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Structures of Proportionality Problems
Krisan Stone, VMP Leslie Ercole, VMP Marge Petit, Marge Petit Consulting (MPC)
Modified October 2008 Original materials created as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structure of Problems Case Study comes first 20 min. Materials: Cards sets in envelops Tab 3 Analyze 6 problems for structures/features of the problems. Sort into 3 categories: easiest, moderate, and most challenging. Record the features that influence your decisions. Discussion: Focus on the features, not on coming to agreement on the sort. 10 min. SG or partners sort the problems and record features 5-10 min. WG Collect features that were identified for each category (chart paper) **Many of the features you’ve identified have been found to influence students’ ability to solve problems. MOVE TO SLIDE 2
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OGAP Proportionality Framework Structures of Problems Mathematical Topics And Contexts
Other Structures
Evidence in Student Work to Inform Instruction Proportional Strategies
Transitional Proportional Strategies
Non-proportional Reasoning
Underlying Issues, Errors, Misconceptions
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) The goal of this session is to explicitly engage participants in the structures of problems that influence students solving problems involving proportionality. It is founded on research that indicates… “ Students move back and forth between proportional and non-proportional reasoning and more or less concrete strategies depending on the structure of the problem and the strength of their proportional reasoning.” (Cramer, Post, and Currier, 1993; Karplus, Pulos, and Stage, 1983; VMP OGAP Pilots, 2006 and 2007) This slide is divided into four parts – 1) the reference to research; 2) title – OGAP Framework; 3) Structures of Problems; and 4) Evidence in Student Work. Number 1 appears as the slide is introduced, the others are inserted as the following is read. The OGAP Proportionality Framework presents structures in problems involving proportionality and the strategies that students use to solve problems. Today we are going to focus on the STRUCTURES section of the framework. From the sorting activity you identified some of the features/structures that influenced your sorting – Identify those that they identified. At this point introduce the participants to the “real” OGAP Proportionality Framework. Give participants a few minutes to make any observations et al – but don’t spend a lot of time – at this point – explaining each component as that is the purpose of the PD. **Explain that the focus of the session today is to familiarize them with the “structures of problems”. The next full day focuses on understanding the evidence in student work.
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Structure of the problems that students solve Structure refers to – how the problems are built
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) When we refer to structures we are referring to --- how the problems are built.
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Structures of proportionality problems: • Multiplicative relationships in a problem (Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
• • • •
Context (Heller, Post Behr , 1985; Karpus, Polus, and Stage, 1983 Different types of problems (Lamon, 1993) Complexity of the numbers (Harel, G., Behr, M. 1993) The meaning of the quantities in the problem as defined by the context and the units
(Silver, 2006Vermont meeting; VMP
OGAP Pilots, 2006 and 2006)
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) Research indicates and VMP pilots support that there are primarily 5 structures that influence the strategies students use to solve problems involving proportionality 1) The multiplicative relationship in a problem 2) Context of problem 3) Types of problems 4) The meaning of quantities 5) Complexity of the numbers **In this session we will be focusing on --1) The multiplicative relationship in a problem 2) Context of problem 3) Types of problems 4) Meaning of the quantities 5) Skipping complexity of numbers --pretty obvious
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Researchers say… When the multiplicative relationship within or between ratios is an integer it is easier for students to solve than when it is a non-integer (Cramer, Post, and Currier, 1993; Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) NOTE: In most cases participants have already identified this structure in the Structures Case Study – not necessarily by stating it in this way – but when they placed problem number 1 (Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?) in the “Easiest” category. It usually is placed there for two reasons – small numbers – and – in participant words – “It is easy to see that 8 is 4 x 2.”
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Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
Within a ratio…
3 boxes x = 2 bushels 8 bushels
Between ratios…
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) The first thing that needs to be established is what is meant by “within” and “between” ratios. The slide illustrates this idea, BUT we have found that we must continuously have teachers think and rethink this idea. In this case the multiplicative relationship “within the ratio (boxes to bushels)” is nonintegral (1.5x), while the multiplicative relationship “between the ratios (4x)” is integral (4 x). Background: Some of the literature refers to “within” and “between” measure spaces, not ratios. For example – “within” bushels as one measure space or “between” boxes and bushels as two different measure spaces. However, this did not resonate with the field. We were advised to use “within” and “between” ratios which has resonated with teachers.
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Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples? Between a ratios…
Within ratios…
3 boxes 2 bushels = x boxes 8 bushels
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) This slide illustrates that the proportion can be set up in multiple forms and can alter the “within” and “between” relationships. In this case the multiplicative relationship within the ratios is integral (1.5 x) and between is non-integral (4 x).
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Researchers say… When the multiplicative relationship within and between ratios are both non-integers then students have more difficulty AND often revert back to non-proportional reasoning and strategies. (Cramer, Post, and Currier, 1993; Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) The most difficult situation is when the multiplicative relationship is non-integral both “within” and “between” the ratios.
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What is the multiplicative relationship within and between the ratios?
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 2-9 (10:30-10:55) End of “between” and “within” discussion. Have participants write the ratios two different ways and then identify the multiplicative relationship “within” and “between” the ratios. Multiplicative relationships within and between for this problem. 3 boxes: 2 bushels = x boxes:7 bushels (“within” – non-integral (1.5 x), “between” – 3.5 x) 3 boxes: x boxes = 2 bushels: 7 bushels (“within” – non-integral (3.5 x), “between” – 1.5 x)
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Multiplicative Relationships VMP Pilot Study (n=153 Seventh grade students)
• Three similar problems administered across one week period (Monday (pilot 1), Wednesday (pilot 2), and Friday (pilot 3)) • Main difference between the problems is the multiplicative relationship “within” and “between” the ratios. PILOT 1: A school is enlarging its playground. The dimensions of the new playground are proportional to the dimensions of the old playground 120 ft.
40 ft. 80 ft.
What is the length of the new playground? October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) TAB 4--Case Study & Samples of Student Work--MULTIPLICATIVE RELATIONSHIPS CASE STUDY Background: To study the impact of the multiplicative relationship within and between ratios the Vermont Mathematics Partnership conducted a pilot study involving 153 seventh grade students. In the pilot study students solved three versions of the same problem at three different times across one week. The main difference between the three problems was the nature of the multiplicative relationships “within” and “between” ratios. Pilot 1:40Aft.school is enlarging its playground. The dimensions of the new playground are 120 ft. proportional to the dimensions of the old playground. What is the length of the new playground? 80 ft.
Old Playground
New Playground
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Student Work Analysis (n= 6 students) Part 1: • Solve each problem • Identify the multiplicative relationship within and between the ratios for each problem • Anticipate difficulties that students might have when solving each problem DISCUSSION Part 2: With a partner: • Identify the multiplicative or additive relationship evidenced in the student response (e.g., X 3 (between ratios), + 6 (within ratios). Place your analysis in the cell that corresponds with the student number and Pilot number in the table (page 3). • Complete Discussion Questions. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) This activity is divided into two parts and is designed to engage participants in the impact in student work that results when the multiplicative relationships “within” and “between” ratios is altered. Part 1 is focused on the problems used in the study. Part 2 is focused on an analysis of student work. Part 1: 1) Hand out the Case Study to participants 2) Follow the directions on the slide. 3) Before going onto to Part 2, engage in discussion with participants. This is an opportunity to assure that participants understand what is meant by – “within” and “between” the ratios. Part 2: 1) Hand out student work sets. 2) Explain that the student work is organized by students and they should be reviewed that way. (e.g., review student 1 – pilot 1, then pilot 2, and then pilot 3. Record the data for student 1 as you review the work. Then go to student 2.) 3) Follow the directions on the Case and the corresponding PP slide. 4) Before reviewing the data from the Vermont Mathematics Partnership pilot study on slide 13, engage in a discussion with participants relative to discussion questions on the Case and on slide 12.
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Study Discussion Questions 1) What did you see that you expected? 2) What surprised you? 3) What are the implications for instruction and assessment? October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) Before conducting a group discussion it is recommended that the pairs share their responses to the questions with the rest of their table. 1) What did you see that you expected? [Most participants will say that they expected better results with problems 1 then with problems 2 and 3.] 2) What surprised you? [Most participants will be surprised by a couple of things. 1) that students can use a multiplicative strategy for problems 1 and 2, but revert to additive for problem 3; 2) students will stay with a “between” or “within” strategy even when one has an integral relationship and their choice doesn’t.] 3) What are implications for instruction and assessment? [This is the most important question. Participants usually get a big “aha’ at this point (or start t0). That is, they realize that they need to pay attention to assuring that students have experience solving problems involving proportionality that vary the multiplicative relationship “within” and “between” ratios. As well as not assume that if students get an 80% on an assessment focused on proportionality, that they are proficient without paying close attention to how students handle problems with varied multiplicative relationships “within” and “between” the ratios.]
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Study Findings OGAP 2006 Pilot (n=153)
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) The results of the VMP Pilot Study supported findings by other researchers.
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Structures of proportionality problems: • Multiplicative relationships in a problem (Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
• Context (Heller, Post Behr , 1985; Karpus, Polus, and Stage, 1983) • Different types of problems (Lamon, 1993) • The meaning of the quantities in the problem as defined by the context and the units
(Silver, 2006Vermont meeting; VMP
OGAP Pilots, 2006 and 2006)
• Complexity of the numbers (Harel, G., Behr, M. 1993
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) PART II
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Context Matters • More familiar contexts tend to be easier for students than unfamiliar contexts (Cramer, K., Post, T., and Currier, S., 1993)
• How proportionality shows up in different contexts impacts difficulty (Harel, G., Behr, M. 1993)
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) The next two slides are designed to call attention to these two points, NOT by providing examples of all the possible contexts, but by showing three different contexts and three different ways in which proportionality “shows up” in problems. Have students take out their OGAP Proportionality Framework. Also, be ready to refer to the graphs made by participants in the Proportionality Activity.
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The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? Nate’s shower uses 4 gallons of water per minute. How much water does Nate use when he takes a 15 minute shower? A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
1) Which contexts might be more familiar to students? 2) How does proportionality show up in these different contexts?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45)--FLY THOUGH THIS SLIDE The most familiar context is Nate’s shower and the least familiar is the scale factor problem. Proportionality shows up in three different ways in these problems. 1) Rectangle problem: Using the scale factor to scale down 2) Nate’s shower problem: Application of a unit rate 3) Toasty Oats problem: Rate comparison
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Different types of problems (Lamon, 1993) • • • • • • •
Ratio Rate Rate and ratio comparisons Missing value Scale factor Qualitative questions Non- proportional OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) Please take out the OGAP Proportionality Framework. Locate the “Problem Types” on the framework. As we review each of these types we will look closely at the structures within the problems.
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Ratios and Rates Ratio – is a comparison of any two like quantities (same unit)
The ratio of boys to girls is 1:2 The ratio of people with brown eyes to blue eyes is 1:4
Rate – A rate is a ratio that compares two quantities measured in different units and describes how one unit depends on another unit. $5.00 per hour $3.00 per pound 25 horses per acre
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) There is often confusion between the difference between ratio and rates. This slide and the examples that follow are meant to clarify those difference. The “big idea” is that ratios are comparisons of like quantities – people to people OR eyes to eyes, while rates compares two different quantities and describes how one quantity depends upon the other quantity.
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Relationships • Part: Part OR Part: Whole Referents • Implied OR Explicit
Ratio Problems OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) THINK PAIR SHARE Refer to the Framework: You will notice two structures related to ratios on the framework: Ratio Relationships and Ratio Referents. The relationships in ratios can be part to part OR part to whole. In addition, the reference to the whole or part may be explicitly stated or implied in the problem. Let’s look at a couple of examples on this slide and the next slide.. Read the problem. With a partner answer the following questions. 1) Is the relationship a part to part OR a part to whole relationship? [This is a part to whole – 7th grades boys (part): 7th grade students (whole)] 2) Is the whole explicitly given or implied in the problem and data given? [The whole is implied. What is given is the two parts – number of girls and number of boys – not the whole – the 7th grade students.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
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Relationships • Part: Part OR Part: Whole Referents • Implied OR Explicit
Ratio Problem
There are red and blue marbles in a bag. The ratio of red marbles to blue marbles is 1:2. If there are 10 blue marbles in the bag, how many red marbles are in the bag?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) THINK, PAIR, SHARE Read the problem. With a partner answer the following questions. 1) Is the relationship a part to part OR a part to whole relationship?[This is a part to part problem – red marbles: blue marbles.] 2) Is the whole explicitly given or implied in the problem and data given? [The parts are explicitly stated.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
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Rate Problems
What are the meanings of the quantities in this problem? What is the meaning of the answer?
Leslie drove at an average speed of 55 mph for 4 hours. How far did Leslie drive?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) 25 min. TALK ABOUT ADDITIVE REPRESENTATION 55, 55,… ASK DANA As mentioned earlier rates are comparisons of two different quantities where one quantity is dependent on the other quantity. Rate problems assume you start with two different quantities and end with an entirely different type of quantity. For example, this problem provides a rate (speed as defined by miles per hour), the time (in hours), and asks for a distance (undefined). Instructionally it becomes important for students to think about the meaning of the quantities, not just the units. One way to help students focus on the meaning of the quantities is to have students model the situation. Review the model that represents the situation. Explain how this model illustrates the meaning of 55 miles per hour. [Note: teachers claim that many students think miles per hour is one word – milesperhour. This model helps students to understand that miles per hour means miles per every hour.]
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Rate comparison problems… A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
What is the general structure of rate comparison problems?
Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) THINK, PAIR, SHARE Have participants work in pairs to analyze these two rate comparison problem. [In general, in rate comparison problems the two quantities that make up the rate are given, but not the rate to be compared.] NOTE: Some participants might say that these problems are built to “trick” students because the order in which the quantities (e.g., horses and acres) are given in the problem statement is not the same as the rates students are asked to compare (e.g., acres per horse). If this occurs, please point out that OGAP questions are designed to elicit fragile understandings – they are formative, not summative. You want to know if your students are paying attention to the quantities, so questions are designed to determine if students are attending to the problem situation.]
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Meaning of the Quantities Case Study In Part I of this case study you will analyze 4 student solutions to Ranch problem. The solutions represent the kinds of “quantity interpretation” errors that students make when they solve rate comparison problems. In pairs, analyze the student solutions and then respond to the following. 1) What is the evidence that the student may not be interpreting the meaning of the quantities in the problem? 2) Suggest some questions you might ask each student or activities you might do to help them understand the meaning of the quantities in the problem and the solution. Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 23 & 24 (11:45-12:00) TAB 5 MEANING OF QUANTITIES CASE STUDY To explore the importance of placing an instructional focus on “meaning of the quantities” in problems complete the “Meaning of Quantities” Case Study. Hand out the Meaning of the Quantities Case Study materials. Use this slide to introduce Part I of the Case Study. Provide about 10 – 15 minutes for participants to work in pairs. Then about 10 minutes to debrief the activity focusing on general instructional strategies that arise from the group. [See Facilitator Notes for an analysis of each of the solutions.]
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Meaning of the Quantities Case Study – Part II What evidence in the student solution below of the student understanding both the meaning of the quantities in the problem and the solution?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 23 & 24 (11:45-12:00) TAB 5
Individually and then as a group analyze the student solution in Part II. [See Facilitator Notes for the Case for an analysis.] LUNCH
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Missing value problems (MVPs) What is the general structure of a missing value problem?
Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) TAKE OUT YOUR OGAP FRAMEWORK The next Problem Type that we will analyze for structures is missing value problems. What is the general structure of a missing value problem? [In general, missing value problems involve finding a missing value in a set of equal ratios. That is, three of the four quantities are given and the solution involves finding the fourth quantity.]
3 boxes x = 2 bushels 8 bushels THESE SHOULD GO QUICKLY.
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Researchers suggest that the location of the missing value may affect performance… (Harel, G., Behr, M. 1993)
Internal Structure
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples? Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. She needs 7 bushels of apples packed. How many boxes will she need? OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) Please refer to the OGAP Framework and find “Internal Structure.” You’ll notice a reference to Internal Structures. In the Ranch problem many of you noticed that the quantities were given in one order (acres and horse), but the rate was asked for in a different order (horses per acre). This is an example of non-parallel structure. In missing value problems researchers also suggest that the location of the missing value matters. The implications for instruction and assessment is that the location of the missing value should be varied in problems that students solve. Review of problems: The quantities in both these problems are the same. In both problems the solution is the number of boxes. However, These two problems are the same except that the missing values are not in the same place in the problem statement. In the top problem the structure is parallel (i.e., boxes to bushels throughout). In the second problem the structure is NOT parallel. The first ratio is given as boxes to bushels. Then the number of bushels is given. Then students are then asked to determine the number of boxes. .
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Change this problem to make it easier, and then harder.
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) AS A WHOLE GROUP, BRIEFLY DISCUSS HOW to modify this problem to be easier, harder. Share a few of the problems that participants write. Things that participants will probably change: 1) the order of the quantities, 2) the magnitude of the numbers, or 3) the multiplicative relationship within and between the ratios.
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A school is enlarging its playground. The dimensions of the new playground are proportional to the old playground. A school is enlarging its playground. The dimensions of the new playground are proportional to the
What isdimensions the measurement of the missing of the old playground. length of the new playground? Show your work. Rectangles are not drawn to scale
Old Playground
New Playground
90ft .
110 ft. 630 ft. .
What is the measurement of the missing length of the new playground? Explain how you found your What answer.
type of problem is this similarity problem?
OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) Similarity problems show up as missing value problems, ratio comparison problems (See slide 29), and scale factor problems (see slide 30). This is a missing value problems.
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What type of problem is this similarity problem?
The dimension of 4 rectangles are given below. Which two rectangles are similar? A)2” x 8” B) 4” x 10” C) 6” x 12” D)6” x 15”
OGAP Proportionality Framework
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Slides (29-31) 12:45 GIVE TIME TO THINK ABOUT STRATEGIES. SPEND TIME ON THIS SLIDE. This is a ratio comparison problem. It can be solved by comparing the ratios between the two dimensions or by finding the multiplicative relationship between the dimensions. 2 DIFFERENT STRATEGIES (1) WITHIN THE FIGURE OR (2) BETWEEN THE FIGURES Ratio comparison solution (B and D because the ratio of one length to the other is the same (2:5)) A) 1:4 B) 2: 5 C) 1:2 D) 2: 5 Multiplicative relationship between dimensions (B and D because the multiplicative relationship with both ratios is 2.5) A) 8” is 4 times 2” B) 10” is 2.5 times 4” C) 12” is 2 times 6” D) 15” is 2.5 times 6”
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Scale Factor Problems
What is the general structure of scale factor problems?
The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides (29-31) 12:45 GO FAST The examples illustrate two different structures for scale factor problems. John Hancock Center Problem: The height of the model and the original are given. The problem asks for scale factor. Rectangle: The scale factor is given with length of the larger rectangle. The problem asks for the length of smaller rectangle
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Scale Factor The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?
If a student was unable to solve this problem successfully, what variables would you change to make it more accessible? Why?
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Slides (29-31) 12:45 In pairs, participants modify this problem. Ways to modify problem: 1) Change scale factor to an integer 2) Give dimension of smaller rectangle and ask to scale-up instead of scaling down
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Researchers indicate that students should interact with qualitative predictive and comparison questions as they are developing their proportional reasoning…. (Lamon, S. (1993))
OGAP Proportionality Framework
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Slides 32 & 33 (12-45-12:50)
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Qualitative Problems
Why do you think researchers suggest these as important stepping stones?
Kim ran more laps than Bob. Kim ran her laps in less time than Bob ran his laps. Who ran faster? If Kim ran fewer laps in more time than she did yesterday, would her running speed be: A) faster; B) slower; C) exactly the same; D) not enough information. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 32 & 33 (12-45-12:50) These are non-numerical problems that involve a proportional situation. These help students think about the relationships, and not take cues from numbers.
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Researchers say… Students need to see examples of proportional and non-proportional situations so they can determine when it is appropriate to use a multiplicative solution strategy. (Cramer, Post, and Currier, 1993)
OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55) Slide 38--NON-PROPORTIONAL STUDENT WORK-SORTING TASK
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Solve these problems (Cramer, K., Post, T., and Currier, S. (1993)
Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?
3 U.S. dollars can be exchanged for 2 British pounds. How many pounds for $21 U.S. dollars?
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Slides 34-39 (12:45-12:55) Slide 38 TAB 6--NON-PROPORTIONAL STUDENT WORK-SORTING TASK These two problems were given to a group of pre-service teachers. [If participants have not already solved Sue or Julie (or a problem like it) then have participants solve both these problems. Then review solutions. It has been our experience that many teachers will apply a proportional strategy to the additive situation in the Sue and Julie problem. This is consistent with what happened with the group of pre-service teachers. See the next slide.]
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Classic Non-proportional Example (Cramer, Post, and Currier cited in Research Ideas, 1993)
“Sue and Julie were running equally fast around a track. Sue Started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?” *
22/33 undergraduates student treated this as a proportional relationship.
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Slides 34-39 (12:45-12:55) [Slide 38--NON-PROPORTIONAL STUDENT WORK-SORTING TASK]
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Proportional example… Three U.S. dollars can be exchanged for 2 British pounds. How many pounds for 21 U.S. dollars? Same group – 100% solved it correctly using traditional proportional algorithm No one in the same group could explain why this is a proportional relationship while the “running laps” is not. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55)
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Vermont Version
Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run? Do student work sort!
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Slides 34-39 (12:45-12:55) TAB 6--NON-PROPORTIONAL TASK SORT To determine if middle school students treated this additive situation as a proportional situation, the VMP OGAP conducted a small study involving 82 sixth grade students. Some of the students had had instruction in solving proportional problems and others did not have instruction prior to solving the problem. Handout the Non-Proportional Student work set of papers. To get a feel for the type of responses found in the study (with a partner) sort this student work into 2 piles – 1) treated the problem as a proportion problem; 2) Treated as an additive problem. What did you find?
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Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?
Vermont Version Grade 6(n= 82)
• 39/82 (48%) solved as a proportion • 33/82 (40%) solved reflecting the additive situation • 10/82 (12%) non-starters What are the instructional implications?
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12:55 These are the data from the study. What are the instructional implications of these data? Have participants discuss in pairs for a minute or two and then as a full group. [Usually the discussion starts with the realization that the results might be an artifact of instruction. That is, most mathematics programs/texts ONLY include proportional problems during units focused on proportions. Students may use the structure of 3 known quantities and one unknown quantity and assume it is a proportional situation. From there participants recognize the importance of embedding non-proportional problems in instruction and assessment so students have to discriminate between the situations.]
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Topics • Ratios and proportions • Percents • Scaling • Similarity • Linear equations • Linear patterns and relationships • Slope • Rates • Frequency distributions • Probability
Different problem types are embedded in the different mathematical topics …within different contexts …which involve different multiplicative relationships …where the meaning of quantities within the problem and answer can vary ... And by the way, the numbers used (integer vs. non-integer) also affect performance. Wow! No wonder proportions are tough to teach and learn.
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Slide 40-41 (12:55) The focus here should also be on the instructional implications assuring that students interact with a variety of structures in both instruction and assessment.
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….A proportional reasoner should be able to • recognize the nature of the proportional relationship, • find a sensible and efficient method to solve problems given the context, problem type, complexity of the numbers, meaning of the quantities, and the number relationships. •Represent the quantities in the solution with units that reflect the meaning of the quantities consistent with the problem situation. Ultimately, a proportional reasoner should not be influenced by context, problem types, the quantities in the problems and their associated units, or numerical complexity. (Cramer., Post, Currier, 1993; Silver, Ed VT cite visit (2006)
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12:55
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Pre-assessment Analysis • • • • •
Problem types Context Multiplicative Relationships Internal Structure Ratio – relationship and referents (if applicable)
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Slide 42 12:55-1:45 (40 min.) TAB 7 PRE-ASSESSMENTS Hand out the OGAP Framework and grade level pre-assessments. Regroup participants into grade level groups (if this makes sense this late in the workshop). Have participants analyze each of the problems for 1) problem types;2) Context; 3) multiplicative relationships within and between ratios; 4) Internal structures; and, 5) ratio relationships and referents
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Administering the OGAP Pre-assessment • Tips for Students • Time • Level of Teacher Assistance • Analysis before December Meeting - NONE [IMPORTANT: Teachers should select 1 classroom of students to pre-assess for the December meeting.]
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Thoughts on Administering the OGAP Proportionality Pre-Assessment An important component of the Vermont Mathematics Partnership’s Ongoing Assessment Project involves gathering information about student understanding of proportionality concepts before beginning instruction through the administration of a pre-assessment. This pre-assessment is designed to elicit developing understandings, preconceptions, misconceptions, strategies that student use, and common errors that students make when solving questions involving proportionality. It is in this spirit of formative assessment that we offer the following thoughts on administering the pre-assessment. Tips for Students Let the students know that this is a pre-assessment on material that they will be learning this year so some or all of the material may be new to them. Encourage them to try their best even if they are unsure. Remind them that the information will help you in your planning, and will not be used as a grade. Time The amount of time students need to complete the pre-assessment will differ depending on the grade level and the number of items in your pre-assessment. The pre-assessment can be administered in numerous ways. Some teachers choose to spread the assessment over several days while others administer the entire assessment in one class period. Again, the purpose is to collect evidence from your students so feel free to choose a schedule that works best for your students. Level of Teacher Assistance The purpose of formative assessment is to collect evidence that will help you best meet the needs of your students. With this in mind, feel free to read any items to students who you feel need this type of accommodation. You may also decide to scribe for students who require assistance with writing. Although no special materials are needed to complete the pre-assessment, students can use tools or manipulatives that are part of regular classroom instruction. By all means assist students in decoding non-mathematical vocabulary. You should not, however, help students interpret any mathematics content. Final Thoughts The ideas above are not intended to be used as a “checklist of do’s and don’ts” but rather as a way to communicate the spirit in which the pre-assessments are best administered to your students. Please bring the completed preassessments to the December session. A major goal of these sessions is to help you learn how to analyze the evidence in your students’ responses and use your findings to influence your upcoming proportionality instruction. Feel free to contact Marge Petit ([email protected]) if you have any questions.
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Krisan Stone, VMP Leslie Ercole, VMP Marge Petit, Marge Petit Consulting (MPC)
Modified October 2008 Original materials created as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF (EHR-0227057) and US DOE (S366A020002)
Structure of Problems Case Study comes first 20 min. Materials: Cards sets in envelops Tab 3 Analyze 6 problems for structures/features of the problems. Sort into 3 categories: easiest, moderate, and most challenging. Record the features that influence your decisions. Discussion: Focus on the features, not on coming to agreement on the sort. 10 min. SG or partners sort the problems and record features 5-10 min. WG Collect features that were identified for each category (chart paper) **Many of the features you’ve identified have been found to influence students’ ability to solve problems. MOVE TO SLIDE 2
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OGAP Proportionality Framework Structures of Problems Mathematical Topics And Contexts
Other Structures
Evidence in Student Work to Inform Instruction Proportional Strategies
Transitional Proportional Strategies
Non-proportional Reasoning
Underlying Issues, Errors, Misconceptions
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Slides 2-9 (10:30-10:55) The goal of this session is to explicitly engage participants in the structures of problems that influence students solving problems involving proportionality. It is founded on research that indicates… “ Students move back and forth between proportional and non-proportional reasoning and more or less concrete strategies depending on the structure of the problem and the strength of their proportional reasoning.” (Cramer, Post, and Currier, 1993; Karplus, Pulos, and Stage, 1983; VMP OGAP Pilots, 2006 and 2007) This slide is divided into four parts – 1) the reference to research; 2) title – OGAP Framework; 3) Structures of Problems; and 4) Evidence in Student Work. Number 1 appears as the slide is introduced, the others are inserted as the following is read. The OGAP Proportionality Framework presents structures in problems involving proportionality and the strategies that students use to solve problems. Today we are going to focus on the STRUCTURES section of the framework. From the sorting activity you identified some of the features/structures that influenced your sorting – Identify those that they identified. At this point introduce the participants to the “real” OGAP Proportionality Framework. Give participants a few minutes to make any observations et al – but don’t spend a lot of time – at this point – explaining each component as that is the purpose of the PD. **Explain that the focus of the session today is to familiarize them with the “structures of problems”. The next full day focuses on understanding the evidence in student work.
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Structure of the problems that students solve Structure refers to – how the problems are built
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Slides 2-9 (10:30-10:55) When we refer to structures we are referring to --- how the problems are built.
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Structures of proportionality problems: • Multiplicative relationships in a problem (Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
• • • •
Context (Heller, Post Behr , 1985; Karpus, Polus, and Stage, 1983 Different types of problems (Lamon, 1993) Complexity of the numbers (Harel, G., Behr, M. 1993) The meaning of the quantities in the problem as defined by the context and the units
(Silver, 2006Vermont meeting; VMP
OGAP Pilots, 2006 and 2006)
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Slides 2-9 (10:30-10:55) Research indicates and VMP pilots support that there are primarily 5 structures that influence the strategies students use to solve problems involving proportionality 1) The multiplicative relationship in a problem 2) Context of problem 3) Types of problems 4) The meaning of quantities 5) Complexity of the numbers **In this session we will be focusing on --1) The multiplicative relationship in a problem 2) Context of problem 3) Types of problems 4) Meaning of the quantities 5) Skipping complexity of numbers --pretty obvious
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Researchers say… When the multiplicative relationship within or between ratios is an integer it is easier for students to solve than when it is a non-integer (Cramer, Post, and Currier, 1993; Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
OGAP Proportionality Framework
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Slides 2-9 (10:30-10:55) NOTE: In most cases participants have already identified this structure in the Structures Case Study – not necessarily by stating it in this way – but when they placed problem number 1 (Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?) in the “Easiest” category. It usually is placed there for two reasons – small numbers – and – in participant words – “It is easy to see that 8 is 4 x 2.”
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Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
Within a ratio…
3 boxes x = 2 bushels 8 bushels
Between ratios…
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Slides 2-9 (10:30-10:55) The first thing that needs to be established is what is meant by “within” and “between” ratios. The slide illustrates this idea, BUT we have found that we must continuously have teachers think and rethink this idea. In this case the multiplicative relationship “within the ratio (boxes to bushels)” is nonintegral (1.5x), while the multiplicative relationship “between the ratios (4x)” is integral (4 x). Background: Some of the literature refers to “within” and “between” measure spaces, not ratios. For example – “within” bushels as one measure space or “between” boxes and bushels as two different measure spaces. However, this did not resonate with the field. We were advised to use “within” and “between” ratios which has resonated with teachers.
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Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples? Between a ratios…
Within ratios…
3 boxes 2 bushels = x boxes 8 bushels
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Slides 2-9 (10:30-10:55) This slide illustrates that the proportion can be set up in multiple forms and can alter the “within” and “between” relationships. In this case the multiplicative relationship within the ratios is integral (1.5 x) and between is non-integral (4 x).
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Researchers say… When the multiplicative relationship within and between ratios are both non-integers then students have more difficulty AND often revert back to non-proportional reasoning and strategies. (Cramer, Post, and Currier, 1993; Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
OGAP Proportionality Framework
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Slides 2-9 (10:30-10:55) The most difficult situation is when the multiplicative relationship is non-integral both “within” and “between” the ratios.
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What is the multiplicative relationship within and between the ratios?
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples?
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Slides 2-9 (10:30-10:55) End of “between” and “within” discussion. Have participants write the ratios two different ways and then identify the multiplicative relationship “within” and “between” the ratios. Multiplicative relationships within and between for this problem. 3 boxes: 2 bushels = x boxes:7 bushels (“within” – non-integral (1.5 x), “between” – 3.5 x) 3 boxes: x boxes = 2 bushels: 7 bushels (“within” – non-integral (3.5 x), “between” – 1.5 x)
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Multiplicative Relationships VMP Pilot Study (n=153 Seventh grade students)
• Three similar problems administered across one week period (Monday (pilot 1), Wednesday (pilot 2), and Friday (pilot 3)) • Main difference between the problems is the multiplicative relationship “within” and “between” the ratios. PILOT 1: A school is enlarging its playground. The dimensions of the new playground are proportional to the dimensions of the old playground 120 ft.
40 ft. 80 ft.
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Slide 10-13 (10:55-11:20) TAB 4--Case Study & Samples of Student Work--MULTIPLICATIVE RELATIONSHIPS CASE STUDY Background: To study the impact of the multiplicative relationship within and between ratios the Vermont Mathematics Partnership conducted a pilot study involving 153 seventh grade students. In the pilot study students solved three versions of the same problem at three different times across one week. The main difference between the three problems was the nature of the multiplicative relationships “within” and “between” ratios. Pilot 1:40Aft.school is enlarging its playground. The dimensions of the new playground are 120 ft. proportional to the dimensions of the old playground. What is the length of the new playground? 80 ft.
Old Playground
New Playground
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Student Work Analysis (n= 6 students) Part 1: • Solve each problem • Identify the multiplicative relationship within and between the ratios for each problem • Anticipate difficulties that students might have when solving each problem DISCUSSION Part 2: With a partner: • Identify the multiplicative or additive relationship evidenced in the student response (e.g., X 3 (between ratios), + 6 (within ratios). Place your analysis in the cell that corresponds with the student number and Pilot number in the table (page 3). • Complete Discussion Questions. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) This activity is divided into two parts and is designed to engage participants in the impact in student work that results when the multiplicative relationships “within” and “between” ratios is altered. Part 1 is focused on the problems used in the study. Part 2 is focused on an analysis of student work. Part 1: 1) Hand out the Case Study to participants 2) Follow the directions on the slide. 3) Before going onto to Part 2, engage in discussion with participants. This is an opportunity to assure that participants understand what is meant by – “within” and “between” the ratios. Part 2: 1) Hand out student work sets. 2) Explain that the student work is organized by students and they should be reviewed that way. (e.g., review student 1 – pilot 1, then pilot 2, and then pilot 3. Record the data for student 1 as you review the work. Then go to student 2.) 3) Follow the directions on the Case and the corresponding PP slide. 4) Before reviewing the data from the Vermont Mathematics Partnership pilot study on slide 13, engage in a discussion with participants relative to discussion questions on the Case and on slide 12.
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Study Discussion Questions 1) What did you see that you expected? 2) What surprised you? 3) What are the implications for instruction and assessment? October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 10-13 (10:55-11:20) Before conducting a group discussion it is recommended that the pairs share their responses to the questions with the rest of their table. 1) What did you see that you expected? [Most participants will say that they expected better results with problems 1 then with problems 2 and 3.] 2) What surprised you? [Most participants will be surprised by a couple of things. 1) that students can use a multiplicative strategy for problems 1 and 2, but revert to additive for problem 3; 2) students will stay with a “between” or “within” strategy even when one has an integral relationship and their choice doesn’t.] 3) What are implications for instruction and assessment? [This is the most important question. Participants usually get a big “aha’ at this point (or start t0). That is, they realize that they need to pay attention to assuring that students have experience solving problems involving proportionality that vary the multiplicative relationship “within” and “between” ratios. As well as not assume that if students get an 80% on an assessment focused on proportionality, that they are proficient without paying close attention to how students handle problems with varied multiplicative relationships “within” and “between” the ratios.]
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Study Findings OGAP 2006 Pilot (n=153)
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Slide 10-13 (10:55-11:20) The results of the VMP Pilot Study supported findings by other researchers.
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Structures of proportionality problems: • Multiplicative relationships in a problem (Karplus, Polus, and Stage, 1983; VMP OGAP Pilots, 2006 and 2006)
• Context (Heller, Post Behr , 1985; Karpus, Polus, and Stage, 1983) • Different types of problems (Lamon, 1993) • The meaning of the quantities in the problem as defined by the context and the units
(Silver, 2006Vermont meeting; VMP
OGAP Pilots, 2006 and 2006)
• Complexity of the numbers (Harel, G., Behr, M. 1993
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Slides 14-22 (11:20-11:45) PART II
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Context Matters • More familiar contexts tend to be easier for students than unfamiliar contexts (Cramer, K., Post, T., and Currier, S., 1993)
• How proportionality shows up in different contexts impacts difficulty (Harel, G., Behr, M. 1993)
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Slides 14-22 (11:20-11:45) The next two slides are designed to call attention to these two points, NOT by providing examples of all the possible contexts, but by showing three different contexts and three different ways in which proportionality “shows up” in problems. Have students take out their OGAP Proportionality Framework. Also, be ready to refer to the graphs made by participants in the Proportionality Activity.
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The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? Nate’s shower uses 4 gallons of water per minute. How much water does Nate use when he takes a 15 minute shower? A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
1) Which contexts might be more familiar to students? 2) How does proportionality show up in these different contexts?
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Slides 14-22 (11:20-11:45)--FLY THOUGH THIS SLIDE The most familiar context is Nate’s shower and the least familiar is the scale factor problem. Proportionality shows up in three different ways in these problems. 1) Rectangle problem: Using the scale factor to scale down 2) Nate’s shower problem: Application of a unit rate 3) Toasty Oats problem: Rate comparison
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Different types of problems (Lamon, 1993) • • • • • • •
Ratio Rate Rate and ratio comparisons Missing value Scale factor Qualitative questions Non- proportional OGAP Proportionality Framework
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Slides 14-22 (11:20-11:45) Please take out the OGAP Proportionality Framework. Locate the “Problem Types” on the framework. As we review each of these types we will look closely at the structures within the problems.
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Ratios and Rates Ratio – is a comparison of any two like quantities (same unit)
The ratio of boys to girls is 1:2 The ratio of people with brown eyes to blue eyes is 1:4
Rate – A rate is a ratio that compares two quantities measured in different units and describes how one unit depends on another unit. $5.00 per hour $3.00 per pound 25 horses per acre
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Slides 14-22 (11:20-11:45) There is often confusion between the difference between ratio and rates. This slide and the examples that follow are meant to clarify those difference. The “big idea” is that ratios are comparisons of like quantities – people to people OR eyes to eyes, while rates compares two different quantities and describes how one quantity depends upon the other quantity.
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Relationships • Part: Part OR Part: Whole Referents • Implied OR Explicit
Ratio Problems OGAP Proportionality Framework
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Slides 14-22 (11:20-11:45) THINK PAIR SHARE Refer to the Framework: You will notice two structures related to ratios on the framework: Ratio Relationships and Ratio Referents. The relationships in ratios can be part to part OR part to whole. In addition, the reference to the whole or part may be explicitly stated or implied in the problem. Let’s look at a couple of examples on this slide and the next slide.. Read the problem. With a partner answer the following questions. 1) Is the relationship a part to part OR a part to whole relationship? [This is a part to whole – 7th grades boys (part): 7th grade students (whole)] 2) Is the whole explicitly given or implied in the problem and data given? [The whole is implied. What is given is the two parts – number of girls and number of boys – not the whole – the 7th grade students.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
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Relationships • Part: Part OR Part: Whole Referents • Implied OR Explicit
Ratio Problem
There are red and blue marbles in a bag. The ratio of red marbles to blue marbles is 1:2. If there are 10 blue marbles in the bag, how many red marbles are in the bag?
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Slides 14-22 (11:20-11:45) THINK, PAIR, SHARE Read the problem. With a partner answer the following questions. 1) Is the relationship a part to part OR a part to whole relationship?[This is a part to part problem – red marbles: blue marbles.] 2) Is the whole explicitly given or implied in the problem and data given? [The parts are explicitly stated.] As participants are working in pairs walk around the room and listen in on the conversations. Bring up any important points based on your observations. Briefly debrief the questions.
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Rate Problems
What are the meanings of the quantities in this problem? What is the meaning of the answer?
Leslie drove at an average speed of 55 mph for 4 hours. How far did Leslie drive?
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Slides 14-22 (11:20-11:45) 25 min. TALK ABOUT ADDITIVE REPRESENTATION 55, 55,… ASK DANA As mentioned earlier rates are comparisons of two different quantities where one quantity is dependent on the other quantity. Rate problems assume you start with two different quantities and end with an entirely different type of quantity. For example, this problem provides a rate (speed as defined by miles per hour), the time (in hours), and asks for a distance (undefined). Instructionally it becomes important for students to think about the meaning of the quantities, not just the units. One way to help students focus on the meaning of the quantities is to have students model the situation. Review the model that represents the situation. Explain how this model illustrates the meaning of 55 miles per hour. [Note: teachers claim that many students think miles per hour is one word – milesperhour. This model helps students to understand that miles per hour means miles per every hour.]
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Rate comparison problems… A 20-ounce box of Toasty Oats costs $3.00. A 15-ounce box of Toasty Oats costs $2.10. Which box costs less per ounce? Explain your reasoning.
What is the general structure of rate comparison problems?
Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 14-22 (11:20-11:45) THINK, PAIR, SHARE Have participants work in pairs to analyze these two rate comparison problem. [In general, in rate comparison problems the two quantities that make up the rate are given, but not the rate to be compared.] NOTE: Some participants might say that these problems are built to “trick” students because the order in which the quantities (e.g., horses and acres) are given in the problem statement is not the same as the rates students are asked to compare (e.g., acres per horse). If this occurs, please point out that OGAP questions are designed to elicit fragile understandings – they are formative, not summative. You want to know if your students are paying attention to the quantities, so questions are designed to determine if students are attending to the problem situation.]
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Meaning of the Quantities Case Study In Part I of this case study you will analyze 4 student solutions to Ranch problem. The solutions represent the kinds of “quantity interpretation” errors that students make when they solve rate comparison problems. In pairs, analyze the student solutions and then respond to the following. 1) What is the evidence that the student may not be interpreting the meaning of the quantities in the problem? 2) Suggest some questions you might ask each student or activities you might do to help them understand the meaning of the quantities in the problem and the solution. Big Horn Ranch raises 100 horses on 150 acres of pasture. Jefferson Ranch raises 75 horses on 125 acres of pasture. Which ranch has more acres of pasture per horse? Explain your answer using words, pictures, or diagrams.
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Slides 23 & 24 (11:45-12:00) TAB 5 MEANING OF QUANTITIES CASE STUDY To explore the importance of placing an instructional focus on “meaning of the quantities” in problems complete the “Meaning of Quantities” Case Study. Hand out the Meaning of the Quantities Case Study materials. Use this slide to introduce Part I of the Case Study. Provide about 10 – 15 minutes for participants to work in pairs. Then about 10 minutes to debrief the activity focusing on general instructional strategies that arise from the group. [See Facilitator Notes for an analysis of each of the solutions.]
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Meaning of the Quantities Case Study – Part II What evidence in the student solution below of the student understanding both the meaning of the quantities in the problem and the solution?
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Slides 23 & 24 (11:45-12:00) TAB 5
Individually and then as a group analyze the student solution in Part II. [See Facilitator Notes for the Case for an analysis.] LUNCH
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Missing value problems (MVPs) What is the general structure of a missing value problem?
Carrie is packing apples. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 8 bushels of apples?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) TAKE OUT YOUR OGAP FRAMEWORK The next Problem Type that we will analyze for structures is missing value problems. What is the general structure of a missing value problem? [In general, missing value problems involve finding a missing value in a set of equal ratios. That is, three of the four quantities are given and the solution involves finding the fourth quantity.]
3 boxes x = 2 bushels 8 bushels THESE SHOULD GO QUICKLY.
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Researchers suggest that the location of the missing value may affect performance… (Harel, G., Behr, M. 1993)
Internal Structure
Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. How many boxes will she need to pack 7 bushels of apples? Carrie is packing apples for an orchard’s mail order business. It takes 3 boxes to pack 2 bushels of apples. She needs 7 bushels of apples packed. How many boxes will she need? OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) Please refer to the OGAP Framework and find “Internal Structure.” You’ll notice a reference to Internal Structures. In the Ranch problem many of you noticed that the quantities were given in one order (acres and horse), but the rate was asked for in a different order (horses per acre). This is an example of non-parallel structure. In missing value problems researchers also suggest that the location of the missing value matters. The implications for instruction and assessment is that the location of the missing value should be varied in problems that students solve. Review of problems: The quantities in both these problems are the same. In both problems the solution is the number of boxes. However, These two problems are the same except that the missing values are not in the same place in the problem statement. In the top problem the structure is parallel (i.e., boxes to bushels throughout). In the second problem the structure is NOT parallel. The first ratio is given as boxes to bushels. Then the number of bushels is given. Then students are then asked to determine the number of boxes. .
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Change this problem to make it easier, and then harder.
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) AS A WHOLE GROUP, BRIEFLY DISCUSS HOW to modify this problem to be easier, harder. Share a few of the problems that participants write. Things that participants will probably change: 1) the order of the quantities, 2) the magnitude of the numbers, or 3) the multiplicative relationship within and between the ratios.
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A school is enlarging its playground. The dimensions of the new playground are proportional to the old playground. A school is enlarging its playground. The dimensions of the new playground are proportional to the
What isdimensions the measurement of the missing of the old playground. length of the new playground? Show your work. Rectangles are not drawn to scale
Old Playground
New Playground
90ft .
110 ft. 630 ft. .
What is the measurement of the missing length of the new playground? Explain how you found your What answer.
type of problem is this similarity problem?
OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 25-28 (12:45) Similarity problems show up as missing value problems, ratio comparison problems (See slide 29), and scale factor problems (see slide 30). This is a missing value problems.
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What type of problem is this similarity problem?
The dimension of 4 rectangles are given below. Which two rectangles are similar? A)2” x 8” B) 4” x 10” C) 6” x 12” D)6” x 15”
OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides (29-31) 12:45 GIVE TIME TO THINK ABOUT STRATEGIES. SPEND TIME ON THIS SLIDE. This is a ratio comparison problem. It can be solved by comparing the ratios between the two dimensions or by finding the multiplicative relationship between the dimensions. 2 DIFFERENT STRATEGIES (1) WITHIN THE FIGURE OR (2) BETWEEN THE FIGURES Ratio comparison solution (B and D because the ratio of one length to the other is the same (2:5)) A) 1:4 B) 2: 5 C) 1:2 D) 2: 5 Multiplicative relationship between dimensions (B and D because the multiplicative relationship with both ratios is 2.5) A) 8” is 4 times 2” B) 10” is 2.5 times 4” C) 12” is 2 times 6” D) 15” is 2.5 times 6”
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Scale Factor Problems
What is the general structure of scale factor problems?
The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle? OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides (29-31) 12:45 GO FAST The examples illustrate two different structures for scale factor problems. John Hancock Center Problem: The height of the model and the original are given. The problem asks for scale factor. Rectangle: The scale factor is given with length of the larger rectangle. The problem asks for the length of smaller rectangle
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Scale Factor The scale factor relating two similar rectangles is 1.5. One side of the larger rectangle is 18 inches. How long is the corresponding side of the smaller rectangle?
If a student was unable to solve this problem successfully, what variables would you change to make it more accessible? Why?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides (29-31) 12:45 In pairs, participants modify this problem. Ways to modify problem: 1) Change scale factor to an integer 2) Give dimension of smaller rectangle and ask to scale-up instead of scaling down
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Researchers indicate that students should interact with qualitative predictive and comparison questions as they are developing their proportional reasoning…. (Lamon, S. (1993))
OGAP Proportionality Framework
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 32 & 33 (12-45-12:50)
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Qualitative Problems
Why do you think researchers suggest these as important stepping stones?
Kim ran more laps than Bob. Kim ran her laps in less time than Bob ran his laps. Who ran faster? If Kim ran fewer laps in more time than she did yesterday, would her running speed be: A) faster; B) slower; C) exactly the same; D) not enough information. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 32 & 33 (12-45-12:50) These are non-numerical problems that involve a proportional situation. These help students think about the relationships, and not take cues from numbers.
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Researchers say… Students need to see examples of proportional and non-proportional situations so they can determine when it is appropriate to use a multiplicative solution strategy. (Cramer, Post, and Currier, 1993)
OGAP Proportionality Framework October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55) Slide 38--NON-PROPORTIONAL STUDENT WORK-SORTING TASK
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Solve these problems (Cramer, K., Post, T., and Currier, S. (1993)
Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?
3 U.S. dollars can be exchanged for 2 British pounds. How many pounds for $21 U.S. dollars?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55) Slide 38 TAB 6--NON-PROPORTIONAL STUDENT WORK-SORTING TASK These two problems were given to a group of pre-service teachers. [If participants have not already solved Sue or Julie (or a problem like it) then have participants solve both these problems. Then review solutions. It has been our experience that many teachers will apply a proportional strategy to the additive situation in the Sue and Julie problem. This is consistent with what happened with the group of pre-service teachers. See the next slide.]
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Classic Non-proportional Example (Cramer, Post, and Currier cited in Research Ideas, 1993)
“Sue and Julie were running equally fast around a track. Sue Started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?” *
22/33 undergraduates student treated this as a proportional relationship.
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55) [Slide 38--NON-PROPORTIONAL STUDENT WORK-SORTING TASK]
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Proportional example… Three U.S. dollars can be exchanged for 2 British pounds. How many pounds for 21 U.S. dollars? Same group – 100% solved it correctly using traditional proportional algorithm No one in the same group could explain why this is a proportional relationship while the “running laps” is not. October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55)
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Vermont Version
Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run? Do student work sort!
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slides 34-39 (12:45-12:55) TAB 6--NON-PROPORTIONAL TASK SORT To determine if middle school students treated this additive situation as a proportional situation, the VMP OGAP conducted a small study involving 82 sixth grade students. Some of the students had had instruction in solving proportional problems and others did not have instruction prior to solving the problem. Handout the Non-Proportional Student work set of papers. To get a feel for the type of responses found in the study (with a partner) sort this student work into 2 piles – 1) treated the problem as a proportion problem; 2) Treated as an additive problem. What did you find?
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Kim and Bob were running equally fast around a track. Kim started first. When she had run 9 laps, Bob had run 3 laps. When Bob completed 15 laps, how many laps had Kim run?
Vermont Version Grade 6(n= 82)
• 39/82 (48%) solved as a proportion • 33/82 (40%) solved reflecting the additive situation • 10/82 (12%) non-starters What are the instructional implications?
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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12:55 These are the data from the study. What are the instructional implications of these data? Have participants discuss in pairs for a minute or two and then as a full group. [Usually the discussion starts with the realization that the results might be an artifact of instruction. That is, most mathematics programs/texts ONLY include proportional problems during units focused on proportions. Students may use the structure of 3 known quantities and one unknown quantity and assume it is a proportional situation. From there participants recognize the importance of embedding non-proportional problems in instruction and assessment so students have to discriminate between the situations.]
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Topics • Ratios and proportions • Percents • Scaling • Similarity • Linear equations • Linear patterns and relationships • Slope • Rates • Frequency distributions • Probability
Different problem types are embedded in the different mathematical topics …within different contexts …which involve different multiplicative relationships …where the meaning of quantities within the problem and answer can vary ... And by the way, the numbers used (integer vs. non-integer) also affect performance. Wow! No wonder proportions are tough to teach and learn.
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 40-41 (12:55) The focus here should also be on the instructional implications assuring that students interact with a variety of structures in both instruction and assessment.
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….A proportional reasoner should be able to • recognize the nature of the proportional relationship, • find a sensible and efficient method to solve problems given the context, problem type, complexity of the numbers, meaning of the quantities, and the number relationships. •Represent the quantities in the solution with units that reflect the meaning of the quantities consistent with the problem situation. Ultimately, a proportional reasoner should not be influenced by context, problem types, the quantities in the problems and their associated units, or numerical complexity. (Cramer., Post, Currier, 1993; Silver, Ed VT cite visit (2006)
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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12:55
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Pre-assessment Analysis • • • • •
Problem types Context Multiplicative Relationships Internal Structure Ratio – relationship and referents (if applicable)
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Slide 42 12:55-1:45 (40 min.) TAB 7 PRE-ASSESSMENTS Hand out the OGAP Framework and grade level pre-assessments. Regroup participants into grade level groups (if this makes sense this late in the workshop). Have participants analyze each of the problems for 1) problem types;2) Context; 3) multiplicative relationships within and between ratios; 4) Internal structures; and, 5) ratio relationships and referents
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Administering the OGAP Pre-assessment • Tips for Students • Time • Level of Teacher Assistance • Analysis before December Meeting - NONE [IMPORTANT: Teachers should select 1 classroom of students to pre-assess for the December meeting.]
October 2008 Version 12.0 Vermont Mathematics Partnership (funded by the National Science Foundation EHR-0227057 and the US Department of Education S366A020002)
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Thoughts on Administering the OGAP Proportionality Pre-Assessment An important component of the Vermont Mathematics Partnership’s Ongoing Assessment Project involves gathering information about student understanding of proportionality concepts before beginning instruction through the administration of a pre-assessment. This pre-assessment is designed to elicit developing understandings, preconceptions, misconceptions, strategies that student use, and common errors that students make when solving questions involving proportionality. It is in this spirit of formative assessment that we offer the following thoughts on administering the pre-assessment. Tips for Students Let the students know that this is a pre-assessment on material that they will be learning this year so some or all of the material may be new to them. Encourage them to try their best even if they are unsure. Remind them that the information will help you in your planning, and will not be used as a grade. Time The amount of time students need to complete the pre-assessment will differ depending on the grade level and the number of items in your pre-assessment. The pre-assessment can be administered in numerous ways. Some teachers choose to spread the assessment over several days while others administer the entire assessment in one class period. Again, the purpose is to collect evidence from your students so feel free to choose a schedule that works best for your students. Level of Teacher Assistance The purpose of formative assessment is to collect evidence that will help you best meet the needs of your students. With this in mind, feel free to read any items to students who you feel need this type of accommodation. You may also decide to scribe for students who require assistance with writing. Although no special materials are needed to complete the pre-assessment, students can use tools or manipulatives that are part of regular classroom instruction. By all means assist students in decoding non-mathematical vocabulary. You should not, however, help students interpret any mathematics content. Final Thoughts The ideas above are not intended to be used as a “checklist of do’s and don’ts” but rather as a way to communicate the spirit in which the pre-assessments are best administered to your students. Please bring the completed preassessments to the December session. A major goal of these sessions is to help you learn how to analyze the evidence in your students’ responses and use your findings to influence your upcoming proportionality instruction. Feel free to contact Marge Petit ([email protected]) if you have any questions.
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