This unit helps students to develop procedural fluency with integers and have conceptual understanding of integers in the real world.
Integers are needed to meet the demands of situations where a larger whole number is subtracted from a smaller whole number. For example, if a person has $5 available in cash but owes $8 then their net situation is 5 – 8 = -3. The origins of integers historically lie in the algebra of whole number subtraction. Integers, as quantities, usually reflect a state of balance between directional forces, such as cash being an asset and debt a liability. Furthermore, integers are sometimes called directional numbers because they represent a magnitude (size) and a direction, e.g. -3 represents a magnitude of 3 units in a negative direction from zero. The directional nature of integers is important to real world applications such as transmission factors of gears and pulleys, and to enlargement (dilation).
A common use of integers in real life is to label and quantify points on a scale, such as temperature and height above sea level. In both cases the location of zero is important to the attribute being measured. For example, both the height above normal sea level (0 m) of a spring tide and the temperature below the freezing point of water (0°C) have significant consequences to the severity of the situation. Zero acts as an important benchmark indicating normality or balance. This is also true in sport or games like Bridge where negative numbers reflect a state relative to expectation, e.g. -6 in golf means six under par when a player has taken six fewer shots than the expected norm.
Integers are an extension of the whole number system. Therefore, the properties of integers under the four operations should be the same as those for whole numbers. With addition and subtraction four main properties hold:
The commutative property of addition
The order of the addends does not affect the sum. If -3 + 4 = 1 then 4 + -3 = 1. Note that the commutative property does not hold for subtraction. For example, 4 - -3 = 7 but -3 – 4 = -7.
The distributive property of addition
This property is really about the partitioning of addends and recombining those addends. For example, if 5 = -1 + 6, then -2 + 5 = (-2 + -1) + 6. This property does not hold for subtraction.
The associative property of addition
This property is about ‘associating’ pairs of addends one pair at a time. For example, (-4 + 3) + -1 = -4 + (3+ -1). This property does not hold for subtraction.
Inverse operations
Addition and subtraction are inverse operations so one operation undoes the other. For example, -2 + -3 = -5 so -5 - -3 = -2.
It is the need for these number laws to hold that establishes the effect of operations, such as subtracting a negative integer has the same effect as adding a positive integer.
This unit combines two of Hans Freudenthal’s (1983) models for operations on integers, the annihilation and vector models. In the annihilation model, positives and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative pair that equals zero does not alter the quantity being represented. The vector model presents integers as magnitudes with direction. +1 is represented by a vector of length one in a positive direction and -1 as a vector of length one in a negative direction. Freudenthal cautioned that a quantity of +1 or -1 was easily confused with the operation of adding or subtracting one and teaching needed to make that difference explicit.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
The unit uses the contexts of money, positive and negative spaces, and scale. Other contexts might better suit the interests and cultural backgrounds of your students. Students interested in sports might enjoy golf as a context, and those who enjoy computers might find points schemes interesting. Some students may enjoy the context of comparing temperatures from locations around the world, and finding out which locations have the most and least variation in a full year. Investigating local places that are above and below sea level could also be an engaging context for students.
Te reo Māori vocabulary terms such as tau tōpū (integer), tau tōraro (negative number), and tau tōrunga (positive number) could be introduced in this unit and used throughout other mathematical learning.
This session introduces negative integers in real life and presents integers as vectors.
Card One | Card Two | Card Three | Card Four | Total |
→ | → | → | → | 4 |
→ | → | → | ← | 2 |
→ | → | ← | → | 2 |
→ | ← | → | → | 2 |
← | → | → | → | 2 |
→ | → | ← | ← | 0 |
→ | ← | → | ← | 0 |
← | → | → | ← | 0 |
→ | ← | ← | → | 0 |
← | → | ← | → | 0 |
← | ← | → | → | 0 |
→ | ← | ← | ← | -2 |
← | → | ← | ← | -2 |
← | ← | → | ← | -2 |
← | ← | ← | → | -2 |
← | ← | ← | ← | -4 |
In this session students explore the ‘Hills and Dales’ context for application integers. The context was used in the Oscar nominated film “Stand and Deliver” about Jaime Escalante, an American teacher working with disadvantaged students in Los Angeles. A short video of him teaching algebra using the ‘Hills and Dales’ model is easily accessed online. The video finishes with Escalante asking his students why a negative number multiplied by a negative number gives a positive answer. Good question!
In this unit the context is about road builders. In real life one of the largest costs of new roads is relocation of earth, particularly if earth must be brought in from off-site. Roads are best flat and both hills and dales present potential costs unless a hill can be used to fill a dale.
In this session students explore the addition of Integers in the context of dollars and debts. The net financial position of a person is the sum of the money they have available and the debts they owe.
In this session the vector model is connected to the Hills and Dales and Dollars and Debts models. The aim is to generalise addition and subtraction of integers. It is important to distinguish the vectors that represent positive and negative numbers and the addition and subtraction as operations, addition as movement to the right and subtraction as movement to the left.
In this session the vector model is developed into a number line model which highlights the direction of change when integers are added and subtracted.
Dear parents and whānau,
This week we have been exploring everyday applications of integers, including using models to show what happens when adding and subtracting positive and negative integers. We have also used models to explain why subtraction of a negative integer has a positive effect.
Ask your student to show you the Hills and Dales and Dollars and Debts models to explain adding and subtracting positive and negative integers.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/integers at 8:35pm on the 26th February 2024