Algebra Skills Required for Entry to a Level Two Course in Mathematics

Transcription

1 Algebra Skills Required for Entr to a Level Two Course in Mathematics This is a list of Level One skills ou will be required to demonstrate if ou are to gain entr to the Level Two Achievement Standard Course in Year. To demonstrate that ou have gained these skills ou will be required to attend school and sit an Algebra test. If ou can demonstrate master of out of the Achieved skills and out of of the Merit skills then this will count as satisfing the Merit in Algebra criterion for entr to the Year course. Level of Achievement Skill Eample Simplif epressions involving indices Simplif Achieved Simplif epressions involving brackets Epand and simplif Epand a pair of brackets Epand and simplif Factorise a quadratic epression Factorise 0 Solve a linear equation Solve Solve a factorised quadratic equation Solve 0 Achieved with Merit Factorise, then solve, a quadratic equation Rearrange, then solve, a quadratic equation Solve simultaneous linear equations Rearrange a formula Solve, b factorising, Solve: Solve this pair of equations: 00 0 Rearrange the formula A a bh to make a the subject of the formula Simplif a fractional algebraic epression Simplif Add or subtract algebraic fractions Simplif Worked eamples, practice questions, and practice assessments with answers are provided in the following pages.

2 Achieved Skill Simplif epressions involving indices Eample a: Method: Answer: Simplif Cancel the fraction part. You can use the fraction ke on our calculator to do this. Or just divide both and b, giving the fraction. Now subtract the powers, so =. Eample b: Simplif Method: Multipl the numbers. So = 0 Now add the powers, so + =. Answer: 0 Eample c: Simplif Method: Answer: Work out. This is. Now multipl the powers. So becomes And becomes Achieved Skill Practice questions Simplif these: 0 0 a b 0 d p p d d a a a a a m n d e Go to List of Required Skills (page )

3 Achieved Skill Simplif epressions involving brackets Eample a: Epand and simplif Method: Multipl each term inside the bracket b the number that is outside. Remember = +. Then gather like terms = Answer: = Eample b: Epand and simplif Method: If there is just a sign in front, make it into Answer: = = Achieved Skill Practice questions Simplif these: ( ) + ( ) ( + ) ( ) ( ) ( ) ( ) ( ) + ( ) 0 ( ) + ( ) + ( ) ( ) ( ) ( ) + + ( ) ( + ) ( ) + ( ) ( ) ( + ) ( ) ( ) ( + ) Go to List of Required Skills (page )

4 Achieved Skill Epand a pair of brackets Eample a: Epand and simplif Method: Multipl each term inside the second bracket first b the and then b the. Remember + =. Then gather like terms. = Answer: = Eample b: Epand and simplif Method: Multipl each term inside the second bracket first b the and then b the. Remember = +. = 0 Answer: = 0 Achieved Skill Practice questions Epand the bracket and then simplif: ( )( ) ( + )( ) ( )( ) ( )( ) ( )( + ) 0 ( )( ) ( )( ) ( )( + ) ( )( + ) ( )( + ) ( + )( ) ( )( ) ( + )( ) ( )( + ) Go to List of Required Skills (page )

5 Achieved Skill Factorise a quadratic epression Eample a: Factorise 0 Method: Draw a pair of skeleton brackets, then put at the start of each bracket. Write down pairs of numbers that multipl to give 0. Choose the pair of numbers that also add together to give. Put these numbers into the brackets. Check that ou get 0 if ou epand the brackets. 0 = Answer: = Eample b: Factorise Method: Draw a pair of skeleton brackets, and put at the start of each bracket. Find the pair of numbers that multipl to give + and add together to give. Put these numbers into the brackets. Check that ou get if ou epand the brackets. = Answer: = Eample c: Factorise (This tpe is called the difference of two squares) Method: Draw a pair of skeleton brackets, and put at the start of each bracket. Our pair of numbers must multipl to give and add together to give zero (there is no term). The numbers are + and Check that ou get if ou epand the brackets. Answer: = Achieved Skill Practice questions Factorise: Go to List of Required Skills (page )

6 Achieved Skill Solve a linear equation Eample a: Method: Eample b: Method: + + Achieved Skill Practice questions Solve these equations: 0... Go to List of Required Skills (page )

7 Achieved Skill Solve a factorised quadratic equation Eample a: Solve: 0 Method: Answer: Let each bracket equal zero. This gives two equations, so there are two solutions. 0 or 0 or Eample b: Solve: 0 Method: Let each part equal zero. Again there are two solutions. 0 or 0 Answer: 0 or Achieved Skill Practice questions Solve these equations: ( + )( ) = 0 ( ) = 0 ( )( + ) = 0 ( )( + ) = 0 ( )( + ) = 0 0 ( ) = 0 ( )( ) = 0 ( )( + ) = 0 ( )( + ) = 0 ( + )( ) = 0 ( + )( + ) = 0 ( ) = 0 ( ) = 0 ( + ) = 0 Go to List of Required Skills (page )

8 Merit Skill Factorise and solve a quadratic equation Eample a: Solve: Method: Answer: 0 or 0 Eample b: Solve: 0 0 Method: 0 0 Answer: or Merit Skill Practice questions Solve these equations: Go to List of Required Skills (page )

9 Merit Skill Rearrange, then solve a quadratic equation Eample a: Solve: Method: 0 0 Answer: or Eample b: Solve: 0 Method: Answer: or Eample c: Solve: Method: 0 0 Answer: or Merit Skill Practice questions Solve these equations: Go to List of Required Skills (page )

10 0 Merit Skill Solve simultaneous linear equations Eample a: Solve: 00 0 Method: Substitute + 0 in place of in the first equation ( +0) + = = 0 = 0 = 00 Now substitute this into the second equation to find out what is. Answer: 0 and Eample b: Solve: Method: The equations have to be arranged this wa if ou are using a graphics calculator. Rearrange the second equation so that it matches the first one Divide the first equation b 0 (or multipl the second equation b 0) Add the equations to eliminate the variable 0 0 Substitute 0 into one of the original equations, and solve for Answer: and 0 Merit Skill Practice questions Solve these pairs of simultaneous equations: Go to List of Required Skills (page )

11 Merit Skill Rearrange a formula Eample a: Make a the subject of the formula A a bh Method: a bh A Answer: a a bh A A a b h A h b Eample b: Make a the subject of the formula v u at Method: u at v at v u Answer: v u a t Merit Skill Practice questions The formula for the perimeter of a rectangle is P l b. Rearrange the formula to make b the subject. Make n the subject of the formula 0n S. Make h the subject of the formula A bh. The formula for the distance around a running track is D l w. Rearrange the formula to make w the subject. The formula for the area of a square with an equilateral triangle on one of its sides is A.. Rearrange the formula to make the subject. The formula for the volume of a clinder is V r h. Rearrange the formula to make (i) h (ii) r the subject. Make d the subject of the formula Make v the subject of the formula A d. E mv. Make d the subject of the formula T a n d. 0 The equation of a straight line is. Rearrange the equation to make the subject. Go to List of Required Skills (page )

12 Merit Skill Simplif a fractional algebraic epression Eample a: Simplif: Method: = = Answer: = Eample b: Simplif: Method: = = Answer: = Merit Skill Practice questions Simplif: pq q p q p 0 Go to List of Required Skills (page ) Factorise top, or bottom, or both. Then cancel.

13 Merit Skill Add or subtract algebraic fractions Eample a: Simplif: Method: = = Answer: = Use the method for adding fractions Eample b: Simplif: Method: = = Answer: = Merit Skill Practice questions Simplif: 0 m m p p 0 z z a a d d 0 k k Go to List of Required Skills (page )

14 Practice Assessment One Simplif a b Solve: 0 Simplif Solve these simultaneous equations: a b 0 a b Epand and simplif Factorise 0 Make h the subject of the formula V r h Solve: Simplif a b a ab b Solve: 0 Solve: 0 0 Simplif Go to List of Required Skills (page )

15 Simplif a a Practice Assessment Two Solve: Simplif p q p q Solve these simultaneous equations: Epand and simplif Factorise 0 Make r the subject of the formula A r Solve: Simplif Solve: 0 Simplif m m Solve: 0 Go to List of Required Skills (page )

16 Simplif a a Practice Assessment Three Solve: Simplif a a Solve these simultaneous equations: Epand and simplif d d Factorise 0 Make t the subject of the formula C k nt Solve: Simplif Solve: 0 Solve: 0 0 Simplif Go to List of Required Skills (page )

17 Answers Achieved Skill Simplif epressions involving indices 00 0 a b 0 d p a 0 d a m n d e 0 Answers Achieved Skill Simplif epressions involving brackets ( ) + ( ) = + = ( ) ( ) = + = + + ( ) = + = 0 + ( ) = + = ( ) + + = + + = + + ( ) + ( ) = = ( + ) ( ) = + = + ( + ) ( ) = + + = + 0 ( ) ( ) = + + = 0 ( ) + ( ) = + = ( ) ( ) = 0 + = ( ) ( + ) = = ( ) = + = + ( ) ( + ) = + = Go to List of Required Skills (page )

18 Answers Achieved Skill Epand a pair of brackets ( )( ) = + = + ( )( ) = + = + ( )( + ) = + = + ( )( ) = 0 + = + ( )( + ) = + = ( + )( ) = + = ( + )( ) = + = + ( + )( ) = + = ( )( ) = + = + 0 ( )( ) = + 0 = + 0 ( )( + ) = + 0 = + ( )( + ) = + = + ( )( ) = 0 + = + ( )( + ) = + 0 = 0 + Answers Achieved Skill Factorise a quadratic epression = ( )( + ) 0 = ( + )( ) 0 = ( 0)( ) = ( )( ) 0 = ( + )( ) = ( + )( ) 0 = ( )( ) = ( + )( ) 0 = ( )( + ) = ( + )( ) = ( + )( ) 0 = ( + )( + ) = ( + )( ) = ( )( + ) = ( )( + ) Go to List of Required Skills (page )

19 Answers Achieved Skill Solve a linear equation Answers Achieved Skill Solve a factorised quadratic equation or or or or or or 0 or 0 or or 0 0 or or or 0 or 0 or Go to List of Required Skills (page )

20 Answers Merit Skill Factorise and solve a quadratic equation or or or or or or or or or or 0 0 or or or or or (twice) 0 0 or 0 0 (twice) (twice) or or 0 0 or or or 0 Go to List of Required Skills (page )

21 Answers Merit Skill Rearrange, then solve, a quadratic equation or or 0 0 or or 0 0 or 0 0 or or or or or or 0 0 or or or or or 0 0 Go to List of Required Skills (page )

22 Answers Merit Skill Solve simultaneous linear equations 0 and and 0 0 and 0 subtract then eqn, Rearrange Now substitute intoeqn 0 and and eqn Rearrange subtract then, Double eqn Now substituteintoeqn and and Rearrangeequation 0 0 eqn b Multipl 0 b eqn Multipl 0 Now substitute intoeqn Subtracting equations Go to List of Required Skills (page )

23 Answers Merit Skill Rearrange a formula b l P l P b l P b P b l 0 n S 0 0 this : or ou can do S n S n S n S n S n S n bh A b A h A bh A bh w l D l D w l D w D w l. A... A A A h r V r V h V h r h V r h V r V h r d A A d A d A d mv E m E v m E v E mv E mv d n a T n a T d a T d n T d n a 0 Go to List of Required Skills (page )

24 Answers Merit Skill Simplif a fractional algebraic epression pq q p q p q p pq q q p 0 Go to List of Required Skills (page )

25 Answers Merit Skill Add or subtract algebraic fractions m m m m 0 m a a a a a k k k 0k k k 0 0 p p 0p p p z z z z z d d 0 d d 0 0 d 0 Go to List of Required Skills (page )

26 Practice Assessment Answers Assessment One Assessment Two Assessment Three a b a a 0 0 or 0 or or a and b 0 h ab 0 V r p q 0 or or or and 0 r m A a d d 0 or or or and 0 t C k n Go to List of Required Skills (page )