One Week in Mr. Ricchuiti’s Math Class – Wednesday

Warm-up: Algebra Tiles

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:

Algebra Tiles 2

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.

Tomorrow’s Goal:

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.


One Week in Mr. Ricchuiti’s Math Class – Tuesday

Warm-up: Area Model


I projected the document above (in my own beautiful hand-writing) and asked the students to compute in their notebooks the area of each figure. The first rectangle got a chorus of “Is this a trick question?” Which, of course, it’s not, I just wanted to make sure they remembered how to compute the area of a rectangle. Then we moved on to the second rectangle, the 17 × 31. The students weren’t allowed to use calculators, so my hope was that they would stumble into distribution on their own (17 × 30 + 17 × 1). Some students did, and I called on one to explain their work to the class. Then I let that student also explain the area of the first figure with a variable, the 17 × (x + 1) rectangle.

We continued on with the examples, each time discussing any additional complexity, reviewing the different ways we multiplied monomials and polynomials yesterday. Finally, on the last example, a student asked my favorite question of the day:

“If you ask for most simplified form, which one do you want?”

On the board, a student had written “2y(3y + 5) = 6y² + 10y”. I answered that neither was really “more simplified” and that either may be called for depending on the context. I used the opportunity to give the forms different names. I called the left side “intercept form” and the right side “area form” and talked about how they could be used different ways. On the left, given that the expressions were describing dimensions of a rectangle, we talked about some limitations on what values y could take. On the right, I graphed the parabola in Desmos so we could both see the possible areas. I wished after the fact that I had used a parabola with intercepts on the positive x-axis opening downward, but I’ll have to save that for next year instead.


We picked up where we left off in the textbook yesterday. There were three more examples to work out of the chapter. I didn’t love any of them, but the first gave another opportunity to review addition and subtraction of polynomials:

4(3d² + 5d) – d(d² – 7d + 12)

I assigned different methods for multiplying out the polynomials to different groups of students, and then chose a representative of each group to put their work on the board. We reviewed the area method (which I started calling box method), distribution, and stacking for the multiplication, and then did the addition and subtraction of like terms together. Tonight for homework they’ll work some problems like this area question which I like a lot more:


I skipped the second of the three examples since it was anachronistic pseudocontext:

Greg pays a fee of $20 a month for local calls. Long-distance rates are 6¢ per minute for in-state calls and 5¢ per minute for out-of-state calls. Suppose Greg makes 300 minutes of long-distance phone calls in January and m of those minutes are for in-state calls…

My students were all born post-2002. They don’t have the vaguest idea why the state in which someone lives would affect the cost of a phone call.

We took long on the warm-up anyway, so I jumped into the third example: solving an equation with polynomials on both sides (but one where the squared or cubed terms conveniently cancel out). The example in question was:

y(y – 12) + y(y + 2) +25 = 2y(y + 5) – 15

After working through it rather quickly, a student asked why it didn’t have two solutions (or rather, told me it should have two solutions) based on our brief discussion yesterday of the fundamental theorem of algebra. I asked them why they thought it might not, and encouraged them to think about the question graphically. I got some vague responses about parallel lines, so I encouraged them to go ahead and graph both sides of the equation in their calculators (TI N-spires), and, behold!, they were kind of parallel outside of the obvious intersection.


Selected problems out of the textbook.

Tomorrow’s Goal:

I think we’ll bust out the old algebra tiles and talk more about box method for multiplying polynomials rather than just monomial-by-polynomial.