Mystery of Integers
One challenging concept that I have confronted in teaching grade 11 Math to students who have struggled in the subject and are behind in their skills is the concept of summing integers. I find myself drawing a number line or thermometer to give the student(s) a physical representation that is more concrete than the numerical representation. In the article Modeling Practices with The Geometer’s Sketchpad (Sinclair and Jackiw, 2010), an idea is presented for using coloured circles to represent the summation. They used red chips to represent negative integers and black chips to represent positive integers. For example, if confronted with -3+5, 3 red chips and 5 black chips would be moved to the ‘Chip Board.’ Then pairs of chips are created and moved to the side. In this case, there would be 3 pairs. Two black chips would be left over which represents +2 and is the sum of these two integers. If the negative number is greater, for instance -5+2, then there would be 3 red chips left over which represent -3 and is the sum of these two integers.
Here’s a T-GEM cycle for exploring and better understanding integers.
Cycle 1:
First, students have some familiarity with integers as they are covered in their previous grades, but many claim a lack of understanding. I would begin by asking them to think of where they have seen the use of negative numbers. They may come up with (or I may offer) temperature, money (debt, refund, stock market), attendance (absence), hockey plus-minus differential, latitude/longitude, elevation, and electrical currents. Then I would have them watch this short video about temperature and integers.
Next, I would ask them to hypothesize a rule for summing integers, i.e. If the negative integer is greater than the positive integer, the result will be negative. Then I would have them use the Geometer’s sketchpad to experiment with a variety of single-digit integer summations using the Chip Board and the chip matching exercise.
Then, I would ask them to revisit their hypothesis to determine if it needs revised or if there is new information to add based on their exploration.
Cycle 2:
Students would experiment with a web interactive to calculate temperature change using double-digits to see if their rule holds true with larger values.
Then, they would revisit their rule to revise or add information.
Cycle 3:
Students would be asked to develop rules for subtracting negative integers, i.e. What happens when you subtract a negative integer from a positive integer? and/or What happens when you subtract a negative integer from a negative integer?
Using Geometer’s Sketchpad, I would have the students use a number line to experiment with this concept (which leads to an understanding of… a positive and a positive equals a positive, a positive and a negative equals a negative, and a negative and a negative equals a positive).
Then they would revisit their hypotheses at the end of experimentation.
Reference:
Sinclair, N., & Jackiw, N. (2010). Modeling Practices with The Geometer’s Sketchpad. In Modeling Students’ Mathematical Modeling Competencies (pp. 541-554). Springer US.