Apprenticeship & Workplace 11 Calculator Basics

1. Display

• you can display both your calculation formula (the first line) and your answer (the second line) at the same time • this allows you to check to see if you inputted the correct information for the calculation

2. Modes • your calculator can be set up to perform a variety of different calculations • to set the calculator to the proper mode, press [SHIFT] [CLR] [2] [=] [AC] • use this setting if you ever have to adjust the calculator to perform a particular calculation

3. Input capacity • the memory area used for calculations can hold 79 “steps” • pressing the [SHIFT] or [ALPHA] key does not count as a step • pressing the [Ans] key recalls the last result obtained which you can use in the next calculation

4. Making corrections • use [◄] or [►] to move the cursor to the location you want in a calculation formula • press [DEL] to delete the character at the current cursor position • press [SHIFT] [DEL] in order to insert a character at the current position • pressing [SHIFT] [DEL] or [=] returns the calculator to the normal cursor

5. Replay function • every time you perform a calculation, pressing the [▲] key displays the formula and result of the calculation you

last performed • pressing [▲] repeatedly back steps through past calculations (from new to old) • pressing the [◄] or [►] key while the replay function is on changes the display to an editing screen • pressing [◄] or [►] immediately after you finish a calculation displays the editing screen for that calculation • pressing [AC] does not clear Replay memory • Replay memory is cleared when you press the [ON] key or turn off the calculator

6. Error locator

• pressing [◄] or [►] after an error occurs displays the calculation with the cursor positioned at the location where the error occurred i.e. Math ERROR, Stack ERROR, Syntax ERROR

7. Exponential display format

• this calculator can display up to 10 digits (numbers) • very large or very small number are displayed in exponential notation e.g. or

• press [MODE] [MODE][MODE][3][2] to change the calculator to the proper setting

8. Decimal point and separator symbols

• your calculator is set to display the decimal point as a dot i.e. (.) and the 3-digit separator as a comma i.e. (,) 9. Inputting sequence

• there are three main types of input sequences performed with your calculator • Type A: the value is entered and then the function key is pressed i.e.

• Type B: the function key is pressed and then the value is entered i.e. • Type C: the value is entered, the function key is pressed and then a second value is entered

i.e. • press [ ( ] and [ ) ] when you have complicated calculations that have “layers”

91 000 000 000 1 10= ´70.000 000 281 2.81 10-= ´

1 3 2, ,x x x-

( )3, ,10 ,x -

, , xb ca Ù

• your calculator will determine what percent one number is of another number if you input the original numbers using the division key i.e. [÷] , and then press the percent key i.e. [SHIFT] [=]

Below are some exercises to introduce the basic functions of the scientific calculator that you need to be familiar with in Essentials of Math 10. 1. To enter an exponent

• Use to square a number, use to cube a number and use to raise a base to any power that you want • to find , press • to find , press • to find , press

2. To find the root of a number • use to find the square root, use (SHIFT/ ) to find the cube root and use (SHIFT/ ^) to find any

other root • to find , press • to find , press

• to find , press

3. To find the power of 10 • use (SHIFT/ log)

• to find , press

4. To calculate using fractions

• when entering either a fraction i.e. , or an improper fraction i.e. , use

• to enter , press

• when entering a mixed number i.e. , the twice

• to enter , press

• you can change a mixed number to an improper fraction i.e. , by pressing (SHIFT/ ) once

the mixed number has been displayed 5. To enter a number in scientific notation

• when entering a number written in scientific notation i.e. , press to enter the power of 10 (it will be simply displayed as E)

• to enter , press 6. To enter a negative number

• press the before entering the number

• to enter , press

29 9 2x37 7 3x63 3 ^ 6

3 3x x

49 4 9

SHIFT 3x 1 2 5

4 SHIFT ^ 2 5 6

10x

SHIFT log 7

34

97

b ca34

3 b ca 4

235

b ca235

3 b ca 2 b ca 5

1 254 to 6 6

d c b ca

55.314 10´ EXP

55.314 10´ 5 . 1 3 4 EXP 5

( )-173- ( )- 1 7 3

2x 3x ^

3 125

4 256

710

Apprenticeship & Workplace 11 0.1 Decimals Lesson Focus: To be able to perform the basic operations with decimals. A. Adding Decimals

• adding decimals is the same as adding whole numbers • you must first be sure the digits are in their proper columns

i.e. tens under tens, ones under ones, tenths under tenths, etc. • this will occur automatically if you keep the decimal points in a line • put in a decimal in any number that does not have a decimal • put in zeroes so that every number has the same number • remember to write the numbers that you carry at the TOP of the column

e.g. Add 35.9 + 7.568 + 230 + 0.8764 e.g. 4.19 + 26.2 + 98 + 1.547

B. Subtracting Decimals • use the same rules as above when subtracting • remember to borrow from the column to the left if you need to

e.g. Subtract 6.213 from 32.4 e.g. Subtract 3.9 from 15.816

C. Multiplying Decimals • to multiply decimals, you multiply the decimals as though they were whole numbers • then place the decimal point according to the total number of decimal places in the numbers you are

multiplying i.e. add the number of decimals and move from the last number on the right to the left e.g. Multiply 8.2 3.04× e.g. Multiply 5.76 4.3×

D. Dividing Decimals

• you need to know the names of the following positions

i.e.

• first put a decimal point in the quotient directly above the decimal point in the dividend • divide as for whole numbers • find out how many times the divisor can go into the dividend without going over • if the division has a non-zero remainder, add zeros at the end of the dividend and keep dividing • stop when you have a digit(s) starting to repeat, When remainder is zero or until you reach the accuracy

indicated in the problem

e.g. Divide 262.4 by 5 ( )262.4 5÷ e.g. Divide 129.84 by 6 ( )129.84 6÷

quotientdivisor dividend

remainder

Apprenticeship & Workplace 11 0.2 Fractions Lesson Focus: To be able to simplify fractions, convert between improper fractions and mixed numbers as

well as perform the basic operations on fractions. A. Simplifying Fractions

• fractions are in simplified form or lowest terms when the top (numerator) and the bottom (denominator) do not have any numbers in common

e.g. Simplify 12 618 9

23

=

=

e.g. Simplify 3042

B. Converting Fractions

• we can change an improper fraction into a mixed number by trying to determine how many groups of the denominator go into the numerator

e.g. Convert 143

into a mixed number. e.g. Convert 174

into a mixed number.

( )4 3 2143 3

243

× +=

=

• we can change a mixed number into an improper fraction by multiplying the whole number in front to

the denominator and adding the numerator (remember to keep the denominator the same!)

e.g. Convert 354

into an improper fraction. e.g. Convert 237

into an improper fraction.

3 5 4 354 4

234

× +=

=

C. Adding and Subtracting Fractions

• if the denominators are the same, add or subtract the numerators (keep the original denominator) • simplify to lowest terms if possible

e.g. 1 3 1 38 8 8

4812

++ =

=

=

e.g. 7 49 9−

• if the denominators are different, find the lowest common denominator (LCD), change all fractions to

the LCD and then add or subtract

e.g. 5 1 5 48 2 8 8

5 4818

− = −

−=

=

e.g. 2 13 6+

• in order to add or subtract mixed numbers:

1. add or subtract the whole numbers 2. add or subtract the fractions 3. combine the results

e.g. ( )1 2 1 21 3 1 35 5 5

345

+⎛ ⎞+ = + ⎜ ⎟⎝ ⎠

=

e.g. 7 15 29 9−

D. Multiplying and Dividing Fractions

• in order to multiply fractions: 1. cancel where possible 2. multiply the numerators together and multiply the denominators together 3. change to a mixed number if necessary

e.g. 5 14 1 27 15 1 3

23

× = ×

=

e.g. 4 119 2×

• in order to divide by a fraction, multiply by its reciprocal (exchange the numerator and the

denominator)

e.g. 16 8 16 35 3 5 8

2 35 165115

÷ = ×

= ×

=

=

e.g. 55 58÷

Apprenticeship & Workplace 11 0.3 Ratios and Proportions Lesson Focus: To introduce the concept of equivalent ratios and use equivalent ratios to solve for missing

terms. • a ratio is a mathematical way to compare two numbers

e.g. 1. The ratio of students in Grade 10 compared to all secondary school students is 2:7 2. The ratio of goals scored to shots on goal is 1 to 15 3. The ratio of homes built this year compared to homes built last year is 4/3

• all of the above statements involve comparisons • a ratio can be written with a colon i.e. “:”, the word “to” or as a fraction i.e. “1/5” • two ratios are said to be equivalent if the first (top) term of each ratio compares to the second (bottom) term

in an identical manner

i.e. each of the following ratios are equivalent 1 2 3 4 5 ...2 4 6 8 10= = = = =

• fractions are equivalent if they simplify down to the same fraction (lowest terms) • an equation showing equivalent ratios is called a proportion • an important relation for any proportion is cross-products

i.e. if , then a c a d b cb d= × = × (top left times bottom right equals bottom left times top right)

• we can illustrate cross-products by putting an actual cross over the equal sign

i.e. 53=

10 then 5 6 3 10 306, × = × =

e.g. Are the following proportions true or false?

a) 128

1812

= b) 2412

3521

=

c) 129

106= d)

8136

2712

=

• proportions can be written with one of the values missing • a simple way to solve a proportion is to use cross-products to multiply number to number and then divide by

the number multiplying to the variable

e.g. Find the value of x in each of the following proportions.

a) 10

30 6x= b)

4735

=x

c) 29 54

x= d)

x24

183=

• if any of the numbers in the proportion are fractions, remember to use your fraction button (a b/c)

Apprenticeship & Workplace 11 0.4 Percent Lesson Focus: To be able to convert percent, decimals and fractions as well as complete percent problems. A. Converting Fractions into Percents

• a percent is a fraction with a denominator of 100

i.e. 4747%100

=

• if we can change the denominator of our fraction into an equivalent fraction with a denominator of 100, we can quickly find the percent a fraction represents

i.e.

10

10

9 90 90%10 100

×

×

= =

64 7 48

14 2 43

e.g. Write each of the following fractions as a percent.

a) 3 %2 100= = d)

7 %20 100

= =

b) 2 %5 100= = e)

16 %25 100

= =

c) 3 %10 100

= = f) 43 %50 100

= =

B. Converting Decimals into Percents

• a decimal can be converted into a percent by changing the decimal into a fraction with 100 in the denominator

i.e.

10 10

10 10

72 5 50 813 81.30.72 72% OR 0.5 50% OR 0.813 81.3%100 10 100 1000 100

× ÷

× ÷

= = = = = = = =

64 7 48 6 4 7 4 8

14 2 43 1 4 2 4 3

e.g. Write each of the following decimals as a percent.

a) 0.46 b) 0.8 c) 0.216 d) 1.45

C. Converting Percents into Fractions

• we can change a percent into a fraction in lowest terms by first changing the percent into a fraction with a denominator of 100

i.e.

4

4

52 1352%100 25

÷

÷

= =

64 7 48

14 2 43

e.g. Write the following percent as fractions.

a) 19% b) 38% c) 76% d) 85%

D. Converting Percents in Decimals • we can change a percent into a decimal by moving the decimal two places to the left

i.e. 18% 0.18 OR 75.3% 0.753= = e.g. Write the following percent as decimals.

a) 29% b) 0.6% c) 55.2% d) 23.98% E. Percent Problems

• we will use proportions to solve problems involving percentd • there are two main types of percent problems

1. Find what percent one number is of another number e.g. What percent is 28 of 40? N.B. a) “Percent” is always a fraction with a denominator (bottom) of 100 b) The number following the word “of” always goes in the denominator of the second fraction 2. Finding a number that is a certain percent of another number e.g. What number is 80% of 45? e.g. 88 is 20% of what number?

Apprenticeship & Workplace 11 0.5 Metric System (Introduction) Lesson Focus: To introduce the basic structure of the metric system. • the system of measurement used extensively in most countries of the world is a version of the Systeme

Internationale (SI), more commonly known as the metric system • this system is based on the metre, which is defined as the distance light travels in 1/299 792 458 of a second • smaller and larger lengths are based on multiples or divisions of the metre • below is a list of prefixes used in the metric system and their meaning

Name Symbol Meaning Numerical value kilo k one thousand 1000

hecto h one hundred 100 deca da ten 10 deci d one-tenth 1/10 or 0.1 centi c one-hundredth 1/100 or 0.01 milli m one-thousandth 1/1000 or 0.001

• the symbols are derived from the first letter of the prefix • Why is deca abbreviated da not just d?

_______________________________________________________________________________________ • the basic units of measurement are:

Quantity Unit Name Symbol length metre m mass gram g

volume litre ℓ • we can indicate different sizes of the above quantities by using different prefixes • we can think of the prefixes as standing for different numbers • the metric system uses a combination of prefixes and unit symbols e.g. Complete the table below.

Abbreviation Name Amount

dm hℓ cg km dag mℓ

Apprenticeship & Workplace 11 0.6 Metric System (Conversions) Lesson Focus: To be able to convert between different metric measurement units. • the metric system is based on multiples of 10 • therefore, it is a decimal system similar to our money system • the following diagram shows the relationships between length units of the metric system

• when converting between metric units, move the decimal the number of places AND in the same direction as the chart above

e.g. Convert the following length measurements. 1. 246 m = ____________________ cm 6. 4803 cm = ____________________ m

2. 3.96 dm = ____________________ hm 7. 37 000 mm = ____________________ dam

3. 3.63 km = ____________________ m 8. 12.1 m = ____________________ dm

4. 2.5 m = ____________________ mm 9. 195 dam = ____________________ mm

5. 1 120 mm = ____________________ cm 10. 36.2 hm = ____________________ km

• you can use a similar method when converting between any two units • use the following chart to convert between different units

k

h

da

m g ℓ

d

c

m

e.g. Convert the following measures as indicated.

1. 2.57 m = ____________________ cm 6. 6027 cℓ = ____________________ ℓ

2. 450 dg = ____________________ hg 7. 2500 mg = ____________________ dag

3. 0.517 kℓ = ____________________ ℓ 8. 6.3 m = ____________________ dm

4. 0.5 g = ____________________ mg 9. 2460 daℓ = ____________________ mℓ

5. 315 mm = ____________________ cm 10. 1.6 hg = ____________________ kg

mile yard feet inches

1760×

12× 3×

12÷ 3÷ 1760÷

Apprenticeship & Workplace 11 0.7 Imperial System Lesson focus: To be able to convert between different imperial units. • the imperial system is a measurement system that is still in use in industry and the trades • as compared to the metric system, there is no consistent multiple between the units • the relationships between common imperial units are given below

a) 1 foot (ft) = ______ inches (in)

b) 1 yard (yd) = ______ feet (ft) = ______ inches (in) c) 1 mile (mi) = ________ yards (yd) = ________ feet (ft) • the above relationships can be summarized in the diagram below • when converting between imperial units, multiply or divide by the conversion factor • complete the following table to indicate the method of conversion between imperial units

Starting unit Finishing unit Conversion factor mi yd mi ft mi in yd mi yd ft yd in ft mi ft yd ft in in mi in yd in ft

• we should ____________________ when converting from a large unit to a smaller unit • we should ____________________ when converting from a small unit to a larger unit

e.g. Indicate the conversions factors used in each question. Leave all answers in to the nearest tenth (when

necessary).

1) 36 ft = ___________ yd 4) 42 in = _____________ ft

2) 2 mi = _____________ ft 5) 1200 ft = _____________ mi

3) 5 yd = _____________ in 6) 312 000 in = _____________ mi

mile yard feet inches

1760×

12× 3×

12÷ 3÷ 1760÷

Apprenticeship & Workplace 11 0.8 Metric & Imperial Conversions Lesson Focus: To be able to convert measurements from metric to imperial units or from imperial to metric

units. • since both systems are currently being used, it is very important to be able to convert back and forth

between metric and imperial units • there are four main length conversions you must know:

1. 1.609

1.609mile kilometre

×

÷⎯⎯⎯→←⎯⎯⎯ 3.

× 0.3048

÷ 0.3048foot metre⎯⎯⎯→←⎯⎯⎯

2. 0.9144

0.9144yard metre×

÷⎯⎯⎯→←⎯⎯⎯ 4.

× 2.54

÷ 2.54inch centimetre⎯⎯⎯→←⎯⎯⎯

• the technique to use is:

a) convert between the metric and imperial unit b) use either the metric or imperial conversion charts to get into the units that you want

e.g. Convert the following units. Leave all answers in TWO decimals when necessary.

1. 120 cm = ____________________ in 5. 15 ft = ____________________ dam 2. 30 ft = ____________________ m 6. 28 hm = ____________________ yd 3. 2.7 mi = ____________________ km 7. 47 in = ____________________ mm 4. 82 m = ____________________ yd 8. 9 300 000 mm = ____________________ mi