Tag: math investigation activities

Posted in GeoGebra worksheets, Geometry

The house of quadrilaterals

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In Investigating an Ordering of Quadrilaterals published in ZDM, Gunter Graumann shared a good activity for developing students mathematical thinking. The activity is about ordering quadrilaterals based on its characteristics. He gave the following list of different aspects of quadrilaterals as possible basis for investigation.

  1. Sides with equal length (two neighbouring or two opposite or three or four sides)
  2. Sum of the length of two sides are equal (two neighbouring or two opposite sides)
  3. Parallel sides (one pair of opposite or two pairs of opposite sides)
  4. Angles with equal measure (one pair or two pairs of neighbouring or opposite angles, three angles or four angles)
  5. Special angle measures (90° – perhaps 60° and 120° with one, two, three or four angles)
  6. Special sum of angle measures (two neighbouring or opposite angles lead to 180°)
  7. Diagonals with equal length
  8. Orthogonal diagonals (diagonals at right angles)
  9. One diagonal bisects the other one or each diagonal bisects the other one,
  10. Symmetry (one, two or four axis’ of symmetry where an axis connects two vertices or two side-midpoints, one or three rotation symmetry, one or two axis’ of sloping symmetry). With a sloping-symmetry there exists a reflection – notabsolutely necessary orthogonal to the axis – which maps the quadrilateral onto itself. For such a sloping reflection the connection of one point and its picture is bisected by the axis and all connections lines point-picture are parallel to each other.

The house of quadrilaterals based on analysis of the different characteristics of its diagonals is shown below. Knowledge of these comes in handy in problem solving.

House of quadrilaterals based on diagonals

Read my post Problem Solving with Quadrilaterals. You will like it.:-)

Posted in Algebra, Geogebra

Teaching with GeoGebra – Investigating coordinates of points

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The most basic mathematics students need to know to understand GeoGebra is the coordinate axes. Must you teach students how to plot points and interpret coordinates of points before they use GeoGebra or the other way around? I think, at the same time. Below is a sample activity on how I think this can be done.  The lesson is about investigating coordinates of points on a Cartesian plane. Its objective is to teach how to use GeoGebra’ s point tool, interpret coordinates of points and make generalizations.

1. Locate the reflections of the points A, B, C, D, E, F, and G if they will be reflected along the y-axis. Use the point button [.A] or the reflect button [.\.] to plot the points.

2. Hover the cursor along the points A to E. These pairs of numbers are called the coordinates of the point. What do you notice about the coodinates of these set of points (A through E)? Will this observation be true to the reflections of A, B, C, D, and E you just plotted?
3. Hover the cursor to the other points. How do the coordinates of the points relate to the values in the x and y axes?
4. In the input bar type P=(5,-2). Before hitting the Enter key, predict the location of the point. Experiment using other coordinates. Use the Move button to drag the grid to see the points you plotted, if they are not visible in the panel.
5. The x and y axes divide the plane into four quadrants. Describe the coordinates of the points located in each quadrant. What about the points along the x -axis and y – axis?

Click here to explore.

Of course, the teacher need to understand a little about GeoGebra first before giving this activity to his/her class.

Posted in Number Sense

Subtracting integers using tables

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In my earlier post on this topic, I discussed why teaching subtraction using the numberline is not helping most students to learn the concept. In this post I describe an alternative way to teaching operations with integers that would help students develop a conceptual understanding of the operation and engage their mind in algebraic thinking at the same time.

The table of operation is one of the most powerful tool for showing number patterns and relationships among numbers, two important components of algebraic thinking. It is a pity that most of the time it is only used for giving students drill on operation of numbers. Some teachers use it to teach operation of integers but more for mastery of skills and to show some beautiful patterns created by the numbers. Below are some ideas you can use to teach operation of integers conceptually as well as engage students in algebraic thinking. I promote teaching mathematics via problem solving in this blog so this post is no different from the rest.  Use the task below to teach subtraction and not after they already know how to do it. Of course it is assumed that students can already do addition.

The question “Which part of the table will you fill-in first?” draws the student attention to consider the relationships among the numbers and to be conscious of the way they work with them. It tells the students that the task is not just about getting the correct answer. It is about being systematic and logical. Engage the students in discussion why they will fill-in particular parts of the table first.

table of integers

Students will either subtract first the same number and this will fill the spaces of zeroes or they can subtract the positive integers. They will of course have to define beforehand which will be the first number (minuend) and which will be the second number (subtrahend).

Surely most students will get stuck when they get to the negatives except with the equal ones which results to zero. You may then ask them to investigate the correctly filled up parts of the table that could be of use to them to fill-in the rest of the table. Students will discover that the numbers are increasing/decreasing regularly and can continue filling-in the rest of the spaces. This is not a difficult task especially if the process for teaching addition was done in the same way. Encourage the class to justify why they think the patterns they discovered makes sense.

The discussion of this topic in continued in Algebraic thinking and subtracting integers – Part 2