“A string is wrapped around four nails arranged as the corners of a rectangle. The string’s length is a constant 80 centimeters.”
We pick up where we left off on Wednesday. I ask the students for their questions. They had some great ones, including many I hadn’t considered yet:
What’s the greatest/least possible area of the rectangle?
What’s the greatest/least possible perimeter?
What’s the range of lengths possible for the sides of the rectangle?
What’s the greatest/least possible diagonal of the rectangle? (This was one of the ones I hadn’t considered, and I loved it because my immediate reaction was that the square would be greatest, which of course turns out to be the opposite; the square actually has the least diagonal.)
I wanted to spend most of the time looking at the area question, as that was a good jumping off point for talking about the zeroes/intercepts of the function, how to use the calculator to find the maximum of a function, and then also segue into our lesson for the day: methods for multiplying polynomials.
Today we returned to the textbook to work through several examples multiplying polynomials. My two goals were to continue demonstrating different methods for multiplying the polynomials, and to continue hinting at the patterns students can start to use to both multiply and factor polynomials. The patterns goal will also feed into tomorrow’s lesson on special products.
The three methods we looked at were box method (the abstraction of the area model), FOIL (abstraction of distribution), and stacking (abstraction of stack multiplication). I had students go to the board for several examples, demonstrating the method of their choice. I made sure we had several demonstrations of each method. I emphasized box method, because I think that’s the best one for factoring polynomials, but I told the students the choice was really up to them (in the long run, anyway; in the short run, they had to use each method at least twice on the homework).
To further make my case for box method, we closed with binomial-by-trinomial multiplication and trinomial-by-trinomial multiplication. In both examples, I demonstrated why “FOIL” (especially if you’re married to what the letters stand for) becomes quite unwieldy, while box method scales nicely. I also continued to point out how the degree of the original polynomials affected the degree of the product, and how like terms tend to end up on diagonals, provided the initial polynomials are arranged by order of monomial degree.
Selected problems from the textbook.
Use the patterns we hinted at today as a jumping off point for the difference of squares pattern and perfect square trinomials.
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.
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.
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.
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:
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.
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.
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:
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:
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.
No homework for tonight. Tomorrow, we’ll finish 7-5 lesson and students will do selected homework problems from the textbook.
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.
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.
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?
Continue using the Pythagorean Theorem, now finding the missing sides of various triangles.
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.
(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.”
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.
Introduce the Pythagorean Theorem and use it to classify triangles.
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!)
Before we get into today’s lesson, which was my favorite lesson all week, can I rant about something for a minute?
Why in the world would someone try to teach about square roots without talking about squares? I’m not referring to “squares” as in raising a number to the second power. I’m talking about “squares” as in those pointy shapes with all the sides that match each other.
You will notice that I didn’t even introduce the notation or the term “square root” until my lesson on Tuesday. This was intentional. I don’t want my students to get hung up on this new vocab term or this symbol that kind of looks like a long division sign. No, I just want them thinking about how to find the length of one side of a square that has an area of 73.
So already, on day 1 of this unit, I have students who are accurately estimating square roots. They just don’t know that they’re doing it yet. They think they’re finding the missing side of a square. Once they know that concept and build a strategy to solve that sort of problem, it’s not a major shift to tell students “Ok, that thing you’ve been doing, where you un-square a number? That’s called a square root. And it looks like this check-mark-with-a-bar-next-to-it symbol you’ve noticed on your calculator.”
Conversely, if you start this unit by projecting a slide entitled “Square Roots” and introduce all the formal vocab and symbols from the start, then students don’t have anything to ground their understanding of the operation. You are asking students to use a new symbol to enact a new operation that they’ve never tried before. And forget about asking them to find the missing side of a square – that would be a seemingly-impossible task to those students.
Introduce the challenge first. Once students understand the challenge, then provide the notation and the vocabulary. Don’t dump them both into students’ laps at the same time.
Ok, so this might have been the most awesome thing I’ve done all year. I made a loooooong number line out of a string that stretches almost all the way from my window to my door. I put the numbers 0 and 10 on either side of the number line.
I told kids that I would be showing the whole class a number and then calling on one person to place that number on the number line. Nobody else could talk, but we would take a poll after the student sat down:
Thumbs Up: Perfect!
Thumbs Sideways: Your answer is in the correct order but needs to slide either right or left
Thumbs Down: Your answer is out of order
Then I held up the number 4. I chose 4 on purpose because it’s incredibly familiar to students and yet a bit tricky to place. It’s closer to 0 than to 10, but how much closer? Not to mention, with such a long number line, it’s going to be hard to get the placement exactly right. I was expecting a lot of “Thumbs Sideways” on this first number, and I wasn’t disappointed.
By the way, I use popsicle sticks with names, often called equity sticks, to choose my participants in class for this activity. I know that some teachers feel that equity sticks cause students anxiety, but I think they are worth it for this sort of activity. First of all, they strongly improve engagement. Everyone knows that they could be called up to place the next number, so they are paying attention to each number I present. Secondly, this activity does not have a clear right-or-wrong answer. In fact, I usually end up polling the class and sliding each answer ever-so-slightly in one direction or the other. Since every answer gets improved or amended, the pressure to be exactly right is lowered. Everyone is just making their best guess.
But back to the game. The first student has just placed the number 4, and we have to decide – is it perfect, or should it slide right or left? In my first class I had a student place 4 verrry close to 0. This is a great opportunity to ask students to critique the reasoning of others in a respectful way. I had lots of great comments from students, such as “If 4 was that close to 0, you wouldn’t have room for 1, 2, and 3, and you’d have way too much room for the numbers bigger than 4.” As the lesson went on, the justifications became more precise.
My next choice, root(49), I chose because 7 is exactly halfway between 4 and 10, and I want to see fi my students will pick up on that.
My third choice, root(20), is where the real fun starts. Now students have to use yesterday’s skill of estimating square roots without any benchmarks. Where should root(20) go? Where is 5 on this number line? Where is 6? These are all questions that my students are silently asking themselves as I hold up the card with root(20) on it. At least, it sure seems that way. My students are rapt. They can’t wait to find out if their popsicle stick will get pulled.
From here, my sequence was 6, root(93), root(40), root 4, root(14), root(-4). I’m sure I could have sequenced them better, but I’m not sure how. How would you sequence this activity? Let me know in the comments.
Anyway, I like this lesson for a few reasons:
Student engagement was through the roof. I felt like everyone was with me in a way that almost never happens. Kids were having fun! I even had a group of students ask me in study hall the next day if they could play “the number line game” again
Because the number line is huge, almost nobody placed their card in exactly the right spot on the first try. We always had to shift someone’s card a little to the right or the left. Conversely, almost nobody put their card in the wrong order. So all the students were participating in an activity where nobody got an answer totally wrong, but nobody got an answer totally right. The pressure that students feel when coming to the front of the class was lessened in this case. Sometimes, students had a legitimate difference of opinion and we had to agree to disagree since this activity has an inherent amount of imprecision. But that’s great! I’d rather my students be disagreeing and debating as long as they back up their ideas with some evidence
By placing radicals on a number line, students are beginning to interact with radicals as objects that have a specific value. It’s not just a problem to be solved. Root(14) is a number that is somewhere between 3 and 4. That approach to radicals will be useful in Algebra 1.
I threw root(-4) in as a challenge because I thought it would spark a great debate. I also wanted to add a little bit of new information into the day’s material. Yes, it’s a trick question. Maybe it will feel more memorable to students because they spent 3 minutes arguing over the location of root(-4) before I admitted that there is no place for this answer on the number line. At least, not this number line.
This took about 30 minutes, which sounds crazy, but it was my first time trying a number line, and we had great discussions between every number.
Once we had placed all the numbers on the number line, I pulled more popsicle sticks and got students to sort the numbers on the number line into the categories “Rational” and “Irrational.” This is my attempt to cement the idea of irrational numbers within our existing activity. Kids could sort the numbers perfectly, but I still don’t know if they truly understand what irrational numbers are, and how they actually differ from rational numbers. Something to think about before next year.
Lastly, I got students to create a foldable for their $1 Textbooks to help them classify triangles by their sides and angles. This is vocab that they should know, but it’s going to be vital for tomorrow’s activity, so it’s worth the time investment to make a good foldable.
Note to self: draw triangles in the foldable so kids don’t draw their own “obtuse” triangles that are clearly acute.
IXL Activity on estimating square roots. I am lagging in my implementation of lagging homework. I’ll get better at this.
Classify triangles by their sides and angles, maybe even discover the Pythagorean Theorem?