Negative Numbers

And Fractions

Connections

Thinking

Learning

A Lesson in Learning and Studying

1. Exposure to theory

2. Re-enforce theory with a paradigm from another theory

3. Relate to a real world situation (manipulatives) Our math is designed to match what we observe, if it

does not, we modify it until it does.

4. Practice the examples

5. Go back over the theory

6. Practice the examples

7. Summarize

8. Repeat the process occasionally

Copyright © 2010 Irvin M. Miller

Variables

A+B=C Substituting A=2 B=3,

2+3= 5 = C +

A+B=C Substituting A=4 B=5,

4+5= 9 = C

Defining Subtraction

A+B=C

B=C – A Substituting C=5 A=2

B= 5 - 2=3 take away 2

or

A =C –B Substituting C= 5 B=3

A = 5 - 3=2 take away 3

1 2 3 4 5

Discovering Negative Numbers

2+B=0

B=0 – 2 Since adding 0 does not change a sum, let’s define a new notation

B=0 – 2 = -2, thus

2 + -2 = 0 really 2 +(0-2) =0

Interesting observation

-2 = 0 - 2 definition of a negative number

2 = 0 - - 2 subtracting (inverting) a negative number

+ = 0

2 + -2 = 0 1 blue + 1 brown=0

2 = 0 - - 2 =

Developing the rules

n + -n = 0

-n = 0 – n I. Subtracting n same as adding –n - = +-

n = 0 - - n II. subtracting –n same as adding n - - = +

(n + -n) + (m + -m) = 0 + 0 = 0

III. negative sum = negative of positive sum

(n + m) + (-n +-m) = 0 + =

-n + -m = 0 - (n+m) =-(n+m)

IV. sign is distributive (rule II.) - (- + +) = + + -

(n + -n) + (m + -m) = 0

(n+-m) + (-n + m) = 0 reassociating

n+-m = 0 – (-n+m)=-(-n+m) 3 – 5 =0-(5-3) = -2

The Quiz

5 + 3 = 8

5 + -3 = 5 – 3 =2 + - = - rule 1

5 - 3 = 2

5 - -3 = 5 + 3 = 8 - - = + rule 2

-5 + 3= 3 – 5 = -2 rule 1

-5 + -3 = -(5+3)=-8 rule 3

-5 - 3 = -5 + -3 =-(5+3) rule 1, rule 3

-5 - -3= -5 + 3 = 3 – 5 = -2 rule 2, rule 4

Essence of Subtraction

Change - - to +

Add if signs are same else subtract

Use sign of largest number

5+3=8 -5+3=-2

5-3=2 -5-3=-8

5+-3=2 -5+-3=-8

5- -3=5+3=8 -5- -3=-5+3=-2

New knowledge 0 = -0 (same position on number line)

Adding zero n+-n, helps solve algebraic problems. - = ( ) =

- = ( ) =

Paralleling Subtraction and Division

x = + / = - 1 = 0

x / 1 + - 0

Definition

n + -n = 0 n x /n=1

Rules

-n = 0 – n - = +- /n=1/n / = x/

n = 0 - - n - - = + n = 1//n // = x

-n + -m = 0 - (n+m) =-(n+m) /n x /m = /(n x m)

n+-m=0–(-n+m)=-(-n+m) - (- +) = + - 1/(m/n) = n/m

ReflectReflect

Discovering Fractions

5 x 3 = 15 3 = 15/5 = 3x5 /5 = (5+5+5)/5

4xB=1

B=1 / 4 Since multiplying by 1 does not change a result lets define a new notation

B=1 / 4 = /4, thus

4 x /4 = 1 really 4 x(1/4) =1

Interesting observation

/4 = 1 / 4 definition of a fraction

4 = 1 / /4 dividing (inverting) a fraction

4 x /4 = 1 1 dollar = 4 quarters

4 = 1 / /4 / = /

Developing the rules

n x /n = 1

/n = 1 / n I. Dividing by n same as multiplying by /n / = x/

n = 1 / /n II. Dividing by /n same as multiplying by n

/ / = x / 1 / /4 = 4

III. Fractional product

(n x /n) + (m x /m) = 1 x 1 = 1

(n x m) x (/n x /m) = 1 , then /n x /m = 1 / (n x m) =/(n x m)

IV. Inversion (rule II.) / (/ x x) = x x /

(n x /n) x (m x /m) = 1

(n x /m) x (/n x m) = 1 reassociating

n x /m = 1 / (/n x m)=/(/nxm) 2 x /4 =1/(4/2) = /2

The Quiz

4 x 2 = 8

4 x /2 = 4 / 2 =2 x / = / rule 1

4 / 2 = 2

4 / /2 = 4 x 2 = 8 / / = x rule 2

/4 x 2= 2 / 4 = /2 rule 1

/4 x /2 = /(4x2)=/8 rule 3

/4 / 2 = /4 x /2 = /8 rule 1, rule 3

/4 / /2= /4 x 2 = 2 / 4 = /2 rule 2, rule 4

The Quiz (old notation)

4 x 2 = 8

4 x 1/2 = 4 / 2 =2 x / = / rule 1

4 / 2 = 2

4 / 1/2 = 4 x 2 = 8 / / = x rule 2

1/4 x 2= 2 / 4 = 1/2 rule 1

1/4 x 1/2 = 1/(4x2)=1/8 rule 3

1/4 / 2 = 1/4 x 1/2 = 1/8 rule 1, rule 3

1/4 / 1/2= 1/4 x 2 = 2 / 4 = 1/2 rule 2, rule 4

Essence of Division

Change / / to x

Multiply denominators if both are fractions else divide

Answer greater than 1 if numerator greater than denominator

4x2=8 /4x2=/2

4/2=2 /4/2=/8

4x/2=2 /4x/2=/8

4 / /2=4x2=8 /4 / /2=/4x2=/2

New knowledge unary fractions n/m = n x /m

Multiplying by one n x /n, helps solve algebraic problems.

1 x = 3x/3 three parts each 1/3 size

Additional Relationships

0=-0 /1 = 1

n - n=0 n / n = 1

n - 0=n n / 1= n

Proofs

0+0=0 1 x 1 = 1

0=0 - 0=-0 1= 1 / 1 = /1

n – n = n + -n=0 n / n = n x /n = 1

n – 0 = n + -0 = n+ 0 =n n / 1 = n x /1 = n/1

Cross Relationships

Multiplying negative numbers

3 x –5 = -5 + -5 + -5 =-(3x5)=-15

2x3=6 2x2=4 2x1=2 2x0= 0 2x-1=-2 2x-2=-4 2x-3=-6 sub. 2

2x3=6 1x3=3 0x3=0 -1x3=-3 –2x3=-6 –3x3=-9 sub. 3

-nx(m + -m) = 0

-n x m +-nx-m=0 3

-(nxm) + -n x-m =0 2

-n x –m = 0 - -(n x m) = n x m

Summary 2

Rule 1: + x - = - x + = minus Rule 2: - x - = + x + = plus

5 3 3

Cross Relationships

Fractions

Same denominator

a/n + b/n+c/n=a x /n +b x /n +c x /n =(a+b+c) x /m=(a+b+c)/n

Used the distributive rule

Different denominator

a/n + b /m= 1x a/n + 1x b/m=(m x /m) x a/n + (n x /n) x b/m

= m x a x /(nxm) + n x b x /(n x m)

= (m x a + n x b)/(n x m)

1 / 2 + 1/ 3 = 3 x 1/6 2 x 1/3 = 5/6

More details on multiplying by 1

Multiplying by 1=n/n merely splits the fraction into n parts where each part is 1/n of the fraction.

1/2 1/3

1=3/3 1=2/2

Three parts 1/3x1/2=1/6 Two Parts 1 /2 x 1/3=1/6

1 /2= 3 x1/6

1/3 = 2x 1/6 1 /2 + 1/3=3/6 + 2/6 = 5/6

Summary

x / 1 + - 0

Negative numbers Definition fractions

n + -n = 0 n x /n=1

Rules

-n = 0 – n - = +- /n=1/n / = x/ definition

n = 0 - - n - - = + n = 1//n // = x inversion

-n + -m = 0 - (n+m) =-(n+m) /n x /m = /(n x m)

n+-m=0–(-n+m)=-(-n+m) - (- +) = + - 1/(m/n) = n/m inversion

Additional Relationships

0=-0 /1 = 1 unity inversion

n - n=0 n / n = 1 itself

n - 0=n n / 1= n unity

ReflectReflect