ICAS MATHEMATICS

SECONDARY SAMPLE QUESTIONS NICK CONNOLLY July 2007
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ICAS Mathematics Secondary Sample Questions
This paper is a set of teachers notes to accompany the ICAS Mathematics Secondary Poster. The notes include the full text of each question, the correct answers and notes on how to answer the questions. Some of the text was previously published to accompany the 2005 ICAS Poster.
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Sarah took this picture of a toy robot.
Worked Answer Answer (A). The key features are the orientation of the head and the left arm of the robot. The head is looking to the robots right and the left arm is raised. This is only true in option A. Comment The correct answer, (A), was chosen by approximately 85% of the students who attempted the paper. The second most common answer was (C) which was chosen by almost 10% of the students. Options (B) and (D) were very unpopular choices presumably because students found it easy to reason about which arm was correct. Overall there was little difference between boys and girls on this item except that girls who got the item wrong were more likely to choose option (C). Assessment Framework Space & Geometry: Space Identifying nets, plans and elevations of 3D figures
Jane took a picture of the robot at the same time but from above. Which of these could be Janes picture?
Jane was tossing a coin but one side of the coin was weighted more heavily than the other. Here are the results she obtained.
The picture shows the angle between two sections of a roof.
Based on her results, which of these is the best estimate of the probability of getting a head in a single toss of Janes coin? (A) (B) (C) (D) 0.4 0.5 0.6 0.7
Which of these is the best estimate of the value of a? (A) (B) (C) (D) 120 140
Worked Answer Answer (C) 0.6 The best estimate of the probability will be based on the maximum number of trials, in this case the final row of 5000 tosses. The number of heads is 2989 so the probability is 29895000=0.5978. Rounded to 1 decimal place this gives a value of 0.6. Comment This question uses relative frequency as a measure of probability. This approach is sometimes called experimental probability. Because the coin was weighted you cant assume that heads and tails are equally likely. For a fair coin we could calculate the probability based on equally likely outcomes. Assessment Framework Chance & Data: Chance Finding numerical probabilities based on relative frequencies
Worked Answer Answer (D). The item is an estimation question. Students could use key angles (such as etc) to help them , estimate. Comment The correct answer, (D), was chosen by approximately 60% of the students who attempted the paper. The second most common answer was (C) which was chosen by almost 30% of the students. The photograph is of the entrance to Trial Bay Gaol in Arakoon NSW. Assessment Framework Measures & Units: Measurement Estimating angles 20 degrees or less
Nina made this cube using small plastic cubes.
When visiting Paris, Ghita and James bought models of the Eiffel Tower and the Arc de Triomphe for friends. The table shows the gifts Ghita and James bought and the total amounts each paid.
What is the least number of small plastic cubes she will need to add to make a larger cube? (A) (B) (C) (D) How many Euros did one tower cost? (A) (B) (C) (D) 9 12
Worked Answer Answer (B). The next larger cube would be a 3 cube, with 27 cubes in total. There are 8 cubes already. So Nina needs = 19 cubes. Comment The correct answer, (B), was chosen by approximately 30% of the students who attempted the paper. The most common answer was (A) which was chosen by almost 40% of the students. Option (A) is the total number of cubes in the larger cube. The correct answer is the difference between that figure and the smaller cube. Assessment Framework Measures & Units: Measurement Calculating the volumes of shapes based on rectangular prisms

Worked Answer Answer (C). The item is a number puzzle. It can be solved either by simultaneous equations or by trial and improvement methods. Eqn1: 3t + 8a = 83 Eqn2: 4t + 5a = 71 15t + 40a = 415 Eqn1 5: 32t + 40a = 568 Eqn2 8: Subtracting: 17t = 153 Divide by 17: t=9 Comment The correct answer, (C), was chosen by approximately 40% of the students who attempted the paper. The second most common answer was (B) which was chosen by almost 40% of the students. Option (B) is the cost of an Arc de Triomphe rather than an Eiffel Tower. The item can be worked out by using each of the options as a starting point. Assessment Framework Algebra & Patterns: Algebra Solving a simultaneous equation with two unknowns Algebra & Patterns: Pre-algebra Solving a complex number puzzle expressed in words
FREE RESPONSE QUESTIONS 6. This is how the ancient Egyptians wrote a particular fraction.
A film produces the illusion of movement by showing many pictures very quickly. Each of these pictures is called a frame.
The symbol has been used in this equation.
There are about 16 frames in every 30 centimetres of film. Films are shown at the rate of 24 frames per second. What is the smallest positive number that could be the denominator of the fraction? How many minutes would it take to watch a film that was 3.24 kilometres long?
Worked Answer Answer 10. The item is a number puzzle which uses fractions. The Year 7 paper is done without a calculator.
15 ? 3 = 15 ? 3 = ? 3 = ? = 10
Worked Answer Answer 36. First work out how many frames are in the film. 3.24 kilometres = 3240 metres = 324000 cm 324000 cm 30 cm = 16 frames = 172800 frames Then find how many seconds it would take to watch that many frames. 172800 frames 24 frame per second = 7200 Finally convert into minutes 7200 seconds 60 seconds = 120 minutes Comment This problem involves several different rates. Students need to work carefully and keep track of which calculations they have done and what the figures they have worked out mean. Assessment Framework Number & Arithmetic: Arithmetic Calculating proportions of amounts using combinations of fractions, decimals, percentages, ratios Measures & Units: Measurement Using time measurement, being familiar with its unusual features (time is not measured in simple decimal units)
Comment This is a difficult number puzzle for younger students. Older students with more knowledge of formal algebra still find problems like this difficult as it requires them to combine their knowledge of solving simple equations with their knowledge of fractions. Assessment Framework Algebra & Patterns: Pre-algebra Solving a complex number puzzle expressed in words Algebra & Patterns: Algebra-Expressions Solving a complex linear equation with one unknown Number & Arithmetic: Arithmetic Using the four operations with fractions

The diagram shows a rectangular prism.
The surface area of the prism is 2808 cm2. What is the value of x?
Worked Answer Answer 18. First find an expression for the surface area:
2 x x + 2 x 3 x + 2 ( x 3 x ) = 2 x 2 + 2 x2 + 6 x = x 3
The surface area is equal to 2808 cm2:

x = x2 = = 324 26

x = 324 = 18
Comment The correct answer, 18, was worked out correctly by nearly 5% of the students who attempted the paper in Year 9. The item was also attempted by Years 10, 11 and 12 with nearly 10%, 20% and 30% of students getting the item correct in each respective year group. The most common wrong answer was 14 with 6% of students in Years 9 and 10 who attempted the question giving that as an answer. The same group gave the intermediary answer of 324 at about 3% of the responses. Assessment Framework Algebra & Patterns: Algebra-Expressions Solving quadratic equations with one unknown Measures & Units: Measurement Finding the surface areas of prisms
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