# A problem solving approach for introducing positive and negative numbers

April 27, 2010 5 Comments

Mathematics is a language and a way of thinking and should therefore be experienced by students as such. As a language, math is presented as having its own set of symbols and “grammar” much like our spoken and written languages that we use to describe a thing, an experience or an idea. Thus, a popular approach for teaching numbers is to use it to describe a property of an object or a set of object. For example, numbers are used to describe the amount or quantity of fruits in a basket.

In introducing integers, teachers and textbooks presents integers as a set of numbers that can be used to describe both the quantity and quality of an object or idea. Contexts involving opposites are very popular situations to show the uses and importance of positive and negative numbers and the meaning of its symbols. For example, a teacher can tell the class that +5 represents going 5 floors up and -5 represents going five floors down from an initial position.

But apart from being a language, mathematics is also a way of thinking. The only way for students to learn how to think is for them to engage in it! Here’s my proposed activity for introducing positive and negative numbers that engages students in higher-level thinking as well.

**Sort the following situations according to some categories
**

**3**^{o}below zero

**52 m below sea level****$1000 net gain****$5000 withdrawal from ATM machine****$1000 deposit in savings account****3 kg weight loss****2 kg weight gain****80 m above sea level****37**^{o}above zero**$2000 net loss**

The task may seem like an ordinary sorting task but notice that the categories are not given. Students have to make their own way of grouping the situations. They can only do this after analyzing each situation, noting commonalities and differences.

Possible solutions:

1. Distance vs money (some students may consider the reading the thermometer under distance since its about the “length” of mercury from the “base”)

2. Based on type of quantities: amount of money, temperature, mass, length

3. Based on contrasting sense: weight gain vs weight loss, above zero vs below zero, etc.

The last solution is what you want. With very little help you can guide students to come-up with the solution below.

Of course, one may wonder why make the students go through all these. Why not just tell them? Why not give the categories? Well, mathematics is not in the curriculum because we want students to just learn mathematics. More importantly, we want our students to think critically and creatively hence we need to give them learning experiences that develops good thinking habits. Mathematics is a very good context for learning these.

Here are my other posts about integers:

- Who says subtraction of integers is difficult?
- Assessing operations involving integers
- Algebraic thinking and subtracting integers – Part 1
- Algebraic thinking and subtracting integers – Part 2
- Subtracting integers using numberline – why it doesn’t help the learning
- What is an integer?
- Teaching absolute value of an integer

I like how you present the concept of introducing positive and negative numbers as it correlates to the conceptual understanding of the given situations…

Admiring the time and effort you put into your blog and detailed information you offer!

bba india

Pingback: Who says subtracting integers is difficult? « teaching K-12 mathematics via problem solving

An alternate and equally valid solution (in my eyes) would be one that uses a double sort: first by units, then by quantity. This would result in a single column of numbers, perhaps with the last entry using each unit being underlined.

On the way to the above, the list could be sorted purely by quantity. I don’t see this as “wrong”, but as something that could still be made even more useful/clear, and as a good vehicle for discussing comparisons of quantities that use different units: some can be compared, and some cannot.

I would suspect that the money comparisons could confuse some folks (adult or child) – it’s the old double-entry problem from accounting! The notion of depositing/withdrawing money is often ambiguous as to whether it should be represented as positive or negative. Depositing money into your accounts means you have less in your wallet (should it be shown as a negative?) but more in the bank (oh, maybe it should be a positive)? So it is probably important to be clear about whether they should focus on their wallet, or their bank account.

I very much like the notion of leaving it up to the students to work out (in small groups) the best way of sorting these values, then justifying their answers to one another (and you). A good open-ended problem, with several possible solutions.

Whit Ford

http://mathmaine.wordpress.com

Thanks for sharing another solution. I think some students might sort it that way, indeed. You are right about the money:-)