Math for Teaching

How to scaffold problem solving in geometry

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Scaffolding is a metaphor for describing a type of facilitating a teacher does to support students’ own making sense of things. It is usually in the form of questions or additional information. In scaffolding learning, we should be careful not to reduce the learning by rote. In the case of problem solving for example, the scaffolds provided should not reduce the problem solving activity into one where students follow procedures disguised as scaffolds. So how much scaffolding should we provide? Where do we stop? Let us consider this problem:

ABCD is a square. E is the midpoint of CD. AE intersects the diagonal BD at F.

List down the polygons formed by segments BD and AE in the square.
How many percent of the area of square ABCD is the area of each of the polygons formed?

Students will have no problem with #1. In #2, I’m sure majority if not all will be able to compare the area of triangles ABD, BCD, AED and quadrilateral ABCE to the area of the square. The tough portion is the area of the other polygons – ABF, AFD, FED, and BCEF.

In a problem solving lesson, it is important to allow the learners to do as much as they can on their own first, and then to intervene and provide assistance only when it is needed. In problems involving geometry, the students difficulty is in visualizing the relationships among shapes. So the scaffolding should be in helping students to visualize the shapes (I actually included #1 as initial help already) but we should never tell the students the relationships among the geometric figures. I created a GeoGebra worksheet to show the possible scaffolding that can be provided so students can answer question #2. Click here to to take you to the GeoGebra worksheet.

Extension of the problem: What if E is 1/4 of its way from C to D? How many percent of the square will be the area of the three triangles and the quadrilateral? How about 1/3? 2/3? Can it be generalized?

Please share with other teachers. I will appreciate feedback so I can improve the activity. Thank you.

Fun with ‘definitions’ in algebra

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WARNING: use the following definitions with great caution.

A number phrase is a mathematical phrase which does not express a complete thought.
An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
A polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
A monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.

I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.

Algebra, GeoGebra worksheets

What is a coordinates system?

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This is the first in the series of posts about teaching mathematics and Geogebra tools at the same time. I’m starting with the most basic of the tools in GeoGebra, the point tool. What would be a better context for this than in learning about the coordinate system. Teacher can use the following introduction about geographic coordinates system and the idea of number line as introduction to this activity.

A coordinates system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric elements. An understanding of coordinate system is very important. For example, a geographic coordinate system enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude and longitude. Sometimes, a third coordinate, the elevation is included. For example:

Philippine Islands are located within the latitude and longitude of 13° 00 N, 122° 00 E. Manila, the capital city of Philippines is 14° 35′ N, 121º 00 E’.

In mathematics we study coordinates systems in order to describe location of points, lines and other geometric elements. The numberline is an example of a coordinates system which describe the location of a point using one number. The coordinates of a point on a numberline tells us the location of a point from zero. But what if the point is not on the line but above of below it? How can we describe exactly the location of that point? This is what this activity is about: how to describe the position of points on a plane.

You would need to familiarize your students first about the GeoGebra window shown below before asking them to work on the GeoGebra worksheet.

Click here to go the GeoGebra worksheet – What are coordinates of points?

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