Building Mathematical Connections : Ratio, Algebra, Geometry

1
Building Mathematical Connections Ratio,
Algebra, Geometry

• Purpose
• Connecting algebraic reasoning to proportional
  reasoning through geometry
• Modeling ways for GAP kids to discover pi as a
  ratio of the circumference of a circle to its
  diameter
• Modeling ways for GAP kids to discover the linear
  relationship between the circumference and
  diameter of a circle

2
How do we find the diameter when no center is
marked?

• Think on your own
• Pair with a partner and tell them your ideas
• Share your ideas with the large group and be
  prepared to complete the sentence starter with
  drawings and/or words

I can find the diameter of a circle by
3
I can find the diameter of a circle by
4
How many diameters fit the circumference of a
circle ?

• Cut a string the length of a diameter.
• Measure the circumference with that string.
• Record your results for 3-5 different circles.

5
Summary How many diameters in the circumference
of a circle?

• How many diameters fit into the circumference?
• Do the same number of diameters fit around a
  circle no matter what size ?
• If the diameter is 1 inch, what would the
  circumference be?

6
The Silo Problem

• My uncle and I were looking at
• buying a grain silo from a
• neighbor. My uncle had
• already prepared a
• square area on which
• to place the silo. We
• needed to know if the silo
• would fit inside the square but
• we didnt want to go inside the
• silo to measure the diameter.

What do you think we did ?
7
Graphing the relationship ofdiameter to
circumference

• Plot relationship between diameter and
  circumference by placing circle of different
  sizes on the x-axis to mark the diameter.
• Roll the circle OR measure with a string the
  circumference of each circle
• Make a point where the diameter and circumference
  intersect.

8
Graphing the relationship ofdiameter to
circumference

• Where are the points in relationship to the lines
  for the rule and ?
• What is the relationship between the diameter and
  the circumference of a circle ?
• Is the relationship between diameter and
  circumference linear? How do you know?

9
ELL strategies

• Sentence starter
• Non linguistic representations
• Explicitly modeling tasks
• Real manipulatives
• Structures for language practice
• Elaboration

10
ELL strategy Elaboration

• Simple, efficient strategy for introducing and
  making meaning of vocabulary
• Purposefully plan and do it throughout lesson
• Repetition
• Replaces need for vocabulary lists and dictionary
  definitions
• Connect to a non-linguistic representation
• High expectations for language acquisition.
• Need to assess in order to leave elaboration
  behind
• Can lead to extension