square Archives - Mathematics for Teaching

Students’ understanding of mathematics is a function of the quality and quantity of the connections of a concept with other concepts. As I always say in this blog, ‘To understand is to make connections’.

There are many ways of helping students make connections. One of these is through activities involving multiple representations. Here is a lesson you can use for teaching the difference of two squares, $x^2-y^2$ .

Activity: Ask the class to cut off a square from the corner of a square piece of paper. If this is given in the elementary grades, you can use papers with grid. If you give it to Grade 7 or 8 students you can use x for the side of the big square and y for the side of the smaller square. Challenge the class to find different ways of calculating the area of the remaining piece. Below are two possible solutions

Solution 1 – Dissect into two rectangles

Solution 2 – Dissect into two congruent trapezoids to form a rectangle

Extend the problem by giving them a square paper with a square hole in the middle and ask them to represent the area of the remaining piece, in symbols and geometrically.

Solution 1 – Dissect into four congruent trapezoids to form a parallelogram

Solution 2 – Dissect into 4 congruent rectangles to form a bigger rectangle

These two problems about the difference of two squares will not only help students connect algebra and geometry concepts. It also develop their visualization skills.

This is a problem solving activity. It’s important to give your students time to think. Simply using this to illustrate the factors of the difference of two squares will be depriving students to engage in thinking. They may find it a little difficult to represent the dimensions of the shapes but I’m sure they can dissect the shapes. Trust me.

How does mathematics define a math concept?

Definitions of concepts in mathematics are different from definitions of concepts in other discipline or subject area. A definition of a concept in mathematics give a list of properties of that concept. A mathematics object will only be an example of that concept if it fits ALL those requirements, not just most of them. Further, a definition is also stated in a way that the concept being defined belongs to an already ‘well-defined’ concept. On top of this, economy of words and symbols and properties are highly observed.

Does a math concept only have one definition? Of course, not. A concept can be defined in different ways, depending on your knowledge about other math objects. In a study by Zaskin and Leikin, they suggested that the definitions students give about a concept mirrors their knowledge of mathematics. Below are examples of definitions of squares from that research. Do you think they are all legitimate definitions?

What is a square?

A square is

a regular polygon with four sides
a quadrilateral with all the angles and all the sides are equal
a quadrilateral with all the sides equal and an angle of 90 degrees
a rectangle with equal sides
a rectangle with perpendicular diagonals
a rhombus with equal angles
a rhombus with equal diagonals
a parallelogram with equal adjacent angles and equal adjacent sides
a parallelogram with equal and perpendicular diagonals
a quadrilateral having 4 symmetry axes
a quadrilateral symmetric under rotation by 90 degrees
the locus of all the points in a plane for which the sum of the distances from two given perpendicular lines is constant. Click this link to visualize #12.

Making (not stating) definitions is a worthwhile assessment task.

Here’s three great references for definitions of mathematical concepts. The first is from no other than Dr. Math (The Math Forum Drexel University). The middle one’s for mom and kids – G is for Google and the third’s a book of definitions for scientists and engineers.