Fraction Applets for Developmental

Mathematics Students

Wade Ellis

West Valley College (retired)wellis@ti.com

Fraction Applets for Developmental Mathematics Students

Developmental Mathematics students have problems learning algebra based in part on their misconceptions about fractions and fraction operations. This presentation will demonstrate the instructional use of fraction and proportional reasoning applets along with inquiry questions that enhance and deepen student understanding of fractions sufficient to improve student performance in algebra.

A Learning TrajectoryAlgebra

FractionsRatios

Rates

Coordinate Axes

by x

a

y mx b

ax b c

OutlineIntroduction

Procedures and Understanding

Research Basis for an Approach to Fractions

Examples of Applets (Lua and Nspire)

Basic Ratio Concepts Require Fractions

Research Basis for Applet & Activity Development

Examples of Activities for Applets

Comment and Questions

Learning FractionsIf you are training someone to be a retail clerk, and you believe that that person will never need to know much more math than a retail clerk knows, then you can teach fractions using standard algorithms for doing common fraction problems.  But, if you think that the person you are teaching might need to know more advanced mathematics later, then you should teach fractions in a different way. 

Jim Pellegrino Distinguished Professor of

Cognitive Psychology

University of Illinois at Chicago

Learning Fractions (Cont’d)In math, you can teach arithmetic by simply teaching the most efficient arithmetical algorithms or you can teach it in a way that greatly facilitates the learning of algebra – so you understand the idea of equivalence . . . , not just what you need to do to execute procedures.  . . . Research shows what kids understand and what they don’t understand depends very much on how we teach the material.

Jim Pellegrino

James Stigler: UCLA Psychology Dept.in May 2011 MathAMATYC Educator

Students who have failed . . .[might succeed] if we can first convince them that mathematics makes sense . . .

. . . key concepts in the mathematics curriculum . . . included comparisons of fractions, placement of fractions on the number line, operations with fractions/decimals/percents, ratio, . . .

. . . the ability to correctly remember and execute procedures . . . is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas.

Finally, when students are able to provide conceptual understanding, they also produce correct answers.

James Stigler: UCLA Psychology Dept.Author of The Learning Gap

Students who have failed . . .[might succeed] if we can first convince them that mathematics makes sense . . .

. . . key concepts in the mathematics curriculum . . . included comparisons of fractions, placement of fractions on the number line, operations with fractions/decimals/percents, ratio, . . .

. . . the ability to correctly remember and execute procedures . . . is a kind of knowledge that is fragile without deeper conceptual understanding of fundamental mathematical ideas.

Finally, when students are able to provide conceptual understanding, they also produce correct answers.

Research Basis for an Approach to Fractions

Fraction Constructs (Behr, Lesh, Post and Silver, 1983)

 

Fractions According to Prof. Wu

A fraction is a point on the number line

Unit fractions are emphasized• (1/b is a unit fraction, b is a positive integer)

Common denominators

Improper fractions presented long before mixed numbers

Fraction Applets

What is a Fraction?

Creating Equivalent Fractions

Adding and Subtracting Fractions withCommon Denominators

Fractions and Unit Squares

Adding Fractions with Unlike Denominators

Division of a Fraction by a Fraction*

Fraction Applets

Basic Ratio Concepts Require Fractions

Basic Concepts

1. Ratio as a ratio relationship between two quantities

2. Ratio and rate

– ratio as a relationship

– rate as a fraction with units

3. Unit rate b/a associated with a ratio a:b with a ≠ 0

4. Equivalent ratios

5. Percent of a quantity as a rate per 100

6. Constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions

Ratio of Quantities a:b

Part to Whole Part to Part

Fraction

Number

Point on the Number Line

Length Area

Percent Unit RateProportion

Rate y kxa c

b d

What do we gain?Ratios are much more than just a different notation for fractions; ratios communicate a relationship between quantities

The story emphasizes paired quantities over and over.

Paired quantities leads naturally to graphs and to proportional relationships

The constant of proportionality relates to both the graphical idea of slope, the physical idea of rate, geometrical notion of scaling

Emphasis on variety of strategies to solve ratio/proportion problems (ratio tables, double number lines, graphs, …)

Algebraic Use of Fractions and Ratios

Students use fractions and ratios in algebra when they study similar figures, slopes of lines, solving linear equations and proportional reasoning problems (and later when they study sine, cosine, tangent, and other trigonometric ratios in high school).

Research Basis for Applet & Activity Development

Evidence from many sources suggests students often do not understand fundamental mathematical concepts.

Our hypothesis: Consider another approach rather than continuing

what has been unsuccessful for many Use interactive dynamic technology to support the

development of understanding, especially of “tough to teach/tough to learn” fundamental concepts.

Building Concepts

Burrill, Dick, & Ellis, 2013

Engaging in a concrete experience

Observing reflectively

Developing an abstract conceptualization based upon the reflection

Actively experimenting/testing based upon the abstraction

People learn by

Zull, 2002

As a tool for doing mathematics - a servant role to perform computations, make graphs, …As a tool for developing or deepening understanding of important mathematical concepts

The Role of Technology

Dick & Burrill, 2009

Principles for effectively integrating interactive dynamic technologies in the classroom:

The Action-Consequence Principle

The Questioning Principle

The Reflection Principle

Burrill, Dick, & Ellis, 2013

Conceptual Knowledge:– Makes connections visible, – Enables reasoning about the mathematics, – Less susceptible to common errors, – Less prone to forgetting.

Procedural Knowledge: – Strengthens and develops understanding– Allows students to concentrate on relationships

rather than just on working out results

NRC, 1999; 2001

Focus of an Activity

On fundamental concepts

One or two ideas per activity

Follow a learning trajectory supported by research

Recognize student misconceptions/difficulties

Ratio Activity Examples

What is a Ratio?

Ratio Tables

Building a Table of Ratios

Connecting Ratios to Equations

Variables and Expressions

Ratio Applets

Teacher Notes

The Mathematical Focus of the Activity

Objectives of the Activity

About the Applet

Sample Questions

A Ratio Problem

7. Suppose the ratio was 5 to 3. If there were a total of 120 circles and squares, how many squares would there be? Explain how you found your answer.

An Algebraic Solution to A Ratio Problem

The Learning TrajectoryAlgebra

FractionsRatios

Rates

Coordinate Axes

by x

a

y mx b

3rd Grade

5th Grade

7th Grade

What you teach.

How you teach it.

1. What is a Fraction?

2. Equivalent Fractions

3. Fractions and Unit Squares

4. Creating Equivalent Fractions

5. Adding & Subtracting

Fractions with Common Dens.

6. Adding Fractions with Unlike Denominators

7. Fractions as Division

8. Mixed Numbers

Building Concepts: Fractions9. Multiplying Whole Nos.

and Fractions10. Fraction Multiplication11. Dividing a Fraction by a

Whole Number12. Division of Whole

Numbers by a Fraction13. Dividing a Fraction by a

Fraction14. Units Other Than Unit

Squares15. Comparing Units

Building Concepts: Ratios1. Introduction to Ratios

2. Introduction to Rates

3. Building a Table of Ratios

4. Ratio Tables

5. Comparing Ratios

6. Connecting Ratios and Fractions

7. Double Number Line

8. Connecting Ratio to Rate of Change

9. Adding Ratios10. Proportions11. Proportional

Relationships 12. Solving Proportions

13. Ratio and Scaling14. Ratio and Similarity

Questions and Comments

wellis@ti.comWu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79

www.education.ti.com

Additional Material

Understand ratio concepts and use ratio reasoning to solve problems.1. Understand the concept of a ratio and use ratio language to describea ratio relationship between two quantities.

CCSSM, 2010

What is a ratio?

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compareratios.

CCSSM, 2010

Ratio Tables/ Connection to Graphs

From ratios to rates to

proportions

Building Concepts: Ratios, 2014

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.c. Represent proportional relationships by equations.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

CCSSM, 2010

Proportions

Connection to GeometryThey can apply a scale factor that relates lengths in two different figures, or they can consider the ratio of two lengths within one figure, find a multiplicative relationship between those lengths, and apply that relationship to the ratio of the corresponding lengths in the other figure.

When working with areas, students should be aware that areas do not scale by the same factor that relates lengths.

ReferencesBehr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press.Building Concepts: Ratios & Proportional Reasoning. (2014). Texas Instruments Education Technology. education.ti.com

Burrill, G., Dick, T., & Ellis, W. (2013). Design principles for interactive learning technology. Presentation at Learning Forward. Dallas TXChance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The role of technology in improving student learning in statistics.” Technology Innovations in Statistics Education 1. pp. 1-26. Retrieved from: http://eschlarship.org/uc/item/8sd2t4rrCharalambos, Y., & Pitta-Pantazi, D. (2005). Revisiting a theoretical model on fractions: Implications for teaching and research. 2005. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 233-240. Melbourne: PMECommon Core Standards. College and Career Standards for Mathematics 2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA).

delMas, R., Garfield, J., & Chance, B. (1999) A model of classroom research in action: Developing simulation activities to improve students’ statistical reasoning. Journal of Statistics Education, 7(3), www.amstat.org/publications/jse/secure/v7n3/delmas.cfm

Dick, T. & Burrill, G. (2009). Technology and teaching and learning mathematics at the secondary level: Implications for teacher preparation and development. Presentation at the Association of Mathematics Teacher Educators, Orlando FL.

Empson, S., & Knudsen, J. (2003). Building on children’s thinking to develop proportional reasoning. Texas Mathematics Teacher, 2, 16–21.

Johanning, D. (2008). Learning to use fractions: Examining middle school students’ emerging fraction literacy. Journal for Research in Mathematics Education. 39(3), 281-310.

Lamon, S. (1999). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Hillsdale, NJ: Erlbaum.

Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte, NC: Information Age Publishing.

Lane, D. M., & Peres, S. C. (2006). Interactive Simulations in the Teaching of Statistics: Promise and Pitfalls. In A. Rossman and B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics. [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.National Assessment for Educational Progress (2013). Released Item. Grade 8. National Center for Educational Statistics.

National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press. www.nap.edu.

Program for International Assessment (PISA). 2012 Release items. Organization for Economic Co-operation and Development. http://www.oecd.org/pisa/pisaproducts/pisa2012-2006-rel-items-maths-ENG.pdf

Progressions for the Common Core State Standards in Mathematics (2011). 6-7, Ratio and Proportional Reasoning.

Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students, Educational Studies in Mathematics, 43(3), 271-292.

Wu, H. (2011). Understanding Numbers in Elementary School Mathematics, American Mathematical Society. http://www.ams.org/bookstore-getitem/item=mbk-79

James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning. Association for Supervision and Curriculum Development, Alexandria, Virginia.