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NATIONAL 5 : BLOCK 3 EVALUATION BOOKLET

1 St Andrew’s Academy Mathematics Department.

Skills R A G Amber/Red Go to

Indices

*Remember Normal rules for multiplying and dividing number still apply, also (4𝑥3𝑦6)3 = 43𝑥9𝑦18

Page 37-38



Factorising

Common Factor: Look for the Highest times table the numbers are in. Any common letters between terms. Common Letters with the smallest power will be brought out as a common factor.

4m + 16 12xy + 9yz 4𝑚2 + 24𝑚 16𝑥3𝑦 − 4𝑥2𝑦5 =4(m + 4) =3y(4x + 3z) =4m(m + 6) =4𝑥2𝑦(4𝑥 − 1𝑦4)

Page 39-40



Difference of 2 squares: Require only two terms. Must be able to be written as something squared and will always have a subtract sign in the middle. Look for common factors first. (Take the square roots of perfect squares) Each bracket must have a different sign one + one - 𝑎2 − 𝑏2 25𝑥2 − 4𝑏2 27𝑎2 − 300𝑏2 𝑥2 − 1 =(𝑎 − 𝑏)(𝑎 + 𝑏) =(5𝑥 − 2𝑏)(5𝑥 + 2𝑏) = 3(9𝑎2 − 100𝑏2) =(𝑥 − 1)(𝑥 + 1) =3(3𝑎 − 10𝑏)(3𝑎 + 10𝑏) Trinomials Involve a squared term a letter term and a constant. There are many different techniques, you should follow the one your teacher has shown you. If the trinomial is of the form 𝑎𝑥2 ± 𝑏𝑥 + 𝑐 The plus with the c term means that the brackets will have the same sign. The middle terms sign (b) will tell you which sign to choose. Numbers must multiply to make c but add or subtract to make b. Examples: 𝑥2 + 5𝑥 + 4 𝑥2 − 5𝑥 + 6 =(𝑥 + 1)(𝑥 + 4) =(𝑥 − 3)(𝑥 − 2) * Note 6 and 1 would also give 5 but only if the signs were different.* If the trinomial is of the form 𝑎𝑥2 ± 𝑏𝑥 − 𝑐 The minus with the c term means that the brackets will have different signs. The middle terms sign (b) will tell you which sign to choose to put with the biggest number. Numbers must multiply to make c but add or subtract to make b. Examples: 𝑥2 + 2𝑥 − 8 𝑥2 − 5𝑥 − 24 =(𝑥 − 2)(𝑥 + 4) =(𝑥 + 3)(𝑥 − 8)



Coefficient with the squared term. Examples: 3𝑥2 + 11𝑥 + 6 *Note the start of our brackets now must multiply to give 3𝑥2 = (3x )(x ) The 3 will also multiply the last term in the second bracket. = (3x + 2)(x + 3) It is a good idea to always multiply out and check your answer. 3𝑥2 + 9𝑥 + 2𝑥 + 6 = 3𝑥2 + 11𝑥 + 6 6𝑥2 − 11𝑥 − 10 We now have options for 6𝑥2 which are 6 and 1 or 3 and 2. =(3x )(2x ) =(3x - 2)(2x + 5) The rules for the signs are still the same.

Algebraic Fractions Simplifying Fractions: In order to simplify a fraction each term must have something in common.

Page 41-43



* We must factorise numerator and denominator first*



Multiplying and Dividing Algebraic Fractions We multiply and divide Algebraic Fractions in the same way we deal with numerical fractions.



Adding and Subtracting : We still need a common denominator, we can use the method of smile and a kiss. (ONLY WHEN ADDING OR SUBTRACTING). Remember to simplify at the end.

Note b and d do not cancel as not every term involves b and d.



Examples:

Example 2: 2

𝑥+

3

𝑥3

=2𝑥3+3𝑥

𝑥4 Each term involves an x and so this can be cancelled as

a common factor.

= 2𝑥2+3

𝑥3



Vectors

Component Form:

Page 44-47



Multiplying a Vector by a Scalar

If vector a= ( 𝟑−𝟐

) then any multiply of a is found by multiplying its components.

4a =(𝟏𝟐−𝟖

) . Everything that can be done to 2D vectors can be applied to 3D.



Different Representations of Vectors

Component Form: a= (56) Coordinate Form: A(3,7,-2) Cartesian Form u = 4i + 3j +6k



Right-Angled Trigonometry

It is most important that you label the triangle correctly. Make sure your calculator is on Degrees. Shift Mode 3. Or press the button that says DRG until it reads DEG at the top. The Hypotenuse is always opposite the right angled. The opposite is across from the angle you are given or trying to find. The adjacent is the side that touches both the angle and the right angle.

SOH-CAH-TOA are abbreviations used to help us remember the following ratios:

Use the technique your teacher has taught you to identify which sides you have and what you are trying to find. Example 1: Find the angle A. Then use properties of triangles to find the angle B.

Page 48-49



As A= 41.2˚ Angle B = 180 - (90 + 41.2) = 48.8 ˚. Remember if you have two angles you can find the third. Example 2



𝒕𝒂𝒏𝒙˚ =𝑶

𝑨

𝒕𝒂𝒏𝒙˚ =𝟏. 𝟖

𝟐. 𝟒

𝒙˚ = 𝒕𝒂𝒏−𝟏 (𝟏. 𝟖

𝟐. 𝟒)



𝒙˚ = 𝟑𝟔. 𝟖𝟕˚

𝒕𝒂𝒏𝒙˚ =𝑶

𝑨

𝒕𝒂𝒏𝟓𝟓˚ =𝒙

𝟏𝟎

𝟏𝟎 × 𝒕𝒂𝒏𝟓𝟓˚ = 𝒙 𝒙 = 𝟏𝟒. 𝟐𝟖𝐜𝐦

Pythagoras Theorem (Including Circles)

c is the hypotenuse and is across from the right angle. This is the longest side on a right-angled triangle. Start by labelling the sides of the triangle and then substitute the values of the sides in appropriately.

Page 50-54



*Note* You could also be asked to calculate the length of the chord, again set up the right angled triangle and find the appropriate length. Finish finding the full chord by doubling your answer.



Scale Factor

3 different types of Scale Factor Linear Scale Factor: Will calculate a missing side on a similar shape.

Enlargement Scale Factor: 𝐵𝑖𝑔 𝑛𝑢𝑚𝑏𝑒𝑟

𝑆𝑚𝑎𝑙𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 Reduction Scale Factor: :

𝑆𝑚𝑎𝑙𝑙 𝑁𝑢𝑚𝑏𝑒𝑟

𝐵𝑖𝑔 𝑁𝑢𝑚𝑏𝑒𝑟

Enlargement Scale Factor if the unknown is on the bigger shape. Reduction Scale Factor if the unknown is on the smaller shape.

Page 55



Identify the appropriate pairs and then multiply the matching side by the scale factor to find the missing length. If you set up the ESF and RSF in this way you will always multiply by the scale factor. Area Scale Factor: Will find the missing area on a similar shape.

Find the ESF or RSF

Square the value of the ESF or RSF

Multiply the appropriate area by the squared Area scale factor. Volume Scale Factor: Will find the missing volume on a similar shape.

Find the ESF or RSF

Cube the value of the ESF or RSF

Multiply the appropriate volume by the cubed Volume scale factor. REMEMBER: To square or cube the linear scale factor first, before multiplying. If your teacher has shown you a different method please use what you understand.



Similar Triangles (Scale Factor Continued)

Similar Triangles must all have the same corresponding angles.

For x: Reduction Scale Factor 100

8. Multiply the corresponding side to x. x = 10 ×

100

8 = 125units.

For y: Enlargement Scale Factor 8

100. Multiply the corresponding side to y. y = 90 ×

8100 =7.2 units

It would help to rotate one triangle, in order to, easily identify the corresponding sides. Y corresponds to 37.25. x corresponds to 3.2 and 2.8 corresponds to 45. Follow the same method as example 1 to find the values of x and y.

Page 56



Angles in a Triangle

Angles around a point add to 360˚. Parallel Lines: Angles that form within an F shape are corresponding angles. They will both sit in the same position to the line whether it be above or below. For example 136˚. Using the opposite angles rule from the first box we can calculate others around the point. You can also look for Z shapes the corners of the Z’s are known as alternate angles and will be on the opposite side of the transversal (line connecting parallel lines).

Page 58



Corresponding Angles Alternate Angles Angles in a Triangle add to give 180˚, Angles on a straight line add to give 180˚.

Angle STR happens at T. 180 – ( 25 + 35 ) = 120˚

Isosceles Triangles: Usually presented in a circle using a chord and two radii.



180 – 116 = 64˚ The remainder must be shared equally between the base angles. 64˚ ÷ 2 = 32˚



Angles in a Semi-Circle Tangents

We

then use the property of angles in a triangle adding to 180⁰. Tangents: A tangent is a straight line that meets the circumference of a circle in one place. A tangent will always be outside the circle. It meets the radius at 90⁰.

Page 57



Indices

Exercise 1



Factorising



Trinomials



Algebraic Fractions: Simplifying, Multiplying and Dividing



Algebraic Fractions: Adding and Subtracting



Vectors



Right-Angled

Trigonometry



1.

2.

3.



Pythagoras



Pythagoras: Space Diagonals



Pythagoras in the Circle



Converse of Pythagoras



Scale Factor



Similar Triangles



Angles in the circle



Angles in a Triangle

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