Integrating Mathematical Integrating Mathematical Tasks in Elementary and Tasks in Elementary and

Middle Grades ClassroomsMiddle Grades Classrooms

Drew PollyDrew Polly

UNC-CharlotteUNC-Charlotte

Mathematical TasksMathematical Tasks

““Activities with mathematical goals”Activities with mathematical goals” Provide opportunities for students to Provide opportunities for students to

apply content and skills in contexts apply content and skills in contexts that are:that are:• ComplexComplex• Meaningful and relevantMeaningful and relevant

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 tasks Higher-level tasks

(Smith & Stein, 1998)

Designing Effective TasksDesigning Effective Tasks

1) Start with a mathematical goal or 1) Start with a mathematical goal or objectiveobjective

2) Select appropriate tasks2) Select appropriate tasks• Procedures with connectionsProcedures with connections• Doing mathematicsDoing mathematics

3) Integrate technology 3) Integrate technology when when appropriateappropriate

Paper HexagonsPaper Hexagons• You and your friends have paper hexagons. You split yours You and your friends have paper hexagons. You split yours

into 2 equal pieces, Shanika splits hers into three pieces, while into 2 equal pieces, Shanika splits hers into three pieces, while Tomas splits his into 6 equal pieces. As you each put your Tomas splits his into 6 equal pieces. As you each put your pieces on the table, Shanika says to Tomas, “It looks like pieces on the table, Shanika says to Tomas, “It looks like some of your pieces can be combined to get my piece.” Is this some of your pieces can be combined to get my piece.” Is this true? If so, how many pieces does Tomas need to combine?true? If so, how many pieces does Tomas need to combine?

• Tomas says to you, “I can also put some of my pieces together 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 to equal one of your pieces.” Is this true? If so, how many pieces must be combined?pieces must be combined?

• Lastly, Shanika says to you, “I can also put some of my pieces 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 together to equal one of your pieces.” Is this true? If so, how many pieces can be combined?many pieces can be combined?

Paper HexagonsPaper Hexagons

Approaches using the pattern blocks??Approaches using the pattern blocks??

http://www.arcytech.org/java/patterns/patterns_j.shtmlhttp://www.arcytech.org/java/patterns/patterns_j.shtml

Task DesignTask 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.shtmlhttp://www.arcytech.org/java/patterns/patterns_j.shtml

Paper HexagonsPaper Hexagons

What content?What content?

What do tasks about this standard typically look like?

Fractions- Grade 4Fractions- Grade 4

Grade 4: 1.03Grade 4: 1.03     Solve problems using models,     Solve problems using models, diagrams, and reasoning about fractions and diagrams, and reasoning about fractions and relationships among fractions…relationships among fractions…

Other pattern blocks tasksOther pattern blocks tasks• Cover the hexagon in as many ways as Cover the hexagon in as many ways as

possible.possible. What solutions are there?What solutions are there?

Solutions to Covering the HexagonSolutions to Covering the Hexagon

Counting CoinsCounting Coins

You have 19 coins in your pocket. You have 19 coins in your pocket. The coins equal a dollar. What are The coins equal a dollar. What are the possible combinations of coins the possible combinations of coins that you have?that you have?

Possible approaches??Possible approaches??

Give it a try on paper.Give it a try on paper.

Counting CoinsCounting Coins

Let’s use Excel!Let’s use Excel!

Crayon ProblemCrayon Problem I have 10 fruits. They are either apples or I have 10 fruits. They are either apples or

oranges. How many of each do I have? oranges. How many of each do I have? Show all the possible combinations.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 If there were an even number of apples how many combinations do you have?how many combinations do you have?

Adapted from Investigations in Number, Data and Space

Crayon ProblemCrayon Problem Appropriate manipulatives?Appropriate manipulatives?

Adapted from Investigations in Number, Data and Space

•Cuisenaire Rods

•Base-10 blocks

•Ten frames

Three DaughtersThree Daughters A census taker walks up to a house and asks the man answering 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," the door to state the number of people in his household. "Five," he says. "There's me, my wife, and my three daughters." he says. "There's me, my wife, and my three daughters."

"How old are your daughters?" the census taker asks."How old are your daughters?" the census taker asks.

"The product of their ages is 72, and the sum of their ages is "The product of their ages is 72, and the sum of their ages is the number on my house." the number on my house."

The census taker leaves, but soon he comes back and replies, The census taker leaves, but soon he comes back and replies, "That's not enough information." "That's not enough information."

"Oh, I forgot, the youngest one likes chocolate pudding." "Oh, I forgot, the youngest one likes chocolate pudding."

How old are the three daughters? Explain how you found the How old are the three daughters? Explain how you found the result. result.

Three DaughtersThree Daughters Approaches?Approaches?

Content embedded in the task?Content embedded in the task?

Clothes ShoppingClothes Shopping• Melanie is  shopping for work clothes.  She Melanie is  shopping for work clothes.  She

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

Millie’s GardenMillie’s Garden

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

Millie’s GardenMillie’s Garden

• Approaches?Approaches?

• Appropriate manipulatives?Appropriate manipulatives?

• Appropriate technologies?Appropriate technologies?

Millie’s GardenMillie’s Garden

• ManipulativesManipulatives Centimeter square tiles Centimeter square tiles

• Cash register tapeCash register tape Base-10 blocks- cubesBase-10 blocks- cubes

• TechnologiesTechnologies ExcelExcel Area ExplorerArea Explorer on the Internet on the Internet

Resources/Contact InfoResources/Contact Info• Technology Integration in Mathematics Technology Integration in Mathematics

http://lpsl.coe.uga.edu/mile3/timhttp://lpsl.coe.uga.edu/mile3/tim

• ElemMathElemMath http://http://www.drewpolly.com/elemmathwww.drewpolly.com/elemmath//

• Drew PollyDrew Polly abpolly@email.uncc.eduabpolly@email.uncc.edu http://www.drewpolly.com/http://www.drewpolly.com/