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Given y = 3/(5√x) . The expression for y2 is:
A. 6/(10x2)
B. 9/(25x2)
C. 6/(10x)
D. 9/(25x)
1. Deni is 36 years old, while Lira is 44 years old. Lira says she and Danny are about the same age in the top ten. a) Is Lira right and why? b) Will Deni and Lira be the same age in the nearest tenth even after one year? c) How old will Deni and Lira be next time they are of the same approximate age in the nearest ten?
Other important numerical concepts often used in algebra are exponents and roots. We mention these two ideas in the same breath as they are each other’s inverses
Do the following:
In 1911, two hat-makers each taught their two sons to make hats.
In 1932, each of these sons taught their two sons to make hats.
In 1953, each of these sons taught their two sons to make hats.
In 1974, each of these two sons taught their two sons to make hats.
In 1995, each of these two sons taught their two sons to make hats.
How many hat-makers were there in this family by the end of 1995?
This situation in algebra:
Inverse operations is a concept that needs to be understood because it is often used in algebra to solve variables.
We explain inverse operations as applied in numerical statements. Inspect the following example of a numerical equation, which is later translated into an algebraic equation:
21 × 5 - 10 = 95
21 × 5 = 95 + 10 (Add 10 both sides of =.)
21 = (Divide by 5 both sides of =.)
Do the following:
Following our example to the left, work through the numerical equation below and then translate it into an algebraic equation, where you solve for the unknown value:
3/5 + 13 = 20
• The identity property has two parts:
– 0 is the identity element for addition (or the additive
identity of number), meaning a number stays the
same if 0 is added to it.
– 1 is the identity element for multiplication (or
the multiplicative identity of number), meaning numbers stay the same if multiplied by 1.
It is also true that 0 subtracted from any number leaves the number unchanged; and that any number divided by 1 leaves the number unchanged.
• The zero product property. The product of numbers multiplied by each other is 0 only if at least one of them is 0.
• The density property. Between any two real numbers there is another real number.
• Distributive property
Example:
4 (2 t + 5) = 8 t + 2 0 Do the following:
3 (3 t + 7) =
• Additive identity 7u + 0= 7u
• Multiplicative identity
5 v × 1 = 5v
• Zero product property
Example:
(2w-3)(3w+2)=0,meansthat either (2 w + 3) = 0 or (3w+2) = 0
Do the following:
3 x (4 x - 3) =0
Learners need to know the properties of numbers and the properties of operations on numbers. Several properties are important here:
• The commutative property of numbers for addition and multiplication.
• The associative property of numbers for addition and multiplication.
• The distributive property of numbers for multiplication over addition.
• Commutative property Example: 4 p + 6 = 6 + 4 p
Do the following: 5 + 3 q = Example: 2 r × 3 s = 3 s × 2 r Do the following: 4 t × 6 =
• Associative property Example: (5 + 4 p) + (6 + 2 p) = (4 p + p 2 )+ (5 + 6)
Do the following: (4 q+ 5) + (3 q + 7) =
Example:
(5 r × 2 r) × ( 3 s × 4 s)
= (5 r × 3 s) × (2 r × 4 s) Do the following:
(4s×6)×(5×3s)=
Activity 2.3 Application of mathematical concepts and procedures in algebra
Mathematical idea (concept or procedure) Example of application in algebra
1. Learnersmustbeabletoswitchcomfortablybetween all operations involving whole numbers and apply these operations.
Two ideas come together here:
• command of number and number sense
• comfortably operating numbers
In the Intermediate Phase these numbers are positive; in the Senior Phase learners start operating on integers (including positive and negative whole numbers).
What is the rule that you can abstract from what you found above?
Explain why this is the case.
Generalise your findings to predict what you will observe in the sum of
(a) fiveconsecutivemultiples of 10 like (30 + 40 + 50 + 60 + 70)
(b) three consecutive numbers like
(7 + 8 + 9)
(c) seven consecutive numbers like
(2 + 3 + 4 + 5 + 6 + 7 + 8)
Activity 2.3 Application of mathematical concepts and procedures in algebra
Mathematical idea (concept or procedure) Example of application in algebra
1. Learnersmustbeabletoswitchcomfortablybetween all operations involving whole numbers and apply these operations.
Two ideas come together here:
• command of number and number sense
• comfortably operating numbers
In the Intermediate Phase these numbers are positive; in the Senior Phase learners start operating on integers (including positive and negative whole numbers).
Do the following:
Calculate and write down the sum of the following:
• 11 + 12 + 13 + 14 + 15 =
• 23+24+25+26+27=200
+ 300 + 400 + 500 + 600 =
What is the fifth multiple of 13? What is the fifth multiple of 25? What is the fifth multiple of 400?
What is the rule that you can abstract from what you found above?
Explain why this is the case.
determine the sum of the geometric series 7+35+75+...+t8
a company packages stickers in sets of 40. if each set costs $80.85, what is the cost of 240 stickers?"
