Tag: integers

Posted in Elementary School Math, High school mathematics, Number Sense

Teaching positive and negative numbers

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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.

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.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 them in it!  Here’s my proposed activity for teaching positive and negative numbers that engages students in higher-level thinking as well.

Sort the following situations according to some categories

  1. 3o below zero
  2. 52 m below sea level
  3. $1000 net gain
  4. $5000 withdrawal from ATM machine
  5. $1000 deposit in savings account
  6. 3 kg weight loss
  7. 2 kg weight gain
  8. 80 m above sea level
  9. 37o above zero
  10. $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:

Posted in Misconceptions, Number Sense

From whole numbers to integers – so many things to “unlearn”

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A lot of online resources on integers are about operations on integers especially addition and subtraction.  Most of these resources  show visual representations of integer operations. These representations are almost always in the form of jumping bunnies, kitties, frogs, …  practically anything that can or cannot jump are made to jump on the number line. Sometimes I wonder where and when in their math life will the students ever encounter or use jumping on the number line again.  If you want to know why I think number line might not work for teaching operations, click link –  Subtracting  integers using number line – why it doesn’t help the learning.

Of course there may be other culprits apart from rote learning and the numberline model. Maybe there are other things that blocks students’ understanding of integers especially doing operations with them.

Before integers, students’  life with numbers had been all about whole numbers and some friendly fractions and decimals. So it is not surprising that they would have made some generalizations related to whole numbers with or without teachers help. I pray of course that teachers will have no hand in arriving at these generalizations and that if indeed students will come to these conclusions, it should be by the natural course of things.  Here are some dangerous generalizations.

over-generalizations about whole numbers

These generalizations are very difficult to unlearn (accommodate according to Piaget) because based on students experiences they all work and are all true. Now, here comes integers turning all of these upside down, creating cognitive conflict. In the set of integers,

  1. when a number is added to another number it could get smaller (5 + -3 gives 2; 2 is smaller than 5)
  2. the sum of any two numbers can be smaller than both of the addends (-3 + -2 gives -5; -5 is smaller than -3 and -2)
  3. when a number is taken a way from another number, it could get bigger (3 – -2 = 5, 3 just got bigger by 2)
  4. you can get an answer for taking away a bigger number from a smaller number (3 – 5 = -2)
  5. when a number is multiplied by another number, it could get smaller (-3 x 2 = -5)
  6. when a number is divided by another number, it could get bigger (-15/-3 = 5)

On top of these, mathematics is taught as something that gives absolute result. So how come things change?

You may be interested to read my article on Math War over Multiplication. It’s also about overgeneralization.

Feel free to share your thoughts about these.

Posted in Number Sense

Subtracting integers using tables

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In my earlier post on this topic, I discussed why teaching subtraction using the numberline is not helping most students to learn the concept. In this post I describe an alternative way to teaching operations with integers that would help students develop a conceptual understanding of the operation and engage their mind in algebraic thinking at the same time.

The table of operation is one of the most powerful tool for showing number patterns and relationships among numbers, two important components of algebraic thinking. It is a pity that most of the time it is only used for giving students drill on operation of numbers. Some teachers use it to teach operation of integers but more for mastery of skills and to show some beautiful patterns created by the numbers. Below are some ideas you can use to teach operation of integers conceptually as well as engage students in algebraic thinking. I promote teaching mathematics via problem solving in this blog so this post is no different from the rest.  Use the task below to teach subtraction and not after they already know how to do it. Of course it is assumed that students can already do addition.

The question “Which part of the table will you fill-in first?” draws the student attention to consider the relationships among the numbers and to be conscious of the way they work with them. It tells the students that the task is not just about getting the correct answer. It is about being systematic and logical. Engage the students in discussion why they will fill-in particular parts of the table first.

table of integers

Students will either subtract first the same number and this will fill the spaces of zeroes or they can subtract the positive integers. They will of course have to define beforehand which will be the first number (minuend) and which will be the second number (subtrahend).

Surely most students will get stuck when they get to the negatives except with the equal ones which results to zero. You may then ask them to investigate the correctly filled up parts of the table that could be of use to them to fill-in the rest of the table. Students will discover that the numbers are increasing/decreasing regularly and can continue filling-in the rest of the spaces. This is not a difficult task especially if the process for teaching addition was done in the same way. Encourage the class to justify why they think the patterns they discovered makes sense.

The discussion of this topic in continued in Algebraic thinking and subtracting integers – Part 2

Posted in Number Sense

Teaching absolute value of an integer

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integers and algebraic reasoning

The tasks below are for deepening students’ understanding about the absolute value of a number and provide a context for creating a need for learning operations with integers. You may give the problems  after you have introduced the students the idea of absolute value of an integer and before the lesson on operations with integers.

Tasks (Set 1)

1) Find pairs of integers whose absolute values add up to 12.
2) Find pairs of integers whose absolute values differ by 12.
3) Find pairs of integers whose absolute values gives a product of 12.
4) Find pairs of integers whose absolute values gives a quotient of 12.

These may look like simple problems to you but note that these questions involve equations with more than one pair of solutions. The problems are similar to solving algebraic equations involving absolute values. Problem 1) for example is the same as “Find the solutions to the equation /x/ + /y/ = 12. Of course we wouldn’t want to burden our pupils with x’s and y’s at this point so the we don’t give them the equation yet but we can already engage them in algebraic thinking while doing the problems. The aim is to make the pupils  be comfortable and confident with the concept of absolute values as they would be using it to derive and articulate the algorithm for operations with integers later in the next few lessons.

Tasks (Set 2)
1) Find pairs of integers, the sum of absolute values of which is less than 12.
2) Find pairs of integers the difference of absolute values of which is greater than 12.

Encourage students to show their answers in the number line for both sets of task.