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Presentation flowchart:

The title of the rule

Mathematical expression of the rule

Example for the rule

Additional comments / observations of the rule

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Group 3

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((X+15).2)/3 - x/3 -->> 10

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Group 2

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[[(X+15) × 2 + x] /3] -×
-> [(2x + 30 + x)/3] - x
-> [(3x + 30)/3] - ×
-> × +10 - x
-> 10

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Removing brackets in algebraic expression is done by:

(a)Multiplying or dividing each term inside the bracket by the term outside the bracket, but

(b)If the term outside the bracket is negative then each term inside the bracket changes sign.

For example:

3x(y-2z)=3xy-6xz

-2y(2x-4z)=-4yx+8yz

(y+x)/8x-(y-x)/4x=3/8-y/8x

Exercises: Expand the following and then refactorize where possible:

(a)   8x(y-z)+2y(7x+z)

(b)   (3a-b)(b-3a)+b^2

(c)    -3{w-7[x-8(3-z)]}

(d)    (2a-3)/4b+(3a+2)/6b

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8. Similar terms, Common factors and factorization

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8. Similar terms, Common factors and factorization

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In the algebraic expression:

ab+bc

Both terms contain the letter b and for this reason these terms, though not like terms, are called similar terms.

Common symbols such as this letter b are referred to as common factors and by using brackets these common factors can be factored out.

Example 6:

ab+bc=b(a+c)

This process is known as factorization.

Exercise: Simplify each of the following by collecting like terms and factorizing:

(a)   4xy+3xz-6zy-5zx+yx

(b)   3rst-10str+8ts-5rt+2st

(c)   2pq-4pr+qr-2rq+3qp

(e)   5lmn-6ml+7lm+8mnl-4ln

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Terms which have the same variables are called like terms.

Like terms can be collected together by addition or subtraction. For example:

4x+3y-2z+5y-3x+4z
=(4x-3x)+(3y+5y)+(-2z+4z)
=x+8y+2z

Exercises: Simplify each of the following by collecting like terms:

(a)   4xy+3xz-6zy-5zx+yx

(b)   -2a+4ab+a-4ba

(c)   3rst-10str+8ts-5rt+2st

(d)   2pq-4pr+qr-2rq+3qp

(e)   5lmn-6ml+7lm+8mnl-4ln

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Algebraic expression notation:
1 – power (exponent)
2 – coefficient
3 – term
4 – operator
5 – constant term
x y c – variables/constants

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Group Discussion and Presentation, based on the Note - Summary of Algebra (The Formal Rules of Algebra),
GIVE EXAMPLES!!! Peer marking trial*

Group 1 - 1.The axioms of equals; 2. Commutative rules; 3. Identity elements.
Group 2 - 4. Additive inverse; 5. Multiplicative inverse; 6. Definition of subtraction.
Group 3 - 7. Definition of division; 8. Inverse of inverse; 9- b-a and a-b.
Group 4 - 10. Rule of signs; 11. Rules for 0; 12. Factoring / Multiplying.
Group 5 - 13. Operation on both sides; 14. Change the sign; 15. Change the sing on inequality.
Group 6 - 16. Equations; 17. Change of sense on inequality; 18. Absolute value.
Group 7 - 19. Equivalent fractions; 20. Multiplication of fractions; 21. Division of fractions.
Group 99 - 22. Addition of fractions; 23. Rules of exponents; 24. Negative exponent.
Team X - 25. Exponent 0; 26. Square root; 27. a^2=b.
Volunteer ?- 28. Multiplying radicals; 29. nth root; 30. Rational exponent; 31. Laws of logarithms; 32. Complex unit.

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Group Discussion: Why everybody achieved the same result? Try to explain mathematically on a paper and then upload to Padlet.

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Example 5: Two light sensors are giving the reading of the ambient brightness, S1 and S2, respectively. Write an expression to calculate the average brightness of the two sensors.

Answer: Assume the average brightness is B, the expression would be: B = (S1 + S2) / 2.

clybot C6: 'hear' the brightness, send the average brightness as the frequency to the C6.

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Example 4: Which is the constant, which is the variable?

Answer:

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A constant is a number.
A variable is a value that can change.
Usually, letters from the beginning of the alphabet, i.e. a, b, c, d, … are used to represent constants, and letter from the end of the alphabet, i.e. …v, w, x, y, z are used to represent variables.

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Answer

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Example 3: Mitosis is the process in which one cell divides into two cells identical to the parent cell. Calculate the cell number after 5 mitoses of two original cells?

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Example 2: What if the f doubles each second?

Answer: f=100, f'=f*2*2*2*2*2=3200(Hz)

clybot C6: listen to the difference of the tone.

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Example 1: A frequency variable f is assigned by f=100 (Hz). The f will increase by 200 Hz every second, express and calculate the f after 5 seconds.

Answer: f=100, f'=f+200*5=1100(Hz)

clybot C6: listen to the beep over 5 seconds.

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A robot kit for learning maths, programming, electronics, computing, microcontroller.........

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Poker Game: Take any poker card, add 15 to it, double the result, add this to the poker card number, divide the result by 3, take away the poker card number, calculate the result.

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An introduction to the basic concepts and techniques of algebra for students who have not studied mathematics at tertiary level.

Number of Credits | 20
Level | L3
Module Lecturers | Dr Xin Yang and Dr Alistair Reid

On completion of the module a student should be able to

Solve linear and quadratic equations

Solve simple simultaneous equations

Understand the terms function, domain, range, one-to-one function and composition of functions

Recognise, sketch and interpret graphs of functions

Understand the definition and properties of ex and ln x

Understand coordinate geometry and apply it to solving equations

Use standard notation of vectors, apply a number of operations to vectors and use them to solve a variety of geometric problems

Solve equations numerically

Assessment Breakdown

Examination - Autumn Semester   80%
Class Test   20%  (4 Nov 2016)

Essential Reading and Resource List

Foundation Mathematics – K.A. Stroud (Palgrave Macmillan)

Foundation Mathematics – L. Mustoe and M. Barry (Wiley)