Develop your ‘Variable Sense’

By | November 1, 2012

Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) xa = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.

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