Drawing to Scale: Designing a Garden

Transcription

1 Problem Solving Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Drawing to Scale: Designing a Garden Mathematics Assessment Resource Service University of Nottingham & UC Berkeley Draft Version For more details, visit: MARS, Shell Center, University of Nottingham Please do not distribute outside schools participating in the initial trials

2 Drawing to Scale: Designing a Garden MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able to solve problems involving scale drawings, including computing actual lengths from a scale drawing, and lengths and areas on a scale drawing from actual lengths. The lesson will help you identify and help students who have difficulties understanding and using graphical, algebraic and numerical approaches to scale drawings. COMMON CORE STATE STANDARDS This lesson relates to the following Mathematical Practices in the Common Core State Standards for Mathematics: 2. Reason abstractly and quantitatively 4. Model with mathematics. 5. Use appropriate tools strategically. 8. Look for and express regularity in repeated reasoning. This lesson gives students the opportunity to apply their knowledge of the following Standards for Mathematical Content in the Common Core State Standards for Mathematics: 7-G: Draw, construct, and describe geometrical figures, and describe the relationships between them. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7-EE: Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7-RP: Analyze proportional relationships and use them to solve real-world and mathematical problems. INTRODUCTION This lesson unit is structured in the following way: Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You review their responses and create questions for students to consider when improving their work. At the start of the lesson, students reflect on their individual responses and use the questions posed to think of ways to improve their work. They then work in small groups to analyze sample responses to the task. Using what they have learnt from the responses, students, in the same small groups work collaboratively on an extension of the same task. In a whole-class discussion, students review the methods they have seen and used. Finally, students reflect on their work. MATERIALS REQUIRED Each student will need a copy of the task: Designing A Garden, My Plan.doc, a sheet of paper, a mini-whiteboard, a pen, an eraser, and a copy of the review questionnaire How Did You Work? Each small group of students will need copies of the Assistants Methods, My Plan.doc, Mandy s Second .doc, and a blank sheet of paper. Provide rules, scissors, glue sticks, card, and graph paper for students who choose to use them. There are some projector resources to support whole-class discussion, and to help introduce activities. TIME NEEDED 20 minutes before the lesson for the assessment task, an 85-minute lesson, and 10 minutes in a follow-up lesson. Timings given are only approximate. Teacher guide Drawing to Scale: Designing a Garden T-1

3 BEFORE THE LESSON Assessment task: Designing a Garden (20 minutes) Have students complete this task, in class or for Designing a Garden homework, a few days before the formative Imagine you are a garden designer. assessment lesson. This will give you an You have received this from a customer: opportunity to assess the work, and to find out the kinds of difficulties students have with it. You should then be able to target your help more effectively in the follow-up lesson. Give each student a copy of the assessment task: Designing a Garden, My Plan.doc, and a sheet of paper. Introduce the task briefly, helping the class to understand the problem and its context. Use just the garden features in the attached file to draw to scale a plan of Mandy's garden. On a separate piece of paper, explain the math. Mandy attached the following files: Garden Features.doc!"#$%&'()'%)'#$'*+'%,-./$0' This task is about garden design.! 1"/.' "#$%!&'(%'%(!)*+,!,*%(-!! ")!+,!.!/%)%',!0+(%1!2-.!/%)%',!3&45!64(!.-7! /%)%',!)633-' What do garden designers do?! What plans do designers draw? [Plans of garden features such as patios, lawns, etc.] 2#-345,-'6)$.' 8%96:,%!&;!)*%!)<=%!&;!;+,*!"!=364!)&!=:)!+4!)*%! =&4(1!"#(!3+>%!)*%!,:';69%!)&!?@%!6)!3%6,)!A!/.-!! 7'8/$3"' B*%!?%49*!+,!7.!9%4)+/%)%',!(%%=!64(!CDE! 9%4)+/%)%',!3&45-! Will these plans be to scale? Why?!! Student Materials Designing a Garden 2012 MARS, Shell Center, University of Nottingham S-1 You may want to show them Slide P-1 of the projector resource. Designing a Garden Projector Resources Designing a Garden P-1 Read through the questions and try to answer them as carefully as you can. At this stage, you only need to add the scale and the three features to the plan It is important that, as far as possible, students are allowed to answer the questions without assistance. If thye are struggling to get started, ask questions that help students understand what is required, but make sure you do not do the task for them. Teacher guide Drawing to Scale: Designing a Garden T-2

4 Students who sit together often produce similar answers, and then when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual seats. Experience has shown that this produces more profitable discussions. Assessing students responses Collect students responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches. We suggest that you do not score students work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics. Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given in the Common issues table on the next page. These have been drawn from common difficulties observed in trials of this unit. We suggest you make a list of your own questions, based on your students work. We recommend you either: Write one or two questions on each student s work, or Give each student a printed version of your list of questions, and highlight the questions for each individual student. If you do not have time to do this, you could select a few questions that will be of help to the majority of students, and write these on the board when you return the work to the students. Teacher guide Drawing to Scale: Designing a Garden T-3

5 Common issues Student is unable to calculate the scale Student scale is incorrect, lacks detail or is not simplified For example: In the scale ratio, the student writes the figures in reverse order, e.g. 50 : 1 instead of 1 : 50. OR: The student mixes the units in the scale ratio, e.g. 1 : 0.5. OR: The student states the scale is 10 m to 20 cm Student has difficulty calculating measurements on the plan For example: The student has difficulty when the actual measurements are in centimetres OR: The student has difficulty when the actual lengths are not whole numbers. Student has difficulty calculating the radius of the pond Student makes a technical error Suggested questions and prompts What do you know? What do you need to figure out? How can you use what you know to do this? Complete this sentence: 1 cm on the drawing represents... What does a scale ratio, such as 1 : 20 mean? In a scale ratio, what do you know about the units? How can you get rid of the fraction in the ratio? How do you convert 3 m to a measurement on the plan? Now apply your method to figure out what the length 1.4 m/0.5 m is on the plan. The formula for figuring for area of a circle is!r 2. Use it to figure out the radius of the pond. Check your work. Teacher guide Drawing to Scale: Designing a Garden T-4

6 SUGGESTED LESSON OUTLINE Throughout the guide the scale is written either as 1 : x or an equivalent description, such as 1 cm on the plan is equivalent to 0.5 m (or 50 cm) in the garden. Individual work (10 minutes) Return the assessment task to the students. Give each student a mini-whiteboard, a pen, and an eraser. Begin the lesson by briefly reintroducing the problem. If you did not add questions to individual pieces of work, write your list of questions on the board. Students select questions appropriate to their own work and spend a few minutes answering them. Recall what we were looking at in a previous lesson. What was the task about? Today we are going to work together to try to improve your initial attempts at this task. I have had a look at your work, and I have some questions I would like you to think about. On your own, carefully read through the questions I have written. I would like you to use the questions to help you think about ways of improving your own work. Use your mini-whiteboards to make a note of anything you think will help to improve your work. Collaborative analysis of Assistants Methods (30 minutes) Distribute copies of Assistants Methods to each group of students. This task gives students an opportunity to evaluate different approaches to the task, without providing a complete solution strategy. Imagine you have three assistants. They also had a go at designing the garden. In your groups you are now going to look at their methods for figuring out the lengths on the scale drawing. All their work is incomplete. You may want to add notes to the work to make it easier to follow. There are some questions for you to answer as you look at the work. Slide P-2 of the projector resource summarizes how students should work together. Encourage students to focus on evaluating the math contained in the assistant's work, not whether the assistant has neat writing etc. Also, check to see which of the explanations students find more difficult to understand. Teacher guide Drawing to Scale: Designing a Garden T-5

7 The methods used by the assistants are incomplete and do include mistakes, however they do illustrate efficient strategies for figuring out lots of measurements for the scale drawing, and also the reverse, obtaining real life measurements from the measurements on a scale drawing. Billie has used a graph. This method means Billie can read off the graph the measurement on the plan of any garden length, to a maximum of 20 meters. Billie needs to add the y-axis title Plan lengths (cm).' He has extended the x-axis to 20 meters, however the maximum length in the garden is 10 meters. To obtain more accurate values for the measurements on the plan, Billie should change the scale and use square paper that is divided into millimeters. Billie has incorrectly used the equal sign (1 m = 2cm.) He has also incorrectly written the scale ratio using two different units of measure. The scale is 2 cm : 100 cm = 1 : 50. Billie has correctly figured out the radius of the pond, but his units are incorrect. The radius is 1.5 m. He then needs to use the graph to figure out the equivalent measurement on the plan. Lisa has provided a rule to use for drawing the features in the plan. It is correct, although she has not specified the scale. Lisa could improve the rule by starting from a garden measurement of 0 m instead of 1 m. There is no need to extend the rule beyond the maximum garden measurement of 10 m. To obtain more accurate measurements Lisa could use square paper that is divided into millimeters. Lisa has incorrectly figured out the radius of the pond. Her mistake is to convert m 2 to cm 2 ; there is no need to do this. Her workings after this are correct, but inevitably result in an incorrect answer. Teacher guide Drawing to Scale: Designing a Garden T-6

8 Jim has chosen an algebraic method. He correctly figures out the width of the garden. The equation for b is incorrect. b = 2a. When defining a and b Jim does not state the units. The disadvantage of this method is that each measurement on the plan needs to be computed. But the advantage is that he has exact measurements. Collaborative activity: Adding Garden Features to the Plan (30 minutes) Have graph paper, scissors, card, and glue sticks available. Give each group Mandy s Second , My Plan.doc and a blank piece of paper. Explain to the class that Mandy has sent a second . Your task now is to design Mandy s garden using all the features. Before doing this, draw a rough sketch of where you think each feature will go. Take turns to select a garden feature. Explain to your partner how you obtained the measurements of the feature for the plan. You may want to use a similar method to the assistants. Your garden can have more than one area of lawn, and more than one flower border. Slide P-3 summarizes this information. While students work in small groups you have two tasks: to note different student approaches to the task, and to support student problem solving Note different student approaches to the task Designing the Yard 1. First draw a rough sketch of where you think each feature will go. 2. Take turns to select a garden feature. 3. Explain to your partner how you obtained the measurements of the feature for the plan. You may want to use a similar method to the assistants. 4. Your garden can have more than one area of lawn, and more than one flower border. Be Creative! Listen and watch students carefully. Note different approaches to the task and what assumptions students make. Do students use a similar approach to one of the assistants? Do students create their own rule or graph? Do students work systematically? Do students check their work? What do students do when they become stuck? Does the garden work? Have students made practical mistakes such as placing the shed right next to the window? In particular, note any common mistakes. You can then use this information to focus a whole-class discussion towards the end of the lesson. Teacher guide Drawing to Scale: Designing a Garden T-7

9 Support student reasoning Try not to make suggestions that move students towards a particular approach to the task. Instead, ask questions that help students to clarify their thinking. In particular focus on the strategies rather than the solution. Encourage students to justify their statements. You may need to tell your students that 1 foot is equivalent to about 0.3 meters. Make sure students understand that they do not have to use up all the gravel or grass seed. How much gravel is Mandy prepared to buy? How can you use this value to figure out the maximum length of path in the garden? Imagine sitting on the decking with five friends. How much decking do you need to sit comfortably? What area does each square cm on the plan represent in Mandy's garden? How can you use this to figure out the area of lawn? Do you like the layout of your garden? Imagine sitting on the decking, what would you see? Imagine looking out of the window, what would you see? You may want to use the questions in the Common issues table to support your own questioning. If the whole class is struggling on the same issue, you could write one or two relevant questions on the board, and hold a brief whole-class discussion. Whole-class discussion: comparing different approaches (15 minutes) Hold a whole-class discussion to consider the different approaches used. Focus the discussion on parts of the task students found difficult. Ask the students to compare the different solution methods. Ask the following questions in turn: Which approach did you use? Why? Did anyone use one of the assistants methods? What is the advantage of using their methods? [Billie and Lisa s methods are efficient for converting lots of measurements, however these measurements may be inaccurate.] Did you change the method in any way? Did anyone come up with a different method for drawing the plan? Which of the assistants methods did you find most difficult to understand? Why? How would you adapt Billie s/lisa s/jim s method for a different scale, say 1 : 20 ( or 1 cm on the plan is equivalent to 20 cm in the garden)? Depending on your class, you may want to ask students to consider the assistant's methods in general terms: Can any of the assistants methods be adapted for any scale, for example if 1 cm on the plan represents a m in the garden? [Lisa s method cannot be adapted.] Show me. Using Billie s and Jim s method: Teacher guide Drawing to Scale: Designing a Garden T-8

10 If a m in the garden is represented by 1 cm on the plan, then 1 m in the garden is represented by 1 a cm on the plan, so y = 1 a x y is a length (cm) on the plan, and x is the equivalent length (m) in the garden. OR, if a cm in the garden is represented by 1 cm on the plan, then 1 cm in the garden is represented by 1 a cm on the plan, so y = 1 a x y is a length (cm) on the plan, and x (cm) is the equivalent length in the garden. For Billie s method, this equation would be plotted on graph paper. This is a challenging problem, so once students have spent a few minutes tackling it on their own, encourage them to discuss it with a partner. Students should test their solutions using real examples. To support the discussion, you may want to use Slides P-4, P-5, and P-6 of the projector resource. Follow-up lesson: Review work (10 minutes) Give out the sheet How Did You Work? and ask students to complete this questionnaire. The questionnaire should help students review their progress. Some teachers give this task for homework. Teacher guide Drawing to Scale: Designing a Garden T-9

11 SOLUTIONS Assessment task: Designing a Garden The scale can be calculated using the length on the plan of 20 cm, this is equivalent to 10 meters. Therefore 1 cm on the plan is equivalent to 0.5 m in the garden. This scale can be written as 1 : 50. There are many ways students could draw the design. Below are some examples of how students could efficiently figure out measurements on the plan of actual measurements in the garden. A graphical method: Length on Plan (cm) Length in Garden (m) The general equation of a line is: If a m in the garden is represented by 1 cm on the plan, then 1 m in the garden is represented by 1 a cm on the plan, so y = 1 a x y is a length (cm) on the plan, and x is the equivalent length (m) in the garden. An algebraic method: 0.5 m in the garden is represented by 1 cm on the plan, then a m in the garden is represented by In general terms : 1 a = 2a cm on the plan. 0.5 If b m in the garden is represented by 1 cm on the plan, then a m in the garden is represented by a b cm on the plan. Teacher guide Drawing to Scale: Designing a Garden T-10

12 A Scale Rule Method: Length on Plan (cm) Lengh in Garden (m) Garden Features Students may use their own method, or one of the above methods to figure out the dimensions on the plan of the garden features: Shed I ve ordered this shed. It is 2 meters wide, 3.2 meters long and 2.8 meters tall. Circular pond Because of the type of fish I plan to put in the pond, I'd like the surface to be at least 7 m 2. Dimensions on Plan 4 cm by 6.4 cm "r 2 = 7 7 r = " = 1.5 m Radius on plan : 3 cm A bench The bench is 82 cm deep and 140 cm long. A gravel path I d like a one meter wide path to go from the shed to the house and from the garden gate to the house. The gravel should be about 8 cm deep. Gravel costs $40 per cubic meter. I don t want to spend more than $60. Borders In the past I ve found having borders more than 4 foot wide are really difficult to manage. Decking for barbeques I d like this near the house. It should be big enough to seat at least six people. Lawn Could you please grass the remaining areas using roll out grass? It costs $12 per square meter. Please let me know the cost cm by 2.8 cm Volume of gravel: = 1.5 m 3. Maximum length of path: 1.5 (1 " 0.08) = m. This is 2 cm by 37.5 cm. 1 foot = 0.3 m; 4 feet = 1.2 m. Maximum width on plan: 2.4 cm, If the decking is rectangular then the dimensions could be at least 3 m by 3 m. This is 6 cm by 6 cm on the plan Students answers will vary Teacher guide Drawing to Scale: Designing a Garden T-11

13 Designing a Garden Imagine you are a garden designer. You have received this from a customer: Use the plan to draw the three features. Add the scale. On a separate piece of paper, explain the math. Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

14 My Plan.doc Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

15 Assistant Billie s Method Explain Billie's method. How could Billie improve the graph? What mistakes has Billie made? Explain how he could correct these mistakes. Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

16 Assistant Lisa s Method Explain Lisa s method. How could the method be improved? What mistakes has Lisa made? Explain how she could correct these mistakes. Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

17 Assistant Jim s Method Explain Jim s method. Jim s formulae use centimeters and meters. Would the formulae be different if a and b represented lengths in centimeters? Explain your answer. What mistakes has Jim made? Explain how he could correct these mistakes. Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

18 Mandy s Second Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

19 Mark the boxes and complete the sentences that apply to your work. 1 Our group method was better than my own work Our group method was better because How Did You Work? 2 Our method is similar to one of the assistants work OR Our method is different from all the assistants work Our method is similar to Add name of the assistant Our method is different from all the assistants work because: I prefer our work / the assistant s work (circle) because 3 What advice would you give a student new to this task about potential pitfalls? Student Materials Drawing to Scale: Designing a Garden S MARS, Shell Center, University of Nottingham

20 Designing a Garden Projector Resources Drawing to Scale: Designing a Garden P-1

21 Evaluating the Design Assistants Work 1. Take turns to work through the method of an assistant. - Write your answers on your mini-whiteboards. 2. Explain your answer to your partner. 3. Listen carefully to explanations. Ask questions if you don t understand. 4. Once you are both satisfied with the explanations, write the answers below the assistants work. - Make sure the student who writes the answers is not the student who explained them. Projector Resources Drawing to Scale: Designing a Garden P-2

22 Designing the Yard 1. First draw a rough sketch of where you think each feature will go. 2. Take turns to select a garden feature. 3. Explain to your partner how you obtained the measurements of the feature for the plan. You may want to use a similar method to the assistants. 4. Your garden can have more than one area of lawn, and more than one flower border. Projector Resources Be Creative! Drawing to Scale: Designing a Garden P-3

23 Assistant Billie s Method Projector Resources Drawing to Scale: Designing a Garden P-4

24 Assistant Lisa s Method Projector Resources Drawing to Scale: Designing a Garden P-5

25 Assistant Jim s Method Projector Resources Drawing to Scale: Designing a Garden P-6