Tag Archives: algebraic expressions

Rich and Robust

Coffee beansI recently had the pleasure of learning from Tim Kanold of Stevenson High School fame. I heard him speak on several occasions last fall, and he kept saying that we need to involve students in “rich and robust tasks.” He was addressing the Common Core‘s call to the Standards of Practice. These practices can be summarized by saying that the Common Core is demanding students to think and to communicate their thinking. This can’t get done by taking notes from an overhead and doing the odd problems in the textbook. It gets done by purposefully deciding that students are going to solve rich and robust problems rather than simply watch their teacher complete examples of algorithms.

There is still a time and place for direct instruction and guided practice; but that should not be the complete experience for students, which is what we unfortunately find in the vast majority of American classroom instruction. For quite sometime, MPJ has been producing what we hope to be rich and robust tasks. Due to the growth of the internet, the availability of such rich and robust tasks has expanded tremendously. There are many exotic islands of innovation among the seas of tradition, but the blogosphere has made these islands less remote. Below I have listed a few, alongside my paraphrasing of the some of the Common Core Practices. This is not a comprehensive list by any means, however, I encourage you to take a few minutes and peruse these lessons in order to get a quick taste of what I think Kanold means by Rich and Robust.

Listed here are some additional sites that offer rich and robust tasks. {Note: I will be happy to update this list with any reader-submitted links, subject to review.}

The activities listed above obviously are not your typical math lessons. For good or for bad, the mathematical frontier created by the experiences highlighted here would make for a far different academic education than the gauntlet of lectures that most of us remember from school.

Now, I am going to assume that while the thought of introducing these large-scale examples into one’s repertoire is exhilarating for many, it may be terrifying for some. Let me ease those hearts by saying that rich and robust can be done on a much smaller scale. For example, we could simply ask the students: “Is x times x equal to two x or x squared. In other words, which of the following statements do you think is always true, if either: x·x = 2x  or x·x = x2?”

The CC Practices call for students to construct viable arguments and critique the reasoning of others. If your students stare back at you in silence with this question, then you will know why the Common Core Practices are so needed. If you answer the question for them, then they will watch you participate in a rich and robust activity, while they again participate in mundane note taking. For those that believe that this prompt is too elementary for any course above Algebra, let me assure you that it is not. I posed this very question to my International Baccalaureate students. A handful chose incorrectly, while several “could not remember.” When I asked the rest of the class, which was comprised of some of the brightest seniors on campus, no one could justify their correct answer. The best I got was that they “remember someone teaching us that once.” A simple question turned out to be far more rich and robust than it should have been, but it was a worthwhile day. {Try this one and get back to me.}

I must say here that I am grateful for my math education; it was far better than not having one at all. However, admittedly it was not rich and robust. The question is: Will we make it so for our students? It will take a conscious decision on our part to give our students a different educational experience than most of us had. So ask yourselves: When was the last time that you immersed your students in a rich and robust task? When is the next one planned? Has the time between those two dates been far too long? Are we up to the rich and robust task of offering rich and robust tasks?

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Number Tricks (student sample)

Number Tricks is a lesson that involves writing and simplifying expressions. It demands the higher order thinking skills called for in the Common Core in several ways. 1) The students are to write a mathematical model for a trick given to them. 2) They are to create their own trick and offer the algebraic expression that represents it. 3) It presses the students to understand the concept of a variable; in this case, the variable represents the number originally chosen. 4) The students are asked to compare their simplified expression to the pattern generated by the various numbers tested. The lesson offers a great opportunity for a high level of critical thinking with a rather low level piece of content.

Here is an erroneous submission from my Algebra class. I want to analyze the mistake and discuss why this lesson was so very good for this student even though the “answer was wrong.”

This was Dewey’s response to creating his own Number Trick, including 3 numbers to generate the pattern, and the algebraic expression it represents:

Pick a number 3 10 -7 x
Add 4 7 14 -3 x + 4
Multiply by 2 14 28 -6 2x + 4
Subtract 3 11 25 -9 2x + 4 – 3
Subtract the
Original Number
8 15 -2 2x + 4 – 3 – x
Simplified: x + 1
Common Result: one more than the number picked

Now of course we can see that the student should have included the parenthesis when multiplying by 2. The final expression should have been:

2(x + 4) – 3 – x, simplified: x + 5

So the positives? The student is showing that he is solid in his operations with negative integers, that he can simplify correctly and that he is interpreting the final expression properly (x + 1 means 1 more than the original number). The Big Negative? The pattern of numbers does not support the students simplified expression. The resultant numbers are NOT one more than the original number; they are 5 more.

My judgement call here was to ask Dewey if his expression matches the pattern. He couldn’t answer right away. There was disconnect between generating the expression and actually knowing what the expression represented. Once I pointed out that the last numbers in each column where not 1 more than the first, I asked him to find his own mistake, which he did. Dewey was then able to correctly simplify and without assistance verify that his new expressions supported the pattern of numbers.

Dewey did a great deal of complex thinking with a topic as simple as simplifying expressions.