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# How To Solve Scale Factor Problems

The Lake Smith basketball team had a team picture taken of the players, the coaches, and the trophies from the season. The team decides to have the picture enlarged to a poster and then enlarged again to a banner measuring 48 inches by 72 inches. Sketch drawings to illustrate the original picture and enlargements. If the scale factor from the picture to the poster is 500%, determine the dimensions of the poster. What scale factor is used to create the banner from the picture? What percent of the area of the picture is the area of the poster? What scale factor would be used to reduce the poster to the size of the picture?

Justify your answer using the scale factor AND by finding the actual areas. Write an equation involving the scale factor that relates the area of the poster to the area of the picture. Assume you started with the banner and wanted to reduce it to the size of the poster.

Have students from both camps explain their reasoning.

Your role may be to help compare the two approaches using some accessible scheme.

The key is whether students understand what they are doing—that they’re reasoning about the situation and not simply following a recipe.

One way to help this happen is to take the time to compare the two approaches to some problem the class has done.

We might reason, “3 miles in 36 minutes means 1 mile in 12 minutes. So he can jog 5 miles.” We’ve used a : 12 minutes per mile. For Dwight, we used “within” reasoning, finding a ratio using his 3-mile numbers.

For Dierdre, we used “between” reasoning, finding a ratio relating the 30-minute to the 60-minute jogs.

Implications for teaching: Middle-schoolers, who are just getting fluent with proportion, are unlikely to recognize whether they’re using “within” or “between” reasoning. You’ll want to recognize that in any given problem there are at least two approaches.Key Points: Overall Conclusion If the scale factor is represented by k, then the area of the scale drawing is k times the corresponding area of the original drawing.Example 1 What percent of the area of the large square is the area of the small square?With maps and scale drawings, you’re dealing with similarity and can go either way—but more often, scaling is what you want: as with a recipe, you figure out the scale factor (e.g., one inch = 10 miles) and use it repeatedly. Strategic Education Research Partnership1100 Connecticut Ave NW #1310 • Washington, DC 20036• (202) 223-8555 • [email protected] funding provided by The William and Flora Hewlett Foundation and S. Video solutions to help Grade 7 students solve area problems related to scale drawings and percent.The main activities ask students to use images for fractions and ratio to establish connections between operations, symbols and language in order to build on existing knowledge and develop understanding of fractions as operators; link fractions and ratio and consider equivalent expressions.The starter activities in phase two are designed such that students practise skills of simplifying fractions and converting between fractions, decimals and percentages.For today’s lesson, the intended target is “I can determine scale factor.” Students will jot the learning target down in their agendas (our version of a student planner, there is a place to write the learning target for every day). Activating Prior Knowledge: To get this lesson started, I like to pose the following problem: The scale of a blueprint is 1 inch = 10 feet. Opener: As students enter the room, they will immediately pick up and begin working on the opener – Instructional Strategy - Process for openers This method of working and going over the opener lends itself to allow students to construct viable arguments and critique the reasoning of others, which is mathematical practice 3. I let tables discuss this problem for a few moments and take volunteers to share out their answers to the group.Learning Target: After completion of the opener, I will address the day’s learning targets to the students.