Do I really need to teach that? It’s common sense!

What you think is common sense may not be for your students,.  Remember, you are an adult!  While you may think they should just know certain things and that students will make connections using number sense, it is doubtful that they all will.  Teach even what you believe should be common sense.

Wouldn't you be doing your students a disservice by not drawing their attention to those simple connections that could really help them?

Here are three examples based on my experience in the classroom:

Example #1

When graphing linear equations, students don't typically look at equations such as x = 3, y = 8, and 3x + y = 6 and think about what type of line each will form.

So, they just graph a line somewhere on the grid without considering whether the line should be diagonal, horizontal, or vertical.

Fixing this issue can be as simple as having students predict how a line will look before putting pencil to paper.

• Have them write "horizontal", "vertical" or "diagonal" before graphing.
• Give a quick check for understanding focused on only this topic.
• As you work through examples in class, ask students to predict what type of line to expect before graphing.

I think of this as having an understanding of the "big picture".  Others describe it as common sense.

 

Example #2

Another topic I've noticed that students struggle with is labeling perimeter, area, and volume.  Many of them just don't make the "big picture" connection.  They perform all sorts of contortions to decide how to label an answer.  Some seem to think that the shape determines the labeling.

All they need to know is this:

• All areas are square units (area, surface area, lateral area, base area).
• All volumes are cubic units.
• Anything else is single units (length, base, width, height, perimeter, circumference).

It is not common sense ... teach it!  Every student will not just figure it out.  If so, I would need to be writing this post.  

Example #3

The same goes for positive vs. negative slope.  A positive slope is a line that slants upward from left to right showing an increase.  A negative slope is a line that slants downward from left to right showing a decrease.  Train your students to think about whether the slope will be positive or negative before counting it or graphing a linear equation.

Verbalize the key "big picture" takeaways.

Remember, common sense is not common!

How to Introduce Negative Exponents

Teaching lessons where patterns are involved is fun!

When I began having students make charts as they were learning about negative integer exponents, it made it so much easier for them to understand rules and concepts.

Let’s face it - to students, it's not logical that negative exponents result in fractions.  Looking at a chart and examining patterns changes promotes understanding.

I begin by having students complete a chart like the one below.

Next, I have them extend the chart as shown.  Then, ask students to study the chart and explain what is happening as they examine the chart which goes from 625 to 125, 125 to 25, and 25 to 5.  Hopefully, they'll recognize that each number is divided by 5 as they work their way down the chart.

I always tell my students that math is all about patterns. 

They love to look for patterns!

We complete the rest of the chart together by continuing with the pattern of dividing by five.  This is a perfect way to review the exponent of zero rule and introduce negative exponents at the same time.

After completing this chart with students, I have them complete a similar one on their own.  Then, I follow that up with a brief class discussion to reinforce the observation that negative exponents result in fractional answers.

Parentheses when Substituting

Do you sometimes think, "I wish my students could just remember this one thing?"

I'd like mine to remember to use parentheses when substituting into an algebraic expression.  I stress outlining the problem before replacing the variables with numbers.  However, I can't force students to do it.  It can be frustrating because if they do outline the problem first, as shown below, they generally solve it correctly.

One little trick I use when working examples for the class is to highlight each space where a number will be substituted.  Little things like this can help put the focus where it belongs, and is especially helpful to students who are visual learners.

Big Picture Equation Solving

 Do you have students who get lost while solving a multi-step equations?

Does it appear that they are randomly writing work without understanding what to do when or why?

A different perspective is needed when this happens.  And ... it can be simple!

In the midst of solving a multi-step equation, students can lose track of the goal - to solve for the variable.

What I call "outlining the equation" is one technique I use in this situation.  It doesn't help all kids, but it does help a number of them.  It is a visual that keeps the end goal in sight.  Students begin by making an outline as shown, using DCV.  

Distribute - Combine Like Terms - Get all Variable Terms on One Side

After completing DCV, it looks like this:  

Success!

Algebra: Teach the Instructions

Have you noticed this when teaching Algebra?

I’ve found that when my algebra students are struggling, the cause is often not the math, but the instructions and vocabulary.

For example, if the instructions say “factor the polynomial completely”, students often write solutions as if solving an equation.  As shown below, the student knows how to factor, but goes beyond that due to misunderstanding how an expression and equation are different. 

Often it seems as though students are understanding the material, yet they don’t perform well on assessments.  The example above shows one reason why this may happen.

Here is another example: “Find the zeros to the quadratic function.” or “Find the roots to the quadratic function.”  (Hands go up … “What are zeros?”  or What are roots?”)

The issue is the vocabulary term.  My students may have mastered using the quadratic formula, yet not have a clue how to begin to answer this question.

How can we fix this you ask? 

When you begin the quadratics unit, teach the vocabulary including terms students may see in instructions.  Stress that “answers, solutions, roots, zeros, and x-intercepts” are all synonyms.  Because you know what?  It is confusing!  Although you may not typically consider these to be vocabulary terms for the unit, being familiar with these synonyms is critical to success.

My advice is to make a conscious effort to teach students about not just the algebra, but about the most common errors they are likely to make and how to understand instructions.  You may think it is common sense, but kids don’t have the experience teachers do.  After completing high school and college, we’ve solved so many problems similar to the one above that we naturally do what we should because we have learned.  Students are still learning!

Secondary Math Plans for a Substitute Teacher

There's an old story about a student seeing a teacher in the grocery store.  The student turns to his/her parent, and says, "You mean, teachers shop, too?"

Yes, teachers do the same things and go the same places as other people!

Also, just like people in any profession, a teacher has to miss work occasionally. Perhaps you or a family member get sick?  Maybe you need to attend a funeral?  Do you have professional development or a case conference to attend?  Once in a while you will need a substitute teacher.  

In an effort to make it a bit easier for you when this happens, I have created some sub packs for 6th Grade Math, 7th Grade Math, Pre-Algebra, and Algebra 1.  

    

These sub packs could be purchased by an individual teacher, a math department, a substitute teacher, or a school.  Pick and choose what lessons to use when. All resources are not only easy to use, but focused on mathematical topics students need to practice.  A day with a substitute teacher should not be a "lost" instructional day!  I've included a number of options in each pack for flexibility.  Activities can be chosen to fit appropriate times of the school year. 

I hope these resources can make missing school a bit easier.  🤞

Free Google Slide Factoring Task Cards

These days, so many of us are searching for digital-friendly activities that provide effective practice.  That's why I designed these FREE factoring task cards.  Use the PDF version to create traditional task cards, or the Google Slides link to easily assign them to students digitally. 

Even "in person" instruction involves more technology right now due to social distancing and minimizing contacts.  For certain, traditional task card activities do not lend themselves to distance learning.  However, these Google Slide task cards work great and can be used to change things up a bit. 

Students will completely factor polynomials that have a GCF that is not equal to one.  The binomial or trinomial remaining will have an "a value" of one.  I would consider this activity to be of average difficulty for an Algebra 1 student.

I'd love to hear about your experiences with these task cards!

 Free Equation Flowchart Notes with Video Example!

You might think that you could just pick up any set of notes and use them.  However, no matter how much you may like the idea of a certain graphic organizer, will it work smoothly for you and your students?  How can you possibly know without trying it out?

I'm giving you the opportunity to see how well this flowchart of "Questions to ask yourself as you solve an equation ..." works!  

The idea is to teach your students to ask the correct questions each time they solve an equation.  That way, they will learn the routine of working through the equation-solving process.

The Equation Flowchart Notes are free in my Teachers Pay Teachers store.  Simply download them if you would like to use them.  I've solved an example using them in the instructional video below.  Whether you like the notes or not, you can make a well informed decision about using the flowchart.  If you use it with your students, let me know how it goes! 

Equation Flowchart in Action!