Program Update: Math with Scott and Virgil

6th – Sixth graders have really been digging into ratios and rates (arguably the most important and useful conceptual understanding that middle schoolers can develop and deepen).  Using number sense, double number lines, and tables to make mathematical comparisons and determine equivalencies, we began by scaling up recipe batches, paint mixes, and drink flavor strengths and have now arrived at comparing prices and converting units.

We will tackle percentages and briefly revisit equivalent fractions before diving deeply into division; its important meaning and its logical practice with whole numbers, fractions, and both positive and negative integers. You may have a chance to reinforce the important question that division asks (and answers): If I start with this many things, how many groups this big can I make?  It sounds obvious, but it’s remarkable how seldom we think about the question and just try to produce “an answer,” and we’re trying to learn how to do both.

We’re also really working on how to be helpful to someone who wants to learn something.  Coaching another mathematician is challenging and it often feels better to just take the thinking opportunity away by showing them what I would do, but learning is likely maximized by asking non-directive questions and letting learners struggle through the thinking themselves.  We’re all still learning how to better do that!

7th – The seventh graders are finishing up their unit on writing and solving equations. They began by examining correspondences between descriptions of situations and tape diagrams, then drew tape diagrams to represent situations in which the variable representing the unknown is specified. They also used hanger diagrams to examine how properties of equality will keep the equation balanced. They used the distributive property to transform an equation of one form into the other and noted how such transformations can be used strategically in solving an equation. Finally, they wrote equations in order to solve problems involving percentage increases and decreases.

8th – I couldn’t help thinking, as my ski group alternately barreled and floundered around the big loop at Cabin Creek, how much the adventure felt like a metaphor for Algebra class: How do we respond to struggle? How do we support a friend who is having a difficult moment?  How can we all benefit and progress even if some folks ski faster than others?  Are the faster skiers actually learning and using techniques that conserve energy or just moving unthinkingly fast for the thrill of covering ground quickly?  Additionally, I noticed the literal positive and negative slopes we encountered and had a few golden opportunities to think about parallel and perpendicular.

Back in the classroom, we returned to functions and the language interpretation required to navigate them.  This knowledge allows us to discuss limitations on domain and range, to find inverse functions, and to efficiently evaluate things like f(3) and g(-½) – all of which are just different ways to practice graphing, solving for multiple variables, and generating and evaluating equations.  This function exploration has also allowed us to move beyond linear equations.  We examined piecewise functions and are ready now to apply our understanding to exponential functions and ultimately to quadratics.  

Eighth graders are thinking about ninth grade course choices and some are preparing for high school placement tests.  They may want to explore some topics on their own (and might need parental support for that) or practice things that feel fundamentally shaky (and might require parental support) in preparation.  I am available for consultations and appointments to discuss strategies for supplementing what we are exploring in class.