Contents. The Double Number Line Double Scale Line 40 City Blocks 42 Weights and Prices 44 Summary 48 Check Your Work 49.

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1 Contents Section D Double Scale Line 4 City Blocks 4 Weights and Prices 44 Summary 48 Check Your Work 49 Additional Practice Answers to Check Your Work Student Activity Sheets Contents v

2 D Double Scale Line In many countries, distances are expressed in kilometers. In the United States, distances are represented in miles. Today, many maps use a double scale line, using both kilometers and miles. This map of Toronto, Canada, has a double scale line. Double Scale. a. Describe how you might use the double scale line. b. Use the double scale line to find three relationships between miles and kilometers. Write your relationships like this.... miles equal about... kilometers... kilometers equal about... miles 4 Models You Can Count On

3 D The Toronto City Centre Airport is located on an island close to the coast. The distance from downtown Toronto to the airport is about 4 km.. a. Estimate this distance using miles. b. About how many kilometers equals 5 mi? Show how you found your answer. You can measure distance two different ways: as the crow flies (in a straight line) or as Taxi Cab distance (along the ground from place to place). The distance as the crow flies is shorter (except for cases in which the two distances are the same). The distance from downtown Toronto to Vaughan as the crow flies is about 5 mi. 3. About how many kilometers is this distance? Show how you found your answer. This is part of a distance table. You can read the distances between Toronto and other large cities. Distances are given in both kilometers and miles. Some information is missing. Dawson Creek, BC Chicago, IL Calgary, AB Anchorage, AK Montréal, PQ Halifax, NS Gaspé, PQ Edmonton, AB Saint John, SB Regina, SK Québec, PQ Prince Rupert, BC Ottowa, ON New York, NY Thunder Bay, ON Sault Ste. Marie, ON San Francisco, CA Toronto, ON How far is the distance between Toronto and Chicago in kilometers? What is this distance in miles? How can you be sure which is which? 5. Find the missing information. You do not have to be precise. Section D: 4

4 D City Blocks Springfield city blocks are made up of streets and avenues, that are very regular and look much like a grid. Each city block in Springfield is usually 8 mile long. Springfield School Gary lives mi from school. He walks to school every morning. miles 6. How many city blocks does Gary walk to school? How did you figure this out? Sharon lives 4 mi from school. She bikes to school every morning. 7. How many city blocks does she bike to school? How did you figure this out? 8. Use the city map on Student Activity Sheet to locate where Gary and Sharon could live. 4 Models You Can Count On

5 D Rene travels blocks from home to school. 9. a. How many miles is this? How did you find out? b. Would you advise Rene to use her bike or to walk to school? Give reasons to support your answer. Marcus wants to find out how far Ms. Anderson lives from school. He knows she travels 9 city blocks to school. He draws a double number line like this blocks miles 8 4 This double number line is drawn to scale, with numbers on top as well as on the bottom. Learning how to use a double number line will help you make precise calculations effortlessly.. a. On Student Activity Sheet, use this double number line to find out how far Ms. Anderson lives from school. b. Use the double number line to find out how many city blocks there are in 4 3 mi. Every morning, Gary takes about minutes to walk mi to school. Sharon s bike is broken, so she is making plans to walk 4 mi to school. She asks Gary how long this might take her.. a. Copy the double number line below in your notebook. Using a grid helps to partition the spaces evenly. b. Use the double number line to calculate how long it will take Sharon to walk to school. State any assumptions you are making in finding your answer. minutes miles Section D: 43

6 D Gary and Sharon like to hike. This weekend they plan to walk a mi lake trail. They estimate how long they will hike. Gary uses a double number line like the one on page 43.. Draw a double number line and use it to find the time needed for the hike. Sharon uses a ratio table to make the same calculation. Minutes Miles a. Explain how Sharon decided on the numbers in each new column in the table. b. Which model do you prefer, the double number line or the ratio table? Explain your preference. Weights and Prices Jack s Delicatessen sells many different kinds of food. Jack is the shop owner. He imports fresh deli meats and more than 8 kinds of cheese. Workers slice the food items, weigh them, and calculate the prices. Susan weighs a piece of pepper salami at. kilograms (kg). 4. Explain how Susan will calculate the price for this piece. What information does she need? 44 Models You Can Count On

7 D Ahmed is shopping for some brie, a kind of French cheese. He wants the piece to weigh about 4 kg. Susan cuts off a piece and puts it on the scale. The scale shows:.5 kg 5. a. Does Ahmed have the amount of brie he wants? b. Calculate how much Ahmed has to pay for this piece of brie. Ahmed often buys brie at Jack s Delicatessen. Lately he has been thinking about a clever way to estimate the price. The scale reminds him of a double number line. He creates the following double number line including both weight and price on it. $ price.5 Kg weight 6. Show how Ahmed can use this double number line to estimate the price for.7 kg of brie. This week, brie is on sale for $9. per kilogram. 7. What is the sale price of.7 kg of brie? Show how you found your answer. 8. a. Select three different pieces of brie to purchase and write down the weight of each piece. b. Draw a scale pointer to mark each weight on a different number line. (You may want to exchange your notebook with a classmate after both of you have drawn your pointers.) c. Estimate the regular price and the sale price of the three pieces of brie. Section D: 45

8 D Jack s Delicatessen also sells fruit. This week, California grapes are on sale for $.89 per kilogram. Grapes $.89/kg Madeleine places her grapes on the scale. This is what she sees..5 kg 9. a. How much do Madeleine s grapes weigh? b. Estimate how much Madeleine has to pay for the grapes. You may want to use a double number line for your estimation and explanation. Ahmed decides to use the money he has left to buy some grapes. He counts his money and realizes he has $.5.. Estimate what weight of grapes Ahmed can buy for $.5. Show how you found your answer. Susan weighed fruit for 5 customers. She wrote the prices on small price tags, but, unfortunately, the tags got mixed up.. Match the information written below to the corresponding price tags. Find out which note belongs to each customer. Show your work. Mounim Claire Frank Nadine Gail.5 kg grapes for $.5/kg.8 kg apples for $.5/kg.5 kg oranges for $.9/kg. kg bananas for $.5/kg.9 kg kiwis for $.4/kg $4.75 $.75 $.75 $.5 $.6 46 Models You Can Count On

9 D Math History Edmund Gunter (58-66), an English mathematician, invented a measurement tool for surveying. It came into common usage around 7 and was the standard unit for measuring distances for more than 5 years. Gunter's Chain is 66 ft long. Its usefulness comes from its connection to decimals; it is divided into links. chains From 83 to 859, the General Land Office conducted the original public land survey of Iowa. 5 links. How many chains make one mile? Because Gunter's chain was used to measure America, the United States did not use the metric system (developed in France in 79). Section D: 47

10 D Double Number Line A double number line is a number line with a scale on top and a different scale on the bottom so that you can organize and compare items that change regularly according to a rule or pattern. Example The price changes $ for every kilogram purchased. $6 Example The time changes by minutes for every half-mile walked. $ price.5 kg weight 3 minutes You can use a double number line to solve problems with fractions, decimals, and ratios. Ratio Table Model miles You can use a ratio table to solve these same problems. Example 3 The time changes by minutes for every half-mile walked. Minutes Miles On a double number line, as on a single number line, the numbers always appear in order. 48 Models You Can Count On

11 Many American cities have a street plan that looks like a grid. Philip read the following on the Internet. A lot of American cities are laid out with grids of 6 mi by mi 8 (metric equivalents: m by m). Major streets are usually at 4 -, -, or -mi intervals. Philip uses this information to set up the following relationships between miles and meters. meters ] 6. Help Philip use the double number line to find the distance between major U.S. streets. Write your answers using the metric system. miles Near the end of the week, Jack sells a basket of mixed fruit for $.5 per kilogram.. a. How much mixed fruit can you purchase for $5.? b. The scale indicates 3. kg. What is the price for this fruit? Sharon s bike is fixed so she and Gary plan a 7 - mi bike ride. They want to estimate how long they will bike. Sharon knows that it takes 7 minutes to bike the 4 mi to school. 3. Use a double number line or a ratio table to find the amount of time they will bike. How are double number lines and ratio tables alike? How are they different? Which do you find easier to use? Section D: 49

12 Additional Practice Section D. Five miles is the equivalent of exactly 8 km. 5 5 miles 8 kilometers a. How many miles equal km? b. In cities in The Netherlands, the speed limit for driving is 5 kilometers per hour (km/h). About how many miles per hour is that?. If Norman bikes to school, it takes him about a quarter of an hour to cover the 3 mi. a. At the same average speed, how many miles can Norman bike in hours? b. How long would a 5-mi trip take at the same average speed? 3. Ahmed buys a piece of cheese at Jack s Delicatessen. This is what the scale shows..5 kg a. What is the amount in kilograms shown on the scale? b. Find out how much Ahmed has to pay if the price of kg of the cheese is $9. You may use a double number line. c. The piece is too expensive for Ahmed. Jack shows him another piece and says, This will cost you $5.4. What is the weight of this piece of cheese? 4. Kendra has a pen pal in Europe named Richard. How tall are you? she asked in an to him. I am.85 meters tall. How tall are you? Richard writes. I m 5 feet 6 inches tall, Kendra answers. Who is the taller of the two? You may use the general rule that there is a little less than 3 feet (ft) in m and there are inches (in.) in ft. Show your work. 64 Models You Can Count On

13 Answers to Check Your Work Section D. Major streets are usually 4, 8, or,6 m apart. Sample explanation: meters miles a. 4 kg. Sample strategy using a double number line: $.5 $.5 $5. $ kg b. The price for 3. kg of apples is $4.. To get 3. kg, you can add. to 3. The price for 3. kg of apples is $ $.5 = $ minutes are needed. Sample strategy using a double number line: miles minutes Calculations: First double, and then times 3. The same calculations can be made using a ratio table. 3 Miles Minutes Answers to Check Your Work 7

14 Student Activity Sheet Use with Models You Can Count On, pages 4 and 43. Name Springfield School miles 96 Models You Can Count On Student Activity Sheet Encyclopædia Britannica, Inc. This page may be reproduced for classroom use.