Using Problem Solving in Understanding Percent

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Using Problem Solving in Understanding Percent
Patrick Francis Healy Middle School

• Jim Rahn
• www.jamesrahn.com
• james.rahn_at_verizon.net

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NJ Ask
Conceptual Understanding
Data Analysis, Probability Statistics, and
Discrete Math
Spatial Sense and Geometry
Number Sense, Concepts and Applications
Patterns, Functions and Algebra
Procedural Knowledge
Problem Solving Skills
POWER BASE Reasoning Connections
Communication Problem Solving - Estimation, Tools
and Technology Excellence and Equity
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National Center for Research on Teacher Learning
(NCRTL)

• Former ideas
• Student learning consisted of rote memorization
  of new knowledge--students listened to lectures
  and read books, their progress measured by their
  ability to recite what they had heard and read.

• New Ideas
• Learning occurs when instruction is
  inquiry-oriented
• Encourage learners to actively think about and
  try out new ideas in light of their prior
  knowledge
• Personally transform the knowledge for their own
  use
• Apply new ideas in other situations.

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Why is teaching for active engagement in learning
important?

• Mere regurgitation of facts and figures, without
  a deep rooting in the reasoning behind such
  information, is not sufficient for in-depth
  understanding.
• Students should learn how to pose questions,
  construct their own interpretations and ideas,
  and clarify and elaborate upon the ideas of
  others.

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What are the goals of teaching for active
engagement in learning?

• To focus classroom activities on reasoning and
  the evaluation of evidence
• This allows students the opportunity to develop
  the ability to formulate and solve problems.
• To empower students to think and problem solve
  themselves through a problem or situation.
• This allows students build their problem solving
  skills and see there are several ways to solve a
  problem.
• To enable students to clarify and explain their
  ideas for a solution.
• This helps students put the whole thing together
  for themselves and make the needed connections
  between previous knowledge and new knowledge.

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• Students should talk with one another, as well as
  in response to the teacher.
• Students should talk and reflect upon their own
  thinking, questioning, negotiating, and
  problem-solving strategies.

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What do your students understand about percent?
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What is 20 of 250?
Think about this question
When your students solve this problem do they
just do the multiplication?
What is their understanding about why they are
multiplying?
Is it because of means multiply?
Is it because they know how to change to
decimals?
Why is 20 changed to .20?
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What is 20 of 250?
Or do they set up a proportion?
What is their understanding about why they are
setting up a proportion?
When they set up the proportion, what happens
next?
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What is 20 of 250?
Do students stop and think what the proportion is
saying?
Do students try to rewrite the proportion?
Are the students caught up in doing a procedure?
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Think about this question
32 is what percent of 96?
What would your students understanding of this
question be?
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Will they simply write an algebraic equation such
as 32 R x 96 and then solve for R?
Do students see that the answer is equivalent to
33 1/3 ?
Do students wonder if the answer is 0.33 1/3,
1/3, or 33 1/3?
In what form will the students give their answer?
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Think about this question
48 is 30 of what number?
What would your students understanding of this
question be?
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Will they simply write an algebraic equation such
as 48.30 x B and then solve for B?
Can students see that the answer is 160 without
cross multiplying?
Will the students simplify the proportion and
think about it?
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In Solving a Percent Problem

• Is solving a percent problem simply working with
  partpercent x base?

• Is solving a percent problem simply writing a
  proportion--then cross multiplying?

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• Are percent ideas broken into 3 types of problems
  and only solved using an algorithm or only solved
  with a proportion?

• Have your students ever thought about percent in
  a visual way and the used problem solving to
  answer any percent question?

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Hundredths Squares and Download Bars can be
engage students in Understanding Percent and
Problem Solving Percent Problems
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• The square on the left is called a Unit Square.
  It can represent any number.

• The square on the right is called a hundredths
  square? Why?

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• Place your communicator on top of the Unit
  Square-Hundredths Square template.

• Trace the unit square on the left.

• Slide the communicator to the right and compare
  the hundredths square to the unit square. What
  does the hundredths square do to the unit square?

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Unit Square Hundredths Square

• Let the Unit Square represent 100

100
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Using the Hundredths Square

• Shade in 1 small square. What does 1 small
  square represent?
• How many names does this square have?

1, 1 out of 100, 1/100, 0.01
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Using the Hundredths Square

• What statement can we make about this one square?

1 is 1 of 100
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What if I shade in more than 1 square?

• If 1 square is 1, what is another name for 5
  squares?

5 out of 100, 5/100, 1/20, .05

• What statement can you write?

• 5 is 5 of 100

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What if I shade in more than 1 square?

• If 1 square is 1, what is another name for 10
  squares?

10 out of 100, 10/100, 1/10, .10

• What statement can you write?

• 10 is 10 of 100

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What if I shade in more than 1 square?

• If 1 square is 1, what is another name for 25
  squares?

25 out of 100, 25/100, 1/4, .25

• What statement can you write?

• 25 is 25 of 100

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What if I shade in more than 1 square?

• If 1 square is 1, what is another name for 50
  squares?

50 out of 100, 50/100, 1/2, .50

• What statement can you write?

• 50 is 50 of 100

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What if I shade in more than 1 square?

• If 1 square is 1, what is another name for 75
  squares?

75 out of 100, 75/100, 3/4, .75

• What statement can you write?

• 75 is 75 of 100

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How much do I shade?

• What will you shade in if you are asked to shade
  in 20 of the hundredths square?

• How many ways can you describe what you have
  shaded in?

20 out of 100, 20/100, 1/5, .20

• Do these names make sense to you?

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Thinking about the squares

• 20 squares was 20 of the whole board.
• Another way to describe this is to say 20 squares
  is 20 of the 100 squares.
• What statement could you make about 55 squares?

55 is 55 of 100
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Lets think beyond a unit of 100
?
200
Let the unit square represent 200
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Lets Change the Unit Square
200

• Suppose we had 200 pieces of candy in the unit
  box. How many pieces of candy will be in each
  small square?

• If 1 square is 1, what is 1 of 200?

• If 5 squares is 5, what is 5 of 200?

• If 10 squares is 10, what is 10 of 200?

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With the whole board representing 200

• Shade in 20 squares.

• Write a statement about the 20 squares.

40 is 20 of 200
200

• Explain why this makes sense.

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What proportion does this visual illustrate?
200
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Try these combinations

• Let the unit square represent
• 1200
• 400
• 150
• 50

• Write a statement about 1 of the unit square.

• On the hundredths square shade in
• 60
• 75
• 90
• Write a statement about each percent.

• On the hundredths square shade in
• 1 square or 1

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Picture these statements

• 20 of 250
• 30 of 150.00
• 49 of 3000 voters

• How much is 1?
• How much does 20, 30, 49 represent?
• How many squares did you color in for each part?
• Explain your reasoning for each statement.

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Represent this problem on the template

• In New Jersey residents pay 7 sales tax.
• We want to find the amount of tax paid on a 50
  item, what shape should be used for the 50?
• What does 1 represent?
• How can you determine the tax?

50
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What have you learned to do?

• The unit square can represent any number larger
  or smaller than 100
• The hundredths square separates the unit square
  into 100 equal parts Divides the unit square by
  100
• You can always find 1 Divide the unit by 100
• You can find 10, 20, etc. Multiply
• You can expand the 1 to find other percents
  Use multiplication, addition, and subtraction

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Changing the situation

• Suppose we give you the unit square and describe
  just part of that number. Can you find the
  percent involved?

135 is what percent of 900?
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Lets change things slightly

• Suppose a farm owns 900 chickens.

• Of these chickens, 135 are red.

• What shape should be represented by the 900?

900

• What fact can you still describe?

• Can you determine what 1 of 900 is?

• 135 red chickens would be represented by percent?

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Think about it

• Does it make sense that 135 is 15 of 900?

• Explain why this statement makes sense.

900
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Try another problem

• A surf team owns 150 surfboards
• Of these boards, 27 surf boards are long boards.
• If you want to know what percent of the surf
  boards are long boards, how can you think about
  the 150 surfboards and 27 long board surfboards
  with the hundredths grid and unit square?

150
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What have you learned?

• What does each square of the board always
  represent?
• Unit square
• Hundredths Square
• If the unit square represent any number other
  than 100 how can you figure out what 1 of the
  number represents?
• Explain what 10 looks like? 20? 30, 40?
• Explain what 15, 25, 75 look like?

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How would you think about this problem?

• Suppose a contractor owns fifty acres of land,
  but she will only be able to build on forty-seven
  of the acres.
• How many squares should you shade in to represent
  47 acres?
• Will you shade in more than half?
• More the 3/4 of the hundredths square?

50
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How would you think about this problem?

• How many acres are represented by each row?

• About how many rows do you need to shade in?

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• What percent of the land cannot be developed?

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Try a problem with larger numbers

• If a store discounts a sofa costing 1250 by
  375, what percent discount did the store offer?

1250

• Where would you place the 1250?

• What percent can you find easily?

375 is 30 of 1250

• What percent is represented by 375?

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Think about what you have just done

• 375 is 30 of 1250

• If we move the unit square on top of the
  hundredths square what do we see?

• What proportion do you see?

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Lets look at a slightly different problem

• If 90 represents a 15 discount on an item, how
  much did the item originally cost?

• How would you represent 15?

• Where will you place the 90? Explain your
  reasoning.

90 is 15 of 600

• Can you determine what the unit square equals?

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Try another one

• Suppose a convention is planning to use several
  different colored balloons in their celebration.
• Suppose 12,000 of the balloons are red. If these
  12,000 balloons represent 75 of the balloons,
  how many balloons are their altogether?

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Try another one

• How will you represent 75?
• Where will you put the 12,000 balloons?
• Write several statements that describe what you
  have just pictured.

12,000 is 75 of the balloons.
Each 4000 is 25 of the balloons.
16,000 unit square
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Represent this problem on your template

• A store marks up an item they are selling by 25.
  If they marked up an item 30, how much did they
  buy the item for?
• Where will you place the 30?
• Can you determine how much 1 represent? 25?
• Explain how you will determine the cost of the
  item?
• How much will they sell the item for?

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So far

• You have used the Hundredths Square and Unit
  Square to represent problems of the form
• What is 10 of 250
• 12 is what percent of 24
• 30 is 15 of what number
• The Hundredths Square has set up the proportion
  visually.

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Representing more than 100

• If one board represent 100, how will we
  represent more than 100?

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Thinking over 100

• Show how to represent 120 of 100

• Show how to represent 150 of 200

• Show how to represent 160 of 400

100
200
400
100
200
400
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Discount Problems

• A store is offering a 20 discount for
  Inauguration Day. How much will a 90 pair of
  sneakers cost?
• Use the hundredths square to solve this problem.
• Explain your reasoning.

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Sales Tax Problem

• You have just purchased some shirts and jeans.
  If the 7 sales tax was 10.50, how much were the
  shirts and jeans? How much is the total bill?
• Use the Hundredths Square to solve this problem.

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How do I represent this problem?

• Suppose I wanted to add a 30 profit on an item I
  purchased for 300. How much would the item now
  cost.
• Explain how you would use two boards to represent
  this situation.

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Be careful with this one

• Suppose an item cost 390 but that day they were
  offering a 30 discount. How much would the item
  cost?
• Picture this on the board.
• How many hundredths square do you need to use?

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Study the last two problems

• When a 30 profit was add to the cost of a 300
  item the item was sold for 390.

• When 30 was discounted off a items costing 390,
  the item cost 273.

Why didnt the cost of the item return to 300?
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How does it work?

• If I add 20 to a price and then remove 20
  explain why doesnt the price return to the
  original price?

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Try this

• Can you picture these questions without marking
  your board?
• What is 40 of 300?
• 30 is 15 of what number?
• 20 is what percent of 50?
• Is 30 more or less than ¼?
• 135 is about what percent of 450?
• What is 120 of 200?
• If 300 is reduced by 10 what is the result?

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Using Download Bars to Picture Percent

• Students are familiar with download bars when
  they download songs or programs.

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2 min
How long for the entire download?
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Using Download Bars to Picture Percent

• Mr. Martinez graded 16 of his papers in 5
  minutes. At that rate, how long will it take him
  to grade the whole class?

16
5 min
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Using Download Bars to Picture Percent

• Dinesh has completed 4 out of 5 miles she runs
  each day. What percent of the daily run has she
  completed?

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Using Download Bars to Picture Percent

• Justins computer indicated it would take 24
  minutes to download a file. How much time is left
  if the task is 75 completed?

6min
24 minutes
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Using Download Bars to Picture Percent

• Shasha took 20 minutes to type the first third of
  his paper. Based on this information, how long
  will it have taken him when he finishes the whole
  thing?

20 minutes
What percent of her paper did Shasha complete?
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At Sadies Ice Cream Shoppe, employees get a 5
discount on all purchases. What was the amount of
the discount Shikya got when she purchased a
2.00 cone?
5
2.00
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What percent problems cant be visualized with a
hundredths grid or a download bar?

• NONE

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Why should you use hundredths squares and/or
download bars to develop understand for percent?

• Just memorizing techniques does not build
  conceptual understanding for percent.
• Hundredths squares and download bars help
  students build a concrete picture of percent and
  problems involving percent.
• Hundredths squares and download bars helps
  students build problem solving strategies they
  can use to solve a problem.
• Hundredths squares and download bars eliminates
  categorizing problems one picture can be used
  to solve all percent problems.

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• Hundredths squares and download bars connects the
  area model for fractions and decimals to the
  meaning of percent
• Hundredths squares and download bars engage
  students in conceptualizing solutions
• Hundredths squares and download bars help
  students build this concrete model for proportion
  and algebraic equations
• Hundredths square and download bars help make
  solving percent problems a sense-making
  experience.

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Using Problem Solving in Understanding Percent
Patrick Francis Healy Middle School

• Jim Rahn
• www.jamesrahn.com
• james.rahn_at_verizon.net