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mathematics
cambridge checkpoint lower secondary mathematics
Cambridge Checkpoint Lower Secondary Mathematics Student's Book 7 3rd Edition Frankie Pimentel, Ric Pimentel, Terry Wall - Solutions
Luisa and Pedro are sister and brother. They decide to save their pocket money in their money boxes as follows:Luisa starts by putting in $15 then adds $3 to it each week.Pedro starts by putting in only $5, but then adds $4 to it each week.After how many weeks will the ratio of their savings be 1 :
P, Q and R represent three whole numbers.The ratio of P : Q is 3 : 5, whilst the ratio of Q : R is 9 : 10.What is the smallest possible value of P + Q + R?
The distance–time graph below shows the motion of a train over a period of time.Choose the statement below which best describes the motion of the train. Justify your choice.(i) The train is travelling at a constant speed.(ii) The train has stopped.(iii) The train is travelling on flat ground.(iv)
The distance–time graph below shows the motion of a cyclist over a period of time.Choose the statement(s) below which describes the motion of the cyclist. Justify your choice(s).(i) The cyclist is travelling at a constant speed.(ii) The cyclist is travelling back to a point.(iii) The cyclist is
In a café, the number of coffees sold is recorded every hour and the total number sold since the start of the day is calculated. The café opens at 08:00. The totals are shown in the table below.a. Plot a line graph of the data.b. Use your graph to estimate the number of coffees sold by 11:15.c.
In a year group of 120 students, students can opt to study either Spanish (S), Mandarin (M), neither language or both.The numbers of students studying each are shown in theVenn diagram below.a. If 42 students study Mandarin, calculate the percentage of students who study Spanish.b. What percentage
The first five numbers of a sequence are:−7 −6 −5 −4 −3a. Write down the next two terms in the sequence.b. Write down the nth term of the sequence.c. What is the 100th term of the sequence?
A swimming pool is left to heat up over a weekend. Unfortunately, the thermostat does not work properly and the pool overheats. The temperature of the water at 09:00 on Monday morning is 56 °C.The heater is switched off and temperature readings are taken every 6 hours over the next three days. The
An octahedron is a 3D shape with eight faces each the shape of an equilateral triangle as shown:a. Sketch the 2D plan view indicated by arrow P.b. Sketch the 2D side view indicated by arrow S. P -S
The graph above shows an object travelling at 8 m/s (a distance of 8 m each second).Using the graph or by any other method, find:a. How long the object takes to travel 50 mb. How far the object travels in 5 secondsc. How long the object takes to travel 70 md. How far the object travels in 8
The five containers below are filled with the same constant rate of water from a tap.The graphs below show the depth of water over time for six containers.a. Match each container above to their graph.b. Draw a possible container which would produce the extra graph when filled with water at a
The coordinates of three vertices of a parallelogram ABCD are shown on the axes.Calculate the coordinates of vertex C. A(-2, 6) 10 19 8 m 2 -5 -4 -3 -2 -1₁° #2 ¹5 D(4, 7) 4 LA B(3,-2) to 00 9 10
A daughter is 27 years younger than her father. Let the daughter’s age be d and the father’s age F.Write an algebraic function machine to work out the daughter’s age from the father’s age.
An electric car travels at 12 m/s in a straight line.Draw a graph of the first 10 seconds of its motion.
The graph below shows a family car journey.a. What time did the family set out?b. (i) How far did they travel in the first hour?(ii) What was their average speed?c. What time did they stop for breakfast?d. How long did they stop for breakfast?e. How far did they travel between 09:00 and 11:00?f.
Forty-nine square tiles are arranged to form a square.Show, by drawing three squares, that 49 can be written as the sum of three square numbers.
A train leaves a station at 07:30. It travels for 1 hour 30 minutes at 100 km/h and then stops for 15 minutes. It travels a further 200 km at 100 km/h and then stops for 30 minutes. It then does the return journey non-stop at 100 km/h.a. Draw a graph of the journey.b. When does the train arrive
The graph below shows part of a cycle journey for a boy.Copy the graph above.At 10:00 the cyclist decides to cycle back to the start point.The average speed of the cyclist travelling back to the start is 5 km/h.He gets back in two phases, each with a different constant speed.a. Draw two possible
A map is printed to a scale of 1 : 25 000.A rectangular field is 820 m × 1460 m in real life.Calculate its dimensions on the map, giving your answers in millimetres.
To make eight bread rolls, the following ingredients and amounts are needed:On checking the kitchen cupboard, Mario realises he has 1200 g of flour and 20 g of yeast.Assuming he has enough of the other ingredients, what is the maximum number of bread rolls he can make? 500g of bread flour 7g of
Two neighbours have identical water butts in their gardens to water their plants. One neighbour decides to water his plants by filling up a watering can several times. The other neighbour attaches a simple pump and hose to his water butt so that he can water his plants in one go. Both water butts
Without a calculator add the following two numbers: 52 + 68.The units 2 and 8 form a number bond to 10. Therefore the calculation can be worked out by adding together 50 + 60 + 10 = 120.This can also be visualised using a number line. 40 50 60 +60 +10 70 80 90 100 110 120 130
Without using a calculator subtract the following numbers: 146 − 36.The units of both numbers are the same, so 6 − 6 = 0. The calculation can therefore be worked out using the remaining digits, i.e. 140 − 30 = 110.On the number line below, 146 − 36 can be seen to be the same as 140 − 30.
Without a calculator add the following two numbers: 79 + 32. 79 is very nearly 80. Therefore an easier calculation would be to add 80 + 32 = 112. But because 1 was added to 79 to make the 80, it must now be subtracted from the answer.Therefore 79 + 32 = 111.
Without using a calculator subtract the following numbers: 118 − 44. 118 can be rounded up to 120 as this will make the calculation easier. It now becomes 120 − 44 = 76. But as 2 was added to 118 it must be subtracted from the answer.Therefore 118 − 44 = 74.
Calculate the area of the compound shape below:The shape is an example of a compound shape as, although it is a pentagon, it can be split into a rectangle A and a triangle B as shown.Area of rectangle A: 3 × 8 = 24 cm2Area of triangle B: The ‘height’ of the triangle can be deduced as it is the
Draw an equilateral triangle, mark on its lines of symmetry and state its order of rotational symmetry.The equilateral triangle has a rotational symmetry of order 3. I
Estimate the answer to the following calculation:286 + 407Rounding 286 to the nearest hundred becomes 300.Rounding 407 to the nearest hundred becomes 400.Therefore an estimate for 286 + 407 is 300 + 400 = 700.
A pupil does the following calculation in his head:1121 + 788 − 210 − (−592) and says that the answer is 1291.Explain why this answer must be incorrect.Rounding each of the numbers gives the following approximation:1000 + 800 − 200 − (−600) = 2200The estimation gives an answer of 2200.
Calculate 7 + 4 × 9 − 8.7 + 4 × 9 − 8 = 7 + 36 − 8 = 35 The multiplication is done first.
Calculate 25 − (2 + 3) × 4.25 - (2 + 3) x 4 = 25 - 5 x 4 = 25 - 20 = 5
A boy is y years old. His sister is 4 years older than him, his younger brother 2 years younger than him and his grandmother 8 times older than him. Write an expression for the ages of his sister, younger brother and grandmother.As his sister is 4 years older than him, the expression for her age is
Simplify the expression2a + 3 × 4a − a.2a + 3 × 4a − a = 2a + 12a − a = 13a
Using the formula P = 2l + 2b, calculate the perimeter of a rectangle if l = 3 cm and b = 5 cm.P = 2 × 3 + 2 × 5P = 6 + 10P = 16 cm
The coordinates of three vertices of a rhombus are given in the diagram.a. Deduce the coordinates of the missing vertex D.The coordinates of D are (6, 6).b. Justify your answer using the properties of the rhombus.The diagonals of a rhombus intersect at right angles. Vertices A and C lie in the same
The axes below show three vertices of a square ABCD.a. What are the coordinates of vertex D?b. Justify your answer by referring to the distance between the vertices. C(-2, 6) B(-2, 1) A(4,1)
The axes below shows triangle ABC and its position to A’B’C’ after a translation.a. Describe the translation that maps ABC on to A’B’C’.b. Deduce the coordinates of B’.c. Deduce the coordinates of C’.d. Explain why the area of triangle A’B’C’ is 21 units2.
Three vertices of parallelogram WXYZ are shown in the axes below.a. A student states that the y coordinate of vertex Z is −5. Explain why this must be wrong.b. Deduce the coordinates of Z. X(-2, 1) Y(-5, -3) W(8, 1)
A square PQRS is shown on the grid below.The square is translated to a new position P’Q’R’S’.a. One of the following coordinates of the translated square P’Q’R’S’ is incorrect. Which one?P’(−4 , 5) Q’(4 , 3)R’(7 , 10)S’(−1 , 13)b. Justify your answer to (a). S(-5,
Draw a grid with centre (0, 0), the origin, and mark the x- and y-axes with scales from −8 to +8. Mark these points on your grid.a. A(5, 2)b. B(7, 3)c. C(2, 4)d. D(−8, 5)e. E(−6, −8)f. F(3, −7)g. G(7, −3)h. H(6, −6)
A triangle ABC is shown on the axes below.The triangle is translated and the vertex A moves to the new coordinates (1, −2).a. Describe the translation.Vertex A has been translated 5 units to the right and 2 units down to its new position.b. Calculate the coordinates of the vertices B and C after
Draw a separate grid with x- and y-axes from −6 to +6. Plot the points, join them up in order and name the shape you have drawn.A(3, 2)B(3, −4)C(−2, −4)D(−2, 2)
The diagonals of the rectangle EFGH intersect at a point M.The rectangle EFGH is translated to a new position E’F’G’H’ and the intersection point of the diagonals is at the origin.a. Calculate the coordinates of the vertices E’, F’, G’ and H’.b. Justify your answers in part (a).
One side of a rectangle LMNO is given in the diagram below. The coordinates of the centre of the rectangle C are (−1, −3.5).Calculate the coordinates of the two missing vertices N and O. L(-4, 0), C(-1, -3.5) M(0, 1)
Draw a separate grid with x- and y-axes from −6 to +6. Plot the points, join them up in order and name the shape you have drawn.D(1, 3)E(4, −5)F(−2, −5)
One side of a square PQRS is given on the axes below.a. If the diagonals of the square intersect at the origin, calculate the coordinates of the two missing vertices R and S.b. If the diagonals of the square intersect at (3, −3) what are the coordinates of the vertices R and S? P(0, -3) Q(3, 0)
a. Draw a grid with x- and y-axes from −10 to +10. Plot the points P(−6, 4), Q(6, 4) and R(6, −2).(i) Plot point S such that PQRS is a rectangle.(ii) Write down the coordinates of S.(iii) Draw diagonals PR and QS. What are the coordinates of their point of intersection?(iv) What is the area
Three vertices of a kite ABCD are shown below.a. What must the x-coordinate of vertex D be? Justify your answer.b. The area of the kite ABCD is 20 units2. Deduce the coordinates of vertex D. A(-5, 1) B(-3, 4) C(-1, 1)
Draw a separate grid with x- and y-axes from −6 to +6. Plot the points, join them up in order and name the shape you have drawn.G(−6, 4)H(0, −4)I(4, −2)J(−2, 6)
The diagram below shows a 3D object made from six cubes.a. Draw 2D drawings of the elevations shown on the diagram.b. Are the two 2D elevations enough to be able to visualise the 3D shape correctly? Justify your answer. No, the two elevations are not enough.If the cube labelled ‘1’ was moved
a. The shape below is made up of five cubes.Explain why some of the faces appear to be triangles.b. Draw a plan view of the shape.
A square-based pyramid is shown below.a. Draw a side elevation from the direction of the arrow.b. Draw a plan view.
The 3D shape below is made from seven cubes.A student decides to draw four side elevations (left, right, front and rear) as shown.a. One of the elevations is incorrect. Which one is it?b. Redraw the incorrect elevation correctly.
A cylinder is shown below:a. Draw a side elevation.b. Draw a plan view.
The diagram below shows a truncated cone.a. Draw the elevation from the direction of the arrow.b. Draw a plan view.
a. The diagram below shows five cubes joined to make a shape.Draw the two-dimensional elevations seen from directions (x) and (y).b. In fact the diagram is made from six cubes.(i) Which of the five visible cubes A, B, C, D and E must the sixth cube be attached to?(ii) Justify your answer above by
Two L-shaped objects, each made of four cubes are shown below.The two objects are then joined so that cube 1 is directly on top of cube 2.a. Draw a 2D diagram of the view from the direction of the arrow of the joined shape.b. Draw a plan view of the joined shape. 1 2
A length of guttering to be attached under a roof is shown below. The back of the guttering is taller than the front.Draw elevations for each of the views labelled a, b and c. b a
A designer wishes to draw a sphere in two dimensions. What shape will he draw?
The shape below is made from cubes as shown:a. All the cubes are visible in the 3D diagram.How many cubes is the shape made from?b. Draw 2D elevations from each of the directions shown.
The numbers 3, 2, 1, 0, −1, −2 are entered into the function machine above.Calculate the output in each case.The information in the table can also be shown using a diagram known as a mapping diagram. Input 3 2 1 01 2 -1 -2 Output 6 4 2 0 -2 -4
The input numbers are listed in the table. Calculate the output values. In Add 5 Out
Chocolate bars are sometimes sold in packs of 6 as shown.a. What shape is the end face of the pack?b. Draw an elevation from the direction of the arrow.c. Will the plan view be different from the elevation drawn in (b) above? Use a diagram to justify your answer. End face Autos Chocolate
A function machine is given as:If the output is 8, calculate the input.This is done by working out the inverse function.Working backwards, we must work out what mathematical operation undoes the original function.The opposite of ‘Subtraction’ is ‘Addition’.The original function can
The input numbers are listed in the table. Calculate the output values. In Multiply by 3 Out
A grandmother is 56 years older than her grandson.Let the grandmother’s age be M and the grandson’s age n.Write an algebraic function to work out the grandson’s age from the grandmother’s age.
Express the equation y = x + 4 as a function machine.Here, whatever value x is given, 4 is added to it to produce the y-value.Therefore x is the input and y the output.As a function machine this can be written as x Add 4 y
The input numbers are listed in the table. Calculate the output values. In Divide by 2 Out
A plumber charges $15 per hour of work.a. Write a function machine to work out the amount he earns p for working h hours.b. On one job he works 22 hours. Calculate how much he earns for that work.
In the following question a function machine is given:a. Write down the inverse function machine.b. Use the function machine and its inverse to complete the following: In Subtract 7 Out
A taxi firm charges 80 cents for each kilo metre travelled.a. Which of the following function machines represents the cost $C of travelling n kilometres?b. Explain why two other function machines are incorrect.c. Use the correct function machine to calculate the cost of taking the taxi for a 55 km
Four inverse function machines are given below:a. Which of the four inverse function machines produce the correct input from the given output?b. Complete an input/output table for each of the inverse function machines answered in part (a). In In In In Add 8 Divide by 3 Subtract 8 Divide by
Write, in function machine form, the relationship between the number of elephants (e) and the number of elephants’ legs (l).a. b. Write your function machine as an algebraic function.l = 4ec. If there are 68 elephant legs, how many elephants are there?So the inverse function machine
Claudia lives in Brazil and is going on holiday to India.She goes to the bank to change Brazilian currency Real (R) to Indian currency Rupee (r). The exchange rate is 1R = 17.4r.a. Write a function machine to convert Reals to Rupees.b. If she changes 500 Reals, how many Rupees will she receive?c.
Below are two function machines P and Q.a. If both function machines have the same input, calculate the value of that input if the output of P is twice the output of Q.b. If both function machines have the same output, calculate the value of that output if the input of Q is three times the input of
Solve the following equations:a. 2g−3=7b. 3h−1=2c. 7i−15=6d. 3j−18=3e. 5k + 7=32f. 9m+11=74g. 6n+12=72h. 7r−8=41i. 6q−12=84j. 3k + 7=46k. 5m+12=72l. 9n + 9=72m. 6r−8=40n. 11q−10=89
These diagrams show the first three patterns in a sequence of growing tile patterns.a. Draw the next two diagrams in the sequence.b. Copy and complete this table.c. Describe the pattern linking the number of white tiles and the number of red tiles.d. Use your rule in part (c) to predict the number
a. Draw the next two diagrams in the sequence.b. Copy and complete this table.c. Describe the pattern linking the number of white tiles and the number of green tiles.d. Use your rule in part (c) to predict the number of green tiles in a pattern with 100 white tiles.
Here is a sequence of numbers.4 9 14 19 24a. Describe the term-to-term rule.The term-to-term rule for this sequence is +5.b What is the tenth term?To calculate the tenth term in the sequence, the pattern can be continued using the term to-term rule:4 9 14
Look at the tile pattern sequence below.a. Copy and complete the table.b. Describe the relationship between the number of pink and the number of blue tiles.c. Write an expression for the nth term of the sequence, where n represents the number of pink tiles.d. If there are 65 pink tiles, how many
For each of the following sequences:(i) Describe the term-to-term rule.(ii) Write down the next two terms of the sequence.(iii) Calculate the tenth term.a. 2 4 6 8 10b. 1 3 5 7 9c. 4 7 10 13 16d. 2 6 10 14
In the following sequences:(i) Write down the next two terms.(ii) Give an expression for the nth term.a. 6 7 8 9 10b. 9 10 11 12 13c. 2 4
Here is a sequence of numbers.1 3 9 27 81a. Describe the term-to-term rule for this sequence The term-to-term rule for this sequence is ×3.b. What is the eighth term in the sequence? To calculate the eighth term in the sequence, the pattern can be continued using the term
Karl wants to multiply 36 × 24.He gets the answer 866.Without doing the calculation explain why the answer must be wrong.
Four cards are numbered as shown. A fifth card is turned over. a. What must be the number on the fifth card if:(i) The median of all five cards is 7(ii) The mean of all five cards is 9?b. Explain whether it is possible to work out the number on the fifth card if:(i) The median of all five
a. Draw the next two diagrams in the sequence. b. Copy and complete this table.c. Describe the pattern linking the number of white tiles and the number of orange tiles.d. Use your rule in part (c) to predict the number of orange tiles in a pattern with 100 white tiles. A
The sequence of patterns below is made of matchsticks.a. Write the number of matchsticks in the first five patterns as a sequence of numbers.b. Generalise by writing down the nth term of the sequence, where n is the pattern number.c. How many matchsticks are there in the 20th pattern?d. Explain why
This table gives the first five terms of a sequence and their positions in the sequence.a. Describe the position-to-term rule.By looking at the sequence it can be seen that the term is always the position number + 4.b. Write the position-to-term rule as a rule for the nth term. The position can be
The shape below is made from five identical squares. Copy the diagram and add three more squares to form a shape with a rotational symmetry of order 4.
a. Draw the next two diagrams in the sequence.b. Copy and complete this table.c. Describe the pattern linking the number of white tiles and the number of blue tiles.d. Use your rule in part (c) to predict the number of blue tiles in a pattern with 100 white tiles.
The grid below shows a pattern of triangles. The first three are shown.The coordinates of the top vertex of each triangle are given and form a sequence.a. What are the coordinates of the top vertex of the 20th triangle? Justify your answer.b. A triangle in the sequence has a top vertex with
This table gives the first five terms of a sequence and their positions in the sequence.a. Describe the position-to-term rule.By looking at the sequence it can be seen that the term is always the position number × 4.b. Write the position-to-term rule as a rule for the nth term. The position can be
A large rectangle is split into two smaller rectangles. The area of the two rectangles are 6a units2 and 10 units2 as shown. Write an expression, in its simplest form, for the perimeter of the large rectangle. 6a units² 10 units² 2
The pyramid opposite has some numbers already written in some of the blocks. The pyramid is constructed in a way such that the numbers in the blocks in the top two rows are the sum of the two fractions in the blocks directly beneath them.Calculate the missing fractions in the blocks labelled (a),
Showing your method, calculate the size of angle x. 105° 48% xo
The dimensions of two rectangular pieces of card A and B are given below. Assuming x is positive, are the following statements always true, sometimes true or never true? a. (i) 4x + 6 is bigger than 3x + 10.(ii) Justify your answer.b. (i) The area of A is smaller than the area of B.(ii)
A box contains 80 sweets of three different colours: yellow, red and green.Two friends decide to try to work out how many of each colour there are without actually counting them. They take a sweet out, record its colour and then put it back in the box.They do this 360 times and their results are as
A kite has dimensions as shown: a. Explain why the area of the kite is given by the expression 32x cm2.b. If the area of the kite is 160 cm2, calculate the value of x. 4x cm -16 cm
Find 25% of 60.We know that as a simplified fraction 25% is 1/4.This rectangle has been divided into 60 squares and split into quarters. One of the quarters has been shaded.From the diagram it is clear therefore that 25% of 60 is 15 as 15 squares are shaded.Doing a drawing though can be time
Find 30% of 250.As a fraction 30% can be written as 30100 which when simplified can be written as 3/10.The rectangle above is made up of 250 squares. It has been divided horizontally into tenths. To calculate 3/10, therefore, three rows have been shaded in.
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