Download pdf - Transfer Exam - 立命館大学 · Transfer Exam . Introduction . ... Topic 3 Fractions: representing fractions, ... Divide by 2. f. Subtract the original number

$Page 1: Transfer Exam - 立命館大学 · Transfer Exam . Introduction . ... Topic 3 Fractions: representing fractions, ... Divide by 2. f. Subtract the original number$
IPS2 Diagnostic Exam Transfer Exam

Introduction Students come into the IPS program from all parts of the world, at all levels of English, as well as at various levels of mathematics proficiency. Rather than picking any country’s standard of math at this grade level, the IPS Mathematics department has decided to test on the content taught in the IPS program before the date of the entrance exam. That content is outlined below.

In addition, examinees are responsible for interpreting mathematical symbols and units where notation and measurements may be different from standards students are used to. Some examples of different issues students may encounter include:

• Comma usage, or their absence, for large numbers (e.g., 1 000 000 = 1000000 = 1,000,000) • Period usage and leading zeroes for decimals (e.g., 0.01 = .01 = 1

100 )

• Imperial and metric units and their abbreviations (“inches” becomes “in”, “centimeters” becomes “cm”, etc). Understanding the units will not affect whether the student can answer the problem (see example 4 below).

Topic 1 – Whole Numbers: the number system, rounding numbers, operating with numbers, estimation, exponent notation, squares and cubes, order of operations

Example problems from this topic include:

1. Write in numeral form the number “fifteen thousand twenty-seven”. 2. Find the product of 51 and 17. 3. Round 145,987 to:

a. the hundreds place b. 3-significant figures

4. Ian ran 4.5 miles in a training session. If the park’s running track is 1.5 miles in distance, how many laps did he run?

5. A door, a thumb, a dog. Which of the previous objects is closest to 2 inches or 5 cm? 6. Estimate 4015 × 98. 7. Evaluate the following:

a. 23 × 32 × 5 b. (18 ÷ 3 + 3) ÷ (4 × 4 − 7)

Topic 2 – Properties of Numbers: divisibility tests, factors of natural numbers, multiples of natural numbers, directed numbers, simple roots of whole numbers


1. Explain how you can know if a number is divisible by 5. 2. Final all possible values of ⌂ if the number “89⌂12” is divisible by 3. 3. Find the highest common factor of 18 and 24. 4. Find the lowest common multiple of 5, 9, and 12. 5. Evaluate:

a. 5 − 14 ÷ −2 + 1 b. √25 c. √273

Topic 3 – Fractions: representing fractions, equal fractions, manipulating fractions, operations with fractions, problem solving, simple square roots of fractions


1. Find the sum of three-fifths and two-thirds. 2. Write a fraction equivalent to 4

5 with a denominator of 20.

3. Which of the numbers, 10, 45, 0.5, and 𝜋𝜋 are not rational?

4. Evaluate: a. 7

3− 7

2+ 1 1

4

b. 13

÷ �25�2

c. �10049

5. Science says we should sleep 8 hours a day. How much of the day should we spend sleeping? 6. Find 3

7 of $434.

7. 313

of a field was searched for mushrooms and 39 were found. How many mushrooms would we expect to find in the remainder of the field?

Topic 4 – Algebra (Patterns and Models): geometric patterns, substituting into formula, number sequences, word problems


1. The first three figures below are made up of 4, 7, and 11 “sticks” respectively. Find the number of “sticks” in the 15th figure.

2. Consider a formula that gives an output number as “seven more than three times the input

number”. a. Calculate the output number when the input number is 2. b. Calculate the input number when the output number is 22. c. Write the formula for the input number 𝑥𝑥 and the output number 𝑦𝑦.

3. If 𝑦𝑦 = 8𝑥𝑥 − 4, find 𝑦𝑦 when 𝑥𝑥 = 10 12 .

4. Michelle has £2000 in a bank account. Each week she deposits another £120. Given that there are 52 weeks in 1 year, how much money does she have in her bank account after 1 year?

5. State the next 3 numbers in the sequence: 243, 81, 27, 9, … .

Topic 5 – Decimal Numbers: place value, operations with decimal numbers, terminating and recurring decimals, decimal approximations


1. Write 0.375 as a fraction in simplest form. 2. Write the following numbers in ascending order (smallest to biggest): 90

100, 0.09, 0.99, 0.099

3. Evaluate: a. 4000 × 0.6 b. 0.08 ÷ 0.004

4. Write 79 as a decimal number.

5. Round 159.3872 to the nearest hundredth.

Topic 6 – Percentages: interchanging fractions / decimals / percentages, percentages of quantities, percentage increase / decrease, percentage change, business applications, simple interest


1. Write 2 12

% as a: (a) decimal, (b) fraction. 2. A rectangular pizza has been cut into 20 pieces. Chili sauce has been put on 4 pieces.

a. What fraction of the pizza does not have chili? b. What percentage of the pizza does have chili?

3. Bob was told she needs to lose 10% of his body weight. This means he needs to lose 9 kg. What’s Bob’s current weight?

4. Find 36% of 4200 yen. 5. A bicycle was bought for $500. Later that same bicycle was sold for $660. Find the percentage

profit or loss. 6. Simple interest is the cost, per year, of borrowing money and is calculated as a fixed (constant)

percentage of the original amount of borrowed money. a. If $10,000 is borrowed for 3 years at 10% per year simple interest, how much do you

need to repay?

Topic 7 – Algebra (Expressions and Evaluation): building expressions, simplifying expressions, evaluating algebraic expressions


1. Simplify 2(2𝑥𝑥 + 𝑦𝑦 + 1) + (𝑥𝑥 + 1) + (𝑦𝑦 − 1) + 𝑥𝑥. 2. If 𝑥𝑥 = 3, 𝑦𝑦 = −2, and 𝑧𝑧 = −1 evaluate:

a. 3(3𝑥𝑥 + 4𝑦𝑦) b. 𝑧𝑧10 + 𝑥𝑥

2𝑦𝑦− y2

3. Write the expression for the given set of rules: a. Let 𝑥𝑥 be any number. b. Double it. c. Add 7. d. Take away 1. e. Divide by 2. f. Subtract the original number.

Topic 8 – Equations: solving equations, equations with a repeated unknown


1. For each of the following equations, state if they are: (a) true for exactly one value of 𝑥𝑥, (b) true for exactly two values of 𝑥𝑥, (c) true for all values of 𝑥𝑥, or (d) never true.

a. 𝑥𝑥 × 0 = 0 b. 𝑥𝑥2 = 𝑥𝑥

2. Solve for 𝑥𝑥: a. 𝑥𝑥

4+ 5 = −8

b. 3𝑥𝑥 + 2 = 𝑥𝑥 + 14 c. 5(𝑥𝑥 − 2) + 3(1 − 2𝑥𝑥) = 6

Topic 9 – Ratio and Proportion: conversions, ratios, proportions, scale diagrams, gradient / slope


1. Fill in the box with the correct conversion: 12 m/min = cm/s. 2. Express 70 mL to 3 L as a ratio. 3. The ratio of teachers to students in a school is 1 ∶ 15. If there are 675 students, how many

teachers are there? 4. Gravel, sand, and cement are mixed in the ratio 5 ∶ 3 ∶ 1 to make concrete. I wish to make 18

tons of concrete. How much gravel, sand, and cement must I purchase? 5. The scale image was drawn at a scale of 1 ∶ 236. What is the actual length of the elephant?

6. Write down the 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ∶ 𝑟𝑟𝑟𝑟𝑟𝑟 ratios of the triangles in the form 1 ∶ 𝑥𝑥.

Topic 10 – Angles, Lines, and Parallelism: points and lines, measuring and classifying angles, angle properties, angle pairs, parallel lines


1. Solve for the missing values:

Find α:

Find x:

Find β:

2. Decide if the figures below contain parallel lines, giving a brief reason for your answer:

3. Find the values of the unknowns:

(a)

(b)

Topic 11 – Area: perimeter, area of polygons, area of composite shapes


1. Find the lengths of the unknown sides and hence calculate the perimeters of the figures below. Assume that all corners are right angles.

2. A right angled triangle has sides of length 3 m, 4 m, and 5 m.

a. Find the area of the triangle ABC. b. A perpendicular is drawn from side [AC] to B. Let this

perpendicular have length 𝑥𝑥 m. Find 𝑥𝑥. 3. Find the area of the shaded region below:

Topic 12 – Polygons: classifying triangles, angles of triangles, quadrilaterals, angles of quadrilaterals, interior angles of polygons


1. Find 𝑥𝑥 on the pictures to the right. 2. Name the polygons given the number of sides:

a. Five b. Eight

Topic 13 – Circles: parts of a circle, circumference and area (in terms of 𝜋𝜋)


1. The diagram shows plans for a garden which is 40 m by 40 m. It consists of 4 quarter circles of lawn with a flower bed in the middle as shown. Find:

a. The perimeter of the garden b. The total area of the lawn c. The total area of the garden d. The area of the bed e. The length of the edging around the flower bed

(Remember to write all of your answers in terms of 𝜋𝜋)

Topic 14 – Three Dimensional Geometry: volume, solids, nets of solids, oblique / isometric projection


1. Find the volume of the solids below:

2. Draw and name the solids which would be formed from the following nets:

3. Redraw the picture below on the provided isometric grid.

Topic 15 – Transformations: reflections, rotations, translations


1. Sketch the image of the to the right if a mirror was placed on the mirror line as shown:

2. Rotate the object below counter-clockwise about point O through the given angle.

3. Translate the kite up two units and to the left 3 units. 4. Translate the kite using the translation vector �−11 �.

Topic 16 – Statistics: data collection, categorical data, numerical data, mean / median / mode


1. Would you rather use a census or a sample to investigate “the height of Japanese students”? 2. In an English middle school, 80 students were asked

to name their favorite fruit. The following data was collected:

a. Construct a vertical column graph to illustrate this data.

b. Can we make a conclusion about the favorite fruit of all middle school students from this survey? Give a reason for your answer.

3. Make a stem-and-leaf plot given the data below: 16 97 32 24 18 73 11 40 23 20 84 31 26 63 97 15 30 14 98 51 21

4. What is the mean, median, and mode of the data set: 0 1 2 3 3 4 5 5 5 6 7 9

Type of fruit Frequency Apple 20 Banana 24 Grapes 3 Orange 11 Mandarin 10 Nectarine 7 Peach 2 Pear 3

Topic 17 – Graphs: scatterplots, travel graphs, direct / indirect variation, domain / range


1. When a bottle of pop was placed in a refrigerator, its temperature was measured at 5-minute intervals. The results are shown on the graph.

a. Determine the temperature of the liquid when it was first placed in the fridge.

b. Find the time taken for the temperature to drop to 20° C.

c. Find the fall in temperature during the second 15 minutes.

2. The graph shows the progress of Joe as he cycles to school. a. How long did it take Joe to cycle to school? b. How far is it from his home to the school? c. On his way to school, Joe has to cross an

intersection with traffic lights. How far are these lights from his home? How long was he delayed at these lights?

3. Sally sells seashells by the seashore. She receives $2 in commission for each shell she sells. Suppose she sells 𝑥𝑥 seashells and the total commission is $𝑦𝑦.

a. Complete the following table:

Number of seashells Sally sells (𝑥𝑥) 0 1 2 3 4 5

Total Commission ($𝑦𝑦)

b. Create a scatterplot with the data from the table. Remember to label your axes. c. Write a relationship between 𝑥𝑥 and 𝑦𝑦 d. Should we join the points together? Is the data

continuous of discrete? 4. Suppose 𝑦𝑦 varies inversely with 𝑥𝑥.

a. Write an equation if 𝑥𝑥 = 3 when 𝑦𝑦 = 4. b. Write the domain and range of the equation in (a).

Topic 18 – Linear Functions: slope and midpoint formulas, slope of horizontal and vertical lines, slopes of parallel and perpendicular lines, domain and range, equations, graphing linear functions and vertical lines (with and without restrictions), applications to linear functions, finding the area enclosed between linear functions


1. Find the midpoint between A = (−5,1) and B = (12

,−4.5). 2. Find the slope of the linear function 2𝑥𝑥 − 3𝑦𝑦 + 1 = 0. 3. Graph the equation −𝑥𝑥 − 𝑦𝑦 + 5 = 0.

a. Graph the same equation but with the restriction −3 < 𝑥𝑥 ≤ 2. 4. Write the domain and range of the function below: