As part of our rational functions unit in Algebra 2, we stumble on the troubling work, mixture and motion word problems.  You know, if train a leaves at 1 and train b leaves at 3 and...or if one pipe fills a tub at a rate of 20 liters per minute and another pipe drains it at 12 liters per minute...  my students groan when we get to these.  Despite having worked with word problems and real world applications for years leading up to this topic, it is still a struggle for many students.

Students can just memorize the formulas, but for the idea to really stick a little inquiry is helpful. We start with motion which seem the most instinctual. We start with basic but silly problems.

I either put myself, my students or famous people in the problem.
• Ed Sheeran walks at a rate of 1.5km per minute. He arrives at the music studio 30 minutes after he left home.  How far is it from his house to the music studio?
• Eleanor Roosevelt rides 10km to the United Nations.  It takes 20 minutes.  What is the rate at Eleanor Roosevelt rides?
• Thaddeus Lowe rides in a hot air balloon from Paris to New York.   The distance is 5834km.  The balloon flies at an average speed of 30km/hour. How long will the trip take?

Too easy? Yes.  But first students solve them.  Then they set up the problems replacing the numbers with variables.

Next step is to work with multiple parts in motion. The two train problem.  There are little tricks that are helpful here.  For example if the trains start at the same station at different times, students can draw the trains distance.  In this way they see that for the train that started later to catch up to the first train, the trains need to travel the same distance. From here setting up and solving becomes easy.  If the trains are meeting, diagram it as well.

With work and motion problems we follow a similar inquiry method.  I set up the various inquiry types at stations and students rotate through inquiry, notes and practice.

Tips:

1. Skip whole class work: the inquiry can be done in partners or individually. If you give students the directions for each part, they can work on their own and you can walk around to help where needed.
2. Make it fun. You absolutely need to include trains for motion and painting for work (that’s what will be on the standardized tests), but spaceships, hydroplanes, and unicycles are a must. Also use the students, yourself, famous people or characters in the problems.  Two girls is not as entertaining as BeyoncÃ© and Katy Perry.
3. Let students struggle.  We start with the motion because they tend to be the most intuitive, but the work and mixture can be more difficult.  It’s okay for them to struggle.  Encourage them to reason and talk it out, diagram it, try and not succeed to eventually get where they want to be.
4. Follow inquiry with notes.  The inquiry isn’t enough. A formal placeholder in the way of notes is necessary for retention.
5. Practice, practice, practice. I am fundamentally opposed to drill and kill but these kinds of problems require lots of practice either in homework or project form.  Even if you end up with just a worksheet or problem set from your book, have students write their own problems and solve them.  It is an easy addition and helps students in their comprehension.

This is one of several posts on teaching Algebra 2.  Topics in the series include:
Algebra 2 Day 2
Linear Equations and Functions
Systems of Equations