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Proportional Reasoning

Grade and Content Area Title GLEs/GSEs Grade 7 Mathematics Proportional Reasoning M(N&O)-7-4 Accurately solves problems involving proportional reasoning; percents involving discounts, tax, or tips; and rates. (State) (IMPORTANT: Applies the conventions of order of operations including parentheses, brackets, or exponents.) Context of the Lesson Prior to this lesson students have used multiple representations of a linear relationships including tables, graphs, and equations in problem-solving situations. Students have also had some experience with ratios; they know that a ratio is a comparison of two quantities. This lesson is the second of a proportional reasoning unit. The first lesson familiarized students with the term constant as the k value in a y = kx relationship, explicitly connected the graphical representation to the relationship and developed the mathematical language for the study of proportionality. In this lesson the students will investigate rate of change and use proportional reasoning to solve problems. The lesson will reinforce the concept of a linear relationship while providing the students with a visual representation of how proportionality works. It will take one class period of one hour. Materials · Worksheets 1-3 · Pencils · Calculators · Chart paper · Markers · Overhead Projector and markers · Blank transparencies for projector Class Organization · The lesson begins with a visual idea of the concept, also building on a previous problem. · There are several opportunities to practice and the problem is broken down into small parts. · Students have practice with several problems, each becoming more complex and building on another. · The concept begins with a concrete example and moves to the abstract.

Opportunities to Learn

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 1

Opportunities to Learn Continued

Differentiation of Instruction · The problems and questions in this lesson are scaffolded for students with varying abilities to successfully solve the problems. · The teacher uses questions to stimulate and push student thinking while assessing students working in groups. · During this lesson students will work individually, in small groups of 3 or 4, and the class will be instructed as a whole. Depth of Knowledge Level 3 Students will be applying the concept of proportional relationships to solve problems.

Objective(s)

Students will be able to do the following 1. Demonstrate a concrete understanding of proportional reasoning by drawing a picture and finding a unit rate. 2. Use the concept of proportionality to solve more abstract proportional problems. Opening (Whole class: 10 minutes) 1. Students will be asked to recall a previous book-stacking lesson where they made a table and wrote a rule to show how they could determine the stack height when given any number of books of the same height. Each group of students was given a different set of books. 2. After a short discussion, the teacher will ask the students, independently, to write a general rule for finding the stack height of identical books when given the height of one book. (See worksheet 1) The students will then use the rule to calculate the stack height for 8 books. 3. Ask students to share their solutions with the class. The teacher or a student can write the work for the solution on the board. In the review of this problem the teacher should make sure the discussion includes the proportional relationship and constant in this problem. The focus of the remainder of the lesson will be proportional relationships like the one found in this worksheet. 4. The teacher will next tell students that this linear relationship between the books is what mathematicians call a proportional relationship. As one quantity changes the other changes also, with each quantity consistently changing the same way. 5. The teacher will then propose the scenario where the stack height and number of books is determined. The students must then work to answer several questions about this stack. Engagement (Students working independently within a group while sharing ideas with each other for 30 minutes)

Instructional Procedures

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 2

Instructional Procedures Continued

PART 1 (20 minutes) 1. Students will work on worksheet 2. The teacher and class will read the information at the top together. Students will then work in groups of three or four to answer the four questions. 2. Groups that finish early can put their work for a question on chart paper or an overhead transparency to share with the class. 3. For students who are having difficulty determining the height of one book the teacher could ask them how they could determine the book height for a problem with simple numbers; 5 books in a stack 15 cm tall. The students can then use the same process to calculate with the numbers given. 4. The teacher could also suggest examining other rules students wrote in earlier problems. Once a student has a rule, the teacher should encourage the student to test the rule to see if it works. 5. Once students have completed the task, groups can share on chart paper or the overhead their answers to the four questions. PART 2 (15 minutes): 1. After discussing the four questions the teacher and students will read together the information provided at the bottom of worksheet 2. 2. Students will complete the unit rate blank. 3. Worksheet 3 is distributed. The teacher and students will read the information at the top together. Students will work together in groups of three or four to answer the problems. Students will be responsible to record their own work. All groups need to be prepared to present their ideas to the class. 4. As the teacher circulates from group to group, the teacher refocuses the groups on the two main questions being asked in this problem: What is proportional to what? and What is the constant of proportionality? The teacher can ask questions like: Why did you do that? How did finding a unit rate help you to find...? 5. Work from each group should be shared for one problem; the teacher can assign groups to share their work for specific problems either before beginning the task or as good work is produced by groups for certain problems. Closure (Whole class: 15 minutes) 1. The student groups share the work they have on the overhead or chart paper. One or two solutions to each problem can be shared if there are different strategies used. 2. The teacher will ask the students what all these problems have in common. Were any of the problems solved in similar ways? How does the constant in a relationship help students solve the problems? Could students use the constant to write other ratios that would also have the same proportional relationship?

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 3

Instructional Procedures Continued

3. As students share ideas the teacher can ask other students in the class if they agree with what someone says and why. The teacher could also ask how their group solved the problem, how was it the same or different? 4. The teacher refocuses the entire class on the objectives of the lesson: finding a unit rate and deepening the understanding of direct proportionality. Formative Assessment · Teacher observation of student work in groups · Student participation/contribution for whole class shared group work · Student responses in the closure discussion Summative Assessment · Before students leave, they will complete a Ticket to Leave (exit) slip for the following question: What does it mean if two quantities have a proportional relationship? ( see Ticket to Leave rubric) · Solutions to homework problems similar to those on worksheet 3.

Assessment

Reflection

Student Work Sample 1: Approaching Proficiency This student work shows understanding at a concrete level with the book stack problems. The student does not identify the constant of proportionality correctly in all problems and does not show what the mathematical calculations performed mean. The student is not able to explain why finding the constant is helpful when solving problems. (Student writes: It helps me because those are the main numbers of the problems, or You divide with it.) Student Work Sample 2: Proficient The student is correct when identifying proportional quantities and the constant of proportionality. Work is shown and labeled indicating the student knows what they are finding when they are performing mathematical operations. The student is not yet able to clearly say why and how the constant of proportionality helps solve problems. (Student writes: The constant helps you because you can use it to solve problems.) Student Work Sample 3: Exceeds Proficiency The student is correct when identifying proportional quantities and the constant of proportionality. Work is shown indicating the student knows what they are finding when they are performing mathematical operations. The student clearly identifies the constant and why it is helpful when solving problems. (Student writes: The constant of proportionality helps solve the problems because it is what you can multiply by to get any height, because it never changes.) Lesson Implementation The objectives and GLEs for this lesson were met. Students were able to draw a picture to represent a proportional relationship and they were able to solve problems involving proportional reasoning (worksheet 2).The assessment was appropriate for the objectives because the students were

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 4

Reflection Continued

asked to show their understanding in multiple forms. They were asked to draw a picture, show mathematically, and write a written response to express their understanding of proportionality. The multiple views of the assessment allow the teacher to examine in multiple ways the depth of understanding a student has. In this lesson all students were able to participate in all the lesson parts and were able to solve the problems given. One part of the assessments asked students to solve and show their work mathematically. The teacher can examine this work to see if they have a good understanding of what their calculations mean. The lesson had many different levels, so students having difficulty could receive help from the teacher on the more concrete level while students who grasped the concept easily were able to solve more abstract problems on their own. All students were engaged through active participation in small group work and discussion. Since each student was responsible for his or her own work they needed to seek the help of the group if they were having difficulty. Each group was also responsible to report their work to the class, therefore holding them accountable to explain their work and thinking. Additionally, each student was asked to complete an exit slip at the end of class. This final part of assessing students understanding helped keep them focused and attentive to the group and class discussion. Drawing a picture to find the constant of proportionality proved easy for all students. They were then easily able to write an algebraic rule in the form of y = kx to solve for any value in the proportional relationship. Once some students began solving for constants that were not represented visually (i.e. the cube problem on worksheet 3) they just began dividing to find a constant. If they were not clear about what they were dividing or what they were trying to find out they often arrived at a constant that was not appropriate. Some students were satisfied with getting an answer and moving on to the next problem. The students need more work on checking the reasonableness of their answers in the context of the problem. In future lessons students will be asked to continue to use the constant of proportionality to solve proportion problems. Students will be given situations where recognizing the proportional quantities and finding the constant of proportionality will be useful such as using unit rates to compare prices. Students will also utilize technology with graphing calculators when they look at proportional relationships in graph form. Students will use the graph representation to see how the constant will affect the rate of change in a proportional relationship. They will need to find the constant and write an equation in y = kx form to input in the graphing calculator.

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 5

Reflection Continued

Students were able to complete the calculations for this lesson, but as I said before, sometimes they would make mistakes in the order of division and use an incorrect constant to solve the problem. In future lessons I must focus on the proportional relationship. The students must continue to practice identifying the proportional quantities and saying what the constant tells them. (The constant tells me how much 1 block weighs. The constant tells me the cost of 1 bottle of water. The constant tells me the height of 1 piece of paper.) After looking at the student work and listening to the students discuss their ideas in groups and sharing their ideas with the class, I found that most students were able to use proportional reasoning to solve problems. They were able to find answers that were reasonable. When talking with students and asking them to write a written reflection I found that most of the students were not able to say what proportionality means. The meaning of a proportional relationship must be clear to students for them to hold on to this learning. If they are not clear about what they are multiplying and dividing for it is too easy for them to make a mistake. In the past when I taught lessons like this I did not ask students to write down what it means for two quantities to be proportional. I was pleased that my students could solve and get the right answer. I now know that I must continuously ask students what it means to be proportional until they have a good understanding. This lesson should probably be spread over two class periods. The first day's focus would be the concrete and visual understanding of the relationships with maybe one abstract problem. For an at home assignment the students would practice similar problems. Day two would be problems that are abstract only. This would allow for more sharing and group discussion and the focus could be on what the constant tells you. A follow-up lesson asking students to compare two ratios where the constant rate of change is different in each might be a way to breakthrough to the students' understanding.

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 6

TICKET TO LEAVE RUBRIC Exceeds Proficiency Clearly articulates their understanding of proportionality. The explanation shows the student knows how the constant will help them solve other problems. Proficient The response shows the student has a good idea of what it means for two quantities to be proportional. The student knows there is a relationship between proportional quantities and that relationship stays the same for other given quantities. Approaching Proficiency Students are beginning to show their understanding of proportionality. They know there is a relationship between proportional quantities but they are not able to explain how it works. Below Proficiency The student shows little or no evidence of understanding a proportional relationship

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 7

WORKSHEETS RUBRIC Exceeds Proficiency Student's work is clearly labeled to show their understanding. The student has indicated they know what information each calculation tells them. The student finds the constant of proportionality and explains why it is important. The student is able to use the constant to solve for other quantities in the proportional relationship. Proficient Student's work is labeled to show their understanding. It is clear the student knows what they are finding when computing with mathematical operations. The student is able to use the constant of proportionality to solve for other quantities in the proportional relationship. Approaching Proficiency The work makes sense and gives the correct answer, but it is not labeled to show they understand what they have found. The student has not shown they understand what they are finding when calculating quantities. The student does not show or identify the constant. The student may make errors because their calculations did not give them the constant of proportionality.

Below Proficiency The student has not labeled their work and has made a calculation error because they weren't sure what they were finding. The student uses an incorrect constant to solve.

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 8

WORKSHEET 1 We found that when we stacked books of the same thickness, the number of books (n) multiplied by the height of one book was equal to the stack height (h). We said the height of the stack is directly proportional to the number of books in the stack. As a review complete the two problems below. 1. If the thickness of one book is 7.2 cm write a formula you could use to determine the stack height (h) for any number of books (n) with that thickness.

2. Use your formula to determine the stack height for 8 identical books with a thickness of 7.2 cm each. Show the work you do to get the answer.

3. What are the variables in this relationship?

4. What is the constant in this relationship?

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 9

Worksheet 2 Now, let's use what we know about the proportional relationship between the height of the stack and the number of books in the stack. When we know the height of one book in the stack we can say how the stack will change if we change the number of books. Imagine there is a stack of 8 identical books that is 27.2 cm tall. Answer the following 4 questions. 1. Draw an end view sketch of the stack. Label the stack height. How can you determine the height of 1 book? Show the work you do to get the answer then label your sketch with the height of one book.

2. Write the rule you could use to determine the height of any stack using this book.

3. In this problem the _____________ of the books in the stack is proportional to the ____________ of the books.

4. What is the constant in this relationship? Explain how you know.

When you found the height of one book you found the unit rate. A unit rate is a ratio where one quantity is 1 unit. In this case you found the height for 1 book (1 unit).

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 10

We started with a ratio comparing the height of the stack to the number of books in the stack. For the stack of books write an equivalent ratio using a unit rate for the height per book. The ratio 27.2 cm: 8 books are equivalent to the unit rate ratio ____________: 1 book. The two quantities, number of centimeters and number of books, are proportional. Once you know the number of centimeters for one book it is always multiplied by the constant height in centimeters of that book. Mathematicians call the height in this problem the constant of proportionality.

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 11

Worksheet 3 Look at the following proportional relationships. Use what you know about ratios and proportional relationships to solve these problems. Show all the work you do to get your answer and explain how you know your answer is correct. 1.) 15 blocks weigh 80 grams How many grams will 4 blocks weigh?

Complete the following sentences using the information from the problem above: In this problem the _________ of the blocks is proportional to the _________ of the blocks. _____________is the constant in this relationship. 2.) 6 bottles of water cost $3.49 How much will five bottles cost?

Complete the following sentences using the information from the problem above: In this problem the __________ of bottles of water is proportional to the __________ of the bottles of water. ____________ is the constant in this relationship. 3.) A ream (package) of printer paper contains 500 sheets and measures 5.2 cm tall. How tall will a stack of 128 pieces be?

In this problem, what are the two quantities that are proportional?

4.) How does the constant in these relationships help you solve the problems?

Lesson Plan by a Rhode Island Educator

Grade 7 Mathematics: Proportional Reasoning - 12

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