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**Using Problem Solving in Understanding Percent**

Patrick Francis Healy Middle School Jim Rahn

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**NJ Ask Number Sense, Concepts Spatial Sense and Geometry**

Conceptual Understanding Statistics, and Discrete Math Data Analysis, Probability Number Sense, Concepts and Applications Spatial Sense and Geometry 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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**Think about this question:**

What is 20% of 250? 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?**

Will they set up a ratio such as ? Do students see that the answer is equivalent to 33 1/3 %? Do students wonder if the answer is /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=**

Will they simply write an algebraic equation such as 48=.30 x B and then solve for B? What will the answer be when they complete ? Will they set up a ratio such as ? 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 part=percent x base? Is percent an is over of memorized procedure? 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**

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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**Let’s think beyond a unit of 100**

? 200 Let the unit square represent 200

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**Let’s 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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**Let’s 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 900 Does it make sense that 135 is 15% of 900?**

Explain why this statement makes sense. 900

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**Try another problem 150 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? 50 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? 1250 1250 What proportion do you see?

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**Let’s 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**

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 didn’t 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 doesn’t 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. 20% 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**

Justin’s computer indicated it would take 24 minutes to download a file. How much time is left if the task is 75% completed? 6 min 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 Sadie’s 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 can’t 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

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