december 14, 2010 proportionality through similarity and geometry

December 14, 2010

Proportionality Through Similarity and Geometry

Objectives

• Look at proportionality vertically through the grades

• Connect the concept of proportionality in similar figures to algebraic thinking

• Examine various scaffolding strategies for all students

• Reflect on use of strategies in classrooms

Engage Activity using Ratio Tables

Build a ratio table and use it to answer the following task:– A person who weighs 160 pounds on Earth will

weigh 416 pounds on the planet Jupiter. How much will a person weigh on Jupiter who weighs 120 pounds on Earth?

Sample Solutions

(a)

Earth weight 160 80 40 120 Jupiter weight 416 208 104 312

(b)


(c)


÷ 2 ÷ 2 × 3

add

÷ 8

add

× 5

Ratio Tables

• In applying this technique, students are using multiplicative relationships to transform a given ratio into an equivalent ratio.

• A recursive pattern (repeated addition) can be used along with a generative pattern or multiplicative relationship between values.

Discussion Point

With a partner, describe the advantages this approach has for students. What should students be able to know and do to use ratio tables?

Algebra Connection

• Input equal ratios in L1 and L2 • Graph the scatter plot– Each axis should correspond to one of the

quantities in the table.

• Find the equation of the line that best fits the points on your graph and plot it in Y1

Algebra Connection Discussion

• What do you notice about the graph?• What do you notice about the slope of the

line?• How are equal ratios (proportions) that are

emphasized in the middle grades connected to algebra readiness?

Connections

• The ratio or rate is the slope of the graph.

• Ratios are a special case of linear situations that will always go through the origin, since they are multiplicative relationships.

• Proportional situations are linear situations.

Within and Between Ratios

When examining two ratios, it is useful to thinkof them as either within ratios or between ratios. • Consider three rectangles A, B, and C. A

measures 2 x 6. B measures 3 x 9, and C measures 8 x 24.

• Find the within ratio for each rectangle.

Discussion

• Is this enough information to convince you that the rectangles are similar?

• Why is it important for students to understand the difference between within ratios and between ratios?

• Why are these ratios different?

• Now examine the between ratios for rectangles A and B and for A and C.

Explore Proportionality through Similarity

• Complete the activities :– Exploring Similarity– Properties of Similar Figures

• Think about the concept of proportionality and the connections to algebra while you’re working on the activities.

Exploring Similarity Extension

• In pairs, work on the activity: Proving Triangles are Similar

• Make connections to proportional reasoning and algebraic thinking as you work through the activity.

Proportionality Through the Grades

• How is the concept of proportionality introduced, and at what grade level?– What skills must be in place for students to be

able to understand the concept of proportionality?

• Why is it important for algebra readiness?

• How is it extended through the middle grades and high school?

6 •Use ratios to describe proportional situations•Represent ratios with concrete models, fractions, and decimals•Use ratios to make predictions in proportional situations•Use tables and symbols to represent and describe proportional and other relationships …

Proportionality through the Grades

7 •Estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit costs, & related measurement units•Use critical attributes to define similarity



8 •Compare and contrast proportional & non-proportional linear relationships•Estimate and find solutions to application problems involving percents and other proportional relationships such as similarity and rates•Use proportional relationships in similar 2D or 3D figures find missing measurements




Alg I •Relate direct variation to linear functions and solve problems involving proportional change.





Geo •Use ratios to solve problems involving similar figures•Develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples…

Alg I •Relate direct variation to linear functions and solve problems involving proportional change.





Proportionality through the Grades• 5th Grade – Equivalent Fractions• 6th Grade – Introduction to ratios and proportions

through ratio tables• 7th Grade – Attributes of Similar Figures, and

Proportional Reasoning: Between and Within Ratios• 8th Grade – Applying Proportional Reasoning,

Attributes of Similar Figures • 9th Grade Algebra – Linear Relationships and Direct

Variation• 10th Grade Geometry – Similarity, Proportional

Reasoning, and Proving figures are similar

Sample Questions

Sample Questions cont…

Reflections

What does proportionality through similarity and geometry look like for:– students– teachers – classroom discussion– questioning– student work

december 14, 2010 proportionality through similarity and geometry

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