MATHEMATICS MAT1L. Grade 9, Essentials

Transcription

1 MATHEMATICS MAT1L Grade 9, Essentials

2 Volume Lesson 16

3 Mathematics MAT1L Unit 4 Lesson 16 Lesson Sixteen Concepts Explore and describe situations from everyday life and the workplace that require calculating or measurement of volume. Investigate the volume of a variety of prism whose bases involve rectangular regions. Predict and explain, from investigations involving the building of prisms, that the volume of a prism is given by multiplying the area of its base by its height. Estimate and calculate the volumes of rectangular prisms drawn from applications in everyday life and the workplace. Select the most appropriate standard unit to measure volume of a figure. Solve problems involving volume in applications drawn from everyday situations. Organize measurement information, using a simple framework Explain their reasoning used in problem solving and in judging reasonableness. Communicate, in writing, the solutions to measurement problems and the results of investigations, using appropriate terminology, symbols and form. Volume Volume is the amount of space an object occupies. Number of cubes in one layer Number of layers Volume = 12 cm 2 x 5 cm = 60 cm Copyright 2005, Durham Continuing Education Page of 47

4 Mathematics MAT1L Unit 4 Lesson 16 Example Find the volume of the rectangular prism given below. a. Answer: Volume = Area of base x Height = 8 m x 7 m x 5 m 2 = 56 m x 5 m Find the area of the base and multiply that answer by the height of the prism. Example Volume always has its units to = 280 m the exponent of. Estimate then find the volume of the rectangular prism given below. b. Estimated Answer: Volume = Area of base x Height = 2 m x 2 m x 1.5 m = 4 m 2 x 1.5 m = 6 m Answer: Volume = Area of base x Height = 2. m x 2.1 m x 1.6 m 2 = 4.8 m x 1.6 m = m Copyright 2005, Durham Continuing Education Page 4 of 47

5 Mathematics MAT1L Unit 4 Lesson 16 Example Find the volume of the object given below. c. Answer: V1 = Area of base x Height V2 = Area of base x Height = 7 cm x 6 cm x 5 cm = 2.2 cm x 2.1 cm x 2 cm 2 2 = 42 cm x 5 cm = 4.62 cm x 2 cm = 210 cm = 9.24 cm Total Volume = V1 + V2 = = cm Copyright 2005, Durham Continuing Education Page 5 of 47

6 Mathematics MAT1L Unit 4 Lesson 16 Support Questions 1. Estimate then calculate the volume for each of the following objects. a. b. 1. Estimate then calculate the volume of the object below. 2. Find the volume of the rectangular prism for each of the following: a. length 5 cm, width 7 cm, height 4 cm b. dimensions 25 m by 10 m by 50 m 2 c. area of base 70 mm, height 10 mm Copyright 2005, Durham Continuing Education Page 6 of 47

7 Mathematics MAT1L Unit 4 Lesson 16 Support Questions (con t). Use the diagram provided to answer the following questions. a) Find the amount of space occupied by the freezer. b) Find the volume of the inside of the freezer if each side is 6 cm thick. 5. Copy and complete the chart for rectangular prisms. Length Width Height Volume 10 cm 8 cm 720 cm m 5 m 120 m 14 mm 7 mm 980 mm Key Question #16 1. Estimate then calculate the volume for each of the following objects. a) b) Copyright 2005, Durham Continuing Education Page 7 of 47

8 Mathematics MAT1L Unit 4 Lesson Estimate then calculate the volume of the object below.. Find the volume of the rectangular prism for each of the following: a. length cm, width 10 cm, height 5 cm b. dimensions 15 m by 7 m by m 2 c. area of base 48 mm, height 9 mm 4. Use the diagram provided to answer the following questions. a. Find the amount of space occupied by the filing box. b. Find the volume of the inside of the box if each side is 1 cm thick. 5. Copy and complete the chart for rectangular prisms. Length Width Height Volume 5 cm 15 cm 75 cm 9 m 4 m 162 m 19 mm 2.5 mm mm 6. What units would you use to find the volume of an above ground swimming pool? Explain your reasoning. 7. Describe two situations from everyday life and/or the workplace that requires calculating and measuring of volume. Copyright 2005, Durham Continuing Education Page 8 of 47

9 Fractions Lesson 17

10 Mathematics MAT1L Unit 4 Lesson 17 Lesson Seventeen Concepts Represent the magnitudes of the fractions 1, 1, 1, 2, and using manipulatives 24 4 and by constructing diagrams and models. Represent the addition and subtraction of 1, 1,, and 1, in the context of fractional parts of an hour, a cup, a dollar, and an inch by constructing diagrams and using models. Estimate and add pairs of simple fractions with the support of an appropriate model. Interpret simple fractions of a dollar in decimal form. Read, interpret and explain, in writing, data displayed in simple tables and graphs. Fractions A Fraction is a number that represents part of a whole. Following are some examples of diagrams that model types of fractions parts are shaded out of a whole of 4 parts. Copyright 2005, Durham Continuing Education Page 10 of 47

11 Mathematics MAT1L Unit 4 Lesson 17 Support Questions 1. Copy the diagram following for each question and shade in the region which models the given fraction. a) 1 4 b) 1 c) 1 2 d) 2 e) 4 Adding and Subtracting Fractions Example Use the diagram following to find the sum of the fractions. Show your answer as a diagram. Copyright 2005, Durham Continuing Education Page 11 of 47

12 Mathematics MAT1L Unit 4 Lesson 17 Example Use the diagram below to find the sum of the times. Show your answer as a diagram. Answer: Example Use the diagram below to find the difference of the glasses of juice. Show your answer as a diagram. Answer: Support Questions 2. Use the diagram following to find the sum or difference of the fractions. Show your answer as a diagram. a. b. Copyright 2005, Durham Continuing Education Page 12 of 47

13 Mathematics MAT1L Unit 4 Lesson 17 Support Questions (con t). Use the diagram below to find the sum or difference of the fractions. Show your answer as a diagram. a. b. Example Estimate then add using a ruler with inches Estimated Answer: = 2 4 Answer: 2 4 is the answer Copyright 2005, Durham Continuing Education Page 1 of 47

14 Mathematics MAT1L Unit 4 Lesson 17 Support Questions 4. Estimate, then add or subtract using a ruler with inches. a b c Fractions of a Dollar 1 cent = cents = 5 = cents = 10 = 1 25 cents = 25 = Support Questions 5. Write each of the diagrams below as a fraction of a dollar. a. b. Copyright 2005, Durham Continuing Education Page 14 of 47

15 Mathematics MAT1L Unit 4 Lesson 17 Support Questions (con t) 5. (con t) Write each of the diagrams below as a fraction of a dollar. c. d. e. Key Question #17 1. Copy the diagram below for each question and shade in the region that models the given fraction. a) 1 4 b. 1 c) 1 2 d) 2 e) 4 Copyright 2005, Durham Continuing Education Page 15 of 47

16 Mathematics MAT1L Unit 4 Lesson 17 Key Question #17 (con t) 2. Use the diagram below to find the sum or difference of the fractions. Show your answer as a diagram. a. b.. Use the diagram below to find the sum or difference of the fractions. Show your answer as a diagram. a. b. 4. Estimate, then add or subtract using a ruler with inches. a) b) c) d) 5 4 e) f) Write each of the diagrams below as a fraction of a dollar. a. b. Copyright 2005, Durham Continuing Education Page 16 of 47

17 Mathematics MAT1L Unit 4 Lesson 17 Key Question #17 (con t) 6. Explain the steps taken to use estimation when answering the following faction problem Copyright 2005, Durham Continuing Education Page 17 of 47

18 Fractions to Decimals and Percent Lesson 18

19 Mathematics MAT1L Unit 4 Lesson 18 Lesson Eighteen Concepts Explore the relationship between fractions 1, 1, 1, 2, and and decimals using a 24 4 calculator, concrete materials and diagrams. Round decimal values appropriately within a given context. Represent and explain the meaning of percent as part of 100, by constructing diagrams, using concrete materials. Explore the relationship between fractions, decimals and percentages using a calculator, concrete materials and diagrams. Identify and use common equivalences or approximations between fractions and percentages. Solve problems involving fractions and percentages in practical situations. Read, interpret and explain, in writing, data displayed in simple tables and graphs. Explain their reasoning used in problem solving and in judging reasonableness. Communicate, in writing, the solutions to proportional reasoning problems using appropriate terminology, symbols and form. Percent Percent means per one hundred or hundredths. 5% means 5 per one hundred and is shown visually below: 4% means 4 per one hundred and is also shown visually below: Copyright 2005, Durham Continuing Education Page 19 of 47

20 Mathematics MAT1L Unit 4 Lesson 18 Example What percent do each of the shaded sections of the diagrams following represent? a. b. Answer: a. Since each column and row represents 10 units then there are 80 units shaded. 80% of the diagram is shaded. b. Since each column and row represents 10 units then there are 27 units shaded. 27% of the diagram is shaded. % is the symbol for percent. Support Questions 1. What percent do each of the shaded sections of the diagrams following represent? a. b. c. Copyright 2005, Durham Continuing Education Page 20 of 47

21 Mathematics MAT1L Unit 4 Lesson 18 Percent as a Decimal Example a. Write 5% as a decimal. Answer: = 0.5 To convert into decimal form, rewrite the percent without the % sign then divide that number by 100. b. Write 47% as a decimal. Answer: = 0.47 Support Questions 2. Write each percent as a decimal. a. 5% b. 8% c. 7% d. 15% e. 87% Writing Fractions as Decimals To convert a fraction into a decimal you divide the numerator (top) of the fraction by the denominator (bottom) of the fraction. b. Write 2 5 as a decimal. Answer: 2 5 =0.4 Copyright 2005, Durham Continuing Education Page 21 of 47

22 Mathematics MAT1L Unit 4 Lesson 18 Support Questions. Write each fraction in decimal form. a. 1 4 b c. 10 d e. 1 2 Example d. Write 5 9 as a decimal. Round to the hundredths place value. Answer: 5 9 = = 0.56 Support Questions 4. Write each fraction in decimal form rounded to the hundredths place value. a. 9 b c. 7 8 d e. 2 Copyright 2005, Durham Continuing Education Page 22 of 47

23 Mathematics MAT1L Unit 4 Lesson 18 Writing Fractions as Percent First convert the fraction into a decimal. Then multiply that decimal by 100. Example b. Write 2 as a percent. Round to the nearest percent. 6 Answer: 2 6 = = = % Support Questions 5. Write each fraction below as a percent. Rou nd to the nearest percent. a. 9 b. 7 7 c d e. 2 Copyright 2005, Durham Continuing Education Page 2 of 47

24 Mathematics MAT1L Unit 4 Lesson 18 Percent of a Number Example b. What is 17% of 10? Answer: = = 22.1 Therefore 17% of 10 is 22.1 Support Questions 6. Answer the following percent of a number, questions. a. What is 24% of 165? b. What is 15% of 175? c. What is 67% of 1000? d. What is 0% of 75? e. What is 50% of 40? f. What is 75% of 100? Copyright 2005, Durham Continuing Education Page 24 of 47

25 Mathematics MAT1L Unit 4 Lesson 18 b. What is the discount amount on an item this 40% off and has a regular price of $125.00? Answer: Discount amount = discount % x cost of item = 40% of $ = 0.40 x The discounted amount is $ Therefore the sale price is $ ( = 75) = $50.00 Copyright 2005, Durham Continuing Education Page 25 of 47

26 Mathematics MAT1L Unit 4 Lesson 18 Support Questions 7. Calculate the tax (1%) on the following items. a. Pair of jeans $49.99 b. Sunglasses $ c. Dish soap $2.29 d. Dog food $ Which sale is the better deal? 1 off or 2% off? Prove with calculations. 9. Calculate the discount amount of each of the following items on sale. a. 45% off a set of dishes that sell at a regular price of $ b. 15% off a pair of shoes that sell at a regular price of $ c. 1 4 off an mp player that sells at a regular price of $ d. 1 off a new NHL replica jersey that regularly sells for $ Key Question #18 1. What percent does each of the shaded sections of the diagrams following represent? a. b. c. Copyright 2005, Durham Continuing Education Page 26 of 47

27 Mathematics MAT1L Unit 4 Lesson 18 Key Question #18 (con t) 2. Write each percent as a decimal. a. % b. 9% c. 1% d. 22% e. 91%. Write each fraction in decimal form. a. 1 5 b c d e Write each fraction in decimal form rounded to the hundredths place value. a. 7 9 b c. 5 1 d e Write each fraction below as a percent. Round to the nearest percent. a. 7 9 b c. 5 1 d e Answer the following percent of a number, questions. a. What is 18% of 205? b. What is 12% of 165? c. What is 6% of 1500? d. What is 2% of 70? e. What is 25% of 60? f. What is 80% of 00? Copyright 2005, Durham Continuing Education Page 27 of 47

28 Mathematics MAT1L Unit 4 Lesson 18 Key Questions #18 (con t) 7. Calculate the tax (1%) on the following items. a. Pair of shoes $19.97 b. DVD player $49.29 c. Bird cage $24.79 d. Shampoo $ Which sale is the better deal? 1 5 off or 18% off? Prove with calculations. 9. Calculate the discount amount of each of the following items on sale. a. 15% off a shirt that sell at a regular price of $ b. 5% off a pair of shoes that sell at a regular price of $ c. 1 off a DVD player that sells at a regular price of $ d. 2 off Christmas dishes that regularly sell for $ What is the price paid after discount on each of the items in question Describe in your own words the process used for calculating an items price after a given discount. Copyright 2005, Durham Continuing Education Page 28 of 47

29 Ratios Lesson 19

30 Mathematics MAT1L Unit 4 Lesson 19 Lesson Nineteen Concepts Identify and use ratios, including equivalent ratios, to express the relationship among quantities represented by models and diagrams. Explore and describe the use of ratios from their personal experiences Solve simple problems using equivalent ratios. Explain their reasoning used in problem solving and in judging reasonableness. Communicate, in writing, the solutions to proportional reasoning problems using appropriate terminology, symbols and form. Ratio A Ratio is a comparison of two like quantities. To make orange juice from concentrate, combine 4 cans of water with every 1 can of concentrate. The ratio of the number of cans of water to the number of cans of concentrate is 4 to 1, 8 to 2, 12 to and so on. 4 to 1, 8 to 2 and 12 to represent the same comparison. These ratios are called equivalent ratios. When ratios are equivalent, they can be written as a proportion. 4:1 = 8:2 or 4 = Equivalent ratios are made by multiplying or dividing the values in the ratio by the same non-zero number. When one ratio can be made by dividing or multiplying the other ratio s numerator and denominator by the same number then the two ratios are equivalent. Copyright 2005, Durham Continuing Education Page 0 of 47

31 Mathematics MAT1L Unit 4 Lesson 19 Example a. Express the following diagram as a ratio of O s to X s. Answer: 4 : or 4 b. Express as a ratio. $0 to 45 kg. Answer: 0 : 45 Support Questions 1. Express the following diagram as a ratio of cars to trucks. 2. Express each as a ratio. a. 12 s to 6 min b. 1 cm to 2 km c. 6 trees to 8 shrubs d. apples to 7 oranges e. 10 men to 14 women f. 14 dogs to 5 cats Copyright 2005, Durham Continuing Education Page 1 of 47

32 Mathematics MAT1L Unit 4 Lesson 19 Support Questions (con t). What is the missing value in each of the pairs of equivalent ratios? 4. To make lemonade from concentrate, combine cans of water with every 1 can of concentrate. Copy and complete the table to show equivalent ratios. Concentrate (cans) Water (cans) Key Question #19 1. Express the following diagram as a ratio of cars to trucks. Copyright 2005, Durham Continuing Education Page 2 of 47

33 Mathematics MAT1L Unit 4 Lesson 19 Key Questions #19 (con t) 2. Express each as a ratio. a. 10 s to 4 min b. cm to 8 km c. 5 trees to 9 shrubs d. 12 apples to 9 oranges e. 4 men to 1 women f. 22 dogs to 4 cats. What is the missing value in each of the pairs of equivalent ratios? 4. To make a light purple colour, 4 ml of red paint is mixed with 8 ml of blue paint and 12 ml of white paint. Copy and complete the table to show equivalent ratios. Red paint (ml) Blue paint (ml) White paint (ml) It takes hours to fill a swimming pool and only 0 minutes to empty. a. Express this comparison as a ratio. b. How long would it take to fill the pool half way? c. How long would it take to empty a pool half full? 6. Describe one situation in your life in which you or someone close to you uses ratios. Explain. Copyright 2005, Durham Continuing Education Page of 47

34 Rates Lesson 20

35 Mathematics MAT1L Unit 4 Lesson 20 Lesson Twenty Concepts Explore and identify rates drawn from their experiences, and the units used in them. Calculate rates in activities drawn from their experiences. Solve problems involving rates. Calculate and compare the unit costs of items found in everyday situations. Explain their reasoning used in problem solving and in judging reasonableness. Communicate, in writing, the solutions to proportional reasoning problems using appropriate terminology, symbols and form. Rate Rate is the comparison of quantities with different units. Unit Rate is the rate that has one as its second term. Copyright 2005, Durham Continuing Education Page 5 of 47

36 Mathematics MAT1L Unit 4 Lesson 20 Calculating the Unit Rate Copyright 2005, Durham Continuing Education Page 6 of 47

37 Mathematics MAT1L Unit 4 Lesson 20 Support Questions 1. Express each as a unit rate. a. 00 words typed in 5 min b. 210 heartbeats in min c. 420 km driven in 7 hours d. $150 in 7.5 hours worked 2. The human heart beats approximately 70 times/min. How many beats are there for each of the time periods? a. 6 min b. 55 min c. 1.5 min. Brianna typed 60 words in 9 min. At this rate, how many words will she type in 0 min? 4. Noah is paid an hourly rate. When he works a 40 h week, he receives $ How much does he receive if he works a 5 h week? Copyright 2005, Durham Continuing Education Page 7 of 47

38 Mathematics MAT1L Unit 4 Lesson 20 Calculating the Unit Price Unit Price is the cost of one unit or item =.4158 Rounded to the nearest cent becomes $0.42 Support Questions 5. Find the unit price to the nearest cent. a. 6 kg of apples for $.99 b. 8 bars of soap for $2.29 c. 12 bagels for $.19 d. 6 cans of apple juice for $ Which is the better deal? a. 25 cookies for $.25 or 80 cookies for $10.00 b. 1 pop for $1.50 or 6 pops for $5.00 Copyright 2005, Durham Continuing Education Page 8 of 47

39 Mathematics MAT1L Unit 4 Lesson 20 Key Question #20 1. Express each as a unit rate. a. 250 words typed in 10 min b. 180 heartbeats in 2.5 min c. 60 km driven in 9 hours d. $200 in 8 hours worked 2. The human heart beats approximately 70 times/min. How many beats are there for each of the time periods? a. 7 min b. 48 min c. 0.5 min. Brianna typed 480 words in 6 min. At this rate, how many words will she type in 45 min? 4. Noah is paid an hourly rate. When he works a 40 h week, he receives $ How much does he receive if he works a 5 h week? 5. The table shows the cruising speeds of several planes. Plane Type Cruising speed (km/h) Boeing Boeing Boeing DC a. If each plane keeps its speed, how far will each travel in.5 h? b. Calgary is 2876 km away. What distance remains after the Boeing 767 is flying for 1.5 h? 6. 1 tickets on a ride at the CNE cost $6.50. How many tickets did people buy if $ was collected for tickets in one day? 7. Find the unit price to the nearest cent. a. 8 kg of potatoes for $4.99 b. 6 cans of soup for $4.9 c. pizza s $.19 Copyright 2005, Durham Continuing Education Page 9 of 47

40 Mathematics MAT1L Unit 4 Lesson 20 Key Question #20 (con t) 8. Which is the better deal? a. 50 cookies for $6.25 or 80 cookies for $11.50 b. pop for $2.10 or 8 pops for $ A can of tennis balls sells for $5.99. A bag of 10 tennis balls sell for $8.97. Which is the better buy? 10. Kristen needs 4 AAA batteries for a portable DVD player. A package of 2 batteries cost $4.18. Packages of 6 are on sale for $7.94. a. Find the unit price for each package. b. Why do you think the more expensive batteries might be more economical? 11. List examples of the use of rates in you everyday life. Copyright 2005, Durham Continuing Education Page 40 of 47

41 Mathematics MAT1L Unit 4 Support Question Answers Answers to Support Questions Lesson Sixteen 1. a. estimated volume = Area of base Height = L x W x H = 10 x 10 x 5 = 500 in volume = Area of base Height = L x W x H = 12 x 8 x 6 = 576 in b. estimated volume = Area of base Height = L x W x H = 0 x 10 x 5 = 1500 cm volume = Area of base Height = L x W x H = 0 x 10 x 6 = 1800 cm 2. estimated V = L W x H 1 = 10 2 x 2 = 40 yds estimated V = L W x H 2 = x 10 = 1000 yds Estimated Total Volume = V + V 1 2 = = 1040 yds Copyright 2005, Durham Continuing Education Page 41 of 47

42 Mathematics MAT1L Unit 4 Support Question Answers V = L W x H 1 = 8 2 x 2 = 2 yds V = L W x H 2 = 12 8 x 9 = 864 yds Total Volume = V + V 1 2 = = 896 yds. a. Volume = Area of base Height = L x W x H = = 420 cm b. Volume = Area of base Height = L x W x H = c. = m Volume = Area of base Height = = 700 mm 5. a. Volume = Area of base Height = L x W x H = = cm b. Length = Width = Height = = 48 cm 0 12 = 18 cm 5 6 = 47 cm Copyright 2005, Durham Continuing Education Page 42 of 47

43 Mathematics MAT1L Unit 4 Support Question Answers Volume = Area of base Height = L x W x H = = cm 5. Length Width Height Volume = 9 cm = 8 m = 10 mm Lesson Seventeen 1. a. b. c. d. e. 2. a. b.. a. b. 4. a. 2 = 1; Copyright 2005, Durham Continuing Education Page 4 of 47

44 Mathematics MAT1L Unit 4 Support Question Answers b = 2; c. 4 = 1; 5. a = = b = c. = = d = e. = = Lesson Eighteen 1. a. 75% b. 58% c. 12% 2. a = 0.05 b = 0.08 c = 0.07 d = 0.15 e = a. 1 4 = 0.25 b = 0.46 c. 10 = 0. d = 0.05 e. 1 2 = 0.5 Copyright 2005, Durham Continuing Education Page 44 of 47

45 Mathematics MAT1L Unit 4 Support Question Answers 4. a. 9 = 0. b = 0.64 c. 7 8 = 0.88 d = 0.05 e. 2 = a. 9 = 0. b = 0.64 c. 7 8 = 0.88 d = x 100 = % 0.64 x 100 = 64% 0.88 x 100 = 88% 0.05 x 100 = 5% e. 2 = x 100 = 67% 6. a x 165 = 9.6 b x 175 = c x 1000= 670 d. 0. x 75 = 22.5 e. 0.5 x 40 = 20 f x 100 = a. 0.1 x = $6.50 b. 0.1 x = $18.84 c. 0.1 x 2.29 = $0.0 d. 0.1 x = $ x 100 = % which is greater than 2% 9. a x = $58.49 b x = $10.42 c. 1 4 = x = $24.50 d x = $72.27 Lesson Nineteen 1. : 5 or a. 12 : 6 or 6 b. 1 : 2 or 2 1 c. 6 : 8 or 8 6 d. : 7 or 7 10 e. 10 : 14 or f. 14 : 5 or 5. a. 2 x 5 = 10 b. 8 2 = 4 c. 15 = 5 4. Concentrate (cans) = 5 24 = Water (cans) x 2 = 6 x 4 = x 0.5 = 1.5 Copyright 2005, Durham Continuing Education Page 45 of 47

46 Mathematics MAT1L Unit 4 Support Question Answers Lesson Twenty 1. a. b. c. d. 2. a. 70 x 6 = 420 beats b. 70 x 55 = 850 beats c. 70 x 1.5 = 105 beats. 40 x 0 = 1200 words typed in 0 minutes Copyright 2005, Durham Continuing Education Page 46 of 47

47 Mathematics MAT1L Unit 4 Support Question Answers 4. 5 x 1 = $ earned in a 5 hour work week 5. a = $0.67/kg b = $0.29/bar c = $0.27/bagel d = $1.42/can 6. a = $0.1/cookie and = $0.1/cookie; Although they round to the same deal, the second deal is the better because it is only 0.125/cookie vs. $0.1/cookie. b = $1.50/pop and = $0.8/pop; The second deal is better. Copyright 2005, Durham Continuing Education Page 47 of 47