of a stage 3 Koch snowflake. Higher level: Students use the Pythagorean Theorem to find missing side lengths. Lower level: Students use a ruler to practice measurement and find the composite area of the shape. This task also includes an extension for higher students.

  • O Christmas Tree: Students are given equations in slope-intercept form and end points to draw on a graph. The finished product is a work of art. Easier level: Students use substitution to find the end points. A challenge is also included as an extension. Graph is included.

  • Colorful Crystals: Students classify real numbers as whole, natural, integers, rational or irrational. Students then color snowflakes according to their answers. This activity is perfect to keep students engaged before a break or a substitution.

What teachers Are Saying About Christmas Activities for Middle School Math

  • It is well thought out, practical for student learning and application, has clear instructions for students, is easy for me to present to students, has wonderful graphics, and etc....

  • Perfect activities for right before winter break!

  • Saved the day when we had no idea how to keep their attention!

  • This is such a fun way to integrate seasonal work!!! Thank you for making fun and challenging work.

  • Lots of options on how to use it.

Click on the image below to check out these Christmas math activities for Middle School Math.

Christmas math worksheets for middle school math product

If you want some fluency developing activities then you will also want to check out these fun Christmas math worksheets for middle school.

Just click on the one that you are interested in.

Use the image below to share these Christmas math activities with other math educators. 

Christmas math activities for middle school math pin

Cultivating a growth mindset in your math classroom is more than the cheerleading of "try harder, you got this!" While that is important, it's not everything.  If you present a problem to a student, and there is only one right way to accomplish the problem with only one correct solution, and all you say is try harder, the student may hit a wall, and if they are truly stuck and they can't get over the wall, then there they will stay . Teaching math as one strategy and one solution promotes a fixed mindset.

Students need more than one path.  Either another strategy to do the problem, which eventually will lead to the same solution.  Or another path to do the problem, which eventually leads to a different (although still correct) solution.  This promotes a growth mindset. I created this lovely visual to better explain.

Fixed Mindset vs Growth Mindset

one path and wall to show a fixed mindset two paths and one solution and two paths and two solutions to show a growth mindset

If you are not sure which type of problems to give your students, I have some that I created with this in mind. These Back-to-School activities were created for students to problem-solve and be successful.  They are back-to-school themed, but would work great for anytime of year.   With a low floor and high ceiling, these would work for a 6th, 7th, or 8th grade classroom.  

Here are some other great activities to incorporate a growth mindset in your classroom.  

math mindset questionnairemathematical growth mindset activitygrowth mindset posters


pin image includes text how to cultivate a growth mindset in your math classroom with a leaf calculator and pencil

My third year teaching we switched over to common core math. I was informed that students would also be tested on the standards for mathematical practice. I thought, "Great! What are those?" They were new to me too, but after studying the standards for mathematical practice in the common core standards I was excited to incorporate them into my teaching.  The best way I found to teach the standards for mathematical practice was using logic and reasoning problems. 

  1. Teach or review a standard(s) for mathematical practice at the start of class. 
  2. Give your students a problem they can reason through and have them focus on the standard for mathematical practice as they work through the problem. 
  3. Incorporate some of the ideas below as they work on their math task.  As you implement these ideas the mathematical practice will often naturally occur.  The key is detailed descriptions and deep conversations. With each strategy below make sure you have the students also explain how they used the standard for mathematical practice.
    • Give each group a large piece of butcher paper.  Have them show their work and solution on their paper.  Make sure that it is clear the strategy they used to solve it.
    • Set up a gallery walk around the classroom, and have your students walk around and notice the different strategies used. 
    • Have a class discussion after the small group work
    • Take note of the different strategies of each group. Have small presentations after to show different strategies that were used.
    • Have a class discussion on which mathematical practices were used.
While the solution to the problem is important, remember to keep the focus on using the standards for mathematical practice. I have found small groups or pair work to be effective. Students can easily practice constructing viable arguments and critiquing the reasoning of others. 

Not sure which problems to use?  You want problems that students can use reasoning skills to work through. Logic puzzles work great, but I would use other types of problems as well.  I created some math tasks to be used to teach the standards for mathematical practice. They are back-to-school themed, but can be used at any time of the year. 

Also, having the mathematical practices displayed in your classroom is the key to using them. I always had them on my front bulletin board, and referred to them during every discussion.  Students benefit from the constant reminder.  I love these posters.

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