Month: November 2014

Class Opener – Day 49 – A Magical Prime?

Post author By Bob Lochel
Post date November 14, 2014
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Today is student choice day, where I look through the Edmodo submissions from my students and choose a short and snappy opener. Chris B. submits this number “magic” as a video worth sharing with the class:

What’s going on here? Is it just plain coincidence? Dumb luck? The devil’s work? 37 is certainly not a number we encounter too often in our daily math life.

A student in my afternoon class quickly picked up on the mystery:

37 times 3 is 111, and they all have 111 in them

Good enough. Converting a division example from the video into multiplication helps verify the claim”

777 = 37 * 21
111 * 7 = 37 * 3 * 7

It’s a quiz day, but I leave the class with the following challenge:

Develop a similar pattern for a longer string of identical numbers

They are pretty easy to find, such as the one below for five-digit strings. And Chris provided a low-stress math challenge to get us into math thinking mode before our quiz.

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Class Opener – Day 48 – Can My Students Apply New Ideas?

Post author By Bob Lochel
Post date November 13, 2014
No Comments on Class Opener – Day 48 – Can My Students Apply New Ideas?

Yesterday’s class featured a discussion of absolute value inequalities, and using Desmos to explore problems through graphs. I knew that not all students were totally on board with this method for comparing two functions, but I felt confident that many in the class could now analyze an inequality problem using a graph.

So, the problem on the board when students walked in was intended to see if yesterday’s discussion could transfer to new ideas. Would my students now be able tackle a more difficult problem, or a problem with a similar theme from another chapter – like the polynomial problem I gave as a bonus? Would they impress me with their ability to analyze the inequality?

Nope.

The first two hands raised to offer solutions gave close, yet incorrect answers, as they used Wolfram to “find” an answer – and incorrectly interpreted the output. Other students attempted a combination of factoring / dividing / shuffling of terms to gain some insight. But as these students have only some limited experience with quadratics, extending to the cubic was difficult.

But I’m not surprised, nor at all disappointed. My students have been trained very well in algebra as mechanical steps. The idea that we can analyze a scenario by looking at its graph is much more foreign to them. I only hope that I have started to chip away a bit and get them thinking about multiple perspectives. And by the end of class, I finally noticed some students toying with Desmos and looking at the given cubic. Tomorrow I’ll help them cross that bridge.

Algebra Class Openers

Class Opener – Day 47 – Visualizing Absolute Value Inequalities

Post author By Bob Lochel
Post date November 12, 2014
2 Comments on Class Opener – Day 47 – Visualizing Absolute Value Inequalities

We’re moving through our algebra review, and today is absolute value day! I’m not sure the students are as thrilled about this as I am –

My students have seen absolute value equations and inequalities before, mostly as a stand-alone unit with a series of special rules to memorize. But I find that students have rarely been asked to think about solutions to inequalities as the comparison of values between two functions. So instead of re-hashing some old rules, let’s fire up the netbooks and look at some graphs!

The Desmos desmonstration here (it’s clickable) allows students to experiment with the parameters of an absolute value function, and compare to a constant function. Before diving into some specific problems, I allowed a few minutes for partnerships to play and try to summarize any relationships they saw. Very few saw an immediate link to what we have already been working on – inequalities – and the best was yet to come.

To start thinking about specific inequalitiy problems, I asked students to set the sliders so that they represent the following problem:

$\left | x+2 \right |+3\geq 6$

The graph then lets us analyze the relationship between the absolute value function (dashed green) and the constant function (dashed blue).

Time to find out if my students see the link between the graphs and the inequality. Groups were given 2 minutes of “table talk” to discuss:

How does this graph allow us to find solutions to the given inequality?

This was not a quick discussion. Many students were eager to participate and provide ideas, but many went back to pencil and paper, rather than analyze the graph. Soon, with some students approaching the board, links beween the green and blue functions were found. But, if scaffolding is needed, think about these prompts:

When is the green “above” the blue? What does this indicate?
When is the green “below” the blue? What does this indicate?
Where do the green and blue intersect?

Finally, students began to understand the meaning of the black and purple lines on the graph – representations of the “greater than” and “less than” solutions sets.

In the end, I find that using technology to analyze the visual relationships between functions allows for a deeper understanding than algebraic maniupulation alone. Yet, I am often surprised when students don’t know that this is a valid (not “cheating” or somehow dirty) method of solving an equation. To assess what parts of this lesson “stuck”, I plan to give the following opener tomorrow. Solve for x:

$x^3-4x^2+x+9<3$