One of the ways to connect concepts in mathematics is to make use of the same representation to teach mathematics. Let me take for an example the visual representation of the square of a sum, . This concept is usually ‘concretized’ using algebra tiles. However, if you have facility for computer technology I would recommend using the GeoGebra applet below: [iframe https://math4teaching.com/wp-content/uploads/2011/09/square_of_a_sum.html 650 550]
1. You can show the grid (click view to do that) to teach Grade 3 pupils about area. You can change the dimensions of the rectangles and squares by dragging D.
2. For older students you can give this task (don’t show grid but you can show lengths then drag D): The square AGHC is dissected into rectangles and squares.If the sides of square AEFD is 2 units and that of square DIJC is 3 units,
a) calculate the area of the other rectangles and square?
b) write two numerical expressions representing two ways of getting the area of the big square.
4. Having worked with numerical expressions, students will be ready to work with variables: If AD is x and DC is y, find two expressions for the area of the square AGHC. This of course leads to the identity . The popular FOIL method should only come after this activity. For the record, I’m not a fan of this method.
5. This representation can also be used to teach how to calculate expressions such as 0.75 x 0.75 +0.25 x 0.75 x 2 + 0.25 x 0.25. This is one way to help students appreciate an application of the square of a binomial.
6. You can also use this figure to teach the idea of function: If you drag D along AC, how will a change in the distance of AD affect the area of the rectangles and squares?
a) Create a table comparing the area of the quadrilaterals as the side AD increases from 0 to 5 units.
b) Do as in a) but this time compare the perimeters.
c) Express the area/perimeter of each quadrilateral as a function of the length of AD.
I will write about square of a difference in the next post.