Category: Algebra

Posted in Algebra

Why negative times a negative is positive

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Among the ‘rules’ for working with negative numbers,  the most counter intuitive is “negative times a negative is a positive”. It is easily forgotten especially if it was learned by rote. It is also not an easy ‘rule’ to make sense of so it needs to be learned with conceptual understanding. Here’s my proposed lesson for teaching multiplication of integers. This lesson takes from the lesson Subtracting integers using tables- Part 1 and Algebraic thinking and subtracting integers – Part 2. Note that this lesson like the rest of the lessons in this blog is not just about students learning the math but more about them engaging in mathematical thinking processes such as searching for patterns, making generalization, reasoning, making connections, etc.

Set the task

Fill up this table  by multiplying the numbers in the first column to the number in the first row. Start filling up the rows or columns you think would be easier to do.

For discussion purposes divide the table into 4 quadrants. The top right quadrant is Quadrant 1, top left is Quadrant 2, bottom left is Quadrant 3, and bottom right is Quadrant 4. This is also one way of leading the students to consider filling-up the quadrants according to their number label.

Explore, Observe, Explain why

Students are more likely to fill-up Quadrant 1 because the numbers to be multiplied are both positive. The next quadrant they are more likely to fill-up is Quadrant 2 or 4. You may want to give the following questions to scaffold their thinking: What do you observe about the row of numbers in Quadrant 1? How can it help you fill up quadrant 2? Do the numbers you put in Quadrant 2 make sense? What does 3 x -2 mean? What about in Quadrant 4? 

From Quadrant 2 students are more likely to fill up Quadrant 3 or Quadrant 4 by invoking the pattern. Questions for discussion:   Do the numbers in Quadrant 4 make sense? What does -3 x 2 mean? This is one way of making the students be aware that commutativity holds in the set of integers. The problematic part are the numbers in Quadrant 3. None of the previous arguments are useful to justify why negative times negative is positive except by following the patterns. But this explanation will be enough for most students. You can also use the explanation below.

Revisit the rule when teaching another topic

We know that 8 x 8 = 64. This means that (10-2)(10-2)=64.  By distributive property, (10-2)(10-2)= 100+-2(10) + – 2(10)+ ____ = 64. Previously students learned that -2(10)= 20. Hence, 100 + -40+___= 64. What should go in the blank must be 4. So (-2)(-2) = 4. This proof was first actually proposed by Maestro Dardi of Pisa in year 1334. In explaining this to students I suggest rewriting (10-2)(10-2) to (10+-2)(10+-2) to reinforce the distinction between the dash sign as minus and as symbol denoting ‘negative’.

Girolamo Cardano sometime in 1545 proposed a geometric interpretation of this operation. He argued that (10-2)(10-2) can be interpreted as cutting off 2 strips of 2 x 10 rectangles from the two sides of the 10 x 10 square. Cutting the rectangles like these meant cutting the  2 x 2 square twice so you need to return back the other square. The figure below shows this. This proof by Cardano is usually used to teach the identity square of a difference (x-y)(x-y)=x^2-2xy+y^2. This is a good opportunity to revisit the rule negative times a negative is positive.

Reference to history is from the paper Historical objection to the number line by Albrecht Heefer.

Posted in Algebra

How to grow algebra eyes and ears

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Math teachers should grow algebra eyes and ears.  To have algebra eyes and ears means to be always on the lookout for opportunities for students to engage in  algebraic thinking which involves thinking in terms of generality and to reason in terms of relationships and structure, etc. In the post Teaching algebraic thinking without the x’s I described some tips on how to engage pupils in algebraic thinking as they learn about numbers. Likewise in Algebraic thinking and subtracting integers and Properties of Equality – do you need them to solve equation?

Here is another example. How will you use this number patterns in your algebra 1 class so students will also grow algebra eyes and ears?

Let me share how I teach this. I like to simply post this kind of patterns on the blackboard without any instruction. For a few seconds students would normally not do anything and wait for instruction but getting none would start scribbling on their notebooks. When asked what they’re doing they would tell me they are generating other examples to check if the the pattern they see works (yes, detecting patterns is a natural tendency of the mind). When I asked what’s the  pattern and how they are generating the examples I sometimes get this reasoning:  the first and the second columns increase by 1 so the next must be 5 and 6 respectively, the third and fourth columns increase first by 6, then by 8 so the next one must increase by 10 so the next numbers must be 30 and 31 respectively. That is, 5^2 + 6^2 +30^2 = 31^2. Of course this is not what I want so I would ask them if there are other ways of generating examples that does not depend on any of the previous cases.

In generating examples, students usually start with the leftmost number. I would challenge them to start from any terms in the equation. After this, if no one thought of proving that the pattern will work for all cases, then I’ll ask them to prove it. It would be easier for me and for them if I will already write the following equation at the bottom of the pattern for students to fill up and prove but this method is for the lazy and lousy teacher. A good algebra teacher never gives in to this temptation of doing the thinking of representing an unknown by a letter symbol for their students.

In proving the identity, I have observed that students will automatically simplify everything so they end up with fourth degree expressions. This is another opportunity to challenge the students: show that the left hand side and right hand side simplifies to identical second degree expressions with only their knowledge of square of the sum (a+b)^2 = a^2+2ab+b^2.

The teaching sequence I just described is consistent with the levels of understanding of equation I described in Assessing understanding of function in equation form.

Posted in Algebra

How to teach the inverse function

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In What  is an inverse function? I proposed a way of teaching this concept starting with its graphical representations using GeoGebra applets. Al-Zboun Lilliana in our Linkedin group shares her idea for introducing the inverse function. She says that the most difficult part in teaching this concept is to make it make sense to students and not so much in making the students understand its definition or teaching them the process of finding the inverse function of a given function(by a graph or by a formula)  or to “verify algebraically” that the functions are inverses.

Here’s her proposed teaching sequence starting with examples that students can relate to in Levels 1 and 2. I would suggest inserting the activities I described in What is an inverse function? before Level 3 which introduces the algebraic solution.

Examples SET(1)-Level 1:
1. If we need to call someone we are asking for her/his name on the list of our phone contacts …
2. If someone of our contacts is calling us “our phone shows who is calling” This is the job of an inverse function: “finding the name corresponding to the number”

Examples SET(2)- Level 2:

1. If George makes $100/day. We know how to answer questions such as “After 7 days, how much money has he made?” We use the function W(t)=100t
But suppose I want to ask the reverse question:
2. “If George has made $700, how many hours has he worked?” The students know the answer : Time : t(W)=W/100. Given any amount of money, divide it by 100 to find how many days he has worked.
This is the job of an inverse function. It gives the same relationship, but reverses the dependent and independent variables.

Examples SET(3)- Level 3:

In this set the teacher includes examples to show simplifying solutions of mathematical questions

Example (1): Solve log (3 x – 2) = 3
• Since logarithmic and exponential functions are inverses of each other, we can write the following.

a = log (b) if and only b = 10^a
• Use the above property of logarithmic and exponential functions to rewrite the given equation as follows.
3x – 2 = 10^3
• Solve for x to obtain.

3x = 1002
x= 1002÷3=334

Example (2): Find the Range of the function ( or any RATIONAL function) :
F(x)= (3x+1)/(3 -x) or [y=(3x+1)/(3 -x)]
• Since the RANGE of a one to one function is the DOMAIN of its inverse. Let us first show that function f given above is a one to one function.
• Hence the given function is a one to one. let us find its inverse.

• Interchange x and y and solve for y.
x =(3y+1)/(3 -y)
And find y = (3x-1)/(3+x)
The inverse g(x) of function f(x) is given by.

g(x) = (3x-1)/(3+x)
• The domain of g(x) is R except x = -3. Hence THE RANGE of f(x) is R/{-3}.

Posted in Algebra

History of algebra as framework for teaching it?

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In many history texts, algebra is considered to have three stages in its historical development:

  1. The rhetorical stage –  the stage where are all statements and arguments are made in words and sentences
  2. The syncopated stage – the stage where some abbreviations are used when dealing with algebraic expressions.
  3. The symbolic stage – the stage where there is total symbolization – all numbers, operations, relationships are expressed through a set of easily recognized symbols, and manipulations on the symbols take place according to well-understood rules.

These stages  are reflected in some textbooks and in our own lesson. For example in in pattern-searching activities that we ask our students to express the patterns and relationships observed using words initially. From the students’ statements we can highlight the key words (the quantities and the mathematical relationships) which we shall later ask the students to represent sometimes in diagrams first and then in symbols. I have used this technique many times and it does seem to work. But I have also seen lessons which goes the other way around, starting from the symbolic stage!

Apart from the three stages, another way of looking at algebra is as proposed by Victor Katz in his paper Stages in the History of Algebra and some Implications for Teaching. Katz argued that besides these three stages of expressing algebraic ideas, there are four conceptual stages that have happened along side of these changes in expressions. These conceptual stages are

  1. The geometric stage, where most of the concepts of algebra are geometric;
  2. The static equation-solving stage, where the goal is to find numbers satisfying certain relationships;
  3. The dynamic function stage, where motion seems to be an underlying idea; and finally
  4. The abstract stage, where structure is the goal.

Katz made it clear that naturally, neither these stages nor the earlier three are disjoint from one another and that there is always some overlap. These four stages are of course about the evolution of algebra but I think it can also be used as framework for designing instruction. For example in Visual representations of the difference of two squares, I started with geometric representations. Using the stages as framework, the next lesson should be about giving numerical value to the area so that students can generate values for x and y. Depending on your topic you can stretch the lesson to teach about functional relationship between x and y and then focus on the structure of the expression of the difference of two squares.

I always like teaching algebra using geometry as context so geometric stage should be first indeed. But I think Katz stages 2 and 3 can be switched depending on the topic. The abstraction part of course should always be last.

You may want to read Should historical evolution of math concepts inform teaching? In that post I cited some studies that supports the approach of taking into consideration the evolution of the concept in designing instruction.

For your reading leisure – Unknown Quantity: A Real and Imaginary History of Algebra.

For serious reading Classical Algebra: Its Nature, Origins, and Uses and of course Victor Katz book History of Mathematics: Brief Version.