I like to think all math concepts are important, but some are so crucial to everything that comes after that to not invest the time for every student to master will make the concepts that follow for years to come more difficult.  It is, I think, why so much time is spent on basic operations, fractions, and percents in elementary school.  Without a deep understanding of those concepts, the next levels become extra challenging.

In sciences such as physics and chemistry, unit conversion is crucial.  In Algebra and beyond, rational expressions are key.  Even though my logical brain says they are just fractions, I am surprised at how rational expressions and equations can seem like a foreign language to students.  So, when we get to rational expressions, I set aside 5 days (2-3 hours) to work with them.  Although some students get it by about halfway through the second day, others need the full week to let the concepts sink in.  Here's how the week looks:

• Day 1*: Simplifying - I give the students a fraction to reduce.  "This is easy," one student says.  They can hardly believe they are in Algebra 2.  We start this way to build confidence.  Then, I give them a rational expression to reduce  - just a simple one with exponents.  This is review.  We worked with exponents in the previous unit.  Then, we move to more complicated expressions - ones that require students to factor in order to simplify.  Students write out 3 steps.  These will seem obvious to many, but these are the basis for all the other operations.
• Day 2: Multiplying/dividing - The simplifying of rational expressions has marinated in students' brains.  Now, they are ready to apply simplifying to multiplying and dividing.  Essentially, this is the same concept but with what we call step 1a: after simplifying each expression, we multiply or divide, and then complete the simplification.  I remind students that these rational expressions are just fractions.  They know how to multiply and divide fractions, so all they need to do is apply the same steps even though we have variables and exponents to work with. I give each student a random problem to simplify at the end of this class.
• Day 3: Differentiating - by day 3 my students are usually working at different levels.  I divide them into three stations.  The ones who clearly "get it" have an inquiry
station on adding and subtracting rational expressions.  I ask them to develop the steps for adding and subtracting rational expressions with like denominators and with unlike denominators.  They prove their steps with problems they develop and then we reinforce it with problems I provide.  Group two works with formal graphic organizers on adding and subtracting rational expressions, similar to the ones they completed for multiplying and dividing.  Group three works with more multiplying and dividing rational expressions in partners.

• Day 4: Wrap-up - Groups 1 and 2 work with rational expressions puzzles that work on all these concepts.  Group 3 is divided into the ones who get it and group 4 the ones who are still less comfortable.  Group 3 does the inquiry exercise from day 3 and group 4 completes formal notes. By the end of this day everyone should be fairly comfortable with working with rational expressions.
• Day 5 (and 6, maybe): Applications - Finally, we take all of that knowledge of rational expressions and apply it to real-world equations, primarily motion, mixture and work problems.
I like to do formal notes on these types of problems.  Certainly, students can problem solve these, but because these problems are so common on the standardized tests my students will see in the future, it's good to give them some tricks.  Students work in groups on notes and write about how the different parts of the equations work.  It's important for students to explain the math not just memorize it.
• Assessment:  one of my favorite assessments is the graffiti activity which students complete in one day,  Students write and solve all the different types of problems related to rational expressions and equations.
From here we are ready to work with rational expressions and equations for everything from probability and statistics to trigonometry.

*NOTE: Prior to beginning this week, my students have worked with variations and graphing rational functions.

Math Mondays is a bi-weekly blog post (2nd and 4th Monday of each month) sharing tips, ideas, resources, and products for teaching math.  If you have questions or think there is something I should include, you can leave me a message in the comments section below or at the store in the question and answer section.