1. Integrating Mathematical Tasks in Elementary and Middle Grades Classrooms Drew Polly UNC-Charlotte

2. Mathematical Tasks “Activities with mathematical goals” “Activities with mathematical goals” Provide opportunities for students to apply content and skills in contexts that are: Provide opportunities for students to apply content and skills in contexts that are: ComplexComplex Meaningful and relevantMeaningful and relevant

3. Trajectory of Mathematical Tasks Memorization Basic facts Definitions Conversions (fraction-> decimals) Procedures without connections Algorithms Computations Limited to one representation (typically symbolic) Procedures with connections Algorithms Computations Multiple representations (table, diagram, picture) Doing Mathematics Complex, non- algorithmic tasks Can involve justification and explaining processes and approaches Lower-level tasksHigher-level tasks (Smith & Stein, 1998)

4. Designing Effective Tasks 1) Start with a mathematical goal or objective 1) Start with a mathematical goal or objective 2) Select appropriate tasks 2) Select appropriate tasks Procedures with connectionsProcedures with connections Doing mathematicsDoing mathematics 3) Integrate technology when appropriate 3) Integrate technology when appropriate

5. Paper Hexagons You and your friends have paper hexagons. You split yours into 2 equal pieces, Shanika splits hers into three pieces, while Tomas splits his into 6 equal pieces. As you each put your pieces on the table, Shanika says to Tomas, “It looks like some of your pieces can be combined to get my piece.” Is this true? If so, how many pieces does Tomas need to combine?You and your friends have paper hexagons. You split yours into 2 equal pieces, Shanika splits hers into three pieces, while Tomas splits his into 6 equal pieces. As you each put your pieces on the table, Shanika says to Tomas, “It looks like some of your pieces can be combined to get my piece.” Is this true? If so, how many pieces does Tomas need to combine? Tomas says to you, “I can also put some of my pieces together to equal one of your pieces.” Is this true? If so, how many pieces must be combined?Tomas says to you, “I can also put some of my pieces together to equal one of your pieces.” Is this true? If so, how many pieces must be combined? Lastly, Shanika says to you, “I can also put some of my pieces together to equal one of your pieces.” Is this true? If so, how many pieces can be combined?Lastly, Shanika says to you, “I can also put some of my pieces together to equal one of your pieces.” Is this true? If so, how many pieces can be combined?

6. Paper Hexagons Approaches using the pattern blocks?? Approaches using the pattern blocks?? http://www.arcytech.org/java/patterns/patterns_j.shtml http://www.arcytech.org/java/patterns/patterns_j.shtml http://www.arcytech.org/java/patterns/patterns_j.shtml

7. Task Design Approaches using the pattern blocks?? Approaches using the pattern blocks?? What fraction does each block represent? What fraction does each block represent? http://www.arcytech.org/java/patterns/patterns_j.shtml http://www.arcytech.org/java/patterns/patterns_j.shtml http://www.arcytech.org/java/patterns/patterns_j.shtml

8. Paper Hexagons What content? What content? What do tasks about this standard typically look like? Fractions- Grade 4 Grade 4: 1.03 Solve problems using models, diagrams, and reasoning about fractions and relationships among fractions…

9. Other pattern blocks tasks Cover the hexagon in as many ways as possible.Cover the hexagon in as many ways as possible. What solutions are there? What solutions are there?

10. Solutions to Covering the Hexagon

11. Counting Coins You have 19 coins in your pocket. The coins equal a dollar. What are the possible combinations of coins that you have? You have 19 coins in your pocket. The coins equal a dollar. What are the possible combinations of coins that you have? Possible approaches?? Possible approaches?? Give it a try on paper. Give it a try on paper.

12. Counting Coins Let’s use Excel! Let’s use Excel!

13. Crayon Problem I have 10 fruits. They are either apples or oranges. How many of each do I have? Show all the possible combinations. I have 10 fruits. They are either apples or oranges. How many of each do I have? Show all the possible combinations. How many combinations do you have? How many combinations do you have? If there were an even number of apples how many combinations do you have? If there were an even number of apples how many combinations do you have? Adapted from Investigations in Number, Data and Space

14. Crayon Problem Appropriate manipulatives? Appropriate manipulatives? Adapted from Investigations in Number, Data and Space Cuisenaire Rods Base-10 blocks Ten frames

15. Three Daughters A census taker walks up to a house and asks the man answering the door to state the number of people in his household. "Five," he says. "There's me, my wife, and my three daughters." "How old are your daughters?" the census taker asks. "The product of their ages is 72, and the sum of their ages is the number on my house." A census taker walks up to a house and asks the man answering the door to state the number of people in his household. "Five," he says. "There's me, my wife, and my three daughters." "How old are your daughters?" the census taker asks. "The product of their ages is 72, and the sum of their ages is the number on my house." The census taker leaves, but soon he comes back and replies, "That's not enough information." "Oh, I forgot, the youngest one likes chocolate pudding." How old are the three daughters? Explain how you found the result. The census taker leaves, but soon he comes back and replies, "That's not enough information." "Oh, I forgot, the youngest one likes chocolate pudding." How old are the three daughters? Explain how you found the result.

16. Three Daughters Approaches? Approaches? Content embedded in the task? Content embedded in the task?

17. Clothes Shopping Melanie is shopping for work clothes. She has found a dress for $75 and a two-piece suit for only $60. She does not have enough money for both, so she must choose only one. Of course, both are "dry clean only." If her dry cleaner charges $4.50 for a dress or $3 for each piece of a suit, which will be a better deal in the long run? How many times will she have to dry clean the purchased item before it is the better deal?Melanie is shopping for work clothes. She has found a dress for $75 and a two-piece suit for only $60. She does not have enough money for both, so she must choose only one. Of course, both are "dry clean only." If her dry cleaner charges $4.50 for a dress or $3 for each piece of a suit, which will be a better deal in the long run? How many times will she have to dry clean the purchased item before it is the better deal?

18. Millie’s Garden Millie is planting a garden. She needs 36 square yards to plant all of her vegetables. Fencing costs $5 per yard. If she wants to spend the least amount of money on fencing what should her dimensions be? Prove that your solution is the cheapest.Millie is planting a garden. She needs 36 square yards to plant all of her vegetables. Fencing costs $5 per yard. If she wants to spend the least amount of money on fencing what should her dimensions be? Prove that your solution is the cheapest.

19. Millie’s Garden Approaches?Approaches? Appropriate manipulatives?Appropriate manipulatives? Appropriate technologies?Appropriate technologies?

20. Millie’s Garden ManipulativesManipulatives Centimeter square tiles Centimeter square tiles Cash register tapeCash register tape Base-10 blocks- cubes Base-10 blocks- cubes TechnologiesTechnologies Excel Excel Area Explorer on the Internet Area Explorer on the Internet Area Explorer Area Explorer

21. Resources/Contact Info Technology Integration in MathematicsTechnology Integration in Mathematics http://lpsl.coe.uga.edu/mile3/tim http://lpsl.coe.uga.edu/mile3/tim http://lpsl.coe.uga.edu/mile3/tim ElemMathElemMath http://www.drewpolly.com/elemmath/ http://www.drewpolly.com/elemmath/ http://www.drewpolly.com/elemmath/ Drew PollyDrew Polly abpolly@email.uncc.edu abpolly@email.uncc.edu abpolly@email.uncc.edu http://www.drewpolly.com/ http://www.drewpolly.com/ http://www.drewpolly.com/