Tag: linking algebra and geometry

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.

Posted in Geogebra, Geometry

Geometric relations – angles made by transversal

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Geometry is a natural area of mathematics for which students should develop reasoning and justification skills and their appreciation of the logico-deductive part of mathematics that build across the grades. Learning tasks therefore should be so designed so that the focus of the learning is on the development of these skills as well and not merely on the learning of facts.

Consider the GeoGebra applets in Figures 1 and 2 below. Which of them will you use for teaching the relationships among the angles made by transversal with parallel lines? Before this lesson of course, the students already learned about linear pairs. Click the figures below to explore the applets before you continue reading.

In the first figure, dragging D or F along the parallel lines, the students will observe that there are angles that will always be equal. Thus from this, they can make the following conjectures:

(1) the alternate interior angles are equal;

(2) the vertical angles are equal;

(3) the corresponding angles (a pair of interior and exterior angles on the same side of the transversal) are equal; and,

(4) the pair of exterior and interior angles on the same side of the transversal sum up to 180 degrees.

In all these cases, the students are reasoning inductively. They will generalize from the measures they observed. Because of this, there seem to be no need for proof since there were bases for the generalizations. The measures of the angles. In this activity students will have learned geometric facts but not the geometric reasoning. Inductive reasoning maybe, but not deductive reasoning.

Contrast the first applet  to the second one. Dragging D or F along the parallel lines, the students will observe that the sum of the pair of exterior -interior angles on the same side of the transversal is always 180 degrees. They will also observe that the other angles also changes. The teacher can then challenge the students to make predictions about the measures of these angles and the relationships among them. These will create a need for proof.

And how should the proof look like? My suggestion is not to be very formal about it like using a two-column proof. For example, to prove that measures of vertical angles are always equal they can set up their proof like these:

To prove p = t:

p + s = 180

s + t = 180

p + s = s + t

p = t.

Students can very well set-up an explanation like this. They have seen it when they learned about solving systems of linear equation. What more, it uses the very important property of equality – the transitive property: If a = c, and b = c, then a = b. Great way to link algebra and geometry.

Posted in Algebra

Teaching algebraic expressions – Counting smileys

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This is an introductory lesson for teaching the concept of variable and algebraic expressions through problem solving. The problem solving task combines numerical, geometric, and algebraic thinking.  The figure below shows the standard version of the task. Of course some easier versions would ask for the 5th figure, then perhaps 10th figure, then the 100th figure, and then finally for the nth figure. This actually depends on the mathematical maturity of the students.

An alternative version which I strongly encourage that teachers should try is to simply show first the diagrams only (see below).

Study the figures from left to right. How is it growing? Can you think of systematic ways of counting the number of smileys for a particular “Y” that belongs to the group? This way it will be the students who will think of which quantity (maybe the number of smileys in the trunk of the Y or the position of the figure) they could represent with n.The students are also given chance to study the figures, what is common among them, and how they are related to one another. These are important mathematical thinking experiences. They teach the students to be analytical and to be always on the lookout for patterns and relationships. These are important mathematical habits of mind.

Here are possible ways of counting the number of smileys: The n represents the figure number or the number of smiley at the trunk.

1. Comparing the smileys at the trunk and those at the branches.

In this solution, the smileys at the branches is one less than those at the trunk. But there are two branches so to count the number of smileys, add the smileys at the trunk which is n to those at the two branches, each with (n-1) smileys. Hence, the algebraic expression representing the number of smileys at the nth figure is n+2(n-1).

2. Identifying the common feature of the Y’s.

The Y’s have a smiley at the center and has three branches with equal number of smileys. In Fig 1, there are no smiley. In Fig 2, there is one smiley at each branch. In fact in a particular figure, the number of smileys at the branches is (n-1), where n is the figure number. Hence the algebraic expression representing the number of smileys is 1+ 3(n-1).

3. Completing the Y’s.

This is one of my favorite strategy for counting and for solving problems about area. This kind of thinking of completing something into a figure that makes calculation easier and then removing what were added is applicable to many problems in mathematics. By adding one smiley at each of the branches, the number of smileys becomes equal to that at the trunk. If n represents the smileys at the trunk (it could also be the figure number) then the algebraic representation for counting the number of smilesy needed to build the Y figure with n smiley at each branches and trunk is 3n-2, 2 being the number of smileys added.

4. Who says you’re stuck with Y”s?

This is why I love mathematics. It makes you think outside the box. The task is to count smileys. It didn’t say you can not change or transform the figure. So in this solution the smileys are arrange into an array. With a rectangular array (note that two smileys were added to make a rectangle), it would be easy to count the smileys. The base is kept at 3 smileys and the height corresponds to the figure number. Hence the algebraic expression is (3xn)-2 or 3n-2.

The solutions show different visualization of the diagram, different but equivalent algebraic expressions, and all yielding the same solution. Of course there are other solutions like making a table of values but if the objective is to give meaning to algebraic symbols, operations, and processes, it’s best to use the visuals.

A more challenging activity involved Counting Hexagons. Click the link if you want to try it with your class.

Posted in Algebra, Geogebra, Geometry, High school mathematics

Teaching with GeoGebra: Squares and Square Roots

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This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator. However, if the students have access to computers then I highly recommend that you use GeoGebra to do this. In my post GeoGebra and Mathematics, I argued that the more the students understand the mathematics behind GeoGebra, the more confident they could become in using this tool. The earlier the exposure to this environment, the better. The way to do this is to integrate the learning of the tool in learning mathematics.

The figure below is the result of the final activity in my proposed teaching sequence for teaching square roots of numbers and some surds or irrational numbers. The GeoGebra tool that the students is expected to learn is the tool for constructing general polygons and regular polygons (the one in the middle of the toolbar).

Squares and Square Roots

The teaching sequence is composed of four activities.

Activity 1 involves exploration of the two polygon tools: polygons and regular polygons. To draw a polygon using the polygon tool is the same as drawing polygons using a ruler. You draw two pints then you use the ruler/straight edge to draw a side. But with Geogebra you click the points to determine the corners of the polygon and Geogebra will draw the lines for you. In the algebra window you will see the length of the segment and the area of the polygon. Click here to explore.

GeoGebra shows further its intelligence and economy of steps in Activity 2 which involves drawing regular polygons. Using the regular polygon tool and then clicking two points in the drawing pad, GeoGebra will ask for the number of sides of the polygon. All the students need to do is to type the number of sides of their choice and presto they will have a regular polygon. Click here to explore.

Activity 3 is the main activity which involves solving the problem Draw a square which is double the area of another square. Click here to take you to the task.

Activity 4 consolidates ideas in Activity 3. Ask the students to click File then New to get a new window from the previous activity’s applet then ask them to draw the figure above – Squares and Square Roots.  You can also use the figure to compare geometrically the values of \sqrt{2} and 2 or  show that \sqrt{8} = 2\sqrt{2}. This activity can be extended to teach addition of radicals.

Like the rest of the activities I post here, the learning of mathematics, in this case the square roots of numbers, is in the context of solving a problem. The activities link number, algebra, geometry and technology. Click here for the sequel of this post.

This is the second in the series of posts about integrating the teaching of GeoGebra and  Mathematics in lower secondary school. The first post was about teaching the point tool and investigating coordinates of points in a Cartesian plane.

GeoGebra book:

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra