Linking Fractions, Decimals, and Percents 6th Grade Lesson Plan Overview: In this lesson, students explain how a shaded area of a rectangle represents a percent, a decimal, and a fraction of the rectangle. Mathematics: To solve this task successfully, students must relate percent, fraction, and decimal in the same representation. Goals: • • • •

Students will solve the problem using a variety of strategies. Students will interpret 6 shaded squares as a percent, a fraction, and a percent of the whole rectangle. Students will describe how their representations show a percent, fraction, and decimal. Students will justify their solutions to the problem.

Number Standards: 1.2 Interpret and use ratios in different contexts. Building on Prior Knowledge: Number Standards: 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation Materials: Linking fractions, decimals, and percents task (attached); grid paper; markers

Phase S E T U P

E X P L O R E

Action Prior to the lesson: • arrange the desks so that students are in groups of 4. • determine student groups prior to the lesson so that students who complement each other’s skills and knowledge core are working together. • place materials for the task at each grouping. • solve the task yourself.

Comments Students will be more successful in this task if they understand what is expected in terms of group work and the final product.

HOW DO I SET-UP THE LESSON? Ask students to follow along as you read the problem. Then have several students explain to the class what they are trying to find when solving the problem. Stress to students that they will be expected to explain how and why they solved the problem a particular way and to refer to the context of the problem.

HOW DO I SET-UP THE LESSON? As students describe the task, listen for their understanding of the goals of the task. It is important that they indicate the goal is to determine and demonstrate what fraction, decimal, and percent the 6 shaded rectangles represent of the larger rectangle. Be careful not to tell students how to solve the task or to set up a procedure for solving the task because your goal is for students to do the problem solving.

It is critical that you solve the problem in as many ways as possible so that you become familiar with strategies students may use. This will allow you to better understand students’ thinking. As you read through this lesson plan, different strategies for solving the problem will be given.

PRIVATE PROBLEM SOLVING TIME Give students 5 - 7 minutes of private think time to begin to solve the problem individually. Circulate among the groups assessing students’ understanding of the idea below.

PRIVATE PROBLEM SOLVING TIME Make sure that the talking of other students does not interrupt students’ thinking. If students begin talking, tell them that they will have time to share their thoughts in a few minutes.

FACILITATING SMALL GROUP PROBLEM SOLVING As you circulate among the groups, press students to show how they know that they have the fraction, decimal, and percent represented by the six shaded squares. After explaining their initial solutions you might say Now look at the squares again. Is there another way to represent those? Could you use a different representation?

FACILITATING SMALL GROUP PROBLEM SOLVING The teacher’s role when students are working in small groups is to circulate and listen with the goal of understanding students’ ideas and asking questions that will advance student work.

What do I do if students have difficulty getting started? Allow students to work in their groups to solve the problem. Assist students/groups who are struggling to get started by prompting with questions such as: - How can you use the diagram to help you figure out the percentage? The fraction? The decimal? - What would you need to show first? - Can you think of a simpler problem you could solve first? - What does each row represent? - What does each square represent?

What do I do if students have difficulty getting started? By asking a question such as “Can you think of a simpler problem you could solve first?” the teacher is providing students with a question that can be used over and over when problem solving. This will help them focus on what they know, what they were given, and what they need to determine.

What misconceptions might students have? Look for and clarify any misconceptions students may have.

What misconceptions might students have? Misconceptions are common. Students may have learned one definition of fractions or not understand that the whole needs to be defined in order to find the fraction, decimal, and percent.

a. Not understanding all 40 squares as the whole. How can you show how much of the whole rectangle is shaded? Is that more or less than one half? How could you determine how much less than one half? How do you know? Is one hundred percent of the rectangle shaded? How do you know?

Some strategies for helping students gain a better understanding include: • Ask students to use other representations of the same problem and find connections among them.

Which problem-solving strategies might be used by students? How do I advance students’ understanding of mathematical concepts or strategies when they are working with each strategy? Students will approach the problem using a variety of strategies. Some strategies are shown below. Questions for assessing understanding and advancing student learning are listed for each.

Which problem-solving strategies might be used by students? How do I advance students’ understanding of mathematical concepts or strategies when they are working with each strategy?

A. Using an algorithm or procedure Can you relate what you did to the diagram? Or how could you use the diagram to explain what you did? Is the fraction related to the percent? The decimal? How?

A. Using an algorithm or procedure Algorithmic solutions are those based on learned procedures; students often implement these without understanding why or what they are doing. Percent. 6/40 = N/100 40XN = 6X100

40XN = 600 N = 600/40 N = 15 The 6 shaded squares represent 15%.

Decimal. Consider that 6 parts out of 40 are shaded and divide the numerator by the denominator. The 6 shaded squares represent .15. Fraction. The fraction is the number of parts shaded compared with the total number of squares in the whole. The 6 shaded squares represent 6/40 or 3/20. _____________________________________________________ -------------------------------------------------------------B. Using the diagram

---------------------------------------------------------------B. Using the diagram The fraction in all four diagrams is evident to be 6 of 40 or 6/40.

First Configuration.

First Configuration.

Percent. Why did you shade one and one half columns? What does a column represent? Where is 10% in your diagram? How do you know what is 100%?

Percent. The entire rectangle represents 100%. Since there are 10 columns in the rectangle, the one column of 4 shaded squares will be 1/10 of the rectangle or 10%. The second column has only 2 squares shaded or ½ of the column. If the whole column is 10%, then half the column is 5%. Thus, the 6 shaded squares that make up the 1½ columns equal 10% plus 5% or 15%.

Decimal. What decimal does one column represent? How do you know? What decimal are the two squares shaded on the second column?

Decimal. The entire rectangle represents one whole. Since there are 10 columns in the rectangle, the one column of 4 shaded squares will be .1 of the rectangle or .10. The second column has only 2 squares shaded or half of the column. Half of .1 or half of .10 is .05.

Second Configuration. Second Configuration.

Percent. How do you know what is 100%? What does a row represent? How could the diagram help you determine what percent each square represent?

Decimal. How do you know what is 100%? What does a row represent? How could the diagram help you determine what decimal each square represent? What decimal of the row do the 6 shaded squares represent?

Third Configuration.

Percent. How many sections like that can you fit in your diagram? If you can fit 6 of those sections into your diagram, what happens to the column that is left over?

Percent. The entire rectangle represents 100%. Since there are 4 rows in the rectangle, one row is ¼ of the rectangle or 25%. Since there are 10 squares in a row, one square must represent 2.5%; 2 squares would represent 5%; 4 squares would represent 10% and 6 squares would represent 15%. Decimal. The entire rectangle represents one whole. Since there are 4 rows in the rectangle, one row is ¼ of the rectangle or .25. Since there are 10 squares in a row, 6 squares represent .6 of the row or .6 X .25 of the whole; .6 X .25 = .15.

Third Configuration.

Percent. The entire rectangle represents 100%. I can partition the rectangle into 6 groups of 2 X 3 sections, leaving one column on the end. The column on the end is 1/10 or 10% because there are 10 columns in the rectangle. Ninety percent divided by 6 is 15%. So, one section of 6 squares would represent 15%.

Decimal. How do you know what is 100%? How could the diagram help you determine what decimal each column represent? What decimal of the whole is one column?

Fourth Configuration.

Percent. How do you know what is 100%? How could the diagram help you determine what percent each square represent? What percent does one square represent? Does it matter which squares are shaded? Why or why not?

Decimal. How do you know what is 100%? How could the diagram help you determine what decimal each square represents? Would the shaded squares need to be next to each other? Why or why not?

Decimal. The entire rectangle represents one whole. I can partition the rectangle into 6 groups of 2 X 3 sections, leaving one column on the end. The column on the end is 1/10 or .1 because there are ten columns in the rectangle. That means the 6 2X3 sections take up the remaining .9 of the rectangle, and .9 divided by 6 is .15. So, one section of 6 squares would represent .15.

Fourth Configuration.

Percent. The entire rectangle represents 100%. Since there are 40 squares, 80% can be distributed across the 40 squares by giving 2% to each square. That leaves 20%. Since 20 is half of 40, the 20% can be distributed across all the squares by giving each square another .5%. Thus each square would represent 10%, and 6 squares would represent 15% (or 6 times 2.5% = 15%).

Decimal. The entire rectangle represents one whole. Each square is 1/40 but I don’t know what the decimal equivalent of that is. If I think about putting the squares together into pairs, then there would be 20 pairs and each pair of squares would be 1/20 or .05% because there are 20 nickels in $1.00. So the 6 squares are 3 pairs or 3 nickels, which would be .15.

Extending to 100.

Extending to 100.

100

Percent and Decimal. Why did you extend to 100 squares? How did you know how many more shaded squares to add? What is the relationship of the 6 shaded squares to the original 40 squares?

Percent. If there were 100 squares, then the number of shaded squares would equal the percent. There are 40 squares to begin with and 6 are shaded. If another 40 squares are added, then 6 more would be shaded. To get 100 squares you need to add 20 more. Since 20 squares is half the original 40 squares, then only half as many squares would be shaded. So 3 of the 20 squares would be shaded. Altogether then, 6 + 6 +3 = 15 of the 100 squares would be shaded. Fifteen out of 100 is 15%.

Decimal. If the whole is made up of 100 squares, then each shaded square represents .01. Since there are 15 squares shaded, this is equal to .15.

S H A R E, D I S C U S S, A N D A N A L Y Z E

FACILITATING THE GROUP DISCUSSION What order will I have students post solution paths so I will be able to help students make connections between the solution paths? As you circulate among the groups, look for solutions that will be shared with the whole group and consider the order in which they will be shared. Ask students to explain their solutions to you as you walk around. Make certain they can make sense of their solutions in terms of their representations.

FACILITATING THE GROUP DISCUSSION What order will I have students post solution paths so I will be able to help students make connections between the solution paths? Even though you may display all solution paths, you should strategically pick specific solution paths to discuss with the whole group. For this particular problem, it could be best to show the first four configurations before showing the one that extends the diagram to 100 squares.

Ask students to post their work in the front of the classroom. The goal is to discuss mathematical ideas associated with finding the fraction, the percent, and the decimal using the diagram to reason about it. What questions can I ask throughout the discussion that will help students keep the context and the goal of the problem in mind? (Driving Questions *)

What questions can I ask throughout the discussion that will help students keep the context and the goal of the problem in mind? Driving questions (*) Driving Questions have been provided because they will help to stimulate student interest, maintain the focus of the discussion on the problem context, and focus the discussion on key mathematical ideas. Many of the questions require students to take a position or to wonder about mathematical ideas or problem solving strategies.

Solutions using drawings. Ask students to present their solution and explain how their drawings help them to get the answers. You might ask How did you know what the whole was? How did the diagrams help you figure out the percent, decimal, fraction? Are the three (the percent, decimal, fraction) related? How? How did you decide to figure out what a column or a row or a cell represented?

Accountable TalkSM Pointing to the drawing. Asking student to use the diagram to explain their thinking. Repeating or Paraphrasing Ideas. Ask other students to put explanations given by their peers into their own words. This is a means of assessing understanding and providing others in the class with a second opportunity to hear the explanation. Position-Driven Discussion. Press students to take a position and to support their claims with evidence from the drawings. Students must say why their drawings represent that fraction of the pan of brownies. In doing so they will have to provide reasons for their claims.

Linking Fractions, Decimals, and Percents Your task: Shade 6 of the small squares in the rectangle shown below. Then determine the percent, the decimal, and the fraction represented by the shaded squares.

Using the diagram, explain how to determine each of the following: a. the percent of area that is shaded. b. the decimal part of the area that is shaded. c. the fractional part of the area that is shaded.

Stein, M.K., Smith, M.S., Henningsen, M.A., and Silver, E.A. (2000). Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. p. 47.