Measuring Length Goals You will be able to choose, use, and rename metric length measurements measure perimeters of polygons solve problems using diagrams and graphs Running in a Triathlon
Racing Snails CHAPTER 5 Getting Started START You will need a ruler or tape measure Angele timed a snail racing down the path between two rows of carrots in her garden.? What other race tracks can you make so the snail travels the same distance? A. How much time did it take the snail to go from one end of the row of carrots to the other? B. The snail moved about 80 cm each minute. How many centimetres long must the row be? C. How many metres long is the row? FINISH 140 NEL
D. Sketch a rectangular race track that covers the same distance as your answer to Part B or C. E. Which measurement description did you use to help you answer Part D: the one in metres or the one in centimetres? Why? F. Sketch two other race tracks that aren t rectangles. They should cover the same distance as your answer to Part B or C. Label the side length measurements. Do You Remember? 1. Copy and complete the relationships. a) 2 km m b) 1.4 m cm c) 1.4 m mm d) 15 cm mm 2. Calculate the perimeter of each shape. a) 5 m b) 6 km 2.5 m 2.5 m 5 m 2 km 2 km 6 km. Sketch two different rectangles with a length of 6.7 cm and a perimeter close to, but not exactly, 18 cm. Record the width of each. 4. How much time does each event take? a) swim class b) checking your e-mail start finish start finish 11 12 1 11 12 1 10 2 10 2 9 9 8 4 8 4 7 6 5 7 6 5 11 12 1 11 12 1 10 2 10 2 9 9 8 4 8 4 7 6 5 7 6 5 NEL 141
CHAPTER 5 1 Measuring Length Goal Select an appropriate measuring unit. James s family went to Lighthouse Park in West Vancouver. The park has many very old and tall fir, cedar, and arbutus trees. James tried to decide which unit he would use to measure the trip and everything he saw. He thinks he can use decametres to measure the length of a path around the base of this tree.? What units of length are appropriate for describing their trip? James s Comparison Dad stood next to a young tree that is just about as tall as he is. Dad s height is 195 cm. decametre (dam) A unit of measurement for length 1 dam 10 m A. What unit would you use to describe the height of the older trees in the park? Explain your choice. B. What unit would you use to describe the distance James s family travelled to get to Lighthouse Park? Why? C. James picked up a piece of bark from a fallen arbutus tree. The bark is known for being very thin. What unit would you use to describe its thickness? D. What unit would you use to describe the thickness of a cedar leaf? E. What other things in a West Coast rain forest might be measured in the same units as your answer to Part D? 142 NEL
Reflecting 1. Why did all of the units you chose include metre? 2. Suppose a leaf was 7.2 cm long. Why can you say that it had been measured to the nearest millimetre?. Why are different units appropriate for different measurements? Checking 4. Select an appropriate unit for each measurement. Explain each choice. a) the height of a room b) the thickness of a window c) the distance across your town or city Practising 5. A nursery is selling young trees. What units are appropriate for these measurements? Explain. a) the heights of the trees b) the lengths of the leaves of the trees 6. For which measurements are centimetres appropriate? Explain. a) the thickness of your math book b) the thickness of your fingernail c) the thickness of 100 sheets of paper 7. Why might you measure the thickness of a wire in millimetres rather than centimetres? 8. Ryan is in a video store. What items in the store, if any, might be measured in these units? a) millimetres c) metres b) centimetres d) kilometres 9. Is it likely that a tall building would be measured in kilometres? Explain. 10. A decimetre is 0.1 metre. A decametre is 10 m. What might be the name of the unit between decametres and kilometres? Explain. NEL 14
CHAPTER 5 2 Metric Relationships Goal Interpret and compare measurements with different units. A group in Mexico City broke the record for the world s largest sandwich in 2004.? How can you describe the measurements of the world s largest sandwich? Denise s Description Each piece of bread is square in shape. Each side of the square is longer than three metre sticks. A. Why might Denise have described the bread this way? B. Why can each side of the square be described as 48 cm long? C. If you measured the length of the side of the square in millimetres, why might you end up with a value between 475 mm and 485 mm? D. The filling of the sandwich is about 10 cm thick. Describe the thickness of each slice of bread in centimetres. E. If you measure the thickness of the filling in millimetres, what are some of the values you might expect to measure? F. Suppose you line up 1000 of these sandwiches and want to know the total distance. Why might you describe the sandwich as between 0.00 km and 0.004 km across? 144 NEL
Reflecting 1. Why might you describe.48 m as 48 cm, but not as 48.0 cm? 2. How do you rename each measurement? a) a metre description in centimetres b) a centimetre description in millimetres c) a kilometre description in metres Checking. Is a slice of bread from the record sandwich taller or shorter than the height of your classroom? Explain. 4. How many millimetres tall is the sandwich? Practising 5. Draw a picture that shows the shape of a regular slice of bread. Describe the measurements of the shape in each unit. a) centimetres b) millimetres c) metres 6. Rename each measurement using the new unit. Explain your thinking. a).45 m to centimetres c).045 m to millimetres b).4 m to centimetres d) 2.1 km to metres 7. Liam designed this triangular flower bed. a) Calculate the length of the third side. b) If you measured the length of the 220 cm side with a millimetre ruler, could you discover that it measures 2195 mm? Explain. 8. Olivia said her pencil is 14.0 cm long. How do you know she measured to the nearest millimetre?.4 m perimeter is 8.5 m 220 cm NEL 145
CHAPTER 5 Goal Perimeters of Polygons Measure perimeters of polygons and draw polygons with given perimeters. You will need a ruler Qi and Raven have each designed giant floor game boards for the class fun day. They ve decided to put a bright red border along the edges of the boards. Both boards require the same length of red border. Qi s Description My game board is a rectangle. It s 2 m wide and m long.? What could Raven s game board look like? A. What is the perimeter of Qi s board? B. Imagine that Raven s board is a square. Sketch this board and label the side lengths. C. Draw the new board so that 1 cm of your picture represents 1 m of edge length on the board. D. Suppose Raven s board is a parallelogram, but not a rectangle. Sketch a possible board and label the side lengths. Remember that the opposite sides of a parallelogram are the same length. E. Draw the board from Part D so that 1 cm of your picture represents 1 m of board length. F. Make a sketch to show that Raven s board could be a triangle. Label the lengths of the sides. G. Sketch two other possible shapes for Raven s board. Label the side lengths. 146 NEL
Reflecting 1. Could you have drawn a different parallelogram for Part D? How do you know? 2. Why are there always many different shapes with the same perimeter? Checking. a) Measure the perimeter of the triangle shown. b) Draw another shape with the same perimeter. Practising 4. a) Measure the perimeter of each polygon. i) ii) iii) b) Draw a polygon with the same perimeter as the octagon. 5. Draw two other shapes with the same perimeter as the parallelogram in Question 4. 6. A baker is piping icing around the edges of these cookies. Each millilitre of icing can pipe about 4 cm. How many millilitres of icing are needed for each cookie? a) b) c) icing 7. Why is the perimeter of the pink half of this design more than half of the perimeter of the whole design? NEL 147
CHAPTER 5 Frequently Asked Questions Q: How do you choose which metric unit to use to describe the length of something? A: Use kilometres for long distances. Use metres for room-size distances. Use centimetres for relatively short distances. Use millimetres for very short lengths. 2 m 1 cm 1 mm Q: How do I rename a length measurement using different metric units? A: Use the facts in the table. For example:.42 m 00 cm 42 cm 42 cm 000 mm 420 mm 420 mm 2.1 km 2000 m 100 m 2100 m 2.4 cm 20 mm 4 mm 24 mm 1 km 1000 m 1 m 0.001 km 1 m 100 cm 1 cm 0.01 m 1 cm 10 mm 1 mm 0.1 cm Q: How is the measurement.2 m different from.20 m? A: A measurement of.2 m means that the length was measured to the nearest tenth of a metre. If the ribbon is later measured to the nearest (to the nearest tenth of a metre) hundredth of a metre, the length might be.2 m reported as.20 m, but it might also be reported as, for example,.18 m (since.18 m.18 rounds to.2). The length measured (to the nearest hundredth of a metre) to the nearest hundredth of a metre might be anywhere between.15 m and.25 m. Q: Can two different shapes have the same perimeter? A: Yes. For example, all of these shapes have a perimeter of 22 cm. 10 cm 10 cm 4 cm cm 4.5 cm 7 cm 8 cm 6.5 cm 2 cm 148 NEL
CHAPTER 5 LESSON Mid-Chapter Review 1 2 1. What unit would you use for each measurement? a) the height of a soup can b) the width of an electrical cord c) the distance from your home to the grocery store 2. Name a length or width you would measure in millimetres. Explain why.. Rename each measurement using the new unit. a) 2.5 cm to millimetres c) 0.5 km to metres b) 6.20 m to centimetres d) 4.200 m to millimetres 4. Ella reports that the height of a tree in her school yard is.8 m. If Ella measured the tree to the nearest hundredth of a metre, what is the least height she might report? 5. Which measurement could describe each length or distance? a) length of a football field b) height of a room c) width of a sidewalk 6. Measure the perimeter of each shape. a) b) 270 cm 1.40 m 100 m 50 m 7. Create a different shape with the same perimeter as the triangle in Question 6. 8. A shape has a perimeter of.00 m. Is it possible that one side length is only 5 cm? Explain. NEL 149
CHAPTER 5 4 Solve Problems Using Logical Reasoning You will need a calculator Goal Use logical reasoning to solve a problem. Jorge and his sister train by running on square paths around their school gyms. Jorge s path is 11.00 m longer than his sister s.? How much longer are the sides of the path at Jorge s school than at his sister s school? Marc s Chart Understand the Problem I know that Jorge s gym is bigger than his sister s. I also know that the running paths are the perimeters of squares and that the measurements were made to the nearest hundredth of a metre. Make a Plan I ll use logical reasoning. I know that when the side of a square gets 1.00 m longer, then the perimeter has to get 4.00 m longer. Carry out the Plan I make a chart to show what happens when the side length increases. I notice that 11.00 m is between 8.00 m and 12.00 m, so the extra side length must be between 2.00 m and.00 m. I ll try an extra 2.50 m in side length. 4 2.50 10.00, so the perimeter is only 10.00 m greater. I ll try 2.75 m longer. 4 2.75 11.00, so the sides of the path at Jorge s school must be 2.75 m longer than at his sister s school. logical reasoning A process for using the information you have to reach a conclusion. For example, if you know all the students in a class like ice cream and that Jane is in the class, you can logically reason that Jane likes ice cream. Extra Change in side length perimeter (m) (m) 1.00 4.00 2.00 8.00.00 12.00??? 11.00 150 NEL
Reflecting 1. How did Marc know the extra side length must be between 2.00 m and.00 m? 2. Why did Marc try a length of 2.75 m after he tried 2.50 m?. How did logical reasoning help Marc solve the problem? Checking 4. When a rectangle s length is doubled and the width stays the same, its perimeter increases by 15.0 cm. How many centimetres longer is the new rectangle than the old one? Explain your reasoning. Practising 5. Two games use cards in the shape of equilateral triangles. The card from one game has a perimeter that is 10.0 cm longer than the card from the other game. About how much shorter is the side length of the smaller card? Explain your reasoning. 7 6. Draw a shape with eight sides with a perimeter of 0 cm. The sides do not have to be of the same length. 7. A square porch and a porch shaped like a regular hexagon each have the same perimeter. a) Which shape has a longer side? b) Express the length of the shorter side as a fraction of the length of the longer side. 8. How many numbers between 100 and 500 have a 4 as at least one of the digits? 9. Sophia opened a book and multiplied the two page numbers she saw. The product was 5550. What was the greater page number? NEL 151
Curious Math Triangle Sides Try this with a classmate. Cut a piece of string about 5 cm long. Tie the ends together to make a loop that measures 0 cm. You will need string scissors a ruler Make a triangle with the string using your fingers. Have your classmate measure and record the side lengths. Repeat to form other triangles, taking turns measuring. 1. What is the longest side that seems possible? Why might that be true? 2. How do you know that the shortest side of a triangle has to be no more than 1 of the perimeter?. List all possible combinations of lengths of triangles with whole-number side lengths and a perimeter of 0 cm. 12 cm 12 cm 6 cm 152 NEL
Math Game Lines, Lines, Lines Number of players: Any number How to play: Estimate lengths in millimetres. You will need a die a ruler Step 1 Step 2 Create a two-digit number by rolling a die twice. The first roll tells the tens digit. The second roll tells the ones digit. Without using a ruler, try to draw a line segment that is the same length as your two-digit number, measured in millimetres. Step Measure your line segment to see how close you were. Score 2 points if you are within 5 mm. Score 1 point if you are between 5 mm and 10 mm away. Take turns. Play until one player has 10 points. Isabella s Turn I rolled a one, then a four. I have to draw a line 14 mm long. That s more than 1 cm but less than 2 cm. I think it s this long. I measure my line. It is 17 mm, so I m mm away. I get 2 points. NEL 15
CHAPTER 5 5 Exploring Perimeter You will need a ruler 1 cm grid paper Goal Explore the relationship between perimeter and area measurements.? How can you keep increasing the perimeter of shapes that you draw inside a square? A. Draw five squares, each with a side length of 10 cm. Calculate the perimeter and area of the squares. B. Inside the first square, draw a polygon like the orange one in the grid. C. Measure the perimeter and area of the new polygon. Are they greater or less than the perimeter and area of the original square? D. In one of the other squares, make a polygon with a perimeter of 50 cm. Use the other squares to make shapes of perimeter 70 cm, 80 cm, and 100 cm. Each time, calculate the area of your shape. E. How did the areas change? Reflecting 1. Does a polygon with more sides always have a greater perimeter? Explain using examples from class work. 2. What strategies did you use to make sure the perimeter was the required value?. Rebecca says she has a polygon with a perimeter of 80 cm. Can you predict the area of Rebecca s polygon? Explain using examples. 154 NEL
Calculating Lengths of Time Mental Math You can use different methods to calculate how much time something took. Akeem s Method Li Ming s Method I can add on to calculate the difference between the two times. 11 12 1 10 2 9 am 8 4 7 6 5 11 12 1 10 2 9 pm 8 4 7 6 5 I can add on too much time and then subtract to calculate the difference between the two times. 1 1:45 AM PM 6:0 AM PM 15 min 6 h 0 min 7 h 15 min 11:45 12:00 The difference is 6 h 45 min. 6:00 6:0 11:45 6:0 6:45 The difference is 6 h 45 min. A. Explain Akeem s method. B. Explain Li Ming s method. Try These 1. Calculate the difference between the two times. a) c) 11 12 1 10 2 9 am 8 4 7 6 5 11 12 1 10 2 pm 9 8 4 7 6 5 4:05 AM PM 8:55 AM PM b) 12 12 d) 11 1 10 2 9 am 8 4 7 6 5 11 1 10 2 9 pm 8 4 7 6 5 8: 15 AM PM :0 AM PM NEL 155
CHAPTER 5 LESSON Skills Bank 1 1. Select the most appropriate unit to measure each length or distance. a) the length of a paper clip b) the thickness of a drum hide c) the height of a lamp d) the distance around the school playground 2 2. Name two distances or lengths you might measure with each of these units. a) millimetres c) metres b) centimetres d) kilometres. Rename each measurement using the new unit. a).42 m to centimetres c) 0.25 m to millimetres b) 5.21 km to metres d) 1.2 cm to millimetres 4. The first column describes a measurement that was rounded to the nearest whole centimetre. Fill in the other columns to describe what the measurement might have been if it had been rounded to the nearest tenth of a centimetre. Rounded Lowest Highest measurement possible value possible value a) 4 cm cm cm b) 89 cm mm mm 5. The first column describes a measurement that was rounded to the nearest tenth of a kilometre. Fill in the other columns to describe what the measurement might have been if it had been rounded to the nearest metre. Rounded Lowest Highest measurement possible value possible value a) 2.1 km m m b).4 km m m 156 NEL
6. Draw a four-sided polygon with each perimeter. a) 26 cm b) 142 mm c) 8.7 cm 7. Measure the perimeter of each polygon. a) c) b) d) 5 8. What is the perimeter of each polygon inside the square? a) b) c) 9. a) Draw two rectangles with the same perimeter but different areas. b) Draw two polygons with the same area but different perimeters. NEL 157
CHAPTER 5 LESSON Problem Bank 2 4 1. A 50 g ball of yarn is about 125 m long. What is the mass of a ball of yarn 2 km long? 2. The perimeter of a square is 96 m. What is the length of the sides?. Draw two non-congruent 5-sided polygons. Use these clues: The perimeter is 40 cm. There are at least two equal sides. At least one side has a length of 10 cm. 4. Draw a 10 cm by 12 cm rectangle. Use these clues to draw 5 different shapes inside it: The shapes do not overlap. The total perimeter for all 5 shapes is greater than 60 cm. 5 5. The perimeters of two squares differ by 8 cm. The total perimeter for the two squares is 96 cm. What is the side length of the larger square? 6. A rectangle has whole number side lengths. The length of the rectangle is tripled and the width is doubled. The new area is 204 cm 2. The new perimeter is 110 cm. What was the old perimeter? 7. You can make a snowflake polygon from an equilateral triangle. At every step, the middle third of each side is bumped out in the shape of a smaller equilateral triangle. Calculate the perimeter of the polygons shown for steps 1 to. 1 m 1 m 1 m step 1 step 2 step 158 NEL
CHAPTER 5 Frequently Asked Questions Q: When might you solve a problem by using logical reasoning? A: When you have some information that you think you could use to figure out something else, you might use logical reasoning. For example, suppose that you know that the length of a rectangular room is times the width and that the perimeter is 48 m. You want to determine the length of the room. You can use logical reasoning: Since opposite sides of a rectangle are equal, the length plus the width is the same as half the perimeter. Half the perimeter is 24 m. Since the length is three times the width, adding the length and width is like adding four times the width. That means four times the width is 24 m. The width must be 6 m, so the length must be 18 m. Q: Why is it possible to create polygons with a greater perimeter, but a similar area? A: Area and perimeter are different measurements. For example, it you took the green rectangle below and cut it in half, you could make the yellow rectangle. You d have the same area, but a much greater perimeter. 20 cm 8 cm 40 cm 4 cm NEL 159
CHAPTER 5 LESSON Chapter Review 1 2 1. Why would you measure the total length of a bus in metres rather than centimetres? 2. What unit would you use to measure the thickness of a cup? Why?. Rename the measurements Owen made of his keyboard so that they are all in centimetres. 0.68 m 19 mm 25 mm 0.940 m 4. In what ways is the measurement.2 m like.20 m? How is it different? 5. Arden spilled maple syrup on her notes. What measurement units make the sentence true?.64 is the same as 64. If I measure in it might be any length between 65 and 645. 6. Draw a square and a triangle that each have the same perimeter as this rectangle. 14 cm 5 cm 7. Draw a rectangle with a perimeter that is 6 cm more than the perimeter of the rectangle in Question 6. 160 NEL
4 8. A square and an equilateral triangle each have a perimeter of 72 cm. How much longer is one side of the triangle than one side of the square? 9. A large rectangular room is 12.4 m long. What increase in width will make the perimeter 5 m longer if the length stays the same? 10. The short side of an isosceles triangle is 16 cm long. Increasing each side length by 24 cm doubles the perimeter. How long are the other sides of the original triangle? Explain your reasoning. 16 cm 5 11. One side of an isosceles triangle is 12 m longer than the other two sides. The perimeter is 84 m. How long are the sides of the triangle? 12. a) Which of the two green polygons has a greater perimeter? How do you know? b) Which of the two has a greater area? How do you know? c) Does a polygon with a greater area always have a greater perimeter? Explain. 1. Draw a polygon with a perimeter of 50 cm inside a rectangle with a perimeter of 0 cm. 14. Start with a square. Suppose you cut it up and rearrange the pieces into a thin rectangle. Is the perimeter of the rectangle greater or less than the perimeter of the square? Why? NEL 161
Mapping out a Biathlon Course A triathlon involves swimming, biking, and running. In youth triathlons, the 11- and 12-year-olds might swim 150 m to 00 m, bike 8 km to 12 km, and run 1.5 km to km. You are planning a cycling and running youth biathlon for Grade 5 and 6 students. CHAPTER 5 Chapter Task? How might you design courses for a youth biathlon? A. Choose a length for each event of your biathlon. B. Would most students be able to complete your biathlon in 40 min? Use the winning times for each event to help decide. Some winning times: Cycling 18 km in 1 h Running 4 km in 21 min C. Redesign your biathlon so that most students can complete it in 40 min. D. Sketch a polygon course for each event. Each course should be on or near your school grounds. E. Make a scale drawing that shows your biathlon courses. Label the side lengths and perimeter of each polygon. Task Checklist Did you explain all of your calculations? Were you reasonably accurate in your measurements? Did you show all your steps? Is your scale drawing clear? 162 NEL