SABA Math Course Program A

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SABA Math Course

Program A

Version 2013

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The learning material is open and free and is developed with the math educational learning objectives and competences for Dutch Secondary Education.

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Math4all Foundation October 3, 2013 page 1

Content

Preface 3

1 Tables and graphs 5 1.1 Tables 6

1.2 Percentages 9 1.3 Graphs 12

1.4 Finding more values 16 1.5 Combine / Compare 19

2 Formulas 23

2.1 Using formulas 24 2.2 Plotting graphs 26 2.3 Equations 28 2.4 Inequalities 30 2.5 More variables 32

3 Linear relations 35

3.1 Directly proportional 36 3.2 Linear functions 38 3.3 Linear models 40 3.4 Linear equations 43

4 Exponential relations 45 4.1 Exponential growth 46 4.2 Calculations with powers 48 4.3 Real indices 50

4.4 Exponential functions 52

5 Change 55 5.1 In graphs 56 5.2 Per step 59

5.3 Average change 62

6 Counting 65 6.1 Possibilities 66

6.2 With or without repeating 68 6.3 Combinations 70

6.4 Pascal's triangle 72

7 Probability 75 7.1 Experiments 76 7.2 Reasoning 78 7.3 Tree diagrams 80

7.4 Probability distribution 82

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8 Normal distribution 85 8.1 Normal curve 86 8.2 Normal probabilities 90 8.3 Standardising 93 8.4 Normal or not 95

9 Probability models 97 9.1 Yes/No probabilities 98 9.2 Binomial distribution 100 9.3 Non-binomial distributions 103 9.4 Probability models 105

Register 108

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Preface

The learning material in this book is based on the content you can ﬁnd at www.math4all.nl. In the text you may ﬁnd references to that website. The header on each page gives you the exact location.

A chapter consists of theory and exercises.

When you encounter references to worksheets or other resources you can ﬁnd these on the website.

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1 Tables and graphs

Tables 6 Percentages 9 Graphs 12

Finding more values 16 Combine / Compare 19

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1.1 Tables

Theory

A table is an array of numbers in rows and columns. Here both the table and the columns have headers. Columns 1 and 2 contain absolute numbers. Column 3 contains relative numbers, namely the ratio of the number of men and women.

Using the table you can compute how many men and women lived in D in a certain year.

There are two types of tables:

> tables with research data, like this one;

> tables containing condensed data such as prices, times and work methods, such as the prices of TPG-post, de-

parture times of the Dutch Railroads and computation tables for insurance premiums.

Although tables are meant to clarify things and you should be able to interprete them properly, this often creates questions...

Exercises

Exercise 1

Below you see a table from Statistics Netherlands, called CBS.

a What is this table about?

b How many rows with numbers of students do you count?

c How can you tell that from 2002-’03 onwards the "speciaal voortgezet onderwijs" merged with the

"voortgezet onderwijs" and the "speciaal onderwijs"? Mention at least two things you ﬁnd in the table.

d What was the average size of a school for "voortgezet onderwijs" in 2002-’03? And in 2007-’08?

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SABA PROGAM A > FORMULAS AND GRAPHS > TABLES AND GRAPHS

e How many universities are there in The Netherlands? Compute the average number of students per university for 2000-’01 and for 2007-’08. What do you notice?

f The number of hbo-students does not seem to have increased from 2000-’01 through 2007-’08. Use a computation to show that the average number of students per institution has actually risen quite strongly.

Exercise 2

This was the ranking in theJupiler-leagueon November 27, 2009. This is the First Division competion in professional soccer.

Abbreviations: GS = played matches, WI = wins, GL = draws, VL = losses, PT = points, V = goals scored, T = goals against.

a Which team is in the lead? How many points did this team acquire? In how many played matches?

b Show that the number of points corresponds to the number of wins, losses and draws. How many points does a team get for a win? And for a draw?

c What is the total number of goals in this competition up to now?

d Although both have the same number of points De Graafschap is ranked above Cambuur - Leeuwarden.

This has to do with the goal diﬀerence. What do we mean by that? Use an example to show that the ranking is correct.

e When teams have the same number of points, you could also compare the goal average. The soccer federation states: "By goal average we mean the number that is obtained by dividing the number of the goals scored by the number of goals against". Would this have changed the ranking?

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Exercise 3

A biologist wants to know how the size of a brood (the number of young birds) inﬂuences the amount of food each of the young gets. He observes breeding titmice. The table contains his data.

a Before starting the biologist presumed that the larger the brood, the smaller the amount of food per young. Did this presumption turn out to be right? Can you give an explanation?

b The total amount of food the parent birds provide is not the same for each brood size. Precisely describe what happens.

c The rate of growth of the young birds, however, is almost the same for each brood size. Can you explain that?

The biologist ﬁnds these data about the heat production of young birds in the professional literature.

d Do these data satisfactorily explain the almost constant growth rate of the young birds?

Exercise 4

In a study about color blindness 12000 people have been tested. Out of the 6500 men participating in the study 612 were color blind, whereas only 53 women were color blind.

a Enter the data from the text above into the proper places in the cross table.

b What fraction of men is color blind?

c What fraction of the color blind is female?

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1.2 Percentages

Theory

The French pour cent means per hundred, so 1 percent is 1 per 100.

That becomes ₁₀₀¹ part of the total. In writing: 1% =₁₀₀¹ = 0.01.

And 12% of 500 is the ₁₀₀¹² part of 500. That is 0.12 × 500 = 60.

> What percentage is 24 of 60?

Answer: ²⁴₆₀≈ 0.369 = 36.9%.

> The number 24 increases to 27, by how many percent is that?

Answer: ²⁷⁻²⁴₂₄ = 0.125 = 12.5%.

> The number 24 increases with 6%, how much will it become?

Answer: 24 × 1.06 = 25.44.

> The number 24 decreases with 6%, how much will it become?

Answer: 24 × 0.94 = 22.56.

> The number 24 is 6% of the total, how much is the total?

Answer: 0.06 × total = 24, so the total is_0.06²⁴ = 400.

> A number increases with 6% to 60, what is the number?

Answer: number × 1.06 = 60, so the number is _1.06⁶⁰ ≈ 56.60.

Exercises

Exercise 1

Marianne goes shopping with her friend Anneke. They pass a store with great bargains. They quickly enter.

a Marianne sees a sweater for € 49.98. What will this sweater cost after the discount?

b Anneke buys two pairs of jeans with a pricetag of € 51.75. What does she pay for them?

c Marianne sees a blouse with a discount of 20%. The shop price is € 33.50 and she has to pay € 27.00 for it. Is the discount percentage correct?

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Exercise 2

You have deposited an amount of € 1000 into a bank account on 1 January 2009. You leave the money untouched and receive a yearly interest of 3.5%.

a How much money is in your account on 1 January 2010?

b And how much on 1 January 2011?

c Is the amount of interest the same every year? Why?

Exercise 3

You get a 40% discount on a stereo set. You do need to pay 21% sales tax, however. There are two possibilities:

> The shop owner ﬁrst computes the discounted price and then adds the sales tax.

>

The shop owner ﬁrst adds the sales tax to the price and then takes of the 40% discount.

Use a computation to show which is the most proﬁtable for you and what the diﬀerence is.

Exercise 4

You can buy a € 295 leather desk chair for € 200.

What is the discount percentage?

Exercise 5

A 250 gram box of chocolate sprinkles costs € 1.75. The producer has the following oﬀer: 20% more for the same price.

What is your discount?

Exercise 6

The water of the river Rhine ﬂows into diﬀerent river branches when it enters The Netherlands.

First 65% enters the river Waal and 35% enters the Nederrijn.

Just before Arnhem the Nederrijn then splits again. There 60% of the water ﬂows into the river Lek en 40% into the river IJssel.

a What percentage of the Rhine’s water ﬂows into the Ijssel?

b What percentage of the Rhine’s water ﬂows into the North Sea via the Lek?

c The Rhine’s water becomes polluted in the Ruhr area by the discharge of a certain amount of coloring agents. Researchers estimate that 640 kg of those coloring agents end up in the IJssel. How many kilograms of coloring agents were discharged?

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Exercise 7

The table contains information about the population and surface area of the Indonesian archipel in 1980. Some numbers are missing.

a What is the total number of inhabitants of Indonesia in 1980? Give the number in millions.

b How big was the total surface area of all Indonesian islands combined?

c Compute the numbers that are missing in the table.

d The population density is the number of people per km². Which island has the largest population density?

e Compute the population density of the whole of Indonesia.

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1.3 Graphs

Theory

When you have a table with two variables, you can draw a graph (using Excel for instance). In ata- ble about the Dutch population provided bij Statistics Netherlands the two variables u�u�u�u�(year) and u�u�u�u�u�u�u�u�u�u�u�u�u�u�u�u�(× 1000) appear. Here u�u�u�u�u�u�u�u�u�u�u�u�u�u�u�u� depends on u�u�u�u�. Therefore you draw a graph with the independent variable u�u�u�u� on the horizontal axis and the dependent variable u�u�u�u�u�u�u�u�u�u�u�u�u�u�u�u� on the vertical axis.

You denote the names of the variables and their units at the axes as captions. And you use tick marks with numbers on the axis. If you don’t do this, you cannot tell the meaning of the axis or the used units. Sometimes a graph gets a graph title. Several types of graphs are denoted:

> in a line graph you connect the points from the table with thin straight lines because you don’t know the values in between the points;

> In a curved graph you connect the points from the table with curved lines because you may assume or know that the values in between increase or decrease gradually;

> In a step graph you assume that the dependent variable changes in steps.

Graphs have certain properties...

A (smooth) graph is called

> increasing if the dependent variable increases when the independent variable increases;

> decreasing if the dependent variable decreases when the independent variable increases.

The graph has

> a maximum when it goes from increasing to decreasing;

> a minimum when it goes from decreasing to increasing.

You often ﬁnd a maximum or minimum at the edges.

The maximum and minimum values are called extreme values.

Sometimes the dependent variable repeats its behaviour over a certain period.

We call such a graph that (kind of) repeats itself periodical.

This graph of a normal heart rhytm is an example of a periodical graph. The period is approximately 0,8 seconds. That means a frequency of 75 heart beats per minute.

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Exercises

Exercise 1

The table contains the number of fatal burn victims in the United States caused by ﬁres between 1950 and 1982.

a Draw a graph using the data from the ﬁrst two columns. Explain which type of graph you have chosen and why.

b Which variable is the dependent variable?

c Can you reach a conclusion about the number of burn victims? Which helps you more, the table or the graph?

d Why is the third column needed to be able to conclude that ﬁre prevention measures started paying oﬀ after 1980?

e Use the table to estimate the number of inhabitants of the United States in 1982.

Exercise 2

Download and printthe school doctor chart for girls, The expression P₁₀₀on this chart means that 100% of the girls is below this line.

a Which two variables have been plotted against each other in the bottom half of this chart?

b Which two variables have been plotted against each other in the top half of this chart?

c What is the meaning of the two preprinted graphs on this chart? Why are they there?

d How can you use this chart to predict the height and the weight of a 20 year old woman?

e The table shows Marleen van Straaten’s data. Her height has been measured every year on her birthday.

draw her growth charts on the school physician’s chart.

f Is Marleen taller than average for her age? And what can you say about her weight?

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Exercise 3

When filling a glass vase from a steady flowing tap, you can see the water level in the vase rising. Below on the left are four vases of the same height, but of different forms. On the right there are four graphs representing the water level ℎ(in cm) charted against time u�(in seconds).

a Why are all four curves so smooth?

b Which graph belongs to which vase?

c Draw the graphs of ℎ depending on u� when the water ﬂows twice as fast.

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Exercise 4

The amount of carbondioxide (CO₂) ande the amount of oxygen (O₂) in the air are kept in balance because the plants on Earth transform carbondioxide into oxygen.

These graphs illustrate this.

a Determine the length of the period for both graphs.

b The amount of CO₂decreases from May through August. Explain why.

c What is the minimum amount of CO₂in cm³in the atmosphere in 1990? And what was the maximum found?

d The maximum values in the upper graph correspond to the minimum values in the lower graph and vice versa. Give an explanation.

e The graph shows a decreasing tendency for the ratio oxygen/nitrogen in the atmosphere. Draw a trend line depicting this.

f Draw a trend line for the carbondioxide graph as well. Is there a correspondence between both lines?

g Predict the amount of carbondioxide in January 2010 if this trend continues.

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1.4 Finding more values

Theory

Many graphs are made from tables with measurement data.

The way you connect the measurements is your choice. In a line graph you generally connect them by straight lines, but sometimes you use a smooth curve. You can only estimate the values that are not in the table.

> interpolation is the estimation of values between two known values. This is often done bij drawing a straight line through two points and reading oﬀ the value in between (mid 2002 there were approximately 870.000 people);

> extrapolation is the estimation of values outside the

measurement area. This is often done by drawing a straight line through the two previous (or the two following) measurements and extending it and reading oﬀ the desired value (early 2008 approximately 968.000 people).

Sometimes you encounter more than one graph with the same variables on the axes. This is called a bundle of graphs. From these you can read oﬀ values that do not exist in the corresponding tables, even in the area between two graphs.

Exercises

Exercise 1

Study again theschool doctor chart for girls. The P₅₀line depicts the height of an average girl.

a Use graphic interpolation to estimate the height of an average 15 years and 3 months old girl.

b On her thirteenth birthday Marleen van Straaten is 164 cm tall and on her fourteenth birthday she is 168 cm tall. Predict how tall she will be at twenty by using the growth charts.

c Why is extrapolation through both points not a good option in this situation?

d Predict Marleen’s weight at the age of twenty.

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Exercise 2

The graph to the right contains the number of local phone calls per person per year and the number of postal packages per person per year in the United States.

a After an initial decrease the graph of the number of local phone calls per person per year increases continually, except for two years. In which years does the graph not increase?

b Did the total number of local phone calls not increase either during those years? Explain your answer.

c In which periods did the number of postal packages per person per year decrease?

d Use extrapolation to determine the number of local phone calls and the number of postal packages per person per year in de United States in the year 2000. Explain what you do.

Exercise 3

The table below shows the number of unemployed in a certain city. The numbers of unemployed have been rounded oﬀ to the nearest hundred.

time mar’04 may’04 jul’04 sep’04 nov’04 jan’05 mar’05 may’05 jul’05 sep’05 number 10700 11400 11100 10300 10000 10700 11900 12600 12300 11500 a Use this data to draw a graph.

b How can you tell from this graph that there is a seasonal inﬂuence on unemployment?

c Assume that this trend continues. Predict the number of unemployed in this city in January 2014.

d How many people will be unemployed in March 2008 according to this trend?

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Exercise 4

The Body Mass Index (BMI) indicates whether people of 16 years and older have a normal weight (20 < BMI ≤ 25), are overweight (25 < BMI ≤ 30), obese (BMI > 30) or underweight (BMI ≤ 20). You can use these graphs to determine a person’s BMI.

a Estimate the BMI of an 18 year old person, with a height of 1.90 m and a weight of 90 kg.

b What is a normal weight for somebody older than 16 years if this person is 1.80 m tall? Give the upper and lower boundaries.

c Sketch the graph for a BMI of 23.

d Why are these BMI graphs valid only for people of 16 years and over?

e If you know somebody’s length, the BMI only depends on this person’s weight. Draw the graph of the BMI for a person with a length of 1.80 m.

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1.5 Combine / Compare

Theory

Sometimes you are dealing with more than one graph.

If you can express two variables into the same units and if they both depend on the same variable, you can draw their graphs in one ﬁgure

> In a point of intersection of both graphs the variables have the same value on the vertical axis.

> You can make a value added graph by adding together the two corresponding values.

> You can make a value subtracted graph by subtracting the corresponding values from each other.

You can also have the situation that two graphs have one common variable. For instance, in one graph the variable u� depends on u� and in the other graph u� depends on t. You can now chain the variables:

for a value of u� you ﬁnd a value of u� and that gives you the corresponding value of y.

Exercises

Exercise 1

By migration balance we mean the diﬀerence between the number of people immigrating to The Nether- lands and the number of people emigrating from The Netherlands. In the graph you see data about the migration of Maroccan and Turkish people to and from The Netherlands in the period 1980 - 1991.

a What is the meaning of points of intersection of the "immigration" and "emigration" graphs for the same population group?

b What is the meaning of the points of intersection of the two immigration graphs?

c For which of these two population groups was the migration balance negative in this period?

d Based on the above, can you say that this population group decreased in numbers in The Netherlands?

e It is remarkable that immigration is much lower in the years 1982 - 1984 than in other years and that emigration is higher during those years. Give some possible reasons for this.

f Draw the migration balance graph for the Turkish.

g In which year does this migration balance decrease the most?

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Exercise 2

Shown are the graphs of the distance covered by two cyclists in a time trial. The graphs have been drawn in such a way that it looks as if the cyclists started simultaneously (in a time trial this is not the case).

a What meaning do the two points of intersection in this graph have?

b During what interval is cyclist A ahead of cyclist B?

c At which moments do the cyclists have the same speed?

Exercise 3

These graphs show how many pieces of a certain article u�(in thousands) a company has sold and how much proﬁt 𝑊(in thousands of euros) it has made doing so. u� is depicted against time u� in months after the introduction of the article.

a How many pieces have been sold after 4 months? And how much proﬁt did this generate?

b After how many pieces did the sale start generating a proﬁt?

c How many months before proﬁt was being made?

d Make a table of the proﬁt 𝑊(in thousands of euros) depending on the time u� in months.

Exercise 4

Have another look at theschool doctor chart for girls. By chaining the two P50-graphs you can make a graph of the weight as a function of the age of an average girl. Explain how you can do this and draw the graph.

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Exercise 5

This ﬁgure shows how horses and mules were replaced by tractors in the 20th century in the U.S.A..

a The graphs intersect in a point. Does this point have a meaning? Explain your answer.

b When was the number of tractors equal to the number of horses and mules?