# Parallelogram. This problem gives you the chance to: use measurement to find the area and perimeter of shapes

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1 Parallelogram This problem gives you the chance to: use measurement to find the area and perimeter of shapes 1. This parallelogram is drawn accurately. Make any measurements you need, in centimeters, and calculate: a. The area of the parallelogram. Show your calculations. The area of a parallelogram = base x height b. The perimeter of the parallelogram. Show your calculations. 2. The diagram below shows the same parallelogram again. a. Find the area of Triangle A. b. Find the area of Triangle B. c. Explain how you found your answers. Triangle A Triangle B Copyright 2007 by Mathematics Assessment Page 57 Journey Test 7

2 3. Which triangle has a larger perimeter, Triangle A or Triangle B? Explain how you can tell without measuring. 4. Sketch a right triangle with the same area as Triangle A. Your diagram does not need to be accurate. Show how you figured it out. 9 Copyright 2007 by Mathematics Assessment Page 58 Journey Test 7

3 Task 4: Parallelogram Rubric The core elements of performance required by this task are: use measurement to find the area and perimeter of shapes Based on these, credit for specific aspects of performance should be assigned as follows 1.a Gives correct answer in the range square centimeters. Shows correct work such as: 7 x 5 or 6 x 6. Accept reasonable measurements shown on diagram. b Gives correct answer in the range centimeters and shows work such as 2(6 + 7). Accept reasonable measurements shown on diagram. points section points 3 2.a Gives correct answer 17.5 square centimetres. Accept half of 1.a 1ft b Gives correct answer: 17.5 square centimetres. Accept half of 1.a c Gives correct explanation such as: They are both equal to half the area of the parallelogram 1ft Gives correct answer such as: Triangle B: both triangles have sides that match the two sides of the parallelogram. The third side of B is longer than the third side of A Sketches a correct triangle and shows correct work such as: The area of the triangle = 1/2 base x height = base x height = 35 So if the base = 7 cm then the height = 5 cm 2ft 2 Note: Deduct 1 point for missing or incorrect units. (Need to show some evidence that are is measured in square units and that perimeter is a linear measure. Total Points 9 Copyright 2007 by Mathematics Assessment Page 59 Journey Test 7

6 Looking at Student Work on Parallelogram Student A is able to draw in heights for the parallelogram and the triangles. The student makes a complete logical argument for comparing perimeters in part 3. The student makes a right triangle, labeling the dimensions of the legs. The student labels dimensions and calculations with appropriate units. Student A 7th grade

7 Student A, part 2 Student B is the only student in the sample to find a correct set of dimensions without using measurements from previous shapes. Student B 7th grade

8 Student C makes a vertical side for the base of the parallelogram and uses appropriate measures. (Notice that the numbers are slightly smaller that expected. The student may have measured from the end of the ruler instead of from zero.) The student seems to be reconstructing the logic of the area formula each time, by showing how the shape can be decomposed and remade into a rectangle. This strategy helps the student design the right triangle correctly. Notes at the bottom of the part 2 seem to indicate that the student calculated the perimeter, but that doesn t help the student think out a convincing argument for why triangle B has a larger perimeter. Student C 7th grade

9 Student C, part 2 7th grade

10 Student D is able to measure the correct height to calculate the area of the parallelogram and the area of triangle A. What error does Student D make in trying to draw a height in triangle B? Find 2 ways to redraw the height. Student D 7th grade

11 Student E uses a height to find the area in part 1. However the height measure is more than 0.5 cm off. Notice that no dimensions are drawn on any diagrams. The student appears to have multiplied base times height in part 2, but makes the common error of forgetting to divide by 2 as evidenced by the size of the numbers and formula. Student E 7th grade

12 Student F does not draw a height to find the area of the parallelogram. However the student does draw in and measure heights for the triangles. The student only uses part of the base, instead of the entire base, to calculate the area of the triangles. What assumption has the student made? Why doesn t it work, isn t the formula half the base times the height? Student F 7th grade

13 In many situations I can decompose part of a number and then recompose it to get an identical answer. Why doesn t this work for the sides of the triangle? Why isn t the area the same? Look at the work of Student G/ Student G Student H has decomposed the parallelogram into two triangles and a rectangle. Why aren t the calculations correct in part 1a? What did the student forget? Student H 7th grade

14 Student I, like 5% of the students, has areas for the triangles in the hundreds. Notice the logic of the justification in part 3. Student I shows a typical drawing for part 4 of a triangle that is not a right triangle and has no dimensions. Student I 7th grade

15 Student J gives side lengths for area. Student J Student K thinks shapes with the same area have the same perimeter. Student K 7th grade

16 While the primary purpose of this task is to look at student understanding about direct and indirect measurement, a rich task can also give insight into other misconceptions. The following set of students is having difficulty calculating with rational numbers. Look at the work of Student L. What error has the student made in multiplying by a fraction? What is the misconception underlying this error? Student L What error has Student M made in multiplying decimals? Student M What error has Student N made in adding decimals? Student N 7th grade

17 Look at the multiplication error in the work of Student O. How is it different from the misconception of Student M? Student O 7th grade

18 7 th Grade Task 4 Parallelogram Student Task Core Idea 4 Geometry and Measurement Use measurement to find the area and perimeter of shapes. Apply appropriate techniques, tools, and formulas to determine measurements. Develop and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence and similarity. Mathematics in the task: Ability to measure using a centimeter ruler Ability to draw and measure an altitude or height in a parallelogram and a triangle Ability to calculate area of a parallelogram and triangle Ability to develop a logical argument on perimeter with statements about the relative sizes of all the sides Ability to draw a right triangle and to find and to label leg size needed to obtain a given area Based on teacher observation, this is what seventh grade students knew and were able to do: Measure dimensions and put quantity on a drawing or diagram Calculate the perimeter of a parallelogram Areas of difficulty for seventh graders: Understanding the difference between height and length of a side in a parallelogram and a triangle Finding the appropriate numbers to substitute into an area formula Dividing by two when finding the area of a triangle Comparing perimeters of two shapes without measurement, thinking about the relationships of the sides Drawing a right triangle Labeling the legs instead of the hypotenuse, to give the dimensions for a given area Applying appropriate measurement units to calculations (cm or cm 2 ) 7th grade

19 The maximum score available for this task is 9 points. The minimum score needed for a level 3 response, meeting standard, is 4 points. About half the students, 52%, could find the perimeter of the parallelogram and describe a correct method of finding the area of a triangle but most did not label answers with correct units. About 25% of the students could find the perimeter, describe how to find the area of a triangle, compare the perimeter of two triangles, and draw a right triangle that with dimensions needed to get a certain area. Again most of these students did not label answers with correct units. Less than 2% of the students could meet all the demands of the task, including drawing and measuring heights of parallelograms and triangles, calculating areas of triangles and parallelograms, and labeling answers with appropriate units. 48% of the students scored no points on this task. 90% of students with that score attempted the task. 7th grade

21 Points Understandings Misunderstandings 8 Students could find the height of the parallelogram and use it to calculate area, find the perimeter of shapes, find the area of triangles, and draw a right triangle that would have a given area. Students used unit labels. 9 Students could find the height of the parallelogram and use it to calculate area, find the perimeter of shapes, compare perimeters, find the area of triangles, and draw a right triangle that would have a given area. Students used unit labels. Students could not compare the perimeters of the two triangles. Students, who didn t get this score, could not find the area of the parallelogram. 57% of the students multiplied two sides together. 6% just put the measurement of 1 side for the area. 5% added two sides together for area. Implications for Instruction Students need more opportunities to investigate shapes and make their own measurements. The reasoning for finding height and understanding how it is measured is different from the reasoning used for substituting numbers into a formula. Students should understand the reasoning for how area formulas are derived. They should be able to think about how a parallelogram can be decomposed are recomposed to form a rectangle. This type of understanding helps students to understand why the measurement of the height is different that of the side length. If possible, students should have the opportunity to investigate with dynamic geometry software programs how the decomposition works and compare with an overlapping picture of a rectangle with the height the same as the side length. Students should also have opportunities to investigate the relationships between a rectangle and a triangle. Again, using geometry software, like Geometer Sketchpad, to investigate these relationships would help students develop a better understanding of these relationships. Students need to be exposed to a greater variety of models for triangles. They also need regular practice making their own diagrams, drawing in heights, and labeling needed measurement for helping them think about problems. Too many students did not draw right triangles in part 4. Students also need more experience developing logical arguments about a variety of topics. Students should be thinking about comparing attributes between figures for similarity and be able to notice and to state that two of the sides of the triangles in part 2 are the same size. Having opportunities to see and compare different arguments in class and discuss which is more convincing helps students develop the logic needed to make mathematical arguments. 7th grade

23 Martha has measured the base and the height of triangle B, but her answer is incorrect. What is the mistake that Martha has made? What would you tell Martha to help her with her diagram? The horizontal measure is 3.9. Students should be able to find 2 different ways for measuring the base and height. Questions could also be posed about how to make a convincing argument. Denise says, Triangle A and B have the same perimeter because they are both half the parallelogram. Logan says, Triangle B has a larger perimeter because it is longer but skinnier than Triangle A. Julia agrees. I can just tell by looking that B is larger. Do you think these are convincing arguments? How could they be improved? What information do you have that could support your opinion? Depending on class interest, you might want to pursue questions about how they made the right triangle for part 4 or you might want to plan some further lessons about geometric shapes. 7th grade

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