Tag: teaching algebra

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

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.

 

Posted in Algebra, Geogebra, High school mathematics

Embedding the idea of functions in geometry lessons

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GeoGebra is a great tool to promote a way of thinking and reasoning about shapes. It provides an environment where students can observe and describe the relationships within and among geometric shapes, analyze what changes and what stays the same when shapes are transformed, and make generalizations.

When shapes or objects are transformed or moved, their properties such as location, length, angles, perimeters, and area changes. These properties are quantifiable and may vary with each other. It is therefore possible to design a lesson with GeoGebra which can be used to teach geometry concepts and the concepts of variables and functions. Noticing varying quantities is a pre-requisite skill towards understanding function and using it to model real life situations. Noticing varying quantities is as important as pattern recognition. Below is an example of such activity. I created this worksheet to model the movement of the structure of a collapsible chair which I describe in this Collapsible  chair model.

Show angle CFB then move C. Express angle CFB in terms of ?, the measure of FCB. Show the next angle EFB then move C. Express EFB in terms of ?. Do the same for angle FBG.
[iframe https://math4teaching.com/wp-content/uploads/2011/07/locus_and_function.html 700 400]
Because CFB depends on FCB, the measure of CFB is a function of ?. That is f(?) = 180-2?. Note that the triangle formed is isosceles. Likewise, the measure of angle EFB is a function of ?. We can write this as g(?) = 2?. Let h be the function that defines the relationship between FCB and FBG. So, h(?)=180-?. Of course you would want the students to graph the function. Don’t forget to talk about domain and range. You may also ask students to find a function that relates f and g.

For the geometry use of this worksheet, read the post Problems about Perpendicular Segments. Note that you can also use this to help the students learn about exterior angle theorem.

Posted in Algebra, Geogebra

Solving algebra problems – which one should be x?

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Every now and then I get an e-mail from a friend’s son asking for help in algebra problems. When it’s about solving word problems, the email will start with “How about just telling me which one is the x  and I’ll figure out the rest”. The follow-up email will open with “Done it. Thanks. All I need is the equation and I can solve the problem”. The third and final e-mail will be “Cool”. Of course I let this happen only when I’m very busy. Most times I try to explain to him how to represent the problem and set-up an equation. Here’s our latest exchange.

Josh: What is the measure of an angle if twice its supplement is 30 degrees wider than five times its complement? All I need is to know which one’s  the x.

Me: How about sending me a drawing of the angle with its complement and supplement?

Josh: Is this ok?

Me: Great. Let me use your drawing to make a dynamic version using GeoGebra. Explore the applet below by dragging the point in the slider. What do you notice about the values of the angles? Which angle depend on which angle for its measure? If one of the measure of one of the angles is represented by x, how will you represent the other angles? (Click here for the procedure of embedding applet]
[iframe https://math4teaching.com/wp-content/uploads/2011/07/angle_pairs1.html 650 435]

Josh: They are all changing. The blue angle depends on the green angle. Their sum is 90 degrees. The red angle also depends on the green angle.  Their sum is 180 degrees. The measure of the red angle also depends on blue angle.

Me: Excellent. Which of the three angles should be your x so that you can represent the others in terms of x also? Show it in the drawing.

Josh: I guess the green one should be x. The blue should be 90-x and the red angle should be 180-x.

Me: Good. The problem says that twice the measure of the supplement is 30 degrees wider than five times the complement. Which symbol >, <, or = goes to the blank and why, to describe the relationship between the representations of twice the supplement and five times the complement:

2(180-x) _____ 5(90-x)

Josh: > because it is 30 degrees more.

Me: Good. Now, what will you do so that they balance, that is make them equal?  Remember that  2(180-x) is “bigger” by 30 degrees? What would the equation look like?

Josh: I can take away 30 degrees from 180-x. My equation would be (180-x) -30 = 5(90-x)?

Me: Is that the only way of making them equal?

Josh: Of course I can add 30 to 5(90-x). I will have 180-x = 5(90-x)+30.

Me: You said  you can do the rest. Try it using both equations and tell me the value of your x and the measures of the three angles.

Josh: x = 40. That’s the angle. It’s complement measures 50 degrees and its supplement is 140 degrees. They’re the same for both equations.

Me: Does it makes sense? Do you think it satisfies the condition set in the problem?

Josh: 2(140) = 280. 5(50) = 250. 280 is 30 degrees wider than an angle of 250 degrees. Cool.

Me: What if you make A’DC your x? Do you think you will get the same answer?

No reply. I guess I’ll have to wait till the teacher give another homework to get another e-mail from him.

I don’t know if the questions I asked Josh will work with other students. Try it yourself. Please share or send this post to your co-teachers. Thanks. I will appreciate feedback.

Problem solving is the heart of mathematics yet it is one of the least emphasized activity. Solving problems are usually relegated at the end of the textbooks and chapters.