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Mathematics: What's the Big Idea?

Workshop #6

Ratio and Proportion:When is a third more than a half?

Content Guide - Beryl Jackson

Supplies Needed for Workshop #6:
worksheets 1-4, pattern blocks, tangram (or tangram cut-outs),
centimeter grid paper (several sheets per participant), tape measure,
square tiles, 1 bag of M&M's® per teacher (20.9 or 47.9 gram bag),

pencils, paper, scissors, rulers, calculators, tape, a variety of colored markers


About the Workshop

What is the theme of the workshop?
Encounters with mathematics in the real world are not always limited to the neatness of the whole number system. We realize that we need a way to express quantities related to a part of a dollar, divisions of land, the likeliness of rain, or the percentage of decrease in crimes against society. We further realize that music, maps, recipes, and architectural and engineering designs provide settings that emphasize the necessity for numbers other than whole numbers. This workshop will address some issues and provide some ideas related to the teaching and learning of rational numbers and proportional relationships.

What issues does this workshop address?
We will see teachers and students at the elementary and middle school levels engaged in activities that range from students representing one-half of the same whole in multiple ways to students representing "messy fractions" on the number line. All of the activities incorporate mathematical tasks designed to reinforce concepts of rational numbers and ratios which build basic concepts while making learning fun.

What issues does this workshop address?
Historically, the study of rational numbers has been a real turnoff for many learners. Many teachers have dreaded pulling out the old unit on teaching fractions. The good news is that there are many strategies and activities that can be used which help students visualize and conceptually understand what fractions are and how they relate to each other. Furthermore, combining the study of rational numbers with that of proportional reasoning illustrates the connectivity of mathematics.

What teaching strategy does this workshop offer?
We will see that communication is key to assessing student understanding. In each of the video clips, the teachers encourage open communication among the students. Students are encouraged to state their answers, provide justification for their answers, and even suggest alternative solutions, where possible. One teacher says, "One of the benefits to having students share their strategies is that. . . they question and challenge each other in a different way so that they can get to a truer meaning of the math."

To which NCTM Standards does this workshop relate?
This workshop relates to Standard 12: Fractions and Decimals, Standard 8: Computation and Standard> 5: Estimation of the K-4 Standards, and extends to include Standard 5 Number and Number Relationships and Standard 6 Number Systems and Number Theory of the 5-8 Standards. Additionally, Mathematics as Communication is essential to building concepts with rational numbers and proportional reasoning. Also interwoven in this workshop is Standard 1: Mathematics as Problem Solving, Standard 3: Reasoning, and Standard 4: Connections.


Suggested Strategies and Classroom Activities

Are you a SQUARE, a LONG RECTANGLE, or a WIDE RECTANGLE?
Have each child, with the help of a partner, determine his/her height (H). Next, have each student determine his/her arm span (A). Students then form a ratio, comparing their height to their arm span (H:A).

  • If H:A is greater than (>) 1, the student is a LONG RECTANGLE.
  • If H:A is less than (<) 1, the student is a WIDE RECTANGLE
  • If H:A is equal to (=) 1, then the student is a SQUARE.
Discuss with the students the conditions that will make each of these relationships true. For example, if H:A > 1, then the height is greater than the arm span. If H:A < 1, then the height is less than the arm span. If H:A = 1, then the height and arm span are the same.

You may have students use non-standard or standard units for measuring. As a follow-up, prepare a class graph classifying the students in the three categories.


The Lilliputians' Measurements
When the Lilliputians measured Gulliver for a suit of clothes, they had their own special way of computing measurements. To determine if their way will work for us, the following relationships should be true:

  • Twice around the base of the thumb = once around the wrist
  • Twice around the wrist = once around the neck
  • Twice around the neck = once around the waist

To do this activity, provide each student with a piece of string (long enough to go around the waist) and centimeter rulers or tape measures. Have students construct ratios which represent the relationships described above. Students can collect class data, construct scatter plots, and view for correlation. This would be a good graphing calculator activity for middle school students.


A Fraction Equivalent to a Third
Challenge students to make a fraction (five digit number over five digit number ) equivalent to 1/3 by using the numbers 0 through 9 exactly once:

	XXXXX		1	 	-----  =	- 	XXXXX		3


Visual Representations of Fractions
The tangram, pattern blocks, square tiles, geoboards and Cuisenaire Rods® are all excellent tools to use to develop fractional concepts. The workshop provided you with a few ways to incorporate their use in the instructional program. Here are a few additional ideas:

  • Geoboards: Use the geoboards to show different ways of representing fourths and eighths.

  • Tangram: Assign a particular value to a piece of the tangram. Based on that value, find the value of all of the other six pieces. The values assigned can be values less than one or greater than one. Since there is a proportional relationship among the pieces, it does not matter.

  • Pattern Blocks: Like the tangram, a proportional relationship exists among the trapezoid, triangle, hexagon and blue parallelogram. Use these relationships to develop fractional situations.

  • Cuisenaire Rods®: Establish one rod as the whole. Have students establish the relationship of the other rods to this whole.


Suggested Strategies

It is more difficult for students to acquire conceptual understanding once they have learned rote procedures. Thus, it is essential to focus initial instruction related to fractions, decimals, ratios, and other multiplicative-based relationships on building conceptual understanding. These play a major role in the development of proportional reasoning which is said to be the cornerstone for much of the mathematics in the secondary years, and merit whatever time and effort it takes to assure careful development. Therefore, whenever possible, use concrete materials, games, pictorial representations, and real situations to assist in the conceptual development processes.



Post-Workshop Questions

  1. How has the way that you have engaged your students in the study of rational number concepts changed over the course of your teaching career? With what results?

  2. The major manufacturers of calculators have all introduced calculators which are able to perform operations with fractions. This even includes calculators with capabilities for providing different representations for a given fraction. What is your opinion on using calculators to perform computations and other operations with fractions? Should they be used in elementary and middle school classrooms?

  3. During the workshop, four ways of interpreting rational numbers were highlighted: part-whole meaning, quotient meaning, ratio meaning, and operator meaning. Where developmentally appropriate, describe activities that you have used in your instructional program which address these rational number levels of meaning.

  4. The Standards suggest that increased attention be placed on the meaning of fractions and decimals, and decreased attention be placed on fraction computation using paper and pencil. Do you agree or disagree with this suggestion? Why or why not?

  5. In the video segments highlighted during the workshop, were there any particular segments which you felt provided a model for instruction that would be most appropriate for your current teaching situation? Describe. If not, what modifications would you make to one of the lessons to make it more conducive to addressing the needs of your students?

  6. Many teachers begin the year with a study of ratio and proportion because there are so many opportunities for capturing the interest of students early in the year, and because there are many rich explorations related to this topic. If this is a topic that you introduce early in the year, how has it helped or hindered the study of other strands taught later in the year?



Pre-Workshop Assignment for Workshop #7

Your Definition of Algebra
How do you define algebra? If you had to write a definition of algebra for an upcoming mathematics dictionary, what would you write?


Algebra Magic: "Think of a Number"
Think of a number, then follow these instructions:

  1. Add 4
  2. Multiply by 3
  3. Subtract 9
  4. Multiply by 2
  5. Divide by 6
  6. Subtract the original number

(Note: Each instruction is to be applied to the previous answer.)

Now think of a different number, and follow the instructions again. Try it several times with several different numbers. What do you find? Is it surprising? Why?

Challenge #1
Try to devise a similar "Think of a Number" magic trick in which the outcome is the same regardless the starting value.

Challenge #2
Try to devise an algebra magic trick in which different starting values yield different outcomes, but you can guess the outcome if you know the starting value. You are a magician!





Mathematics: What's the Big Idea?

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