Proportionality Through
Similarity and Geometry
December 14, 2010
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
Jupiter weight
160
416
×3
÷2
÷2
80
208
40
104
120
312
(b)
add
Earth weight
Jupiter weight
160
416
80
208
40
104
(c)
add
÷8
Earth weight
Jupiter weight
120
312
160
416
×5
20
52
100
260
120
312
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
• Proportional situations are linear
situations.
• Ratios are a special case of linear
situations that will always go through
the origin, since they are
multiplicative relationships.
• The ratio or rate is the slope of the
graph.
Within and Between Ratios
When examining two ratios, it is useful to think
of 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?
• Now examine the between ratios for
rectangles A and B and for A and C.
• Why are these ratios different?
• Why is it important for students to understand
the difference between within ratios and
between ratios?
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?
• How is it extended through the middle grades
and high school?
• Why is it important for algebra readiness?
Proportionality through the Grades
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
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
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
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
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
Alg I
•Relate direct variation to linear functions and solve problems involving
proportional change.
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
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
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
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.
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
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
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
• 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