Double Scale
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.
1. 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
Reaching All Learners
Intervention
To help students use a scale line on a map to find distances, you can have them take a piece of paper and mark the distance that has to be determined at the edge of the paper. Then align the edge of the paper with the scale line, and you can read the distance.
Vocabulary Building
Have students add the term double scale line to the vocabulary section of their notebooks.
40 Models You Can Count On
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Section D: The Double Number Line 40T
Hints and Comments
Materials
Transparency of a map of US and Canada, preferably a map that has also a double scale line, optional
Overview
Students use a double scale line on a map to find relationships between miles and kilometers.
About the Mathematics
A double scale line has two proportional scales. The double scale line on this map helps students see the relationships between miles and kilometers. It is an example of a double number line. A double number line, in general, is a visual model used to represent the equivalencies between two units. The double number line is explained on Student Book page 43.
Students will investigate scale lines and maps more in depth in the units Figuring All the Angles and Ratios and Rates.
Comments About the Solutions
1. b. You may draw a large double number line on the board and use this to collect students’
answers. Ask students to place their solutions on the double number line. Then you can ask, What do you notice about the numbers?
What numbers can be placed on the double scale line when it is extended?
Students may now see how the double number line is like a bar and a ratio table. The numbers are organized proportionally. Students should discover this concept on their own. Do not make the comparison for them at this point.
Solutions and Samples
1. a. Answers may differ. Sample answer:
You can use the double number line to find distances on the map; you can find them both in kilometers and in miles. You can also use the double scale line to see how kilometers and miles relate. For instance, you can see that 15 km is shorter than 15 miles.
b. Answers will vary. Sample answers:
5 km equals about 3 miles; 5 miles equals about 8 km; 15 km equals about 9 miles. It is unlikely that students will give examples exceeding 15 miles and 15 km since the lines stop at these distances, but they may do so anyway.
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Notes
2 and 3 You may need to suggest that students use the double scale line from problem 1.
Discuss these problems in class, focusing the discus-sion on the strategies students have used.
Models You Can Count On
D
The Double Number Line DToronto, ON Anchorage,
AK Calgary
, AB Chicago,
IL Dawson
Creek, BC Edmonton,
AB Gaspé,
PQ Halifax,
NS Montréal,
PQ New
York, NY Ottowa,
ON Prince
Rupert, BC Québec,
PQ Regina,
SK Saint
John, SB San
Francisco,
CA Sault
Ste.
Marie, ON Thunder
Bay , ON
6607 4106
3491 2170
829 515
3985 2477
3390 2107
1422 884
1929 1199
554 344 516
399 4856 3018
745 463
2669 1659
1414 879
4229 2627
655 407
1397 868
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.
2. 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 15 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.
4. 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.
Reaching All Learners
Intervention
For students struggling with problem 4, point out that Chicago is as close to Toronto as New York, and you can use this information to estimate the distance in kilometers for New York.
For problem 5, you could mention that the distance from Toronto to Ottawa is about__12 the distance from Toronto to Chicago. This may help them know to halve the distance in miles.
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Section D: The Double Number Line 41T
Solutions and Samples
2. a. 4 km equals about 2.5 miles. Students may use the given double scale to find this.
b. 5 miles is about 8 km. This can be found using the double scale line.
3. 15 miles is about 24 km. Allow for other numbers close to 24 km. Students can use different
strategies. Sample strategies:
• The scale line for kilometers can be extended until it is as long as the one for the miles.
• Students can make a rough estimate based on the double scale line or the previous answers.
• It is possible to use the answer from 2b. 5 miles equals about 8 km. Multiply this by 3, so 15 miles equals about 24 km.
4. The distance in kilometers is 829 km; the distance in miles is 515 miles. The numbers that indicate the distance are 829 and 515. Since one mile is larger than one kilometer, the smallest number is for miles and the largest for kilometers.
5. Solutions and strategies will vary.
Sample answers:
• Toronto–New York is about 830 km. Found using other numbers in the table: the distance Toronto–New York (516 mi) is about the same as the distance Toronto–Chicago (515 mi). So the same number of kilometers can be used.
• The distance Toronto–Ottawa is about 250 miles.
Sample strategy using other numbers in the table:
Toronto–Dawson Creek is about 4,000 km (2,477 mi); the distance Toronto–Ottawa is about 10 times smaller, so about 250 mi. Or using the double scale line from problem 1.
Hints and Comments
Overview
Students estimate distances and convert miles into kilometers, and kilometers into miles. They read a distance table and find distances that are missing in the table.
About the Mathematics
Distance can be calculated as a straight line between two points, called as the crow flies. A distance can also be calculated along a path or grid, called taxicab distance. Students will investigate these concepts more extensively in the unit Figuring All the Angles.
Comments About the Solutions
4. Note that the distance table is a ratio table.
Students may not see this because of the large numbers.
5. If students have problems finding the missing information, you may encourage them to use the double scale line from problem 1. If this is not helpful, you may encourage them to look for other numbers in the table that are almost equal.
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Notes
Ask students if they can recognize the rectangular grid in the photograph on this page.
It may be helpful to make a transparency of Student Activity Sheet 11 or the map on this page so that students can show their solutions and strategies on the overhead.
6 and 8Observe how students determine their answers, noting which students use notations that are more formal. Also, observe whether they use a tool: a double number line or a ratio table.
6 If students start counting the blocks on this drawing, ask them to find another way to solve the problem.
8 If students wonder where the entrance to the school is, you may want to discuss this.
Models You Can Count On D
Gary lives12mi from school. He walks to school every morning.
6. How many city blocks does Gary walk to school? How did you figure this out?
Sharon lives 114mi 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 11 to locate where Gary and Sharon could live.
The Double Number Line
D