Warm-up: Algebra Tiles
Students began with a warm-up in which they placed algebra tiles into the appropriate slots based on the length and width (two binomials). This called back to yesterday’s area warm-up, but in a more formal way that will lead into the “box method” for multiplying polynomials.
Following the warm-up that students worked on their own, we do two more examples together. The first example contains two binomials with negative constants, so we continue to use the algebra tiles, but talk about the negative x-terms and the positive constant. As we do this, I’m looking out for students who start to grok the patterns (product of the constants as the third term, sum of the constant as the coefficient of the middle term). We will revisit these patterns both when we look at special patterns on Friday, and in the next chapter when we use box method to factor trinomials.
The second (and final) example we do together contains a non-1 coefficient of x in one of the binomials, and a difference in signs. We discuss how both of these items affect the value of the coefficient of the middle term, using stack multiplication of the binomials to illustrate it.
From here, students have about 15 minutes to work at their table groups on a set of six problems. They can continue to use the algebra tiles (for those still struggling to make the connection to the more abstract box method), but do not have to. Several students have already started their own form of FOIL using distribution.
With about 20 minutes to go, I re-direct them to a homework problem that gave several students difficulty the day before:
The question actually begins by describing boxes the post office will and will not accept, including information about “girth” and some other vocab that seems to throw students off. I simplify things by sticking to “length”, “width”, and “height” and throw out the post office’s limitations. I ask students only for limitations based on the piece of cardboard we’re forming the box out of. We get a lower bound (0) and two upper bounds (20 and 30). We discuss what it really means to have two “upper” bounds, and I ask the students for the volume of the box as a function of the edge of the square x. After a few minutes, we get a function together:
V = (60 – 2x)(40 – 2x)(x)
…and graph it in our calculators (I do mine on a smart board). We don’t multiply it out, which will be more along the lines of tomorrow’s lesson, but I call back to what I had called “intercept form” the day before, and when we graph the volume, the students can see why it’s called intercept form. We discuss how the function behaves between the bounds we had discussed before. We discussed what kinds of area don’t really make sense in the context of the problem (namely, negative areas and the growing-to-infinity areas you get after x = 30). I ask them to use the calculator to find the box’s maximum volume.
Students were assigned an open-ended problem based on the one we did in class together: “A string is wrapped around four nails arranged as the corners of a rectangle. The string’s length is a constant 80 centimeters.” They are to write as many questions as they can think of about the scenario down, and then attempt to answer those questions.
Formalize what students have been doing implicitly with the algebra tiles into a more abstract “box method.” In other words, I want students to be able to use an area model without actually counting out algebra tiles or individual rectangles.