lesson 6 problem solving practice changes in dimensions

• Precalculus
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Area and Perimeter of Similar Figures (Lesson 6.6)

Unit 1: reasoning in geometry, day 1: creating definitions, day 2: inductive reasoning, day 3: conditional statements, day 4: quiz 1.1 to 1.3, day 5: what is deductive reasoning, day 6: using deductive reasoning, day 7: visual reasoning, day 8: unit 1 review, day 9: unit 1 test, unit 2: building blocks of geometry, day 1: points, lines, segments, and rays, day 2: coordinate connection: midpoint, day 3: naming and classifying angles, day 4: vertical angles and linear pairs, day 5: quiz 2.1 to 2.4, day 6: angles on parallel lines, day 7: coordinate connection: parallel vs. perpendicular, day 8: coordinate connection: parallel vs. perpendicular, day 9: quiz 2.5 to 2.6, day 10: unit 2 review, day 11: unit 2 test, unit 3: congruence transformations, day 1: introduction to transformations, day 2: translations, day 3: reflections, day 4: rotations, day 5: quiz 3.1 to 3.4, day 6: compositions of transformations, day 7: compositions of transformations, day 8: definition of congruence, day 9: coordinate connection: transformations of equations, day 10: quiz 3.5 to 3.7, day 11: unit 3 review, day 12: unit 3 test, unit 4: triangles and proof, day 1: what makes a triangle, day 2: triangle properties, day 3: proving the exterior angle conjecture, day 4: angle side relationships in triangles, day 5: right triangles & pythagorean theorem, day 6: coordinate connection: distance, day 7: review 4.1-4.6, day 8: quiz 4.1to 4.6, day 9: establishing congruent parts in triangles, day 10: triangle congruence shortcuts, day 11: more triangle congruence shortcuts, day 12: more triangle congruence shortcuts, day 13: triangle congruence proofs, day 14: triangle congruence proofs, day 15: quiz 4.7 to 4.10, day 16: unit 4 review, day 17: unit 4 test, unit 5: quadrilaterals and other polygons, day 1: quadrilateral hierarchy, day 2: proving parallelogram properties, day 3: properties of special parallelograms, day 4: coordinate connection: quadrilaterals on the plane, day 5: review 5.1-5.4, day 6: quiz 5.1 to 5.4, day 7: areas of quadrilaterals, day 8: polygon interior and exterior angle sums, day 9: regular polygons and their areas, day 10: quiz 5.5 to 5.7, day 11: unit 5 review, day 12: unit 5 test, unit 6: similarity, day 1: dilations, scale factor, and similarity, day 2: coordinate connection: dilations on the plane, day 3: proving similar figures, day 4: quiz 6.1 to 6.3, day 5: triangle similarity shortcuts, day 6: proportional segments between parallel lines, day 7: area and perimeter of similar figures, day 8: quiz 6.4 to 6.6, day 9: unit 6 review, day 10: unit 6 test, unit 7: special right triangles & trigonometry, day 1: 45˚, 45˚, 90˚ triangles, day 2: 30˚, 60˚, 90˚ triangles, day 3: trigonometric ratios, day 4: using trig ratios to solve for missing sides, day 5: review 7.1-7.4, day 6: quiz 7.1 to 7.4, day 7: inverse trig ratios, day 8: applications of trigonometry, day 9: quiz 7.5 to 7.6, day 10: unit 7 review, day 11: unit 7 test, unit 8: circles, day 1: coordinate connection: equation of a circle, day 2: circle vocabulary, day 3: tangents to circles, day 4: chords and arcs, day 5: perpendicular bisectors of chords, day 6: inscribed angles and quadrilaterals, day 7: review 8.1-8.6, day 8: quiz 8.1 to 8.6, day 9: area and circumference of a circle, day 10: area of a sector, day 11: arc length, day 12: quiz 8.7 to 8.9, day 13: unit 8 review, day 14: unit 8 test, unit 9: surface area and volume, day 1: introducing volume with prisms and cylinders, day 2: surface area and volume of prisms and cylinders, day 3: volume of pyramids and cones, day 4: surface area of pyramids and cones, day 5: review 9.1-9.4, day 6: quiz 9.1 to 9.4, day 7: volume of spheres, day 8: surface area of spheres, day 9: problem solving with volume, day 10: volume of similar solids, day 11: quiz 9.5 to 9.8, day 12: unit 9 review, day 13: unit 9 test, unit 10: statistics and probability, day 1: categorical data and displays, day 2: measures of center for quantitative data, day 3: measures of spread for quantitative data, day 4: quiz review (10.1 to 10.3), day 5: quiz 10.1 to 10.3, day 6: scatterplots and line of best fit, day 7: predictions and residuals, day 8: models for nonlinear data, day 9: quiz review (10.4 to 10.6), day 10: quiz 10.4 to 10.6, day 11: probability models and rules, day 12: probability using two-way tables, day 13: probability using tree diagrams, day 14: quiz review (10.7 to 10.9), day 15: quiz 10.7 to 10.9, day 16: random sampling, day 17: margin of error, day 18: observational studies and experiments, day 19: random sample and random assignment, day 20: quiz review (10.10 to 10.13), day 21: quiz 10.10 to 10.13, learning targets.

Explain why sides and perimeter of similar figures grow by the scale factor and area grows by the square of the scale factor.

Use relationships between perimeter and area of similar figures to solve for missing sides and areas.

Activity: Why Do Large Prints Cost So Much?

Lesson handouts, media locked, additional media.

Our Teaching Philosophy:

Experience first, formalize later (effl), experience first.

In this final lesson of Unit 6, students look at the relationships between side lengths, perimeter, and area of similar figures. We love this concept because it ties in important Algebra concepts about relative growth rates of linear and quadratic functions, and it allows students to make sense of solving simple quadratic equations in an accessible way. Furthermore, this lesson plants seeds for what students will learn in Unit 9 about the volumes of similar figures.

In this lesson, students will look at pricing structures for photo prints as a context for exploring how area changes in relation to the change in sides. Students will need colored pencils and the graph paper provided in the “Additional Media” section to complete this activity.

Students compare ratios of side lengths and areas of two sets of squares and may be surprised to find out the ratios are not the same. Students should be able to explain why this is the case. A simpler example could be used where the scale factor is a whole number. If the base of a larger rectangle is three times longer than the base of the original rectangle then the height is also three times longer, and if multiplying base and height to find the area, the new area will be 9 times bigger. Students should also be able to informally explain why the ratio of perimeters matches the ratio of the sides. Though the perimeter and the side lengths are not equal, their ratios are equivalent. In the case of a square, the perimeter of each figure is simply four times bigger than the side length, and this is true in both squares. Using what students know about equivalent fractions, they should be able to tell that a/b=4a/4b.

Formalize Later

The key takeaway from today’s lesson is that the ratio of sides and ratio of perimeters of two similar figures is not equal to the ratio of the areas of the figures. In fact, the ratio of the areas is the square of the ratios of the side lengths, regardless of what the shape of the figure is, since area requires using two dimensions which each have been enlarged or reduced by the scale factor.

When students are solving problems related to ratios of sides and areas, we recommend students clearly label what ratio they are talking about: whether it is a Ratio of Sides (ROS) or a Ratio of Areas (ROA). This will help students determine what they must do to get the other ratio.

Math Medic Help

Acc6.2 Ratios, Rates, and Percentages

In this unit, students learn to understand and use the terms “ratio,” “rate,” “equivalent ratios,” “per,” “at this rate,” “constant speed,” “constant rate,” “unit rate,” “speed,” “pace,” “percent,” and “percentage.” They recognize when two ratios are or are not equivalent and that equivalent ratios have equal unit rates.

They represent ratios as expressions, and represent equivalent ratios with double number line diagrams, tape diagrams, and tables. They represent percentages with tables, tape diagrams, and double number line diagrams, and as expressions. They use these terms and representations in reasoning about situations involving unit price, constant speed, measurement conversion, color mixtures, and recipes.

What are Ratios?

• 1 Representing Ratios with Diagrams
• 3 Defining Equivalent Ratios

Representing Equivalent Ratios

• 4 Introducing Double Number Line Diagrams
• 5 Creating Double Number Line Diagrams
• 7 Comparing Situations by Examining Ratios
• 8 Representing Ratios with Tables
• 9 Navigating a Table of Equivalent Ratios
• 10 Solving Equivalent Ratio Problems
• 11 Part-Part-Whole Ratios
• 12 Solving More Ratio Problems
• 13 The Burj Khalifa
• 14 Measuring with Different-Sized Units
• 15 Converting Units
• 16 Comparing Speeds and Prices
• 17 Interpreting Rates
• 18 Equivalent Ratios Have the Same Unit Rates
• 19 Solving Rate Problems

Percentages

• 20 Percentages and Double Number Lines
• 21 Percentages and Tape Diagrams
• 22 Benchmark Percentages
• 23 Solving Percentage Problems
• 24 Finding the Percentage

Let’s Put it to Work

• 25 A Fermi Problem
• 26 Painting a Room