Unit 3: Number Sense and Systems


Number Sense and Systems

Key Concept: 
Logic

Related Concept: 
Generalisation, Simplification

Global Context:
Orientation in time and space



Factual Concept Questions

What applications are there to negative numbers? 


How do you subtract a negative number from a positive number? 


How do you find the difference between two negative numbers? 


How do you order integers?



Conceptual Concept Questions


How do you represent a number below zero?



Debatable 
Concept Questions

Which unit of temperature is better - Kelvin, Celcius, or Fahrenheit? 

What is there more of: positive numbers, or both the positive and negative numbers together?


Content

Directed numbers and the number line

Order Integers (positive and negative)

The operations of addition and subtraction with integers (positive and negative)

Understand and use the definitions of expression, formula and equation

Basic algebraic notation

Using variables

Operations used in algebra

Expand and simplify expressions with single brackets

Collect like terms

Substitute numbers into expressions and formulae


Y7 Number Sense and Systems: Directed number, collecting like terms and distributive rule




Integers  are all the positive and negative whole numbers, including zero.

True or False?

Write down the following statements in your exercise book and state whether they are true or false, If they are false show this by giving an example:

A: 

(-5) + (-6) 
If you add two negative numbers you get a negative answer

B:

(-5) + (+7)
If you add a negative number and a positive number you get a positive answer.

C:

(-5) - (+4)
If you subtract a positive number from a negative number you get a negative answer.

D:

(-5) - (-8)
If you subtract a negative number from a negative number you get a positive answer.

E:

(+10) - (+5)
If you subtract a positive number from a positive number you get a positive number.

F:

(+8) - (-6)
If you subtract a negative number from a positive number you get a positive answer.

G:

5 + (-8) = 5 - (+8)
Adding a negative is like subtracting a positive.

H:

5 - (-8) = 5 + 8
Subtracting a negative is like adding a positive.



Order of Operations

When there is more than one mathematical operation in a calculation we have to work in a special order. 

                                                                                                             step 1:         Brackets
                                                                                                             step 2:         Indices
                                                                                                             step 3*:        Multiplication &
                                                                                                                                Division
                                                                                                             step 4**:       Addition &
                                                                                                                                Subtraction

*step 3: work out the division and multiplication in order left to right
**step 4: addition and subtraction can be calculated in any order BUT you must remember the + and - signs only applies to the number directly after it. To keep it simple and avoid mistakes work left to right.

Remember: 
show all your working
work clearly
work vertically - show 1 step per line
use equals signs (one per line!)



Algebra

Where does Algebra come from?

by Peter, Elie and Joonas

The word and ideas about algebra comes from a book called “Compendious book on Calculation and by Completion and Balancing”.It was written by a Persian mathematician called al-Khwarizmi who lived in Baghdad in 820 AD. 

Algebra comes from Al-Gabr which is the arabic word which means restoration or completion and he associated it in his book with a very specific operation taking something from one side of an equation to another side of an equation. There were other people who also thought of ideas like this, like the ancient babylonians from 2000 BC, Diaphantus in 200 to 300 AD in Alexandria, then there was also Brahmagupta in india at around 600 AD.

(left: picture of the book from wikipedia)






Algebra is the use of letters and symbols to represent numbers. 

An algebraic expression is a mathematical statement using algebra. An algebraic expression can contain any number of letters and arithmetic operations. 
An algebraic equation is a mathematical statement where two algebraic expressions are equal, expressed using an equals sign. 
A formula is a general rule to calculate something, expressed algebraically.

A variable is a number that is represented by a letter or symbol, often called the unknown
term is a part of an algebraic expression that is separated by + and - signs.
A constant is a number that is on its own and not attached to any variable (a number that is a term on its own).
A coefficient is a number that is placed directly before a variable (multiplies the variable in the term).



Simplifying algebraic expressions

examples:
 un-simplified notationsimplified algebraic notation 
 3 x m3m 
 2 x 3 x m 6m
 m x n mn
 n x m mn
 n x 7 x m 7mn
  
a + a2a 
 b + b + b3
 b + b - bb 
 3c + c4
3c - c  2c
 5d + 6d11d 
5d + 6d - 2d  9d
 5d - 7d-2d 
  
 3x + 2y + x 4x + 2y
  
  
  
  


Algebraic Substitution

Algebraic  substitution is when you replace the letters with given values.

a + b = ?

you need to know the values of a and b to be able to work this out.

so, if a = 3 and b = 11 then 
we substitute the values for a and b into the expression:

a + b = 3 + 11   
= 14

3a + 4b = 3(3) + 4(11)     
= 9 + 44
= 53       

4ab = 4(3)(11)
= 132


Questions:
If a = 4, b =5 and c= 10 evaluate the following expressions:

(i) 3a + 2b - c            (ii) 6(a+b)            (ii) a(3-b)        (iv) ab + bc        (v) c - 2b         (vi) abc - a - b - c













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