fractions and decimals - wordpress.com · 2019. 8. 19. · name: bam indicator: solve problems...

8M4 © Kangaroo Maths 2014

FRACTIONS AND DECIMALS

Name:

BAM Indicator: Convert between terminating decimals and fractions

1. a) Write each of these fractions as a decimal

i. 7

25 =

____________

ii. 3

50 =

____________ iii. 11

20 =

____________

iv 9

40 =

____________

b) Write each of these decimals as a fraction in its lowest terms

i. 0.8 =

____________

ii. 0.23 =

____________ iii. 0.16 =

____________

iv 0.005 =

____________ 2. Put a ring around the fractions that are equivalent to terminating decimals. Explain how

you know, without using a calculator.

17

20

23

75

89

130

32

125

31

40

3. This statement shows an equivalence between a fraction and a decimal:

⊗⊗⋒

⊗⋒⋒⋒=⋒.⊗⊗

Find the digit represented by each symbol.

⊗ = ________

⋒ = ________ 4. Write these numbers in order of size, starting with the greatest

2

5, 45%, 0.405,

12

25, 0.35

________, ________, ________, ________, ________

F R P A


5. Match the equivalent fractions and decimals. Complete the blanks to create six matching pairs.

3

2 1.8

1

8

3

5 0.125

9

5 0.15

0.625

3

20 1.5

M

Overall, I think my success level is: Low High

F = Fluency R = Reasoning P = Problem-solving A = Application M = Misconception

Q FRACTIONS AND DECIMALS

I can identify when a fraction can be scaled to tenths or hundredths

I can convert a fraction to a decimal by scaling (when possible)

I can identify if a fraction is terminating or recurring

I can write a decimal as a fraction

I know some decimal and fraction equivalents (e.g. tenths, fifths, eighths)

I can write a fraction in its lowest terms by cancelling common factors

Improvements I could make:

Mathematical presentation Method Accuracy Units


PERCENTAGES

Name:

BAM Indicator: Solve problems involving percentage change, including original value problems

1. Match the percentage calculation to its multiplier. Fill in the blanks to create six correct matchings.

An increase of 2.5% × 0.25

A decrease of 75%

2.5% of an amount × 1.25

A decrease of 25% × 1.025

× 0.75

A decrease of 2.5% × 0.975

2. a) Increase 429 grams by 31%

____________ b) Decrease 345 litres by 12%

____________ c) The height of a sunflower increases from 82 cm to 1 metre in a week. Find the

percentage increase in height.

____________ 3. A jacket is reduced by 40% in a sale. The cost of the jacket is now £126. Pauline says,

“40% of £126 is £50.40. So the jacket originally cost £176.40.”

Pauline is wrong. Explain why.

M F R


4. a) A plumber’s bill is £463.20 plus VAT. The current rate of VAT is 20%. What is the actual cost of work done?

____________

b) The value of sales at a shop decreases from £145,000 a year to £129,000. Find the

percentage decrease in the value of sales.

____________ c) A pair of shoes is in a ‘35% off’ sale. The recommended retail price (RRP) is £86. How

much do the shoes cost in the sale?

____________ d) On the final weekend of the ‘35% off’ sale the prices are cut by a further 15%. The

manager of the shop advertises a ‘50% off’ sale. Is she correct? Explain your answer.

5. A vehicle costs £17,754 including VAT at 20%. What was the price of the vehicle before VAT was added?

____________

A P



Q PERCENTAGES

I can identify and use a multiplier to increase an amount by a percentage

I can identify and use a multiplier to decrease an amount by a percentage

I can find a percentage change for a given situation

I can identify a reverse percentage problem (finding the original value)

I can find the original value in a percentage problem



FORMULAE

Name:

BAM Indicator: Change the subject of a formula when two steps are required

1. Rearrange these formulae to make x the subject

a) y = 3x

x = ____________________

b) y = x + 4

x = ____________________

c) y = x2

x = ____________________

d) y = 𝑥

7

x = ____________________

e) y = 3x – 2

x = ____________________

f) 𝑦 =𝑥+2

3

x = ____________________

2. Alice is given the formula 𝑎 =𝑏

3− 2

She is asked to make b the subject of the formula. Alice writes b = 3a + 2. Explain why Alice is wrong. 3. Match each formula to an equivalent statement. Complete the blanks to create six

matching pairs.

y = 2x + 1 𝑦 + 1

2= 𝑥

𝑦 =𝑥

2+ 1

𝑦 − 1

2= 𝑥

y = 2 – x y + 2 = x

2y – 2 = x

y = 2x – 1

𝑦 =𝑥 + 1

2 2 – y = x

F M P


4. The formula v = u + at describes connections between

u: starting velocity

a: acceleration

t: time

v: final velocity

Rearrange the formula so that acceleration can be found if the other three values are known.

a = ____________

5. Stan and Fran are using the formula p = 2q + 3.

They need to find the value of q when p = 9.5. Their solutions are shown below.

Stan’s solution p = 2q + 3

𝑝−3

2= 𝑞 (change subject of formula)

p = 9.5

𝑞 =9.5−3

2= 3.25

Fran’s solution p = 2q + 3

p = 9.5

9.5 = 2q + 3 (set up equation)

6.5 = 2q

q = 3.25

Stan and Fran are both correct. Which approach do you think you would use? Give reasons for your answer.

A R



Q FORMULAE

I can identify inverse operations

I can identify the correct order of ‘undoing’ a formula

I can change the subject of a formula involving one step

I can change the subject of a formula involving two steps

I can change the subject of a formula when the new subject is being subtracted




EQUATIONS

Name:

BAM Indicator: Solve linear equations with unknowns on both sides

1. Solve these equations

a) 8k + 40 = 16

k = ____________________

b) 3(x + 6) = 3

x = ____________________

c) 4p – 6 = 2p + 10

p = ____________________

d) 10v + 6 = 4v + 15

v = ____________________

e) 9 – 3y = 4y + 2

y = ____________________

f) 3x – 2 = 23 + 8x

x = ____________________

2. Ethan solves the equation 8y – 40 = 3y – 10. He gets the answer y = 10. Do you agree with Ethan? Explain your answer.

3. Write numbers in the boxes to create an equation with a two-digit integer solution.

x + 2 = 4x +

F M P



Q EQUATIONS

I know how to solve an equation by balancing both sides

I can solve an equation with unknowns on both sides

I can solve an equation with unknowns on both sides when brackets are involved

I know how to deal with negative numbers as solutions

I know how to deal with fractions as solutions

I can construct an equation from given information



4. Look at the following equations:

9x + 1 = 6(x + 1) 3x + 2 = 4 – x 5(x + 2) = 3x + 6 3x – 16 = 4 – 2x

Which is the odd one out? Put a ring around the one that is. Give reasons for your answer.

5. A rectangle is 6 cm wide.

Two different expressions are used to state its length. a) Write down an equation that can be used to find the length of the rectangle.

______________________________ b) Solve the equation and find the area of the rectangle

________________ cm2

R A


(3x + 7) cm

(5x – 17) cm

6 cm


MANIPULATING EXPRESSIONS I

Name:

BAM Indicator: Manipulate algebraic expressions by expanding the product of two binomials

1. Expand the following

a) (𝑥 + 5)(𝑥 + 6)

____________________

b) (𝑥 + 3)(𝑥 − 2)

____________________

c) (𝑥 − 1)(𝑥 − 3)

____________________

d) (𝑥 + 7)(𝑥 − 7)

____________________

2. Velma expanded (𝑥 + 3)2 and said,

“When you expand the bracket you get the answer 𝑥2 + 9"

Explain why Velma is wrong

3. By expanding and simplifying each expression, identify the odd one out:

𝑚2 − 𝑛2

2𝑛(𝑚 − 𝑛) + (𝑚 − 𝑛)(𝑚 − 𝑛)

𝑚(𝑚 − 𝑛) − 𝑛(𝑚 − 𝑛)

(𝑚 − 𝑛)(𝑚 + 𝑛)

F M P


4. Eric draws a plan of his garden.

Eric’s shed is square-based and covers an area of 𝑥2 metres squared. Write a simplified expression in the form 𝑥2 + 𝑏𝑥 + 𝑐 for the total area of the garden.

________________ 5. When (𝑥 + 𝑎)(𝑥 − 3) is expanded the answer is 𝑥2 + 𝑏𝑥 + 12

a) Explain why 𝑎 and 𝑏 must be negative numbers.

b) Hence find out the value of 𝑎 and 𝑏

𝑎 = ______

𝑏 = ______

A R



Q MANIPULATING EXPRESSIONS I

I can apply the four operations with negative numbers and expressions

I know an approach to multiply two linear expressions

I can multiply two linear expressions

I can simplify an expression by collecting like terms




MANIPULATING EXPRESSIONS II

Name:

BAM Indicator: Manipulate algebraic expressions by factorising a quadratic expression of the form x2 + bx + c

1. Factorise the following

a) 𝑥2 + 7𝑥 + 12

____________________

b) 𝑥2 + 2𝑥 − 48

____________________

c) 𝑥2 − 13𝑥 − 48

____________________

d) 𝑥2 − 13𝑥 + 12

____________________

2. Haydn writes:

“When you factorise 𝑥2 + 5𝑥 + 6 you get 𝑥(𝑥 + 5) + 6”

Comment on Haydn’s statement.

3. Look at this identity:

𝑥2 − 16𝑥 + 𝑎 ≡ (𝑥 + 𝑏)(𝑥 + 2)

Find the values of a and b. Explain each step of your solution.

F M R


4. Shabneez is designing a new house. She draws a plan of the lower floor. The dining room is a square with side length of d metres. The living room and the kitchen have a total area of 7d m2. Given that the hallway has an area of 10 m2, find the dimensions of the house in terms of d.

Dining Room Living Room

Kitchen Hallway

________________ 5. Killian thinks of a number, n. He multiplies this number by 3, and then takes away 10.

Next, he adds the square of his original number. His answer is the product of two integers. What is the difference between the two integers?

___________

A P



Q MANIPULATING EXPRESSIONS II

I can apply the four operations with negative numbers and expressions

I know an approach to factorise a quadratic expression

I can factorise an expression of the form x2 + bx + c




GRADIENT

Name:

BAM Indicator: Understand and use the gradient of a straight line to solve problems

1. Find the gradient of each of these lines:

2. Which of the following lines are parallel? Show how you reach your conclusion.

a) _________ b) _________ c) _________ d) _________

F M R

𝑦 − 2𝑥 = 1

𝑦 = 2𝑥2 + 3 𝑦 = 2𝑥 + 3

𝑦 = 3𝑥 + 2 2𝑦 = 𝑥 + 6

𝑦 = 𝑥 + 3


3. A line goes through the two points (3,7) and (–2,–8). Louis thinks that the equation of the line is y = 3x – 2.

Do you agree? Explain your answer.

4. The world’s steepest residential street is Baldwin Street in New Zealand. It has a gradient

of 0.35. As the road rises the horizontal distance covered is 330 metres. How much vertical height is gained if you walk from the bottom of the street to the top?

5. An aeroplane takes off at sea level and ascends in a straight line until reaching an altitude

of 3000 metres. The distance travelled during the ascent is 13.5 kilometres. What is the gradient of the ascent?

A P



Q GRADIENT

I know how to find the gradient of a straight line

I can calculate the gradient of a straight line

I can use the gradient of a line to help find the equation of a line

I can identify parallel lines of the form y = mx + c

I can identify parallel lines when algebraic rearrangement is required

I can use the gradient of a line to solve problems




QUADRATIC FUNCTIONS

Name:

BAM Indicator: Plot and interpret graphs of quadratic functions

1. a) Fred plots the graph of 𝑦 = 𝑥2 as shown here. Explain what he has done wrong.

b) Fred then was asked to fill in the table of values for 𝑦 = 𝑥2 + 1. His table of results is below. Comment on Fred’s solutions.

x -3 -2 -1 0 1 2

y -8 -3 0 1 2 5

2. Plot the graph of 𝑦 = 𝑥2 − 𝑥 + 2.

3. The graph here shows the time a ball is in the air (t) in

seconds, and the height of the ball above the ground (x) in metres.

a) What is the maximum height of the ball?

______ m

b) How long was the ball in the air? ______ s

c) Find the times when the ball was 2 metres above the ground.

______ s and ______ s

M F A


4. Find the function that is represented on these graphs. Explain how you know.

_____________ _____________ 5. Here are eight graphs.

A B C D

E F G H

Write the letter of the graph that shows:

𝑦 = 𝑥2 − 4 : ____ 𝑦 = 2 − 3𝑥 − 𝑥2 : ____ 𝑦 = 𝑥2 : ____

R P



Q QUADRATIC FUNCTIONS

I know the characteristic shape of a quadratic graph

I can interpret a graph of a quadratic function

I can plot the graph of a simple quadratic function (e.g. y = x2 + 3)

I can plot the graph of a quadratic function (e.g. y = x2 + 3x – 1)



10M2 v2 © Kangaroo Maths 2017

DIRECT AND INVERSE PROPORTION

Name:

BAM Indicator: Solve problems involving direct and inverse proportion

1. a) y is directly proportional to the square of x. Find the missing values in the table.

x 2 7

y 16 64

b) p is inversely proportional to q. Find the missing values in the table.

p 5 10

q 48 20

2. 6 builders take 80 days to build a house.

a) How many builders must be employed to build the same house in 16 days?

____________

b) State an assumption you have made in your answer to part (a).

3. A box of 33 crackers weighs 185 grams. The information on the box states an energy

value of 498 calories per 100 grams. How many calories are in 4 crackers?

4. Jesse is told that y is inversely proportional to the square of x.

He also knows that y = 1

9 when x = 6.

Jesse is asked to find x when y = 4. He gives the answer x = 0.25

Jesse is wrong. Explain why.

F

R

A

M


5. Draw lines to match each proportional relationship to its graph

y is directly proportional to the

square of x

y is inversely proportional to x

y is directly proportional to x

y is directly proportional to the

square root of x

P



Q DIRECT AND INVERSE PROPORTION

I can distinguish between situations involving direct and inverse proportion

I can use proportional relationships to find missing values

I can connect proportional relationships to graphs




RATES OF CHANGE

Name:

BAM Indicator: Interpret a gradient as a rate of change

1. Here is a distance-time graph for a journey. The journey starts at 11:00 am.

a) Calculate the speed over the first 30 minutes

____________

b) Calculate the speed at 12:15 pm.

____________

2. A scientist measures the temperature of

a liquid every minute for 7 minutes. The results are plotted on a graph as shown here.

a) At what time is the liquid cooling

most quickly. Explain how you know. b) At what rate is the liquid cooling when time = 1.5 minutes?

________________________

3. Natalie is sketching graphs. Natalie says:

“This graph shows that the rate of change of velocity is constant throughout”

Is Natalie correct? Explain how you know.

F A R


4. Here is a distance-time graph of Isaac’s journey.

Find the average speed for his journey.

s____________

5. Here is some information about the movement of an object

The initial velocity is 0 m/s

Between 0 seconds and 5 seconds, the acceleration is 2 m/s2

Velocity is constant for the next 3 seconds

Acceleration then changes to 1.5 m/s2

When the object reaches a velocity of 16 m/s, it starts to decelerate at 5 m/s2. This

continues until the object stops moving.

Plot a velocity-time graph to show this movement

M P



Q RATES OF CHANGE

I know that speed is one example of a rate of change

I can calculate average speed

I can plot and interpret kinematic graphs

I can interpret the meaning of the gradient at a point in the context of the problem




SOLVING QUADRATIC EQUATIONS

Name:

BAM Indicator: Solve quadratic equations by factorising

1. Solve the following equations by factorising: a) 𝑥2 + 3𝑥 = 0

b) 𝑥2 − 3𝑥 + 2 = 0

c) 𝑥2 − 49 = 0

d) 𝑥2 − 6 = 5𝑥

e) 3𝑥2 − 18𝑥 − 21 = 0

2. Melanie says that the solutions to 𝑥2 − 5𝑥 + 4 = 0 are 𝑥 = -4 and 𝑥 = -1. Explain why she is wrong.

3. Jane and Elliot each choose a whole number.

Jane multiplies her number by three and subtracts 2. Elliot multiplies his number by 2 and adds 3. When they multiply their results together, the product is 110. They both chose the same number to start with. What was their number?

F M P


4. This football pitch has an area of 7140 m2. a) Show that 12𝑥2 − 13𝑥 − 7175 = 0 b) Find the values of p and q given that 12𝑥2 − 13𝑥 − 7175 = (𝑥 − 𝑝)(𝑞𝑥 + 287). c) Find the dimensions of the pitch. 5. How many solutions are there to the equation 𝑥2 + 12𝑥 = −36 ?

A R



Q SOLVING QUADRATIC EQUATIONS

I can solve a quadratic where the coefficient of x2 is 1

I can solve a quadratic where the coefficient of x2 is greater than 1

I know I may have to rearrange a quadratic to be able to solve it

I recognise and can use the difference of two squares

I can identify quadratics that cannot be solved by factorising.



4x + 5

3x – 7


TRIGONOMETRY

Name:

BAM Indicator: Apply trigonometry in two dimensions

1. a) Find the missing lengths marked 𝑥 in these diagrams:

i) ii)

___________ ___________

b) Find the missing angles marked θ in these diagrams:

i) ii)

___________ ___________

2. There are two mistakes made in the working below.

Describe the mistakes that have been made. Find the correct solution.

a = 13 ÷ sin50 = 16.97029…

a = 16.97 cm (2 d.p.)

F M

x cm

11 cm

28

x cm

35

7 cm

7 cm 12 cm

3 cm

8 cm

a cm

adj

13 cm

hyp 50


3. A ladder, 6 metres long, rests against a wall. The foot of the ladder is 2.5 metres from the base of the wall. What angle does the ladder make against the ground?

____________ 4. The diagram shows the cross-section of a building. CD is a ledge half way up.

Amy is standing at point A looking at the building. She measures AB = 100 metres,

CAB = 31o and EAB = 42o.

Calculate the length of CD.

CD = ____________ m

5. Jason says:

I only need to know two pieces of information about a triangle and I can find any missing side or angle.

Do you agree or disagree? Explain why.

A P R



Q TRIGONOMETRY

I can correctly label a triangle so as to select the appropriate trigonometric ratio

I can find missing sides of right-angled triangles using trigonometry

I can find missing angles of right-angled triangles using trigonometry

I can apply trigonometry to problems involving multiple triangles




STATISTICS

Name:

BAM Indicator: Analyse data through measures of central tendency and spread, including quartiles

1. The box plots show the lengths of reigns of the last 19 popes and (English) monarchs.

Compare the lengths of reigns of monarchs and popes.

2. Two cricket players both have a median score of 38, based on their last 20 matches.

Player A’s data has an interquartile range of 41. Player B’s data has an interquartile range of 20. Which player would you prefer to select for the cricket team and why?

3. The cumulative frequency graph shows the

time taken to respond to online help requests on a website for 120 customers.

a) Find the median time taken to respond to

an online help request.

_____________

b) The website has a target to ensure that 80% of online help requests receive responses within 30 seconds. Did the website meet its target in this sample? Explain your answer fully.

F R A


4. The box plots show information about results in tests. Match each statement to a box plot. Complete the missing box plot.

The middle 50% of both subjects’ results were equally spread

Results in English were more consistent than results in Maths

25% of the students achieved a higher score in English than anyone achieved in Maths

The maximum mark in English was half that in Maths

On average, students achieved a higher score in English than in

Maths

5. Will says,

“The interquartile range is less accurate than the range because not all data is included”

Do you agree with Will? Explain why.

P M



Q STATISTICS

I can find statistics from a cumulative frequency curve or box plot

I can compare the central tendency and spread of two data sets

I can use quartiles and other measures to make estimates about the population



fractions and decimals - wordpress.com · 2019. 8. 19. · name: bam indicator: solve problems...

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