This document summarizes key concepts about proportional reasoning. It defines proportional reasoning as a mathematical relationship between two quantities that involves a constant multiplicative relationship. It discusses proportional reasoning as developing between concrete and formal operations. It also provides examples of using proportional relationships to solve problems and discusses research on how to best teach proportional reasoning concepts to students.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
Vedic Mathematics is a system of mathematics that allows problems to be solved quickly and efficiently. It is based on the work of Sri Bharathi Krishna Thirthaji Maharaja (1884 – 1964), who devised the system from a close study of the Vedas. The Vedas are ancient scriptures of India that deal with many subjects. It is based on 16 sutras (aphorisms) from the Vedas that provide a principle or a rule of working to solve a problem. These sutras may be ancient in origin, but are still relevant to modern day mathematics.
Vedic Mathematics is a system of mathematics that allows problems to be solved quickly and efficiently. It is based on the work of Sri Bharathi Krishna Thirthaji Maharaja (1884 – 1964), who devised the system from a close study of the Vedas. The Vedas are ancient scriptures of India that deal with many subjects. It is based on 16 sutras (aphorisms) from the Vedas that provide a principle or a rule of working to solve a problem. These sutras may be ancient in origin, but are still relevant to modern day mathematics.
Scientix 8th SPNE Brussels 16 October 2015: Functional thinking in students a...Brussels, Belgium
Presentation of the project "Functional thinking in students at elementary education as an approximation to algebraic thinking"- Spain, held during the 8th Science Projects' Networking Event, Brussels, 16 October 2015
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How has the artist used colors in the work(s)?
- What sort of effect do the colors have on the artwork?
- How as the artist used shapes within the work of art?
- How have lines been used in the work(s)? Has the artist used them as an
important or dominant part of the work, or do they play a different roll?
- What role does texture play in the work(s)? Has the artist used the illusion of texture or has the artist used actual texture? How has texture been used within the work(s)?
- How has the artist used light in the work(s)? Is there the illusion of a scene with lights and shadows, or does the artist use light and dark values in a more abstracted way?
- How has the overall visual effect or mood of the work(s)? Been achieved by the use of elements of art and principles of design.
- How were the artists design tools used to achieve a particular look or focus?
Critique of image/golden retriever 1 5x7.jpg
Critique of image/marilyn.jpg
Assignment 2
The goal of the next few questions is to help you intuitively understand omitted variable bias and too-many
variables bias, multicollinearity, and heteroskedasticity using a simulation in Stata.
Introduction. There's an important debate over how we can get more children to be able to read by 3rd
grade. I have heard anecdotal evidence that the state government plans the number of prisons to build based on
regressions that use current-year 3rd grade reading scores on the RHS, which might be suggestive of the importance
of this goal. But we won't worry about the causal e�ect of literacy on crime in this assignment.
As we saw in class, children from poor families are already behind in terms of reading ability in fall of Kinder-
garten. This is a problem. Local governments have two main policies to try to solve this problem: free full-day
preschool, and family income supports.
Omitted variable & too-many variables bias.
Question 1. Suppose the process that determines child test score in Kindergarten is given by
test scorei = β0 + β1preschooli + β2incomei + �i
where β1,β2 > 0 and preschooli is a continuous measure of �preschool quality.�
Preschool quality can be purchased with cash, and is purchased in cash according to the linear model
preschooli = α0 + α1incomei + ηi
You should interpret both of these linear models as structural models of human behavior. In other words, if you
gave a random family another dollar, the family would indeed purchase α1 more units of preschool quality, and the
family would purchase other children's stu� that has β2 additional e�ect on test scores. This means that the total
e�ect on test scores of giving the family another dollar is:
(a) α0β1
(b) α1
(c) β2 + β1α1
(d) α1β1 + β2
Question 2. Let's try to simulate this model in Stata. Run Part I in the �le �assignment2.do.� This will
generate a fake dataset of 1000 students for this problem a.
This learner's module discusses or talks about Quadratic Equations and Quadratic Inequalities. It also discusses the definition of Quadratic Equations and Quadratic Inequalities and also discusses the formula of Quadratic Equations and Quadratic Inequalities and it also includes an example of Quadratic Equations and Quadratic Inequalities.
39. Prototype for lesson construction Touchable visual Discussion: Makes sense Of concept 1 2 Learn to Record these ideas V. Faulkner and DPI Task Force adapted from Griffin Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity Concrete display of concept
46. Direct Variation, Linear Relationships and a connection Between Geometry and Algebra! Y = mX + 0
47. $ 225 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Lawns Mowed You have a lawn mower and your parents will pay for maintenance and gas if you make use of your summer by starting a lawn mowing business. You will make $25 for each lawn you mow. This means that, if I tell you how many lawns you mow you can figure out how much you earned OR if I tell you how much your earned, you can tell me how many lawns you mowed. Each point has TWO pieces of information! Plot the following points with Your small group: You mowed 0 lawns (0, ___) You mowed 1 lawn (1, ___) You mowed 2 lawns (2, ___) You mowed 3 lawns (3, ___) You earned $100 dollars (___, 100) You earned $200 dollars (___, 200) Look at the point that I plotted. Did I plot it correctly? How do you know?
48. $ 225 200 175 150 125 100 75 50 25 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Lawns Mowed I have a lawn mower and will mow Lawns this summer for$25.00 a lawn Money earned = $25 * Lawns mowed Y = mX
49. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 DIAMETER Pi is a proportion and the Diameter Varies Directly with the Circumference. This is just like mowing the lawns! Do you see that these two parts of the circle Vary Directly and therefore, when you plot them they form a Line? y = mx Circumference = ~3(Diameter) CIRCUMFERENCE 30 27 24 21 18 15 12 9 6 3 0 SLOPE ~ 3/1 3 1
50. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 RADIUS What is the relationship between The RADIUS and the Circumference? Circumference = ~6(Radius) y = mx CIRCUMFERENCE 30 27 24 21 18 15 12 9 6 3 0 Slope ~ 6/1
57. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 RADIUS CIRCUMFERENCE 250 225 200 175 150 125 100 75 50 25 0 Plot these points: Radius of 5 (5, ~___) Radius of 10 (10, ~ ___) Circumference of 72 (~___, 72) On the Graph we have plotted (~16, 1 00) and what other point? (____, _____). Look at the two points that are plotted. Using these two points answer the following: For every 100 change in Circumference, about what will be the change in Radius? What is the slope of this linear relationship? Δ X = Δ Y = 100 (~16, 100)
58. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 RADIUS CIRCUMFERENCE 250 225 200 175 150 125 100 75 50 25 0 Plot these points: Radius of 5 (5, ~ ___) Radius of 10 (10, ~ ___) Circumference of 72 (~___, 72) On the Graph we have plotted (~16, 100) and what other point? (____, _____). Look at the two points that are plotted. Using these two points answer the following: For every 100 gain in Circumference, about what will be the gain in Radius? What is the slope of this linear relationship? Δ X Δ Y (~16, 100) (32, 200) = ~16 = 100 100 ~16 = ~6
64. There is a linear relationship between C and r. Whenever C is increased by 100 cm, r will be increased by 100/(2 π )cm.
65. No matter what size our sphere is (initial radius), if the circumference is increased by 100 centimeters, the distance added from the center of the sphere will always be 100/6 or about 16 centimeters!
66. Pi and the earth Pi is a ratio Pi is a constant 1 2 V. Faulkner and DPI Task Force adapted from Griffin Symbols Record keeping: Generalization Mathematical Structure Sense-making of the concrete Quantity Concrete display of concept c/2*Pi + 100/ 2* Pi
72. Algebraic Thinking Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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79. Fundamental Algebraic Ideas Algebra as abstract arithmetic Algebra as the language of mathematics Algebra as a tool to study functions and mathematical modeling Dr. Shelley Kriegler, UCLA, Teacher Handbook , Part 1 Mathematical Thinking Tools Problem solving skills Representation skills Reasoning skills COMPONENTS OF ALGEBRAIC THINKING
80. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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85. Where the Research Meets the Road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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90. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
100. Prototype for Lesson Construction Touchable Visual Discussion: Makes Sense of Concept 1 2 Symbols: Simply record keeping! Mathematical Structure Discussion of the concrete Quantity: Concrete display of concept Learn to Record these Ideas V. Faulkner and DPI Task Force adapted from Griffin
101. Even Algebra can fit this mold! (av. cost)(items) + gas = grocery $ mx + b = y 1 2 Symbols Simply record keeping! Mathematical Structure Discussion of the concrete Quantity: Concrete display of concept Faulkner adapting Leinwald, Griffin How do we organize data to make predictions and decisions? Why is the slope steeper for Fancy Food then Puggly Wuggly? What about gas cost? Is there a point at which the same items would cost the same at both stores?, etc.
104. Dollar Deals Learning Objectives Dollar Deals 1 2 Symbols Simply record keeping! Math Structure Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y x=y 1x = y 1x = y+0 Ratio; Forms of same info; 1:1; Identity function
107. Puggly Wuggly vs. Dollar Deals Learning Objectives 1 2 Symbols Simply record keeping! Math Structure: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y ? Coefficient and slope How does cost affect slope? Why does PW have a steeper slope? How do you model PW? 2x=y
110. Puggly Wuggly vs. Fancy Foods Learning Objectives 1 2 Symbols Simply record keeping! Math Structure: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y ? Coefficient and slope How does cost affect slope? What does it mean that the slope lines meet at 0? How do you model FF and PW? 2.5x=y 4x = y
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113. Other Forms of the Number Relationship! 20 12.5 5 16 10 4 12 7.5 3 8 5 2 4 2.5 1 Y=4x FF Y=2.5x PW X (Items)
115. Feeding the DumDee Learning Objectives 1 2 Symbols Simply record keeping! Verbal: Mathematical Structures Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y Why do the lines intersect? When is it cheaper to shop At fancy foods? What is a Y intercept and what is a constant? 2.5x=y 4x = y
116. Feeding the DumDee What Does it Look Like? Communicating with “cell phones” and marking their points on the Chart as they go through that point in the script.
121. Feeding the DumDee--Jenn V. Jermaine Learning Objectives 1 2 Symbols Simply record keeping! Verbal: Discussion of the concrete Quantity: Concrete display of concept 1 2 3 3 2 1 x f(x) y Why don’t the lines intersect? Same slope, different Y-intercept—Will there ever be the same cost for the same number of items What is a Y= intercept and what is a constant? 2.5x=y 4x = y
134. Geometric Thinking Defining the Concept Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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136. Diagnosis Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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138. Where the research meets the road Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
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140. Classroom Application: Defining the Concept Diagnosis Where the Research Meets the Road Classroom Application
141. Foundation Level Training Understanding the Relationship between Area and Perimeter Using Griffin’s Model Mayer’s Problem Solving Model Components of Number Sense
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143. Prototype for Lesson Construction Area (Squared Measure) vs. Perimeter (Linear Measure) 1 2 Symbols Record Keeping! Math Structure: Discussion of the Concrete Quantity: Concrete Display of Concept A=L x W=10 P= 2L + 2W=14 V. Faulkner and DPI Task Force adapted from Griffin
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159. Challenge 1 What other perimeter could be shown that has the same area? Area = _____ P = ____
160. Challenge 2 A = ______ P = _____ Rearrange the area to show a diagram with a different perimeter .
161. Challenge 3 Add a square so that the area is 6 and the perimeter remains 12. A = ____ P = ____ Can you find other arrangements with an area of 6 and a perimeter of 12?
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167. How Can We Organize the Data? A = 9 sq. units P= 20 units 1 9 A = 16 sq. units P= 20units 2 8 A = 21 sq. units P= 20units 3 7 A = 24 sq. units P= 20units 4 6 A = 25 sq. units P= 20 units 5 5 A = 24 sq. units P= 20 units 6 4 A = 21 sq. units P= 20 units 7 3 A = 16 sq. units P= 20 units 8 2 A = 9 sq. units P= 20 units 9 1 A = L x W P= 2L + 2 W Width Length