The Parentheses Challenge

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Objective

SWBAT use grouping symbols to change the values of otherwise equivalent expressions.

Big Idea

Let's play a game: given an expression, can you place one set of parentheses that will give that expression the greatest possible value?

First 10 Minutes: Finish and Submit Number Line Problems

10 minutes

Yesterday's classwork and last night's homework was to complete the Number Line Problem Set, which was our first problem set of the year.  I tell the class that I'm going to collect it after the first 10 minutes of class.  The main reason I provide this time is so I can answer questions.  Some students will have questions about problems sets in general, and how to hand this in.  Others will have specific mathematical questions, and I am definitely open to giving a solution or two on the board.  Even though I'm about to collect and grade this problem set, I'm much more interested in using this work as a learning opportunity than I am as a hard-line assessment.  If giving away and explaining an answer or two will help toward that goal, that's great!

Today's opener is on the board for everyone as soon as they've submitted their work, and as I circulate to collect work, I also circulate to explain the opener.

Opener: What do Parentheses do to this Expression?

5 minutes

Today's opener simply instructs students to write the expression 2+3*4+5*6 four times in their notes.  When I see that students have copied the opener, I give them the instructions one piece at a time.  The first is to evaluate this expression.  I make sure that they get 44, and if they don't we troubleshoot.  

The next instruction is to place one set of parentheses in this expression that will give it the greatest possible value.  I try to start little competitions between table-mates, and I encourage opponents to check each other's work.

This part is fun.  If you haven't tried a problem like this on your own, I recommend it!  It's also another great opportunity to find and correct errors in student thinking.

Here, for example, is some work that one of my students put on the board.  

What do they know?

What do they think?

What would you tell them?

The Parentheses Challenge

28 minutes

Until we settle into the CCSS Assessments that will be implemented over the next few years, my ninth grade Algebra 1 course serves as the first year in two-year sequence that culminates with the Massachusetts Comprehensive Assessment System (MCAS) for 10th Grade Mathematics.  There are a lot of great problems on these exams, and my idea for today's activity comes from one: problem #17 from the 2011 Grade 10 MCAS.  I call this activity the "Parentheses Challenge".

Here are my instructions and problems for the Parentheses Challenge on a Prezi.  It's pretty self-explanatory, and kids at all levels have a lot of fun with this.  Essentially, it's a competition done in teams of two, in which students try to place one set of parentheses in each expression to get the greatest possible value.  The competition aspect - when used sparingly and in good fun - is good for getting kids involved, and the team aspect gets them thinking about and arguing for their own ideas.  The fact that they have to verify their work - and will often try a few different possibilities for each problem - means that they're getting some practice.  The running of the clock lends urgency, while the light spirit of the endeavor keeps it from being threatening.

I may adjust the amount of time I give each class on each problem depending what I've seen of their skills so far, but I've found that three minutes per problem and a few extra minutes at the end is usually enough for students to get something done.

When I see students struggling, I'm quick to provide some hints, my goal is for this to feel fun.  By providing some solutions after students turn in their work, I make this learning opportunity by sharing two key ideas to minds that are excited to hear about them.

(Spoiler Alert - Stop reading if you'd like to try these on your own!)

 

 

 

 

After I collect work, I show two solutions that are surprising to most students.  First, on #4, I place parentheses around the 3*2 in the bottom of the fraction.  I explain that no matter how great you can make the top part of this expression, nothing has a more more powerful effect here than dividing by 1 instead of 6.  On #5, show students how, just because there's only one set of parentheses, that doesn't mean it can't span a long part of the expression.  We want to try to multiply that 17 at the end by the greatest possible number.  How can we use parentheses to accomplish that?

 

Source for MCAS Release Task (Accessed October 10 2013)