## Off the Scale

• Lesson
6-8
1

Students use real-world examples to solve problems involving scale as they examine maps of their home states and calculate distances between cities.

To assess students' prior knowledge, have the students brainstorm ideas about where they might use a scale to enlarge or reduce the size of something. List these ideas on the board or on an overhead projector.

To begin the lesson, give the students a copy of their state map and have them locate the legend. One recommended site is located here. Alternatively, students can find their own state map.

Give pairs of students a ruler and have them figure out distances between given cities. Use the Teacher Resource Sheet, Map Activity Sheet, as a guideline to creating your own worksheet.

 Map Activity Sheet

Students should use the map's scale to complete information about the miles and the inches. Work with the students on setting up proportions to compute unknown distances using their scale factors. Be sure that when the students are setting up the proportions, they do actual distances over each other, and the scale distances over each other (the scale distances or one of the actual distances will be represented by x). Once the proportions are set up, the students can simply solve for x in their proportions.

• Maps of birth states of students
• Rulers
• Map Activity Sheet

Assessments

1. At this point in the unit, students should be able to do the following:
• Set up and solve proportions
• Apply proportions in a variety of instances
• Apply what they have learned about making conversions and use distances on maps to convert their units to actual distances

2. You can assess these objectives by verbally asking the students to glance at an atlas and use their conversion factors to approximate the distance between certain cities. Once they make an estimate, have students set up their proportions to verify their guesses.

Extensions

You can extend the activity by having the students select a comic strip from the newspaper. They should mark off their comic in ¼-inch squares. If they write light enough with their pencil, they can erase the grid after the comic is drawn.

Then have students mark off an 8 1/2 × 11 inch sheet of paper in one-inch squares. Have students draw each box from their comic onto their paper, enlarging the comic strip. You may consider drawing out a one-inch by one-inch grid on a transparency to show the students exactly how to measure and mark the grid with their rulers.

Questions for Students

1. What mathematics are involved in enlarging something? Reducing something?

[Proportions, similarity, scale factor.]

2. What mathematics do you use to convert inches to miles (on the map) using the scale on the map?

[Scale, conversion factor].

3. What steps do you take to convert miles to feet? How about miles to yards?

[To convert miles to feet, divide miles by 5,280. To convert miles to yards, divide miles by 1,760.]

Teacher Reflection

• Are the students able to make reasonable estimates using the scale in the map's legend?
• Are the students able to connect the use of scales to real-world situations?
• Is further review of the metric and customary conversion factors necessary?

### Measuring Up: Measurement Terms

6-8
This lesson introduces relationships between measurement and geometry. The activities build on students' prior knowledge as students work with partners and as a whole class to identify and classify terms to develop their understanding of measurement.

### The Golden Ratio

6-8
Students learn about ratios, including the “Golden Ratio”, a ratio of length to width that can be found in art, architecture, and nature. Students examine different ratios to determine whether the Golden Ratio can be found in the human body.

Students learn to write and solve proportions by gathering data and calculating unit rates.

### Do You Measure Up?

6-8
Students learn the basics of the metric system. They identify which units of measurement are used to measure specific objects, and they learn to convert between units within the same system.

### Discovering Gallon Man

6-8
Students experiment with units of liquid measure used in the customary system of measurement. They practice making volume conversions in the customary system.

6-8
Students extend their knowledge of proportions to solving problems dealing with similarity. They measure the heights and shadows of familiar objects and use indirect measurement to find the heights of things that are much bigger in size, such as a flagpole, a school building, or a tree.

### Learning Objectives

Students will:

• Pose and refine questions that can be addressed with numerical data
• Consider aspects of data collection such as how to obtain measurements and record the data they collect
• Represent, then analyze, the ordered numerical data by describing the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed
• Measure with standard units and carry out simple unit conversions, such as from centimeters to meters or feet to inches