Unit 2 sec 5.3

Unit 2 sec 5.3 Using feedback.

30 October 2015

23:28

So you probably have a fairly good idea of how you are getting on with the module, and you may already be making changes to improve your studying.

Doing the assignments will help you to consolidate the main ideas and provide further evidence of your progress.

This subsection outlines the kind of feedback you will receive and how to use it. However, to gain full benefit from each assignment, you should spend some time, say 20 - 30 minutes, looking in detail at the feedback you receive. Immediately after receiving your score, you may be feeling thrilled with your school or disappointed by, or even frustrated by any mistakes you made.

Icma feedback

for each question that you answered incorrectly, you should study the solution provided check that you understand it, particularly Larry where it differs from your solution. When you have worked through the feedback, spend a few moments thinking about whether you need to make any changes to the way you tackle Icma questions in the future.

ª    Do you need to try more practice quiz questions before attempting each Icma?

ª    Do you need to allow yourself more time to complete each Icma?

ª    Would it be better to tackle Icma questions as you work through each section of the unit or to work on them only after you have completed each unit?

TMA FEEDBACK

it may include preys on what you have done well or suggestions for alternative techniques that you may find helpful in the future, as well as constructive comments to help you improve your mathematics and the way you present it stop. The form will highlight the main points that you should try to address before you submit your next TMA. It is a good idea to make use of the general advice straight away as you work on the units, so that those ideas will be familiar to you when you tackle the next TMA. You can practice improving your skills in this area as you work on the activities in the units.

You should check through these comments carefully, making sure that you understand them, and that you would be able to tackle a similar question successfully in the future, if required to do so. It is worth making a note of the main points that you need to remember for the next TMA. Alternatively, you could highlight the row about statements on the TMA form all make some more formal notes. It is important to find a helpful way of recording these comments so that you can refer to them easily. When you tackle the next TMA.

Learning checklist

after studying this unit, you should be able to:

¨    use a map scale to estimate distances

¨    use formulas relating speed, distance and time

¨    understand how the modelling cycle can be used to solve problems,

¨    appreciate that mathematical models simplify reality, take account of some features, and ignore others

¨    appreciate that mathematical ideas can be communicated in different ways (for example, numerically or graphically), and the best way to communicate them may depend on the intended audience

¨    draw and interpret graph

¨    use formulas and find your own formulas to describe simple situations

¨    understand some conventions of writing formulas

¨    using equality signs to describe limits and intervals

¨    if you are studying and make changes to improve it.

Unit 2 sec 5.2

Unit 2 sec 5.2 reading Mathematics

30 October 2015

23:25

Reading mathematics is a different from more general reading you need to concentrate on each word and symbol, learning and using new vocabulary and notation as you go. This means that it takes longer, and you may sometimes feel stuck, you could not see how to get from one line to the next. 4 different sections, you may find it helpful to annotate the text by including some extra working to explain the steps of the notation.

Unit sec 5.1

Unit 2 sec 5.1 some problems - solving strategies

30 October 2015

23:10

In this unit you have seen how some everyday problems can be investigated by using mathematics. One way of tackling such real life problems is to use the modelling cycle. In fact, the steps in the modelling cycle of fairly familiar to those that you use when you tackle mathematics problem, whether it is a practical problem or something more abstract, as summarised in the box below.

Tips for tackling mathematics problems

v check you understand the problem - and if you are not sure, talk to people (your tutor, fellow students, friends) until you do.

v Collect information that will help you to solve it - this may be data, but it can also include techniques that you have used before. That may help you in this case too. What do you want to find out, and what do you know already?

v Simplify the problem. If you can - this may include trying some numerical examples. First of breaking the problem down into smaller, achievable steps.

v Carry out the mathematics. Remember, there are often several different ways of tackling the problem, including numerically way and graphically, which may give different insights. Drawing a diagram often helps too.

v Check that your answers are reasonable and rounded appropriately.

Drawing a diagram is a good way of obtaining different you all a problem. You use diagrams to help with formulas and inequalities. Diagrams can also be used as part of your notes, to hand you to connect different ideas together and to remember key ideas related problems.

A diagram that is very useful for remembering the following formulas.

Speed =, time = , distance = speed × time.

You should decide which quantity you need on the left-hand side of the formula, and cover up that quantity, then look at the precision of the remaining two quantities.

Unit 2 sec 4.2

Unit 2 sec 4.2 illustrating Inequality on a number line

30 October 2015

22:48

And inequality can be represented on a number line by making this section of the number line where the inequality is true. Section of the number line without any gaps is shown as an interval. So illustrates the inequality s ≤ 112. The small solid circle at the limit 112 indicates that 112 is contained in the interval and is a possible value for s.

A straight inequality can be represented on a number line by using a small empty circle at the limit. The possible value for u line it to the right of 4.

Example 15 using a double inequality

the child is 5 years or older, but not yet 16 is eligible for child fare on the train. Children under 5 travel free. Suppose that a represents the age of the child in years.

(A)       draw a number line to illustrate the ages eligible for a child fare.

(B)         Give double inequality to describe the age restriction for child fares. Which whole number satisfy this inequality?

Solution

(A)       mark the limits at 5 and 16 on the number line first, and then join the limits with the line.

The ages of children who are eligible for a child there, as shown on the number line.

(B)         the restrictions of the age for a child there are

a is greater than or equal to 5, and a is less than 16.

Using inequality signs,

a ≥ 5 and a ≤ 16

 Now, a ≥ 5 and a ≤ 16

therefore, the inequalities are

5 ≤ a and a < 16.

These two inequalities can be combined as the double inequality

 5 ≤ a < 16.

The whole numbers satisfy this inequality are

5,6,7,8,9,10,11,12,13,14,15.

The double inequality 5 ≤ a < 16 is read as

‘5 is less than or equal to a, which is less than 16’, or as

‘a is greater than or equal to 5, and worse than 16’.

Inequalities can also be used to illustrate the range of possible numbers that round to a particular value.

unit 2 sec 4.1

Unit 2 sec 4.1 notation for working with inequalities

30 October 2015

21:52

This section considers some notation that can be used for describing a range of possible values that variable can take. In order to describe known restrictions on the variables, and when stating your conclusions. The restrictions on a variable that represents the speed of a car might be that it is greater than or equal to zero and then or equal to the speed limit that applies.

 The nature of this inequality can be expressed by using the phrases less than an greater than or the inequality signs < and >.

   If a number way to the life of another number on the number line, then it is said to be less than the number. -5 lives to the left of -2, as shown in figure 14, so -5 is less than -2. This statement can be written more concisely by using the inequality sign < for less than;

 -5 < -2

#rise to the right of another number on the number line, then it is said to be greater than the other number. -1 lies to the right of -3, -1 is greater than -3. This statement can be written using the inequality sign > for greater than.

-1 > -3

in a statement involving inequality signs is cold and inequality. Each inequality can be written into different ways. For example 4 is greater than 2 so you can write

4 > 2

but also 2 is less than 4, so you can write.

2 < 4

each way of writing and inequality is obtained from the other by swapping the numbers and reversing the inequality sign. This is called reversing the inequality.

As well as the 2 inequality signs introduced above, there are 2 other inequality signs, ≤ and ≥. Before inequality signs and their meanings are given in the following box.

Inequality signs

< is less than

≤is less than or equal to

>      is greater than

≥ is greater than or equal to

here are some examples of correct inequalities

Ø 1 <1.5, because one is worse than 1.5

Ø 1 ≤ 1.5, because one is less than or equal to 1.5 (it is ‘less than’ 1.5).

Ø 1 ≤ 1, because one is less than or equal to 1 (it is ‘equal to’1).

It may seem strange to write 1 ≤ 1.5 and 1 ≤ 1, when the more precise statements 1<1.5 and 1 = 1 can be made, and you would not usually right. The former statements. The inequality signs ≤ and ≥ are useful, however, for specifying the range of values that the valuable intake, as in the following example.

Example 14 specifying the range of variable

suppose that the speed of a car on the UK motorway is s km/h. Write down to inequalities that specify the range of possible legal values of s.

Solution

first, decide what you want to say in words.

This speed must be greater than or equal to 0 and should be less than or equal to the speed limit on the UK motorway, that is, 112 km/h (70 mph).

Replace the words by the appropriate inequalities.

So the two inequalities are

S ≥ 0 and s ≤ 112.

Most inequalities that you will meet involve variables. The value of the variable for which the inequality is true, is said to satisfy the inequality.

Unit 2 sec 3.3

Unit 2 sec 3.3 Constructing your own formulas

27 October 2015

23:26

There are many well-known formulas that you can use to solve problems, but sometimes you need to find your own formula. You can construct a formula by following the 3 steps below.

ª    First, identify the subject of formula and the other variables, and units of measurements. This means that you have to decide the purpose of the formula and what the formulae depends on.

ª    Second, find the relationship between the variables. Here, you need to think about how to work out R from d. The length of the return journey R is twice the distance between the 2 places, that is, two lots of d, which can be written as 2d. So the formula is R=2d.

Third, write down all the details of the formula. The formula is R= 2d, where R is the length of the return journey in km, and d is the distance between the 2 points in km. Alternatively, more concisely: the length of the return journey R km is given in terms of the distance d km by the formula R=2d. Not that you should never include units a formula.

Example 12  finding a formula -driving to work

(A)       hearing a working, Anya year drives from home to her office and back 5 times, and she also makes a number of trips from her office to head office and back. Her office is 12 miles from her home in 7 miles from the head office. Find formula for d, where d is the total number of miles that Anya drives in a week when she makes n trip to head office.

(B)         Use your formula to find the distance driven by Anya in week when she makes 3 trips to head office.

Solution

(A)       diagram if it helps you to understand the situation. Then tackle the problem. Step-by-step, by considering separately to return journeys from her home to the office and from the office to the head office.

The distance in miles that Anya drives from her home to her office and that is 12×2=24. So in 5 days, the distance in miles that she drives from her home to her office and back is 5×24=120.

The length of the return journey from the office to their head office is 2×7 miles = 14 miles. So the distance you drive to the head office and back in n trips is in n lots of 14 that is, 14n miles.

The total distance in miles that she drives is the sum of the total distance she travels between home and the office, and the total distance between head office and the office.

Hence the formula for d is

D=120+14,

Where d is the distance in miles, and n is the number of trips from the office to head office.

(B)         Substituting n = 3 into the formula in part (A) gives

D=120+14×3=120+42=162

   Find the relationship between the variables hard to spot, then try some particular numbers first as that might help you to identify the operations involved.

    Then think about which parts of the calculations stay the same and which change. That might help you to discover that the length of the return journey from home to the office is always 120 miles, and that this distance always needs to be added to the distance for the trips to the head office and back.

Example 13, finding a formula - the car ferry

a car ferry can transport both cars and vans stop the van requires a space of 9m, and a car required a space of 5m. find formula for the length L required for c cars and v vans.

Solution

consider the space needed for the current 1^st, try some particular numbers to start with.

One car needs a space of 5m, 30 cars need 2 × 5m = 10m, 3 cars need 3×5m=15m, and so on.

So to find the space needed for c cars, c lots of 5 metres are needed, that is, distance of 5c metres.

Similarly, for the vans.

1 van needs a space of 9m, so 2 vans need 2 × 9m, 3 vans need 3×9m, and so on.

So the space needed for v vans is v × 9 meters, that is 9v meters.

To find the total length, and the distance for the cars and the distance for the vans together.

So a formula for L is L=5c+9v, where L is the total length in meters, c is the number of cars and v is the number of vans.

(As a chair, though, went for, say, 2 cars and 3 barns without using the formula:

the length is 2×5m+3×9=10m+27=37m

if you substitute c=2 and v=3 into the formula, then you obtain the same answer.)

strategy finding formulas

1)   identify the subject of the formula and the other variables, and their units in measurements. If possible, newsletters for the variables that remind you of the context.

2)   Find a relationship between the variables.

3)   Write down all the details of the formula, defining the variables and starting their units (as appropriate).

(It is a good idea to try particular numbers just to suggest what the formula is one, and then check that your formula works.)

unit 2 sec 3.2

Unit 2 sec 3.2 writing formulas concisely

27 October 2015

22:25

  As containing a lot of mathematical symbols can look quite complicated. To make them more concise, change multiplication are usually omitted.

   However, when you substituting numerical values into a formula, you usually have to part the change multiplication sign back in, to make the meaning clear. So 3 × y can be written as 3y, the 3 × 2 cannot be written as 32. One way to check that you understand what the given Formula means is to try describing in words how to use the formula.

Conventions for writing formulas

these are several conventions that are usually followed when writing formulas concisely stop

¨    in products, numbers are usually written 1^st; for example, the formula k= 1.6 × m, or equivalently K = M × 1.6, is erecting concisely as k= 1.6m.

similarly, (2a+b) ×3 is written concisely as 3(2a+b). However (2a+b)c and c(2a+b) are both acceptable ways of writing (2a+b) ×c.

¨    In products, letters are often written in alphabetical order; for example, d= s×t, or equivalently d=t×s, is usually written as d= st

¨    finally, divisions are usually written in the form of a fraction; for example s= d÷t is written as s=  and read as ‘s equals d over t’.

     This helps to distinguish between say, the distance 5 metres, which is printed as 5 m and the expression 5 × m (that is, 5 × the variable m), which is printed concisely 5m. These work identical and the meaning is obtained from the context. This is one reason why units are usually not included in mathematical calculations that involve variables.

Example 10 estimating the volume of a log

foresters can estimate the binding of a log of wood by using the formula. V=  

where V is the volume of the log ins cubic metres, L is the length of the log in metres, D is the distance around the middle of the log in metres, and Ԉ is approximately 3.14159.

Estimate the volume of the log that is 1.5 m long and 92 cm around the middle, giving your answer to 2 significant figures.

Solution

check that the given information is in the correct units.

The length is 1.5 m, so L= 1.5. The distance around the middle is 92 cm, but the formulae requires the measurement in metres. Since 92 cm = (92÷ 100) m =0.92, we have D= 0.92.

Substitute and do the calculation.

Substituting, L= 1.5 and D=0.92 into the formula V =  gives

V =  =  = 0.101

date the conclusion, including the correct units. Hence the volume of the log is 0.10m³ (to 2 s.f).

     Several differing calculator sequences can be used to calculate the final answer, and some of these sequences involved using the memory and other function keys on your calculator.

Substituting negative numbers

when you replace a letter by a negative number, it is usually helpful to include the number in brackets to avoid confusion.

Example 11 substituting a negative number

consider the formula A= c² - 5c + 3. Find the value of A when

c = -2.

Solution

potting brackets around -2 and substituting it for c gives

A= (-2) ² - 5(-2) +3

   = (-2) × (-2) –(-10) +3

   = 4÷10÷3

   =17.

-2 has been enclosed in brackets when it is substituted to ensure that the - is not separated from the buying mistake.

  Then remember that the calculation above – (-10) means subcontract -10, and subtracting the negative number -10 is the same as adding the corresponding positive number 10. So in this calculation, -(-10) is the same as + 10.

However in the variable being substituted appears 1^st in the calculation on its own, then no brackets are required. For example, if A= C+ 3 and C= -2, then A= -2+3=1.

Sometimes when you are substituting into a formula you have to find the negative of a number. This is the number that is produced by putting a - in front of the number.

Unit 2 sec 3.1

Using formulas Unit 2 sec 3.1 from words to letters

23 October 2015

22:32

   Solving a problem in mathematics often involves using a formula. I used extensively in everyday life.

   Although some formulas are easy to remember when expressed in words most formulas are written in a more concise form. This is particularly true when they are used in computer programs or spreadsheets, or when they are more complicated.

   Many things are represented by a symbol; for example the symbol like a T on a road sign, the symbol P on a map often indicates a car park. Symbols are concerns - they say writing out a whole word or sentence and consequently they make it possible to see information more clearly. In mathematics, you are already familiar with some symbol, such as ÷ and √

   used to estimate the distance travelled by car. If we use the letters

s to represent the average speed,

t to represent the time taken,

d to represent the distance travel.

Then this word formula can be written more concisely as the “letter formula”

  D= s×t.

The letters are used instead of words in a formula, then it is essential to say what quantities the letters represent.

Using a formula

   The letters in a formula stands for numbers that are related in the way given by the formula. Thus you can think of a formula as a way of summarising the calculation process.

  When you use a formula, you replace the letters to the right of the equals sign (in the case, s and t) by numbers, and then carry out the calculation to find the value of the letter to the left (in this case, d).

   This process is known as substituting values into the formula.

Rather than using the word formula to find the distance, breaking news the more concise formula d=s×t, where d, s and t are defined as before.

  Replacing s by 50 and t by 1.2 gives

D= 50×0.2 = 60 hence the distance travelled is 60km.

    Since the values of s, t and d can vary and represent different numbers in different scenarios, they are known as variables. In general, and letter can represent different numbers is called a variable.

So formula is an equation in which one variable, called the subject of the formula, appears by itself on the left-hand side of the equation and only the other variables appear on the right-hand side. Thus, enables you to calculate the value of the subject when you know the values of the other variable stop

   In many formulas the variables represent measurements, and it is important to check that the values you substitute are measured in appropriate units.

Formula with set units

   Some formulas the units are already set and cannot be changed. No other units can be used in this formula. Before you substitute in a formula like this, you must check that the information that you use is express in the correct units, and make any conversions.

Examples 7 substituting value into a formula

  a European car hire company charges €50 per day for that higher of the small car, plus a booking fee of €20. So, the total cost of hiring the car is given by the formula

T= 50×n +20

Where T is the total cost in € and n is the number of days for which the car is hired.

How much does it cost to hire the car for 2 weeks?

Solution

check that the given information is in the correct units.

In formula, the time n is measured in days, so 1^st convert to weeks in 2 days.

There are 7 days in one week, so in 2 weeks there are 2× 7 days= 14 days hence n =14

                

 substitute in do the calculation

substituting n = 14 into the formula gives

t = 50× 14+ 20

  = 700+ 20

  = 720

State the conclusion, including the correct units.

Hence the cost of hiring the car for 2 weeks is €720.

Once you have substituted numbers into formula, you perform the calculation by using the usual rules of arithmetic. The mnemonic BIDMAS helps you to remember the order of operations.

Brackets, then indices (powers and roots), then divisions and multiplications, then additions and subtractions.

Example 8 calculating the time to walk uphill

  Naismith’s rule estimates that the time taken for a walk uphill is given by the formula

T= +

Where

T is the time for the war in hours,

D is the horizontal distance walked in kilometres,

H if the height climbed in metres.

(A)       estimate how long a walk will take the horizontal distance is 20km and their height is 1200m

(B)         why might you need to allow longer than this estimate?

Solution

(A)       check that the given information is the correcting units.

In this case, the horizontal distance is 20km and climbed is 1200m. The units here are those specified for the formula, so no conversion is needed.

Checked and do the calculation.

Substituting D =20 and H = 1200 into the formula gives

T=  +  = 4+2=6

State the conclusion, including the correct units.

Hence Naismith’s rule predicts a 6 hour walk.

(B)         you may need to allow longer than 6 hours to accommodate to rest breaks, all because the terrain is difficult, the walkers are on the or the weather is bad.

Formulas you can choose the units

The formula for the area of a rectangle is A= l×w, where a is the area l is the length and W is the width. You can choose which units to use as long as these units are consistent. The area in square centimetres; if you measure the length in kilometres, then the width should also be measured in kilometres.

   The units given are not consistent with each other, then you should convert the measurements into appropriate units before substituting them into the formula.

In formula d= s×t, mentioned earlier, the units could be consistent; the units of the speed are km/h, then the unit for the time are h and the unit for the distance are km

Example 9 substituting values into a formula

a car travels at an average speed of 95km/h. Use the formula

d= s×t

defying the distance the car travelled in each of the following signs.

(A)       2.5 hours

(B)         40 minutes

give your answers to 2 significant figures

Solution

(A)       unit for speed is km/h and the unit 4 time hours, then the distance in km, so no conversions are required.

When s= 95 and t= 2.5,

D=95×2.5=237.5

so the distance travelled is 240km (to 2 s.f.).

(B)         first, converted the given time into hours. Since

40 min = h = 0.666h,

the time is 0.666h

when s = 95 and t = 0.666

d= 95×0.666= 63.33

so the distance travelled is 63km (to 2 s.f.).

The consistency of units is to substitute the values for the variables together with the units into the formula. The area of the a rectangle in length 3m and width 50cm could have been written as

A= 3m×0.5m

   = (3×0.5)m²

   = 1.5m²

UNIT 2 sec 2.4

Unit 2 sec 2.4 Developing the models further

21 October 2015

23:45

        One of the most significant factors could be the time it takes the driver to react to a hazard. If the driver is not alert or is distracted in some way, then thinking distance could increase substantially, making the overall stopping distance much greater than that suggested by the models in the highway code.

 

     Taking account of these additional factors would involve going around the modelling cycle again, developing the mathematical description to include these new assumptions and checking how the new model matches reality.

   It may not be easy to remember or visualise the recommended stopping distances, so the role that it easier to apply is desirable. Whereas at speeds greater than this, the distance model gives the longer gap.

  

    To make the rule based on time that produced a similar gaps to those of the distance model at the typical motorway speeds, a three- second rule would produce much larger gaps than the distance model at lower speeds. This time model is an approximation to the distance model.

Unit sec 2.2

The highway code might have been developed, by following the 4 stages of the modelling cycle.

   Stage I: clarify the question

Both models have been constructed in order to answer the question:

“what gap between your vehicle’s should be recommended for drivers travelling at different speeds?”

    Stage II making assumptions and collect data:

the distance model is based on typical stopping distances at various speeds. It has been assumed that the stopping distance is determined by 2 factors:

v the thinking distance (the distance travelled from when the driver first see hazard until he or she applies the brakes.)

v The braking distance (the distance travelled from when the brakes are first applied to the point when the vehicle stops.)

      The experiments would probably produce a range of possible times, depending on the individual and their state of alertness. It is possible to determine a typical reaction time.

   The model ignores other features of the situation such as the road surface, the make and weight of the car, the weather conditions and the tiredness of the driver.

Stage III using mathematics to obtain results

the next stage is to use some mathematics, nest case of working out the distance by using formulas.

The car is likely to be travelling at a consistent speed. If you know the speed of a vehicle and the time during which it travels at that speed, then you can calculate the distance it travels. In general if an object needs for a certain period of time, then the distance it covers in this time is given by the following formula.

Distance = average speed × time

This speed by using the data collected to Derive a more complicated formula. Roughly speaking, the effect of the formula is that the speed it doubles, then the braking distance quadruples.

Stage IV interpret and check the results

the distance model could be checked with the reality by, o of whether drivers managed to stop their vehicles within the “typical stopping distance” and also whether collisions occur less frequently when driving to keep this gap between their vehicle and the next.