Year 8. Semester 1 Revisions

Transcription

1 Year 8 Semester One Revisions 0 Students are advised to review previously completed tests, assignments and homework followed by this semester one revision booklet. Be sure to seek extra help early to clarify any difficulties with content before the exam. Page of

2 Unit : Integers Adding & Subtracting Integers o The zero principle states that when you add opposite integers, the result is 0. o The positive sign is often not used for positive integers but you must show the negative sign to indicate a negative integer. o When an integer is positive you move to the right along a number line and when an integer is negative you move to the left along the number line. o To get rid of double signs between two numbers, same signs become a positive and unlike signs become a negative. ie: - (+) = - = - and - (- ) = - + = Multiplying & Dividing Integers o Regardless of whether the operation is multiplying or dividing, if the two numbers have the same signs the answer will be positive. ie: (+)(+) = + and (- )(- ) = + o Regardless of whether the operation is multiplying or dividing, if the two numbers have the opposite signs the answer will be negative. ie: (- )(+) = - and (+)(- ) = - Order of Operations o The order of operations must be followed every time we do a maths calculation. o The acronym used to help us remember the order of operations is BIDMAS. o Brackets, Indices, Division, Multiplication, Addition and Subtraction. Practice. Represent each operation on a number line to demonstrate your solution (- ) + (- ) (- ) Determine each sum. - + (- ) + (- ) (- ) + (- 6). Determine each difference. (- ) - (- ) (- ) - (- ) 6 (- 6) -. Calculate (- ) (- ) - (- ) (- ) +. Calculate each product. (- )() (- )(- 9) ()(- ) (- )(- ) ()() (- )() 6. Calculate each quotient. - 8 (- 6) - 6 (- ) - (- ) 60 (- ) - 0 (- ). Evaluate. ()(8) (-)(-) (-)()(-) 6 [(-)(-)(-)] 8. Evaluate each of the following. + x (- ) (- ) - x - + ( 8) + (- 6) (- ) - x - 8 [ + (- ) ] (- + ) (- ) 6 h)- ( - ) 0 i) x -9 - Page of

3 Unit : Fractions & Decimals Adding & Subtracting Fractions o In order to add or subtract fractions, they must have a common denominator. o All mixed numbers must be changed to improper fractions before any operations can be calculated. o If they have a common denominator, you add or subtract the numerators and keep the denominator. o If they do not have a common denominator, you must find the lowest common denominator, don t forget to adjust the numerators to keep the fractions equivalent. Multiplying & Dividing Fractions o All mixed numbers must be written as improper fractions before beginning. o When multiplying fractions, you should simplify (reduc first. Don t forget you can simplify vertically in both fractions or diagonally from one fraction to another. You are not allowed to simplify horizontally. o Once fractions are simplified, multiply numerator x numerator then denominator x denominator. o Always check to see if your fractions are fully simplified. Reduced to lowest terms. It is not necessary to put your final answers back into a mixed number. o When dividing fractions, take the reciprocal (ie: flip) the second fraction and change the operation to multiplication. Now follow the rules from above for multiplying fractions. Terminating & Recurring Decimals o When changing fractions to decimals, divide the numerator of the fraction by the denominator and round the answer to decimal places if it is not otherwise specified. o Terminating decimals have a fixed number of decimal places. They stop. o Recurring decimals that have one recurring digit are written with a dot above the recurring digit. o Recurring decimals with more than one recurring digit are written with a line above the recurring digits. Addition & Subtraction of Decimals o When adding and subtracting decimals, be sure that the decimal points are lined up in a column one underneath the other. Multiplication & Division of Decimals o When multiplying decimals, count the number of digits after the decimal point in each of the numbers being multiplied and add these together to find the total number of decimal points in the answer. o When dividing decimals, make sure the divisor is a whole number. o When the divisor is a decimal instead of a whole number, multiply both the divisor and the dividend by an appropriate power of ten. Estimation o Estimation is a method of checking the reasonableness of an answer or a calculator computation. o Clustering around a common value can be employed when a basic calculation involving similar values is required. o Rounding involves increasing the value of the desired digit if the following digit is or greater. If the following digit is less than, the value of the desired digit remains the same. o When rounding to the first digit, apply the process of round to the first digit of the number. o Making the dividend a multiple of the divisor is another useful technique for estimating an answer involving division. Page of

4 Practice 9. Add h) + i) j) +8 9 k) + l) + 0. Subtract h) 6 i) 6 8 j) 8 9 k) l). Multiply h) i) j) k) l). Divide h) 8 Page of i) j) k) l). Convert the following fractions to decimals, giving exact answers or using the correct notation for recurring decimals where appropriate. 6 8

5 . Find the following h) i).6. j) k).99. l) m) n) ) 8.9 p) Determine each product. Round to the nearest hundredth if necessary h) Determine each quotient. Round to the nearest hundredth if necessary h) For each of the following numbers, round to the first digit Find an estimate for each of the following Provide estimates for each of the following by first rounding the dividend to a multiple of the divisor Unit : Percentages Converting Percentages, Fractions & Decimals o What does percent mean. o Percentage, fractions and decimals are different ways of expressing the same quantity. o When comparing different forms of a number, first convert them all to the same form. Common percentages and shortcuts o Finding 0% of a number is the same as dividing by 0 or move the decimal point one place to the left. o Other common percentages such as %, 0%, % and % can be found by using some portion of 0% s. o Finding % of a number is the same as dividing by 00 or moving the decimal point two places to the left. If you have already found 0% you can determine 0% of that answer to find your %. This can be helpful in finding other percentages such as % and % for example. Discount is a reduction in price or a savings. o The new sale price of the item can be solved by calculating the amount of the discount then subtracting the discount from the marked price. o To calculate the percentage discount, make a fraction of the cash discount over the original price and multiply by 00. Page of

6 Profit & Loss o When the selling price of an item is more than the cost of that item you will have a profit. If you sell something for less than you paid for it then you have a loss. o To determine the selling price of an item if you know the percent profit, markup the cost price by (00% + percent profit). o To determine the selling price of an item if you know the percent loss, multiply the cost price by (00% - percent loss). o Profit or loss is usually reported as a percentage of the cost price. To find the percentage profit, write a fraction of the profit over the cost and multiply by 00. To find the percentage loss, write a fraction of the loss over the cost and multiply by 00. Practice: 0. Write the following percentages as fractions in simplest form. (Leave your final answer as an improper fraction where appropriat. % 9% 8% % 9% 89% % h) 60% i) % j) 0% k) % l) 00%. Write the following percentages as fractions in simplest form. (Leave your final answer as an improper fraction where appropriat. % %.%.%. Write each of the following percentages as decimals. 6% % 8% 9% % % 00% h) 60% i).% j) 9.% k).% l) 0.0%. Change each of the following fractions to a percentage h) 8 0 i) j) k) 0 l). Put the following in ascending order. %,, 000, 8 00, %,, 9 00 Page 6 of

7 . Change each of the following decimals to percentages h) 0.00 i) 0. j) 0.06 k) 8.6 l) Find the following. 0% of 80 % of 60 0% of 0 0% of % of 90 9% of 00 % of h) 0% of 0 i) % of 9 j) 9% of 00 k) 8% of 0 l) % of 6. A pair of Calvin Klein jeans that normally cost $89.0 have been reduced by %. By how much have the jeans been reduced (to the nearest cents)? How much will the jeans cost at the reduced price? 8. After buying a new tennis racquet you are offered a % discount on your next purchase. A tracksuit is $9. How much would the tracksuit cost you to purchase after the discount. 9. Theo works in a shop that sells electrical appliances. He earns % commission on every sale he makes. If he sells a stereo system for $0, how much commission will Theo earn? 0. Catherine bought an old chest of drawers from a garage sale for $0. She bought paint for $0 and her sister Deb helped to paint and repair it. When they sold it they made a profit of 60%. How much profit did the girls make? For how much did they sell the chest of drawers?. Sandra had $00. After an hour of playing the poker machines she had $90. How much money has Sandra lost overall? What is Sandra s percentage loss after one hour?. Rina had a mass of 90 Kg. She stopped eating junk food and walked every day. After one month her mass was 8 Kg. How much mass did Rina lose? By what percentage did Rina s mass decrease.. Jasmine bought a one- bedroom unit for $ After five years she sold the unit for $0000. By how much has the value of Jasmine s unit increased? By what percentage has the value of Jasmine s unit increased?. Ninety- nine of Canada s 00 plant species are under threat of extinction. What percentage is threatened?. Eighty percent of a class got question correct on their percentages test. How many got the question correct if there are 0 students in the class? 6. Forty percent of primary school children can sing the national anthem. In a group of 6 primary school children, how many would you expect to be able to sing the national anthem? Page of

8 Unit : Statistics Understanding the terminology of the unit. How can you collect data? How can data be organized? o Stem and Leaf Plots o Frequency Tables How can data be displayed? o Bar Graph, Histogram, Circle Graph, etc. How can data be analysed? o Calculating mean, median, mode and range. Practice. The graph below represents the favourite pets of a particular Year 8 class. a. How many students preferred a dog as a pet? b. How many students in the class had a favourite pet? c. Which was the least favoured pet? d. How many times more popular than horses are dogs? e. If there are 8 students in the class, how many do not have a favourite pet? Year 8 Favourite Pets Number Prefering (Frequency) 0 0 Dog Cat Bird Mouse Horse Pets 8. These results were obtained when a Year 8 class was surveyed on their favourite leisure activity. Leisure Activity Number Preferring (Frequency) Reading Television 9 Sport Hanging Out Other a. Select a suitable title and draw a column graph to display the data. Label the horizontal axis Leisure Activity and the vertical axis Number Preferring (Frequency). Scale the vertical axis from 0 to 0 so as to include the highest score. b. How many students chose sport as their favoured leisure activity? c. What is the difference between the most and least favoured activity? d. Which activity received votes? e. If everyone in the class completed the survey, how many students were in the class? Page 8 of

9 9. The following data are the results of testing the lives (in hours) of 00 torch batteries. 0,,, 9, 6, 6,,, 8,, 8, 9,,,,, 0,,, 6 9,, 6,, 0, 0,, 9, 8,,, 6, 9, 8,, 6,,,,,, 9, 0, 9, 8, 8, 9,, 9, 8,, 8,,,, 8, 8,, 0 8,,,,,, 8,,, 8, 0, 0,, 9,,,, 8,, 0, 9, 0,, 6, 6, 6,, 6, 8,,,,,,, 6,,8, Organize the data is a frequency distribution table using intervals of 0-, - 0, 0-, - 0, 0-, - 0. Draw a histogram of this data using your frequency distribution table. 0. The following data gives the number of fruit that have formed on each of 0 trees in an orchard: 9,,, 6,,, 8, 6,, 6,,,,, 9,, 6,,, 8 9, 8,, 6, 9,, 6,, 8, 8,,, 9, 68,,6,, 9, 8, 0 Organize the data in a stem- and- leaf plot.. Find the mean (averag of each set of the following scores. Give the answers correct to decimal places. a.,,,,, 9 b.,, 8, 0, 6, 9,,, 9 c.,,, 0,, 0, 6, 9,, 8 d., 8., 6.6,.0,., 8.0, 6.9. An Olympic figure skater was given these scores by the panel of judges:.8,.6,.,.,.8,.9,.,.0,.8 Find the average score to decimal place.. Calculate the mean of this frequency distribution to decimal place. Score Frequency 0. Find the median of the following scores:,,,, 8,,,,8,,. Find the median of the following scores:,, 6,,,, 8,.,.,.0,.8,.,.,.9,. 6. Find the mean, median and mode of the following sets of data:,, 8,,, 6,, 9,,, 6,,,, 6, 0,,, 6,,,,,,,,,,, 6, 0, 0, 8, 9, 9. Find the range of the following scores:,,,, 8,,,,8,,,, 6,,,, 8,.,.,.0,.8,.,.,.9,. Page 9 of

10 Unit : Ratio & Rates Ratios are used in many aspects of everyday life. They are used to compare quantities of the same kind. They do not have a unit of measurement. o Before ratios are written, the numbers must be expressed in the same unit of measurement. o The order of the numbers in a ratio is important. o Ratios can be written in different ways. o Ratios contain only whole numbers. To form a ratio using fractions, convert the fractions so that they have a common denominator and then write the ratio of the numerators. To form a ratio using decimals, change the decimals into whole numbers by multiplying by powers of 0 (ie: 0 for decimal, 00 for decimals, 000 for decimals, etc.) A proportion is a statement of equality between two ratios. o Cross- multiplication is a technique that can be used to solve for an unknown quantity in a proportion. To share a certain amount in a given ratio, find the total number of shares (parts). Then determine the size of one part. If you know how much one share is worth, you can determine how the given amount can be broken up into a given ratio. Rates are used to compare the changes in different quantities. o Rates are usually written using per or a slash (/). o Rates are considered to be in simplest form if they are expressed with a denominator of one. This is also referred to as a unit rate. o Rates must include the units of measurement. Practice: 8. On a farm there are dogs, cats, cows and horse. Write the following ratios. cats to dogs horses to cows cows to cats dogs to horses dogs to other animals cats to horse to dogs 9. Express each of the following ratios in simplest form. 8 : 6 : 6 mm compared with 0 cm $ compared with 60 cents 0 seconds compared with. minutes to :0 h) 6 : 80 i) hours compared with 0 minutes j). km compared with 00 meters 0. Find the value of n in each of the following proportions. n = 0 n 8 = = 8 n = n 6 n = 8 0 = n. Which is the larger ratio?,, 8 Page 0 of

11 . Divide $ in the ratio :.. Share $0 in the ratio :.. Three people share a Lotto prize of $6600 in the ratio ::6. What is the difference between the smallest and largest shares?. A car travels 80 km on litres of petrol. Find the fuel consumption of the car in L/00 km. 6. David s car has a fuel consumption rate of km/l, and Susan s car has a fuel consumption rate of km/l. How far can David s car travel on 6 L of fuel? How much fuel (to the nearest litr would Susan s car use in travelling 60 km?. A Kg packet of flour costs $.80 and a 0 g packet costs $.0. Which is the cheaper way of buying flour? 8. A jet travels a journey of 0 km in hours and 0 minutes. Calculate its speed in: km/h m/s Unit 6: Geometry Congruent Triangles o Two triangles can be proven to be congruent by SSS, SAS, ASA o If ΔABE ΔMRE then corresponding sides and angles are equal. Similar Triangles o Two triangles are proven to be similar by AAA. o If ΔACB ~ ΔZXY then corresponding angles are equal and corresponding sides are proportional. Angle Properties o The sum of the interior angles of a polygon with n sides is found using the formula (n- ) x 80 0 o The sum of the exterior angles of any polygon is o Supplementary angles add to Complementary angles add to o Vertically Opposite angles are equal. o The sum of the interior angles of a triangle are o An isosceles triangle has two equal sides and the corresponding angles across from each of these equal sides are also equal to each other. o An equilateral triangle has three equal sides and thus all of the corresponding angles are equal. Since there is 80 0 in any triangle and all three angles are equal, each angle in an equilateral triangle has a measure of o A transversal is a line that intersect two or more parallel lines creating alternate, corresponding and co- interior angles. o Alternate angles form a Z pattern. o Corresponding angles form an F pattern. o Co- interior angles add to o We can use these patterns to determine the measure of other angles if one measure is given. Page of

12 Practice: 9. Name each pair of congruent triangles giving a reason. Hence, find the length of the side marked with a variable. 60. The following triangles are similar. Find the missing length. Page of

13 6. Determine the measure of the unknown angles in the figure. 6. Given the measures of angle # and angle # 6 for the figure below, line t and line s are parallel, find all the missing angles. 6. Find all of the missing angles from the figures below; Page of

14 Answers: = ( 6) = h) - i) h) i) 9 j) 9 k) 9 0 l) h) 6 i) j) 6 k) 0 l) h) i) 8 j) k) 8 l) h) i) 6 j) 6 9 k) l) h).9 i). j). k).66 l).8 m).89 n).08 o) 6.0 p).966 Page of

15 h) h) = = = = = = = 0 = 6 =6 9 = 6 =8 6 6 = h) = i) 00 = 0 j) 0 00 = 0 k) 00 = l) = = = h).6 i) 0. j) 0.09 k) 0. l) % 8% 8% 6% 0% % 0% h) 80% i) % j) 80% k) 0% l) 66.66%. %, 000, %,,, 8 00, % % 6% % 8% 0% 0.% h) 0.% i) 0% j) 6.% k) 86% l) 0.0% h) i).9 j) 8 k) 9.6 l) 0.8. reduced by $., costs $ save $., costs $ $.0 0. profit is $8, sold for $8. lost $0, % loss. loss of Kg,.6% loss. increased by $ 000,.% increase. %. students 6. 0 students. 8 Horse 6 times Page of

16 Number Preferring 0 0 Yr 8 Favourite Leisure Activity Leisure Activity 8. Most = 9, Least =, Difference is Hanging Out 6 students 9. & Hours Tally Frequency Torch Light Batteries Life Span Frequency Life Span (in hours) 0. key: 9 = 9 fruit on a tree Stem Leaf Page 6 of

17 6. mean = 6, median =, mode = mean = 8, median = 9, mode =. Range = = 8 Range = 8 = Range = - = Range =. =. 8. : : : : : : : 9. : : : 0 : : 9 : : h) : 0 i) : j) : 0. n = n = 0 n = n = n = 8 n = 6. > 0 >. $0 and $. $0 and $00. $60, $00, $60, difference = $ L / 00 km 6. km.8 L. kg for $.80 is the same as $8.0 / kg, 0 g for $.0 is the same as $8.80 / kg, therefore the kg packet of flour for $.80 is the cheaper way of buying flour km/h 0 m/s 9. ΔBAC ΔQPR (ASA) x = cm ΔRPQ ΔSQP (ASA) x = cm ΔTSP RQP (SSS) x =0 mm y = mm z = mm ΔABC ΔADC (SAS) x = mm 60. KJ = UT = SB = 9 PW = 6. w = 0, x = 0, y = 9 0 x = 0, y = 0, z = 8 0 x = 0 w = 9 0, x = 0, y = 0, z = 0 6. Angles #,, 0, 6 = 9 0 and Angles #, 9 = Angles a, b, c and d all equal 0 a = 06 0, b = 0, c = 0 Page of