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Ratio and Proportional Relationships

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Ratio and Proportional Relationships Mathematics Instructional Cycle Guide Concept (7.RP.2) Rosemary Burdick, 2014 Connecticut Dream Team teacher Connecticut State Department of Education 0 CT CORE STANDARDS This Instructional Cycle Guide relates to the following Standards for Mathematical Content in the CT Core Standards for Mathematics: Ratio and Proportion 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.RP.2 Recognize and represent proportional relationships between quantities, fractional quantities, by testing for equivalent ratios in a table or graphing on a coordinate plane. This Instructional Cycle Guide also relates to the following Standards for Mathematical Practice in the CT Core Standards for Mathematics: Insert the relevant Standard(s) for Mathematical Practice here. MP.4: Model with mathematics MP.7: Look for and make use of structure. MP.8: Look for and express regularity in repeated reasoning. WHAT IS INCLUDED IN THIS DOCUMENT? A Mathematical Checkpoint to elicit evidence of student understanding and identify student understandings and misunderstandings (p. 21) A student response guide with examples of student work to support the analysis and interpretation of student work on the Mathematical Checkpoint (p.3) A follow-up lesson plan designed to use the evidence from the student work and address the student understandings and misunderstandings revealed (p.7)) Supporting lesson materials (p.20-25) Precursory research and review of standard 7.RP.1 / 7.RP.2 and assessment items that illustrate the standard (p. 26) HOW TO USE THIS DOCUMENT 1) Before the lesson, administer the (Which Cylinder?) Mathematical Checkpoint individually to students to elicit evidence of student understanding. 2) Analyze and interpret the student work using the Student Response Guide 3) Use the next steps or follow-up lesson plan to support planning and implementation of instruction to address student understandings and misunderstandings revealed by the Mathematical Checkpoint 4) Make instructional decisions based on the checks for understanding embedded in the follow-up lesson plan MATERIALS REQUIRED o Large poster paper for students to work on and then hang around the room. o Needed for students to complete their Gallery Walk o If possible, a smart board TIME NEEDED Insert Checkpoint Name: Which Container? Insert time needed: 20 minutes Follow-Up Lesson Plan: Insert time needed: 2-3 Class periods Timings are only approximate. Exact timings will depend on the length of the instructional block and needs of the students in the class. Connecticut State Department of Education 1 Step 1: Elicit evidence of student understanding Mathematical Checkpoint Question(s) Purpose (See attached document on page 18) 7.RP.1 Compute unit rates associated with ratios Two liquid storage containers of the same size are being filled. of fractions, including ratios of lengths, areas and CT Core other quantities measured in like or different units. Liquid enters container A at a rate of 2/3 gallon per 1/4 minute. Standard: 7.RP.2 Recognize and represent proportional Liquid pours into container B at a rate of 3/5 gallons per 1/6 minute. relationships between quantities. (Fractional quantities) by testing for equivalent ratios in a table or Container A: Container B: graphing on a coordinate plane. How do students approach a Proportional context involving Fractions, when the unit rate is given as a complex fraction? To what extent do they? • Draw a model that correctly represents the division of time and the amount of container filled. • Connect their model to the situation and how it helped to determine the solution. Use the model to make connections to creating a table of equivalent ratios and to writing an equation. Solve the equation correctly. Target question addressed by this checkpoint: Determine which container is being filled faster. Justify your answer. Support your reasoning by using evidence from the models above. Connecticut State Department of Education 2 Step 2: Analyze and Interpret Student Work Student Response Guide Got It Developing Getting Started What will a response include from a student who has What will a response include from a student who What will a response include from a student who demonstrated conceptual understanding and mastery? demonstrated some understanding and possibly demonstrated minimal understanding and possibly some misunderstandings or undeveloped misconceptions? understanding? . Connecticut State Department of Education 3 Getting Started Student Response Example Indicators What possible indicators may be included in a student response who What will a response include from a student who demonstrated minimal has demonstrated minimal understanding of the standard? understanding and possibly misconceptions? o Lack of understanding between the relationship of the time and gallons flowing into the two tanks. o Lack of a ratio table modeling the relationship between time and gallons. o Lack of understanding a unit rate and how that is connected to this situation. 1 1 o The student calculated and of a minute into seconds 4 6 incorrectly. What strategies, and representations will or will not be used? What understandings or procedural fluency does the student response reveal? o No ratio table to show the progression of time and gallons in each container. o The student tried to calculate the unit rate by using percentages. He showed 3% of the tank filled per second. That cannot be determined, as the amount of water each tank holds was not given. What undeveloped understandings, misconceptions, and common mistakes may be revealed in the student response to this item? o Student made no connect between the amount of time and number of gallons flowing into the tanks. o Student could not recognize a unit rate. o Student did not make the connection between two equal representations of a ratio table. o There was not clear understanding that a comparison had to be made comparing the rates to select which container would be filler faster. Connecticut State Department of Education 4 In the Moment Questions/Prompts Closing the Loop (Interventions/Extensions) What questions could you ask, or feedback could you provide in the moment to LZ video lesson links that may help develop conceptual understanding and develop student understanding, create disequilibrium, or advance student procedural skill needed thinking? What relationship is represented in this problem http://learnzillion.com/courses/43?collection_id=552 Looking at Container A – how much water is flowing into the container and in what time period. Let’s break down the time periods and see how much water will be in the I also use with the student a visual model showing the relationship. This would tank at each time: be found at: http://www.thinkingblocks.com 1 hour – how much water _____________________________ 4 Here students will visually see the connection and relationship between A second 1 hour – how much water ___________________________ 4 numbers. After the student worked through the guided lessons here, I would use Thinking blocks to model this problem 1 After a third hour – how much water _______________________ 4 With this students I would draw the bar model to have them fill in with time – and then the gallons that entered the tank after each 1 hour. 4 I would ask the student how many 1 hours are in 1 whole hour. 4 Then I would ask for them to mark off that time on a Bar model – and see if they can determine the amount of water in the tank. 1 1 1 1 ℎ표푢푟 ℎ표푢푟 ℎ표푢푟 ℎ표푢푟 4 4 4 4 If a student continues to have a hard time with the concept – I would then move to model using half hours – or ONLY whole numbers to see if the student can then see the relationship between the numbers. Connecticut State Department of Education 5 A double number line could also be used to help the students the relationship between time and gallons. ¼ hour ____gallons Connecticut State Department of Education 6 Developing Student Response Example Indicators What possible indicators may be included in a student response who has What will a response include from a student who demonstrated some demonstrated some understanding of the standard? understanding and possibly some misunderstandings or undeveloped o Student showed no ratio table for the relationship between time understanding? and gallons. o Student did not write the unit rate for each tank. o No equation was given to show the relationship of this situation. What strategies, and representations will or will not be used? What understandings or procedural fluency does the student response reveal? o Student did understand that the filling of the containers was in a linear relation – they show the water-level moving in a straight line. What undeveloped understandings, misconceptions, and common mistakes may be revealed in the student response to this item? o For Contain A the student showed a doubling of time increments. He was not consistent with his measurements. o The container that would fill the fastest was not indicated. o Student did not show recognition of unit rate. Connecticut State Department of Education 7 In the Moment Questions/Prompts Closing the Loop (Interventions/Extensions) What questions could you ask, or feedback could you provide in the moment to LZ video lesson links that may help develop conceptual understanding and develop student understanding, create disequilibrium, or advance student procedural skill needed thinking? http://learnzillion.com/courses/43?collection_id=552
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