This unit helps students to develop procedural fluency with integers and have conceptual understanding of integers in the real world.
- Understand everyday application of integers.
- Add and subtract positive and negative integers.
- Use models to explain why subtraction of a negative integer has a positive effect.
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 subtraction of whole numbers. Integers, as quantities, usually reflect a state of balance between directional forces, such as cash being and asset and debt a liability. Furthermore, integers are sometime 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 less shots than the expected norm.
Specific Teaching Points
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 positive and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative pairs that equal 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.
Session One
This session introduces negative integers in real life and presents integers as vectors. Begin with PowerPoint One that shows some situations where negative numbers are used. The contexts include temperature above and below zero degrees, water level above and below sea level, time before and after the birth of Christ, and financial well-being as a balance of available money and debt.
For the next activity you will need sets of cards made from Copymaster One. The cards will be used throughout the unit so investing time in laminating sets is worthwhile. Go outside and draw a horizontal number line on the pavement like this.
Ask students how the number line might be extended. The real-life contexts should encourage them to consider the placement of zero and the negative integers. In the end you want a number line from ^{-}8 to 8. Mention that usually we do not write ^{+}8 for 8 though we could.
Ask one student to act out the addition of integers as vectors. Addition is the operation of combining quantities. Ask him or her to begin on zero and act out the cards they are given. Draw a set of three cards randomly (made from Copymaster One). The ^{+}1 card represents one unit in a positive (right) direction and ^{-}1 represents one unit in a negative (left) direction. As the student walks out the movement given by the cards explain to the whole class why they are moving in that direction and the distance of that move.
Act out several combinations of cards then ask the students if it is possible to predict the finishing location if the cards are known. Choose a set of five cards and invite predictions.
What do you notice about what happens when^{ +}1 and ^{-}1 cards combine?
Is that also true of ^{-}1 and ^{+}1?
How can the fact that positive and negative cards form zeros be used to predict the finishing location?
Return inside for students to play a game of “Which way Wally Weta?” in pairs or threes. Photocopy the gameboard (Copymaster Two). Teams will need counters, preferably transparent, and a set of cards with equal numbers of ^{+}1 and ^{-}1. Each turn the cards are shuffled and the top four cards turned over one at a time. With a counter the player acts out the moves, starting at zero. They finish each play by leaving the counter at the finishing number on the board. Play continues like that for 16 turns.
The ‘house’ gets any counters on ^{-}2 or ^{+}2 and the team gets any counters that are left.
Is the game fair?
Invite students to discuss whether the game is fair, that is their team and the house have an equal chance of winning. You might look at the results of some games to see what seems to happen.
House wins | Draw | Team wins |
|||| | ||| | ||| |
Work ‘outside in’ to work out the ways of the counter finishing on a particular number.
The game is fair as zero can be arrived at in six different card orders, meaning that the house has eight out of sixteen outomes for them and the player has the other eight outcomes.
A nice way to show the possibilities is to consider the ‘ordinal positions’ of the → cards only. The ← cards must take up the in between positions. Order of the cards being collected is important in determining the different ways a counter can finish at a particular place. You may need to drop back to a simpler form of the game, e.g. two or three cards to support the students with ways to find all the outcomes.
This table shows all the possible outcomes for the four card game:
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 |
Session Two
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. PowerPoint Two A shows some scenarios in which a hill (^{+}1) might fill a dale (^{-}1) to create flat land (0). With each slide ask the students two questions:
For example, scenario one is three hills (^{+}3) and two dales (^{-}2). You might connect that scenario to the arrow cards by getting three ^{+}1 cards and two^{ -}1 cards. Students should recognise that you are combining integers and offer the equation 3 + ^{-}2 = 1 or ^{-}2 + 3 = 1. In fact, noting the commutative property holds with Integers is important. Work through the four scenarios before giving the students Copymaster Three to complete independently.
For early finishers pose this problem:
Process the students’ answers to Copymaster Three with emphasis on how one positive and one negative cancel out each other to form a zero. Ask what other situations in real life are like that. Situations might include having money but owing money, scoring below par and above par in golf, getting points and penalties in video games, putting hot air in the balloon and adding weights, going up in an elevator and going down, eating a hamburger and exercising hard.
To further practice addition of Integers ask the students to play games of Integer Coverup.
More practice with the Hills and Dales model is available in the Using integer tiles activity.
Session Three
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. Pose this scenario to the students:
Layla has $7 in her bank account. She owes her parents $3. How well off is Layla?
Students should tell you that Layla actually is worth $4 which would be the result of paying her parents off and being debt free. Her net position can be shown in many ways:
- Using toy money and IOUs
- Use vectors on a number line with^{ +}1 representing $1 and ^{-}1 representing a bill for $1.
Whatever model is used the key point is annihilation of a one dollar debt by a one dollar coin and vice versa. Pose similar problems staying with the Layla scenario and changing the amounts. Be sure to include dollars and debts that result in a negative net position, e.g. $10 money and $13 in debts.
Ask the students to practice this idea using the Disappearing dollars Figure It Out activity.
Extend the dollars and debts model to subtraction of Integers. Begin with Layla’s scenario with a subtle change.
Layla has $7 in her bank account. She owes her parents $3. Her parents say they will not need her to repay $2 of the debt. How well off is she now?
This story can be represented by the equation 4 - ^{-}2 = 6. Originally Layla’s position was $4, accounting for her money and debt. Removal of a two dollar debt is represented by - ^{-}2.
Is Layla better or worse off by her parents’ decision?
Model the scenario using vectors as below.
Initial situation
After the removal of $2 of debt.
Pose similar problems to get a balance of starting position (positive or negative) and subtraction of both debts and dollars (spending). Adapt the Layla scenario. For example:
Layla has $8 in her bank account. She owes her parents $13. Her parents say they will not need her to repay $5 of the debt. How well off is she now? (^{-}5 – ^{-}5 = 0)
Initial state ^{-}5
Final state 0
To practise subtraction using the money model students can work individually or in pairs through the e-ako called "AS4.60 More Integers (Positive and negative numbers)".
Session Four
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.
Begin with slide one of PowerPoint Four A.
What do these models have in common?
Invite students to give the balance of ^{-}2. Ask “Where is ^{-}2 is in each model?”
If we start with a balance of ^{-}2 and subtract 3, what does that look like in each model?
Slides 2-4 of PowerPoint Four A are animated to show what subtracting one looks like on each model. Record the operation as:
^{-}2 – 3 = ^{-}5
Ask the students to anticipate the results on the three models of removing different amounts, particularly 2, 1 and 0. Arrange the equations in order.
^{-}2 – 3 = ^{-}5
^{-}2 – 2 = ^{-}4
^{-}2 – 1 = ^{-}3
^{-}2 – 0 = ^{--}2
What happens if -1 is subtracted (removed)?
Slides 5-7 of PowerPoint Four A model the operation that leaves the balance at^{ -}1. Continue the pattern to ^{-}2 – ^{-}3 = ^{-}1. Discus the positive effect of removing a negative amount, that is the balance is greater than before (There is more earth and more money).
Provide the students with a set of similar equations but starting with a positive balance, such as:
^{+}3 – 2 = ^{-}_
^{+}3 – 1 = ^{-}_
^{+}3 – 0 = ^{-}_
^{+}3 – ^{-}1 = _
^{+}3 – ^{-}2 = _
Ask them to justify their answers using one of the models that have been used.
Students can practice addition and subtraction of Integers using the learning object called Integer Invaders. This learning object is developed n Flash and will not work on some devices. Note that this object uses Freudenthal’s original annihilation model in which zeros are used in the form of one positive and one negative so that calculations are possible. For example, to solve the problem below a player needs to put ^{+}1 and four zeros into the centre so that ^{-}4 is available to subtract (by splitting the zeros).
After the students have practised sufficient problems draw their attention to the number line model that is displayed at the end of each successful calculation.
Why is this subtraction going in the opposite direction to when a positive number if taken away?
Session Five
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. Start with these True/False statements. Ask the students to discuss both statements in small groups.
After a suitable period of discussion share ideas. Look for students to state the conditions under which the statements are true or false. The first statement is true when the second addend is greater than zero (a + b > a iff b>0) and is false if the second addend is equal or less than zero (a + b ≤ a iff b≤0). Find similar conditions for the second statement, i.e. a - b < a iff b>0 and a - b ≥ a iff b≤0.
Ask the students to work in pairs through the e-ako called "AS4.50: Integers (Positive and negative numbers)". They will learn about the direction of change through the context of Claw the Crab.
Summarise the addition and subtraction of integers using this diagram.
To practise the direction of change play the game Walk the Plank with students . You may like to amend the game so the dice are labelled + (for the direction of the shark, S) and – (for the direction of the boat, B). The activity could be used as a probability task where the fairness of the game is examined both experimentally and theoretically. The usual length of a game in number of rolls before the pirate is safe or wet could also be investigated.