A Mathematics/Science Partnership Project

Kalamazoo Area Algebra Project

Western Michigan University

Kalamazoo Math/Science Center

School Districts in Southwest Michigan

KA2P Partners

Mathematics and Science PartnershipCompetitive Grant Program –

Michigan Department of Education

Financing for Project Provided By

Dr. Jane-Jane LoDepartment of MathematicsWestern Michigan University

Dr. Dana CoxDepartment of Mathematics

Miami University

Understanding Proportion–I

Thinking and Modeling Multiplicatively

Introduction: Goals, Objectives, Content Expectations

Lesson 1: Scaling Images

Lesson 2: Human Proportions

Reflections and Evaluation

Understanding Proportion-I

Engage teachers by utilizing proportional reasoning in various contexts

Provide teachers with pedagogical knowledge for proportional reasoning concepts leading to greater understanding of the concepts and improved skills by their students

Introduction:Purpose of Module

Explore two different contexts for proportional reasoning: Scaling Images and Human Proportion

Apply geometric, numeric, statistical, and algebraic reasoning when exploring proportional contexts

Objectives of Understanding Proportion – I Session

Michigan Content Expectations relevant to this module on understanding proportion are presented in the next two slides.

Each slide corresponds to one of the proportional reasoning contexts explored in this session.

Take a few minutes to review these content expectations.

Relevant Content Expectations

Michigan GLCEs Related toScaling Images

G.TR.07.03 Understand that in similar polygons, corresponding angles are congruent and the ratios of corresponding sides are equal; understand the concepts of similar figures and scale factor.

G.TR.07.04 Solve problems about similar figures and scale drawings.

Michigan GLCEs Related to Human Proportions

A.RP.07.02 Represent directly proportional and linear relationships using verbal descriptions, tables, graphs, and formulas, and translate among these representations.

Grade 7:

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Recognize and represent proportional relationships between quantities.

Common Core Standards

Grade 8:

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Common Core Standards

In this section you will investigate the following:

1. What is the difference between additive and multiplicative change?

2. What does proportional growth look like? What does it not look like?

3. Why is similarity so difficult for students to understand?

Lesson 1:Scaling Images

Let’s begin with an informal definition of what it means for two shapes to be similar to one another.

To be similar, two images must look identical, but may be different sizes. Only one of the hearts on the right is similar to the one on the left. The red one on the far right is too tall (or not wide enough) to be similar to the original. The pink one is just right.

Original Heart

Consider the images on the next few slides. Gauge for yourself which pairings are similar and which are not.

On each slide, make your decision first.

Discuss your decision with a colleague. Do you both agree? Justify your decision with mathematical reasoning.

There is a red arrow just like the one below on each slide. When you are ready, click on the red arrow to view the answer and to investigate students’ possible thinking processes.

Click for solution

Student Reasoning: Although middle school students may describe the reason as above, they may not. Students have had experiences with digital cameras and framing pictures in different ways. They may interpret the shape on the left as “more zoomed in” than the one on the right. They may say this means the shapes are not the same. They may also reason that these are ‘pictures’ of the same shape.

When we simply compare the parallelograms, it is possible to prove that these are similar by looking at the relationship of the angles (corresponding angles must have the same measure) and by establishing that the ratio of the corresponding side lengths are equal. However, by adding the star to the center of the shape, we must also consider less obvious lengths in our analysis. For instance, we must consider the lengths and angles that create the star, but also the position of the star inside the parallelogram. In other words, the gaps between the star and the parallelogram frame. It is difficult for students to see gaps or spaces as lengths to measure and compare.

These shapes are not similar. The parallelogram portion of the shape has been doubled in size, but the star remains the same size.

Click for solution

Student Reasoning: By describing the size and position of the girl within the parallelogram, students get closer to perceiving the importance of not only drawn line segments, but also implicit distances such as the height of the girl, the length of her ponytail, and the distance of her cheek from the right hand side of the frame. Once they have perceived these distances, it is possible to guide them to a deeper understanding of proportion and scale.

These shapes are not similar. The parallelograms are similar and related by a scale factor of 2. The girls are similar and related by a scale factor of 2. However, the girl is closer to the left side of the larger shape and more toward the center of the right.

Student responses to the complex pairs that you saw on the previous slides indicate that students struggle to understand similarity—even after formal instruction. In fact, students struggle to apply what they have learned about simple shapes like rectangles and triangles to more complex scaling problems. Our next activities will continue to investigate these difficulties.

Before we begin,1. On a blank sheet of paper, sketch a large

heart shape. Go ahead and fill the page.2. Draw another heart on the same sheet that

is similar to the original, but smaller.

The link below shows the movie, “How the Grinch Stole Christmas” (1966).

http://www.myspace.com/video/sneetches/how-the-grinch-stole-christmas/5378610

The clip from 1:43-2:08 shows the Grinch’s heart.

Discuss with a partner:Are the hearts on the Grinch’s x-ray similar to one another?

What does it mean if the Grinch’s heart is ‘two sizes too small?’

Warner Brothers Pictures ©1966

In the Grinch video, we see concentric hearts; one heart is drawn inside another and they share a center point. Examine the following student reasoning about the scaled hearts that she drew:

Ellie: Sometimes I try to draw hearts like this. I draw another heart around the original like a frame. I try to keep the distance even all the way around. I can make them bigger or smaller. They always turn out looking puffy, though. If I were a better artist, I might be able to do it and have all the hearts look identical to one another.

Discuss: The biggest heart looks very different from the smallest in this picture. Are they similar? Why do Ellie’s hearts turn out looking so different?

One way of examining Ellie’s method more closely is by trying it out ourselves. Hearts are complex figures. They have very few straight, measurable lines. A rectangle, in comparison, has only straight, measureable lines. For now, let’s consider Ellie’s method with a rectangle.

Use Ellie’s framing method to scale a rectangle. 1. Using grid paper, Grid Paper, draw a rectangle that is 3 units by 6 units. 2. ‘Frame’ this rectangle in another, keeping the distance between the two

shapes constant. Note the dimensions of your new rectangle. Perhaps it is a 5 unit by 8 unit rectangle.

3. Draw a few more—both larger and smaller. Be as accurate as you can.4. Create a data table to keep track of the dimensions of your rectangles.

Discuss: What is happening to your rectangles? Are some or all of them similar? Make an argument to support your answer using the dimensions you have recorded in your table.

Long side length:

6 …

Short side length:

3 …

Click forHelpful

Suggestions

Before you continue, revisit: Does Ellie’s Framing Method create similar

rectangles?Discuss your thoughts with your group.

Scaling Images

4

3 5

8

610

Thinking numerically may help us consider our rectangles. When two figures are similar, the ratios of all pairs of corresponding lengths are equal. This ratio is called the scale factor.

To illustrate, consider these two similar rectangles and labeled measurements. The scale factor is 2 since

63

84

105= = = 2

Examine the relationship between these similar hearts closely. What does the ‘frame’ look like? It is colored blue here. If Ellie’s framing technique creates similar hearts, then the distance from B and B’ should be the same as from C to C’. Is this true?

C’

B

C

B’

A

Consider the distance of points from the center point, A. Since the hearts are similar, we know that the ratio of corresponding lengths are equal.

We can see that AB ≠ AC. What does that mean about the relationship of distances AB’ and AC’?

Does Ellie’s method create similar hearts?

Ellie’s method indicates an important idea behind proportion. Proportion is a multiplicative concept. That means that in order for two quantities to be proportional, there must be a constant multiplicative relationship between them.

When we use Ellie’s Method, we are creating figures additively because the difference around the shape is constant—we are adding the same amount of space at all points.

Thus, Ellie’s method does not preserve the proportion of the shapes—no matter how accurately you frame the picture.

Ellie’s method of scaling shapes may have seemed to work on the complex heart shape. It may even have appeared to scale the rectangle proportionally until you looked more closely.

Students struggle to apply to complex shapes what they learn about proportion and similarity.

Technology can often be utilized in the classroom to enhance comprehension and provide practice with scaling images.

Seeing Math by the Concord Consortium has developed the “Proportioner” software that allows students to explore proportions among linear and area measures.

Spend some time exploring this tool and consider how you might incorporate its use in your classroom to help your students develop their proportional reasoning in the area of scaling images.

http://seeingmath.concord.orgresources_filesProportioner.html

Using Technology: Proportioner

To Review:

Geometry gives us an opportunity to visualize a multiplicative relationship and make sense of proportion.

Simple shapes like triangles and quadrilaterals may limit the way students perceive and describe proportional change. By introducing more complex shapes and encouraging students to detect more complex distortion, students may perceive more varied lengths within shapes and develop a deeper understanding of proportional growth and similarity.

You were able to use geometric reasoning to analyze what shapes look like when they are scaled by adding a constant amount of space and when they are scaled by multiplying the dimensions of that space by a constant amount.

You were also asked to reason about proportion numerically. In the next section you will take a different look at proportion using different representations and both numerical and statistical reasoning.

Lesson 2:Human Proportions

In this section you will investigate the following:

1. Do multiplicative relationships exist in other contexts?

2. How can I engage my students in collecting data to investigate proportion?

3. What is the difference between a data table and a rate table?

In the movie Pretty Woman, Vivian (Julia Roberts) suggests to Edward that there is a correlation between the length of your foot and your forearm. This was not the first time that this idea occurred to someone. In the late 1400s, Da Vinci studied the proportions of the male human body. In fact, there are many relationships just like this one.

“Did you know that your foot is as big as your arm from your elbow to your wrist?”

Here are a few other relationships noted by Da Vinci: a palm is the width of four fingers a man's height is 24 palms the length of a man's outspread arms is equal to his height the distance from the hairline to the bottom of the chin is

one-tenth of a man's height the distance from the elbow to the tip of the hand is a

quarter of a man's height the length of the hand is one-tenth of a man's height the length of a man's foot is one-sixth of his height

Discuss with others: How could you engage your students in exploring the validity of these claims?Describe the role of proportion in these relationships. Where can you find multiplicative relationships?

This data set was collected from both male and female adults (age>18). All measurements are in inches.

Gender Height Forearm Foot Length

Wingspan Length of hand

F 69 10 10 68 8

M 74 11.25 12.25 74 9

F 67.5 9 9 67 7.5

M 66 9 10 61 7

M 68 10.5 10 68 8

F 66 10.25 10 67 7.75

M 71 11 12 73 9

M 72 11 11.5 72 8

F 65 9 9.5 64.75 7

M 67.5 9 10.5 68 7

F 64 9.25 9 65.75 6.6

M 70 9 10 69 8

F 68 9.75 10 67.5 7.25

M 71 10.5 13 74 8.75

F 65.25 9 9 65.25 7

Complete the following questions.Discuss your answers with others.1. Consider just the data for Forearm

(measured from inner elbow to inner wrist) and Foot Length. Do you think that Vivian had a point? Is everyone’s foot as long as her or his forearm?

2. Using a graphing calculator, make a scatterplot of the data. What does the graph look like? Does it matter which variable you graph on the x- and y-axis?

3. Draw the line y=x, which illustrates all the perfect cases where Forearm Length=Foot Length. How close do the data points get to this line?

Forearm Foot Length

10 10

11.25 12.25

9 9

9 10

10.5 10

10.25 10

11 12

11 11.5

9 9.5

9 10.5

9.25 9

9 10

9.75 10

10.5 13

9 9

This table is a lot like the table you created a few slides back to keep track of the length and width of some rectangles you drew. Each row in this case is a different person’s data. In the previous table, each column was a different rectangle’s data.Discuss: Did it matter in what order we put the rows/columns? Would anything be gained by organizing the rows or columns in numerical order?

Forearm Foot Length

10 10

11.25 12.25

9 9

9 10

10.5 10

10.25 10

11 12

11 11.5

9 9.5

9 10.5

9.25 9

9 10

9.75 10

10.5 13

9 9

Long side length:

6 8 9 4 ?

Short side length:

3 5 ? 1 ?

Let’s use Da Vinci’s claim that a man’s height is equivalent to 24 palm lengths to make some predictions about some famous people.The relationship can be described with the following rule:

Height = 24 × Palm Length

1. Enter the rule (y=24x) into your graphing calculator and look at the generated table of values. Select an appropriate range of values from the Table Set menu.

2. Yao Ming is 90 inches tall. Use your table to predict how long his palm is.

3. Ryan Seacrest is rumored to be 67 inches. How does his palm compare to Yao Ming’s?

Discuss your answers with the group.

Palm Length

Height

2.50 60.00

2.75 66.00

3.00 72.00

3.25 78.00

3.50 84.00

3.75 90.00

4.00 96.00

Here is a table that is like the one you generated on your calculator. We can refer to this type of data table as a rate table because it is generated to reflect a specific rate of change.

Discuss:1. How can you change the Table Setting on

your calculator so that it will display this table?

2. As the palm lengths go up by .25 inches each entry, by what do the heights increase?

3. Where is Da Vinci’s “times 24” relationship visible in this table?

4. Would a height of 60 and a palm length of 2.7 ever appear on this table? If so, where?

5. How is this table different from the data tables we made earlier?

6. How might you encourage your students to explore these relationships in real life?

S. Megan Che (Mathematics Teaching in the Middle School, March 2009) has created an activity utilizing measurement which could be used to further your students’ comprehension of proportional reasoning.

Her article, “Giant Pencils: Developing Proportional Reasoning,” allows students the opportunity to discover that proportional relationships are multiplicative rather than additive by comparing their own body measurements to a giant's body measurements.

If you have access to this article (available online to NCTM members), spend some time perusing this article and consider how you might implement it into your classroom.

Interventions: Mathematics Teaching in the Middle School Article

Several follow-up activities that would be appropriate applications of human proportions can be found on-line:

On NCTM’s “Figure This!” students are asked to solve a proportional reasoning problem that involves using their body measurements to find those of the Statue of Liberty.

http://figurethis.org/challenges/c61/challenge.htm

Take some time to explore this site and consider how you could incorporate the ideas shown in your own classroom.

Interventions: Figure This!

The Modeling Middle School Mathematics project website displays a “Body Ratios” unit from the MathThematics program (McDougal Littell). Included on the site are videos of students completing the activities, teacher pages, and student work.

http://www.mmmproject.org/br/mainframe.htm

Take some time to explore this site and consider how you could incorporate the ideas shown in your own classroom.

Interventions: MMM Project

To Review: Historically, the concept of proportion was developed to

solve real life problems. Mathematical history and real life contexts are good sources for statistical investigation.

Data collected from real life situations are called experimental data. They may suggest certain numerical relationships between variables, but they DO NOT conform to the relationship precisely.

Data generated from the hypothesized numerical relationship (expressed in symbolic form) is called theoretical data. The relationship between the experimental theoretical data can be analyzed with a graphing calculator.

Human ratios can be used for estimation, and are good

examples of direct measuring with non-standard units (e.g. palm length).

Looking Ahead to Module II

In the next module, we will utilize proportional reasoning to determine length and distance that can not be measured directly and examine direct and inverse proportional reasoning problems.

Module Reflection

If you are now teaching proportional reasoning or have in the past, think about any strategies you have previously used for teaching scaling images and human proportions. Compare and contrast your strategies with those discussed in this session.

Many middle school teachers have their own favorite application problems for teaching proportional reasoning using scaling images and human proportions effectively. Share yours with others at this time.

Evaluation

Write a statement or two that summarizes what you have learned during this session.

What would you like to have added to this first session on the teaching and learning of proportional reasoning using scaling images and human proportions?

E-mail your responses for the above to ruth.a.meyer@wmich.edu

References

Jones, C. & Washam, B. (Directors). (1966) sneetches (Poster) . How the Grinch stole Christmas [Motion picture]. Los Angeles: MGM Television. Retrieved March 30, 2011, from http://www.myspace.com/video /sneetches /how-the-grinch-stole-christmas/5378610.

Marshall, G. (Director). (1990). Pretty woman [Motion picture]. Los Angeles: Touchstone Pictures.

The Math Lab.(1999-2005). Large grid paper. Retrieved March 31, 2011, from http://www.themathlab.com/toolbox/ general%20stuff/ newpage6.htm.

NCTM.(2000-2011). Figure this: Status of Liberty. Retrieved March 31, 2011, from http://figurethis.org/challenges/c61/challenge.htm.

Tuckey, C. (No date). MathThematics: Body Ratios. Retrieved March 31, 2011, from http://www.mmmproject.org/br/mainframe.htm.

Unknown author. (no date). Vitruvian man. March 31, 2011, from http://www.absoluteastronomy.com/topics/Vitruvian_Man.

Appendix

Facilitator Notes◦ Proportional Reasoning - I - Facilitator Notes.pdf