Higher Level Tasks[1]

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Why Should

Teachers AssignHigher Level Tasks?

North Carolina State University

Student Teachers

Fall 2010


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NCTM Problem Solving Principals

Instructional programs from prekindergartenthrough grade 12 should enable all studentsto

build new mathematical knowledge throughproblem solving; solve problems that arise in mathematics and

in oth

er contexts; apply and adapt a variety of appropriate

strategi

es to solve problems; monitor and reflect on the process of

mathematical problem solving.


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NCTM Communication Principal

Instructional programs from prekindergartenthrough grade 12 should enable all students to

organize and consolidate their mathematical

thinking through communication; communicate their mathematical thinking

coherently and clearly to peers, teachers, andothers;

analyze and evaluate the mathematical thinkingand strategies of others;

use the language of mathematics to expressmathematical ideas precisely.


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Common Core Standards

Make sense of problems and persevere in

solving them.

Reason abstractly and quantitatively. Model in mathematics.

Look for and make use of structure.


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Types of Tasks

Novice skill/procedural knowledge

Apprentice performance assessments

Expert multi-day/complex/portfolio


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Types of Mathematics TasksStein, Smith, Henningsen, & Silver, 2009

Low-Level Demands Higher-Level Demands

Memorization TasksExamples:Recall or Memorizing facts, rules or

definitionsTask follows a specified reproductionof work

Procedures with Connections TasksExamples:Focused on the use of the procedure to

develop the sense of the conceptStudent must engage in the idea tomake sense of the problem

Procedures without ConnectionsTasks

Examples:AlgorithmsFocused on the procedure/correctanswerRequires only limited cognitivedemand

Doing Mathematics TasksExamples:

Requires in-depth, conceptualthinkingRequires students to rely onexperiences and previous knowledgeto develop an answer


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Depth of Knowledge

Level 1-Recall

Level 2-Basic Application of Skill/Concept

Level 3-Strategic Thinking Level 4-Extended Thinking


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Mathematical Tasks

What is cognitive demand?

Focus is on the sort of student thinking required.

Kinds of thinking required:

Memorization

Procedures without Connections Requires little or no understanding of

concepts or relationships.

Procedures with Connections

Requires some understanding of thehow or why of the procedure.

Doing Mathematics


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Examples of Mathematical TasksLevel 1

MemorizationWhich of these shows the identity property of multiplication?

A) a x b = b x a

B) a x 1 = a

C) a + 0 = a

Procedures without ConnectionsWrite and solve a proportion for each of these:

A) 17 is what percent of 68?

B) 21 is 30% of what number?

Too much of a focus on lower level tasks discouragesstudent involvement in learning mathematics.


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Examples of Mathematical TasksLevel 2

Procedures with Connections

Solve by factoring: x2 7x + 12 = 0

Explain how the factors of the equation relate to the roots ofthe equation. Use this information to draw a sketch of the

graph of the function f(x) = x2 7x + 12.

Doing Mathematics

Describe a situation that could be modeled with the equation y= 2x + 5, then make a graph to represent the model. Explain

how the situation, equation, and graph are interrelated. Higher level tasks, when well-implemented, promote

involvement in learning mathematics.


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Characteristics of Higher Level Tasks

Higher-level tasks require students to

y do more than computation.y extend prior knowledge to explore unfamiliar tasks

and situations.y

use a variety of means (models, drawings, graphs,concrete materials, etc) to represent phenomena.

y look for patterns and relationships and check theirresults against existing knowledge.

y make predictions, estimations and/or hypotheses and

devise means for testing them.y demonstrate and deepen their understanding of

mathematical concepts and relationships.


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Traditional Problem

Nicoles Carpeting Task

Nicole was redecorating her house. She hasdecided to recarpet her bedroom, which is 15

feet long and 10 feet wide. How many square

feet of carpeting will she need to purchase?


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Advanced Problem

Nicoles Carpeting Problem

Nicole wants to redecorate her bedroom. Shedecides to recarpet. If her room is 5 feet longer

than it is wide, write an equation to represent the

area of her room. If you know her room is 10 feet

wide, how many square feet of carpet will sheneed? If the carpet is sold by the square yard,

how many square yards will she need?


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Requirements for the Estimation Center

Engaging and creative.

Students can work independently.

Makes a connection to real-world or practical applications.

Encourages thoughtful classroom discussion.

Uses digital cameras and other multimedia tools.

Presented in a power point presentation.

Includes the title of the estimation center, mathematical concepts and

connections addressed, and materials and set-up needed

All sources are cited

Worksheet for students to use

Grading rubric for students submissions


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Implementing the Estimation Center

Students must make use of a wide variety of problem solving skills

Students are required to write a thorough description/explanation of the techniques

used while attempting to solve the problem

These explanations form a basis for classroom discussion, with the main focus being

on process and strategies, not on the final answer


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Discussion

Results of the estimations are discussed, not to determine who got

the answer right, but as an examination of effective strategies

The thoroughness of the various approaches and the clarity of the

written summaries are also discussed. Although a definite answer

may not be possible, some strategies may yield more accurateresults than others.

The class data can be reviewed to determine what generalizations

and assessments can be made about the problem. The class helps

answer the question, What did we learn from the activity?


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Estimation CenterMr.WonkasDilemma An Estimation Center

Mathematical Concepts addressed: Estimation

Exponents

Area

Volume

Mathematical Connections addressed: Connections between area and volume

Connection between area formula, volume formula and exponents Materials and equipment needed:

Estimation Center Power Point

Student worksheet

Calculator

Pencil

LCD Projector

White Board

Gobstoppers

Set up needed: Students can either work independently or in pairs

Computer, LCD projector and board at front of room so that all students can see theestimation center

Access to a computer and internet

Access to the library


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An Estimation Center

Megan CoatesCherelle Cole


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The Background

Willy Wonka and his candy factory have been

doing quite well ever since the invention of

the everlasting gobstopper. Kids from all over

the world come to the factory just to see if

they can get a glimpse of the everlasting

gobstopper machine!


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There is such a demand for the everlasting

gobstoppers that Mr. Wonka has to put the

gobstopper producing machine on overdrive

for days!


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The Dilemma

But oh no!!!! The machine cant keep up with

the demand and goes haywire! The machine

explodes and there are everlasting

gobstoppers spewing everywhere! The entire

floor of the everlasting gobstopper room is

covered by gobstoppers! The area of the floor

of the room is the size of two and a halffootball fields.


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The Challenge

Mr. Wonka has to clean up the mess and he

needs your help to determine how many

gobstoppers are littered all over the floor! He

also needs your help to determine how big of

a candy dispensing machine he needs to store

the displaced gobstoppers.


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Can you take on this honorary task? The

rewards are great!


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Remember..

Be sure to explain your process and show all

work. Mr. Wonka will not be able to reward

you if no work is shown. Be prepared to

discuss your findings and explain how you

came to your conclusions since Mr. Wonka

may need to employ your method if

something like this ever happens again.


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Sources

http://www.grubbyhalogallery.com/mklacy/ga

llery/theatre_sets/images/P1010216.large.jpg

http://picsdigger.com/image/176d7fdf/ http://steelkaleidoscopes.typepad.com/steel_

kaleidoscopes/2007/09/the-everlasting.html

http://rubistar.4teachers.org


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Possible Solution Area of floor: Size of 2 and a half foot ball fields

The American football field is approximately 120 yards long by 53.3 yards wide.

The area of one foot ball field is approximately

The area of the floor of the Gobstopper room is approximately Figure out how many gobstoppers fit in one square inch, foot, etc.

Approximately 225 gobstoppers fit in one square foot (15 gobstoppers times 15 gobstoppers).

Convert units to find out how many gobstoppers are in one square yard. (This is not the onlyconversion possible). I will convert feet to yards. There are 3 feet in one yard. There are 9 square feet in one square yard.

So 9 square feet is equivalent to one square yard.

Multiply the number of Gobstoppers in one square foot by 9.

So approximately 2025 Gobstoppers are in one square yard.

Find the number of Gobstoppers that are covering the floor in the Gobstopper room.

So 32,379,750 gobstoppers are covering the floor in the Gobstopper room.

Now find the size of the container that will be needed to store the gobstoppers (there is more thanone way to do this, one could find the volume of a gobstopper and go from there). So find the amount of gobstoppers in one cubic foot : gobstoppers/ft3

Now divide 32, 379,750 gobstoppers covering the floor by the 3,375 gobstoppers in one cubic footto find the size of the container needed to store the gobstoppers.

So Mr. Wonka needs a container that is 9594ft3 to store the gobstoppers.


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Rubric


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Digital Scavenger Hunt

Chelsea Lewis

Matt Hovis

Mary Katherine Miller


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What is the difference between intersecting

lines and perpendicular lines?


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Why can you tessellate a hexagon and not a

pentagon?


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What equations graph would look most like

this picture?


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Gavin and the Giants Button

Stephanie Wood

Jenny Randall


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One day while Gavin and his grandma were

playing in a park, they came across a huge red

button. Gavin wondered aloud, How big is

the giant who lost that button?

Activity


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Questions for Gavin to consider

How can we determine the size of the giant

button? What attributes of the giant and the

button are important in deciding the giants

height based on the size of the button?What

attributes of Gavin and his buttons are

important in deciding the giants height?


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More Questions

What else do we need to know in order to

determine the height of the giant? What if we

also want to know the amount of fabric

needed to make the giant a coat? If the giant

needed a drink of water, how much water

would be equivalent to that in terms of a

humans glass(es) of water? How large wouldthe giants pack of gum be?


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Use the student page Gavin and the Giants

Button to explore relationships between

enlargements and reductions called sizechanges, and the measurements of length,

area, and volume using cubes.

Exploratory


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Gavin and the Giants Worksheet1. Use cubes to investigate what happens to the

surface area and volume of a prism when eachdimension is magnified by 200%, 300%, or

other factors. Use the table to organize and

record work

Magnification

factor

Dimensions of

prism (units)

Surface area of

prism (sq. units)

Volume of prism

(cube units)

100% 1 x 2 x 3 22 6

200%

300%

50%

150%

100n%


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2. Conjecture a rule telling how area and volumechange when length is changed by amagnification factor.

3. Note that some of the magnification factors in

the table are not integer multiples of 100%.On a copy machine, you are also able toreduce the size of a copy or make non-integermagnifications. Do area and volume change in

the same way when the magnification factor isnot an integer multiple of 100%? Why or whynot?


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More Exploratory Questions

1. Assuming giants have pencils, what are possibledimensions of a giants pencil? Compare the giantspencil to some human-sized object.

2. What are possible dimensions of a pair of eyeglasses for

the giant?3. What are the dimensions of the giants footprint? Could

the giant step inside our classroom?Why or why not?What human sized object is approximately the size of thegiants footprint?

4. Using a humans paper cup as a model, determine thedimensions of a giant-sized paper cup. How much fluidwill the paper cup hold? Compare the size of the cup tosome human-sized object.


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Share and Summarize

What assumptions were made about the giant?What assumptions were made about the items

used to decide on the sizes of the giants items?

Keeping all other factors constant, how do your

results change if:

A. the giant is another gender?

B. The giant is from another

generation?C. The button is a jacket button?

D. The button is a shirt button?


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If you were the giant, how tall would a human

be?

Suppose your button is the giants button.

How big is a humans button when compared

to your giant button?

Look back at the table you completed in Gavinand the Giants Button. How do proportions

arise from the table? How can we use

proportions to help solve problems like these?


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Bibliography

Rubenstein, Rheta N., Charlene E. Beckmann,

and Denisse R. Thompson. Teaching and

Learning: Middle Grades Mathematics.

Hoboken, NJ: John Wiley & Sons, Inc., 2004.

Print.

http://www.made-in-china.com/showroom/lishun-

button/product-detailbQmEUTRMtnrp/China-Shirt-Buttons-C08031111-.html

http://gofifo.com/playground.htm


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Thank you so much for listening toour presentation!

Are there any further questions?

Documents

Higher Level Tasks[1]