Integer Operations

The Meanings of Minus

For the past 10 years I have been obsessed with the minus sign and its many meanings and misconceptions.  I have gathered together some of the research and resources that have shaped my own thinking.  I hope that you find it useful.

-3 - 5 = 8

Common points of confusion

The minus sign is a potential minefield of misconceptions because it has so many different meanings.  The meaning can even change part way through a problem!

3 – x = 1  the minus sign represents subtract x

-x = -2         the same minus sign now represents negative x

Vlassis (2008) argues that the minus sign can take on at least three meanings in mathematics.  Difficulties arise from students’ lack of awareness of the functions of the minus sign, namely the unary, binary, and symmetric operators.

Existing research finds that that the multiple roles of the negative sign, as a unary, binary, and symmetric operator, present a fundamental challenge for learners (Vlassis, 2008; Bofferding, 2010). The unary conceptualisation is the idea of a negative number as a position on the number line below zero; the binary conceptualisation corresponds to the subtraction operation; and the symmetric conceptualisation is the idea of a directed magnitude: the opposite of positive.

Developing a concrete-pictorial-abstract model for negative number arithmetic Jai Sharma and Doreen Connor

unary operator -3 "negative 3"

binary operator 7-3 "7 subtract 3"

symmetric operator -3 "opposite of positive 3"

-(-3) or --3 "opposite of negative 3"

-(x + 3) "opposite of x + 3"

I have swung between opposite and inverse over the years.  For the purpose of the minus sign I have chosen to use opposite because inverse is not appropriate in all cases.

Common Manipulatives and Representations

I have focussed on commonly used manipulatives and free online virtual manipulatives.

The main resources I have used are MathsBot.com and Desmos resources that I have created.

Zero Pairs

Algebra tiles and discs, Dienes blocks and double-sided counters can be used to model a number of mathematical concepts.  They are particularly useful to manipulate negative numbers and terms.  They can be used effectively to address common misconceptions.

The key concept common to all these representations is the zero pair.

The zero pair is an example of a symmetric operator.

a + -a = 0

a + opp a = 0

“symbolise manipulation leads to manipulating the symbols”

Pete Griffin, Assistant Director (Secondary), NCETM

pos 1

opp pos 1

pos x

opp pos x

pos x

opp pos x

(x-1)

opp (x-1)

pos x2

opp pos x2

1     --1        ----1         ------1              (-1)2n

-1      ---1         -----1             (-1)2n+1

Opposites on a number line/axis

Is -x always negative?

What is -x when x = 3?

What is -x when x = -4?

What is y when -y = 2?

What is y when -y = -5?

Click on the image on the left to open the graph.  Move the counters to help answer these questions.  You can view a single number line by switching the axis on and off in the settings on the top right of the screen.  Turn the counters on and off using the control panel on the left.

Using zero pairs

These groups of counters show -2

3 + -5 4 + -6

3 + (-3 + -2) 4+ (- 4 + - 2)

(3 +- 3) + - 2 (4 + - 4)+ - 2

The number sentences are an important representation of partitioning the expression.

In the first example -5 is partitioned into -3 + -2 revealing the zero pair 3 + -3.

Manipulating the counters alongside manipulating the number sentence exposes the mathematical structure.

Stem sentence (language structure):

There are  …3….. zero pairs.

The zero pairs sum to …0….. and the set of counters simplifies to ……-2….

Stem sentences make explicit the mathematical structure represented in the models above.

Number lines and vectors

Number lines and vectors show integers as directed numbers with a magnitude and direction.  They can be used to represent the binary and symmetric meaning of minus.

Can you see the zero pairs in the vector and number line representations?

How are these models different to the counters model?

Activity

Shake 5 double sided counters and throw them.  Write down or say what you see.

two positives and three negatives 2 + -3 (+2) + (-3)

three negatives and two positives -3 + 2 (-3)+ (+2)

This activity is taken form the NCETM PD material.

3 + -5 = -2

3 + -5 + 1 = -1

Emphasise subtraction as take away.

The minus sign  here has two meanings

- -1

subtract negative 1

3 + -5 - -1 = -1

Adding negative 1 and subtracting positive 1 have the same overall effect on the expression.

Does this always happen?

a - 1  a + -1 is this identity always true?

See subtracting integers below for more details

3 + -5 + -1 = -3

3 + -5 - 1 = -3

Adding integers with counters and compensation

Adding integers with counters and compensation https://www.desmos.com/calculator/1e1xb3lvke

Compensation is the process of reformulating an addition, subtraction, multiplication, or division problem to one that can be computed more easily mentally.

In the example, left, 10 is replaced with 6 + 4.  This can be seen in the model and the number sentence.

How does compensation reveal the mathematical structure of adding integers?

Integer addition with counters, lines and bar model.

Show all three models or switch each counters, lines or bars on and off by clicking the circle on the control panel (left >>)

In the model above, counters are placed from left to right with negative under positive.

In the model to the left (a+b=c),  a starts at 0 and b follows a.

Negatives are below positives and move to the left.

The bar model allows students to work with larger integers and paves the way for abstract manipulation.

Positive, negative or zero?

57 + -57 57 + -28 28 + -57

a + -b = 0 when a ............ b

a + -b < 0 when a ............. b

a + -b > 0 when a ............. b

This activity is taken form the NCETM PD material.

Try using  a zero pair model for each question.

Adding integers using opposites (symmetric operator)

3 - 5 = 3 + -5

3 subtract 5 is replaced with 3 + the opposite of 5

-5 is partitioned into -3 and -2

3 add opposite 3 make a zero pair and the expression simplifies to opp 2

Like Terms with Integers

3x - 2x + 4x simplifies to 5x

collect the like terms 3x and opp 2x and 4x which simplifies to 5x.

Collect like terms and simplify 3a + 7b -2a - 4b

Collect the like terms in a table.

a     b

3     7.

-2    -4

1      3 1a + 3b

Compare the counters and table with (3 - 2)a + (7 - 4)b

Simplify using zero pairs                 1a + 3b

The diagram and table represent the collecting of like terms.  This is quickly replaced by collecting like terms in brackets.

Do not forget to include constant terms

3x -4y + 8 -5x + 2 -7y

(3 -5)x  +  (-4 -7)y   +   ( 8 + 2)1

-2x + -11y + 10

-2x - 11y + 10

Students make the connections between manipulating number and algebra using consistent representations and language.

Multiple representations allow students to gain a deeper understanding of the mathematical structure.

Comparing multiple methods allows students make connections and use the most appropriate method with fluency.

Comparing different representations uncovers the structure of the mathematics and allows students to make generalisations.

This activity is taken form the NCETM PD material.

a + b = -4

Subtracting integers using additive opposite (inverse)

minuend – subtrahend = difference

These examples highlight the problem with subtraction when the subtrahend is greater than the minuend.

For the next question we need to take away 4 positives from 3 positives.

This is not possible using the subtraction model.

If we think of – as additive opposite and not subtraction:

Minus as opposite reduces the chance of misconceptions with - - = +

3 – 3 = 0   3 add opp 3 = 0

3 – 2 = 1   3 add opp 2 = 1

3 – 1 = 2   3 add opp 1 = 2

3 – 0 = 3

3 - -1 = 4   3 add opp opp 1 = 4

3 - - 2 = 5  3 add opp opp 2 = 5

3 – x ≡ 3 + -x​

This cannot be done using subtraction as take away but it can be represented as 3 add opposite x

x - 3 ≡ x + -3

This cannot be done using subtraction as take away but it can be represented as x add opposite 3

Minus Minus

Click the image above for the interactive version of this model.

This activity is taken form the NCETM PD material.

In this model  - -4 is represented as a vector with magnitude 4 and directed right.

a - b = 3   compare subtraction with adding the inverse/opposite (a - b a + -b)

a - b = -3    compare subtraction with adding the inverse/opposite (a - b a + -b)

Multiplying integers

3 x 2

3 groups of 2

3 x -2

3 groups of opp 2

Double sided counters can be used to represent opposites by flipping the counters.

For example,

-3 x 2

next read -3 x 2 as opposite of 3 x 2 and flip the counters to get the diagram on the near right.

-3 x 2

opp of 3 groups of 2

-3 x -2

opp of 3 groups of opp 2

Mathematical Thinking Activity

Find three values of a and b so that

a × b = − 24

Find three values of a and b so that

a × b = 24

Dividing integers

6 ÷ 2

3 groups of 2

make 6

-6 ÷ -2

3 groups of opp 2 make opp 6

6 ÷ -2

I need to flip 3 groups of opp 2 to make 6

-3 x -2 = 6

6 ÷ -2 = -3

6 ÷ -2

opp of 3 groups of opp 2 make 6

-6 ÷ 2

opp of 3 groups of 2 make opp 6

The use of language is a key representation.  We want students to internalise the language and remove the counters.

4 x -3  “4 groups of opp 3”               -12

8 ÷ -2  “how many groups of opp 2 make 8”      -4

-5 - 2   “opp 5 add opp 2 simplifies to opp 7”     -7

-5 - - 2 “opp 5 add opp opp 2 simplifies to opp 3”  -3

3a – 5a    “3a add opp 5a simplify to opp 2a”     -2a

3x – 4 – (2x -1)     “3x add opp 4 add opp 2x add opp opp of 1”  x - 3