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

Weekly Goals:

Currently, my group of Algebra I students is in the midst of a chapter on polynomials. We’ve just finished our major unit on linear functions, which ranged from graphing and solving linear equations (individually and as parts of systems) to graphing and solving linear inequalities (also individually and as part of systems). This chapter is part of our dive into higher-order polynomials, the primary piece of which will be on quadratic functions and their solutions.

The chapter we’re working on now (Chapter 7, a little over half-way through the Glencoe Texas Algebra I text) is partially a review of what students finished last year looking at: the classification of polynomials (by name and degree) and operations on polynomials. Most students remember the names, but we’ve needed to spend some time reviewing degree of a monomial and then degree of a polynomial. I introduce and review degree as it applies to graphs in the coordinate plane, so we do a little preview of the fundamental theorem of algebra graphically.

I teach at a private classical school, so we don’t necessarily follow Common Core standards. But I make sure our lessons are loosely aligned with standard curriculum, so that as students graduate and move on to different high schools (my school only goes through 8th grade) there’s not a drastic difference in what students encounter. My goals are for students to meet the Common Core standard shown below:

CCSS.MATH.CONTENT.HSA.APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Specifically, I want students to be able to multiply monomials, binomials, and trinomials by the end of this week, and to start to gain familiarity with some special products like perfect squares and the difference of squares.

Monday

Warm-up: Visual Pattern #51 from http://www.visualpatterns.org.

pattern 

Since I’ll be using the area model to demonstrate both how to multiply polynomials and what their products represent, I chose a visual pattern that reduced to a simple rectangle with dimensions n by n + 1 if you rotate the top of the right-most column counter-clockwise. Students who don’t rotate that extra bit of column mostly see the pattern as an n by n square with an n by 1 “antenna,” so we are able to discuss the equivalence between n(n + 1) and n² + n. I relate this equivalence to the distributive property, which will be one of three “methods” we use to multiply monomials and polynomials in today’s lesson.

Activity:

We work primarily out of the textbook most days. Students have a composition notebook meant to be structured like a $1 textbook. They keep a table of contents for each lesson and all their warm-ups in the notebook. Each lesson contains between three and five examples we work together, interspersed with practice problems assigned either for group work and discussion, individual work, or modeling if a particularly ambitious student wants to run through an example. Today, the lesson we’re covering begins with an area model:

textbok

We used algebra tiles last week to model adding and subtracting polynomials, so students are familiar with the representations. I demonstrate three methods for our first example problem: –2x²(3x² – 7x + 10). The first method is the area model. I have students tell me what kind of box I’ll need based on the number of terms (1 by 3), then we quickly review exponent rules as we multiply:

3x² –7x +10
–2x² –6x4 14x3 –20x²

There’s some good class discussion about how the powers of x go in order within the boxes. I ask the students to think through why that might be, and if it will always be true, depending on how we initially write the polynomials. My hope is that they retain this pattern once we branch into binomial or trinomial products, so that they can begin to anticipate the like terms along diagonals. Although it doesn’t come up today, I know when we get to, for example, binomial multiplication, students will trip over why certain boxes get combined while others don’t, and whether to combine those boxes through addition/subtraction or through multiplication. Hopefully, the time we’ve spent today on the difference between the area of the boxes and the lengths that produce those areas will help alleviate those bumps.

Using the same problem, we then do simple distribution, showing the intermediate steps for those students who weren’t quite sure where the “areas” were coming from in the previous model. Then we finish up with a stacking method, demonstrating the similarity between a problem like 372 × 2 and our polynomial problem.
Some students ask why we don’t “carry” like we do with stack multiplication. I put the question back on them and relate stack multiplication back to the distributive property.

Homework

No homework for tonight. Tomorrow, we’ll finish 7-5 lesson and students will do selected homework problems from the textbook.

Tomorrow’s Goal:

Combine what we did today with multiplication of monomials and binomials with what we did last week in adding and subtracting polynomials. So, they’ll be multiplying multiple monomial/binomial or trinomial pairs and then combining like terms. Then move on to equations in which any squared/cubed terms cancel out, and students solve what remains.

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One Week in Mr. Haines’s Math Class – Friday

Friday

Warm-up:

I ask students to look at their worksheet from yesterday and pick one acute triangle, one right triangle, and one obtuse triangle. I then call on 3 kids at random to share their side lengths.

Activity:

I’ve gone through a great deal of discovery work yesterday, so it’s time for a little guided instruction. I give a small lecture about the Greeks, who were faced with a similar problem. Except the Greeks noticed something pretty cool that happens when you square the sides…

I square the sides for each triangle and ask students to look for patterns for 2 minutes. In each class, someone noticed that the right triangle’s short sides added up to the long side when all sides were squared.

So we use that as a springboard to analyze acute and obtuse triangles. Pretty quickly, we see that acute triangles’ short sides add to more than the longest side, whereas obtuse triangles’ short sides add to less than the long side. (I keep saying, over and over, “after you square all the sides” like a broken record because I am terrified of students forgetting that step.)

So now we have a hypothesis. How do we test it? Three new triangles! I get three new triangles from my students and try out our hypothesis, which seems to work!

Time for notes in our $1 Textbooks. We write down the Pythagorean Theorem, a diagram with the legs and hypotenuse defined and labeled, and the rule for classifying triangles using only their sides.

Finally, I give students a final triangle worksheet. In this worksheet, they are given the sides but not provided a ruler. They must use this new classification tool to classify the triangles.

If we have time, I’ve embedded an extension task in this worksheet. In the first problem, we had sides of 6, 8, and 11, which resulted in an obtuse triangle. The second problem, 6, 8, 8, resulted in an acute triangle. What would we need the hypotenuse to be to make a right triangle?

Homework

None

Monday’s Goal:

Continue using the Pythagorean Theorem, now finding the missing sides of various triangles.

One Week in Mr. Haines’s Math Class – Thursday

Thursday

Warm-up:

None – we have a lot to do today!

Activity:

I hand out two worksheets – a set of triangles that I have drawn and a worksheet where students will collect their work for the day. I also give out a ruler to each student. The instructions are simple: Using the centimeter side of the ruler, measure all three sides of each triangle. Then classify each triangle by its sides and by its angles.

Triangles 2

(For the file, click the link above or the Resources section at the bottom of the month)

I had to hand-draw these triangles using a compass and ruler to ensure that the measurements were precise, but that’s fine – I love constructing geometric figures. (In fact, I think kids should spend WAAAAY more time in geometry constructing figures of their own, but that’s a side issue.)

This section of the class whips by pretty quickly, and I was able to help out any students who were struggling with using a ruler. They can all use rulers, but only if I really, truly force them.

On the back of their classification worksheet, I list a bunch of triangles by their sides and ask students to classify these triangles by their sides and angles. Pretty quickly, they realize that all the triangles are scalene, but how can you tell whether they are acute, right or obtuse? Mr. Haines? Mr. Haines? Can you come here?

At this point, I am walking from table to table distributing scratch paper and encouraging students to try to draw each triangle. There is a LOT of trial and error as students draw, then redraw, then redraw their 8, 9, 10 triangles or their 2, 8, 9 triangles. I don’t worry too much about this. After all, I am giving the students a headache so they appreciate the aspirin.

Also, these are some really clever students I’m teaching! In two of my three classes, I had a student come up with the following strategy, which I will paraphrase:

“I pretend the triangle is a right triangle. So I draw the short side and the medium side with a right angle, and I try to connect them with the third side. If the third side reaches perfectly, it’s a right triangle. If it’s too short, the triangle is acute because the short side has to bend down to reach it. If the long side is too long, the triangle is obtuse.”

Triangle Strategy
Pretty cool, right?

By this point, we are edging right up against the bell, so I bring the students together, run through a quick check of their classifications, and then ask them why it took them so long. Lots of grumbling about erasing and redrawing. I mimic every announcer from every infomercial eve: “There’s got to be a better way!”

The bell rings. Tomorrow, we meet Pythagoras.

Homework

None

Tomorrow’s Goal:

Introduce the Pythagorean Theorem and use it to classify triangles.

Resources:

Triangle Worksheet (I tried very hard to get the scale of the triangles to remain after scanning and converting to PDF. I hope this sheet works, but it may depend on your printer. Or you could construct your own!)