Each plank of wood measures 1 inch by 9 inches by 48 inches. Each brick measures 3 inches by 4.5 inches by 9 inches.

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1 Shelves This problem gives you the chance to: solve problems in a spatial context identify and distinguish the four point graphs related to this situation Pete is making a bookcase for his books and other stuff. He already has plenty of bricks and can get planks of wood for $2.50 each. Each plank of wood measures 1 inch by 9 inches by 48 inches. Each brick measures 3 inches by 4.5 inches by 9 inches. For each shelf, Pete will put three bricks at each end then put a plank of wood on top. The diagram shows three shelves. 3 inches 1. Pete wants five shelves in his bookcase. a. How many planks of wood does he need? b. How many bricks does he need? c. How high will the shelves be? d. How much will the bookcase cost? Copyright 2007 by Mathematics Assessment Page 70 Shelves Test 8

2 The diagram below shows graphs with the following descriptions: Description One: The cost of the bookcase against the number of shelves. Description Two: The number of bricks against the number of shelves. Description Three: The height of the bookcase against the number of shelves. Description Four: The width of the bookcase against the number of shelves. The equations of the graphs are y = 48, y = 10x, y = 6x, y = 2.5x A B C D Number of shelves 2. Complete this table to match each graph with its description and its equation. Graph letter Description number Equation A B C D Copyright 2007 by Mathematics Assessment Page 71 Shelves Test 8 8

3 Task 4: Shelves Rubric The core elements of performance required by this task are: solve problems in a spatial context identify and distinguish the four point graphs related to this situation points section points Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answer: 5 Gives correct answer: 30 Gives correct answer: 50 inches Gives correct answer: $ Four points for eight correct answers. 1 1ft 1ft 1ft 4 4 Partial credit 7 or 6 correct 3 points 5 or 4 correct 2 points 3 or 2 correct 1 point (3) (2) (1) 4 Total Points 8 Copyright 2007 by Mathematics Assessment Page 72 Shelves Test 8

4 Shelves Work the task. What are the big mathematical ideas being assessed? Students could interpret part 1 in a couple of different ways how much is needed altogether or how much more is needed (the task explains that they are looking at a diagram not at an actual bookcase). Look at student work for interpretation and consistency. How many of your students interpreted the task as written, total amount of materials needed? How many of your students interpreted the diagram as a picture of an already existing bookshelf (needing 2 more planks, 12 more bricks, height 20 in., and a cost of $5)? Now look at other answers for planks: How many students put 6? How many put 3? How many put 10? Other answers Look at the number of bricks: How many only considered bricks for one side of the book shelf How many made other errors: other? Can you figure out how or why students made some of these errors? Look at height of the bookshelves. How many of your students put: Other Why was this difficult for students? How often do students in your class have opportunities to interpret diagrams? 8 th grade

5 Now look at work on matching graphs to equations and descriptions. How do your students compare with the sample population: Graph Description Number Equations Letter Correct Answer Other responses Correct answer Other response y=10x y=6x y=48 y=2.5x A 12% 14% 7.5% 7.5% 5% 5% B y=6x y=10x y=48 y=2.5x 9% 11% 11% 7% 2% 9% C y=48 y=10x y=6x y=2.5x 7.5% 5% 13% 5% 4% 6% D y=2.5x y=10x y=6x y=48 10% 7.5% 9% 6% 4% 9% Did any of your students try to write their own equations? Why do you think it was easier for students to match the graph with the equation than with the description? How often do students work with variables in context? 8 th grade

6 Looking at Student Work on Shelves Student A draws the bottom shelf to get a sense of dimensionality. Calculations are shown in the margin. Notice how the descriptions are labeled on the side of the graph to make sense of each line. Student A 8 th grade

7 Student A, part 2 8 th grade

8 Student B is able to think about the situation including adding in the height for the boards. The student forgets to think about the size of the brick, assigning each a unit value of one, instead of the 3 inches. The student received full marks on the graph. Student B 8 th grade

9 Student C adds an extra board to the bottom of the bookshelves. In finding height, the student finds the height of one section of bricks rather that the height of all the shelves. Like Student A, Student C labels the sides of the graph with the descriptions. Student C 8 th grade

10 Student D is one of the few students to draw a diagram of the whole bookcase. The student thinks about adding on the two boards and the bricks needed for 2 more boards. The student may have lost the scale for bricks and found the height of the twelve additional bricks. However for cost the student finds the correct cost for all 5 shelves. So there is inconsistency in the chain of reasoning. How often do students get practice working on tasks where they have to use information for a series of steps? Where they need to continue to visualize and to quantify the same situation? Student D draws lines on the graph, but leaves the table blank. Student D 8 th grade

11 Student D, part 2 8 th grade

12 Student E adds an extra board to the bottom of the bookshelves when figuring height but not when finding number of planks. The thinking is inconsistent. The student assigns the cost of the boards to the bricks. The student is able to match all the equations to the graph, but incorrectly marks 3 of the 4 descriptors. Student E 8 th grade

13 Student F also struggles with inconsistency. Sometimes the student thinks about the entire bookshelf and can describe it. Then the student only thinks about the additional bit. Do students get enough opportunity to make their own diagrams to help them think about the entire situation? What other strategies might have helped students to think about this situation? Student F Student G originally thinks about the entire bookshelf, but answers the other 3 questions for only the part modeled in the diagram. However the student has a full grasp of the algebraic representations of the situation. Student G 8 th grade

14 Student G, part 2 8 th grade

15 Student H finds the height for 30 bricks, but only the 3 pictured boards instead of the 5 needed. The student struggles with the algebraic representations. Student H 8 th grade

16 Student I is not able to make sense of the first part of the task. The student even has trouble multiplying with decimals in find the cost of the planks. However the student is able to correctly match each graph with the correct equation. What do you think the student understands? What is your evidence? Student I 8 th grade

17 8 th Grade Task 4 Shelves Student Task Core Idea 3 Algebra and Function Solve problems in a spatial context. Identify and distinguish the pour point graphs related to this situation. Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. Identify functions as linear or nonlinear, and contrast their properties from tables, graphs, or equations. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope. Based on teacher observation, this is what eighth graders know and are able to do: Find the cost of the bookshelves by multiplying decimals Find the number of planks needed for the bookshelf Matching equations to graphs Areas of difficulty for eighth graders: Finding the height of the bookshelves by including both the height of the bricks and the height of bricks Keeping a consistent picture of the situation in their head. Students would think about the total picture for one part, the needed amount for another, and then often calculate what was in the diagram for another. They don t seem to have a strategy for thinking about the information in an organized way or doing an extended chain of reasoning. Few students made a diagram to think about the entire structure. Matching descriptions to either a graph or an equation. 8 th grade

18 The maximum score available on this task is 8 points. The minimum score needed for a level 3 response, meeting standards, is 4 points. Most students, 83%, could either find the cost of the boards or match two or three items on the table. Less than half the students, 44%, could find the boards, bricks, and cost and match two or 3 items on the table. About 12% could match all the representations between graph, equations and descriptors and find boards, bricks, and cost of the bookshelf. Almost 5% could meet all the demands of the task, including finding the height of the bookshelves. Almost 17% of the students scored no points on this task. 75% of the students with this score attempted the task. 8 th grade

19 Shelves Points Understandings Misunderstandings 0 75% of the students with this Students had difficulty matching equations score attempted the task. to graphs. See table in the questions for 1 Students could either match 2 or 3 items in the table or find the cost of the boards. 4 Students could match 2 or 3 items on the table and find the number of planks and the cost. 6 Students could find all the measures on part 1(planks, bricks, height, cost) and match the equations to the graph. 7 Students could find planks, bricks, and cost and match all the parts of the table. 8 Students could move from a partially completed diagram to reason about all the measures in a completed bookshelf (planks, bricks, height, cost). Students could match a graph to its equation and a description of the context being represented. reflection. Common errors for cost of boards included: $7.50, $10.50, $5.00. Common errors for number of planks: 2 (19%), 6 (10%), 3 (5%). Common errors for number of bricks: 15 (13%),12 (14%), 18 (7.5%), 6 (5%) Students struggled matching the descriptions to the graph. See table in the questions for reflection. The struggled with the height of the bookcase. Common errors: 45 (18%), 9 (12%), 54 (5%) 15 (5%), 20 (4%) and 24 (4%) 8 th grade

20 Implications for Instruction Students need more opportunity with reading and interpreting diagrams. Many students are treating the diagram as a picture of a bookcase rather than a schematic for how to build it. Students did not think about drawing the complete diagram as a thinking tool. Adding lines or marks to diagrams becomes critical to solving many geometric problems as well. This is a skill that should be encouraged. Students need many experiences working tasks with diagrams or where diagrams or models are tools for finding the solution. Many students did not include the board dimension in calculating the height of the bookcase. Students also forgot to include enough bricks to hold up both sides of the bookshelves. They don t have enough experience thinking about what parts of a diagram are relevant. Students seemed to understand how to match equations to points on a graph but had difficulty with matching descriptions to equations or points on a graph. Students need to have more experiences working with algebra in a context to see it as a tool for sensemaking and as a way of representing practical relationships. Ideas for Action Research Investigation Professional Research How do we help students learn to use tools like building diagrams or relating symbolic notation to descriptions of situations? What needs to happen in students brains to connect between different representations, such as graphing, equations, and diagrams? We often talk about what needs to be taught or how to teach, but do not spend much time thinking about how students learn. Plan a department meeting to focus on the use and learning of representations. Ask colleagues to bring articles or materials to share and discuss. A good source to start for looking at articles is the NCTM Yearbook, The Roles of Representation in School Mathematics. Article 1 looks at some of the theoretical issues on building internal and external systems of representation. Even a high level of (skillfully performing arithmetic and algebraic computations) does not imply an understanding of mathematical meanings, the recognition of structures, or the ability to interpret the results. Article 7 looks at how students build diagram literacy. What are the stages a student needs to go through to learn to use and to make diagrams? Article 14 looks at promoting multiple representations in algebra. Article 19 explains some of the research used by the Freudenthal Institute in developing their Realistic Mathematics Education and how they make use of representation, including instruction on equations from research on developing algebra units for middle grades students in Mathematics in Context. Read and discuss the articles. What are the implications for how you teach? For the kinds of tasks you choose? For the kinds of questions you ask? Combined with the reading and examining student work, can you pick one goal that you want to investigate in your classroom? What might be some of the first steps for conducting an investigation about the role of representations? 8 th grade

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