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TITLE: Step by Step, Inch by Inch GRADE: Grade 8 CONNECTIONS OF Source: LESSON TO SMART SMART Consortium-MIGG CONCEPTS AND OHIO STANDARDS: Common Misconceptions and/or Errors: * Students may be confused measuring steps in inches so SMART Concept: you may need to review how to read a ruler for measurements less than one full inch. * Students may not understand that the ratio of riser to tread in inches is equal to the ratio of riser to tread in centimeters. * Students often reverse the measurements for riser and State of Ohio Academic tread, thus inverting the ratio and impacting the slope. Content Standard: Patterns, Functions and Algebra Lesson Summary: Standard Students will examine the relationship between the tread width and riser height of a variety of steps to chart Benchmark: measurements and understand the concept of slope. • Analyze and compare Estimated Time duration: functions and their graphs 90 minutes using attributes¸ such as rates of change¸ intercepts and zeros. Materials/Equipment Needed: Teacher: • Solve and graph linear 1 set of photos for group discussion equations and inequalities. 1 set of overhead slides of the tables used 1 projection device for whole group Grade Level Indicator: 8A4 Extend the uses of Student: variables to include covariants 1 set of cut out photos of steps (see appendix) per group where y depends on x 1 copy of Table 1 per group 8A6 Describe the relationship Tape measure (customary units) between the graph of a line and Graph paper its equation, including being Pencils able to explain the meaning of slope as a constant rate of Pre-Assessment: change and y-intercept in real- Pre-Assessment Scoring Rubric: world problems. Prior to beginning instruction, facilitate a full group 8A15 Describe and compare discussion about "steps" in which you share some of this how changes in an equation background information: affects the related graphs; e.g., *Steps come in a variety of different lengths and patterns of for a linear equation changing rise. the coefficient of x affects the *Steps have two distinct parts - the tread and the riser. slope and changing the constant *Most steps have a 9" tread and an 8.5" riser. affects the intercepts. *Ask students to think about what causes some steps to be "steeper" than others. Mathematical Processes Standard Post-Assessment: Benchmark: Post-Assessment Scoring Rubric: The drama club is currently working on the construction of the set for the spring play. They have asked for your help in building stairs that are needed to permit people to step up onto the dance floor. The dance floor is 3 feet off the ground. They would like plans presented at their next TITLE: Step by Step, Inch by Inch GRADE: Grade 8 meeting for consideration. Use cardboard to illustrate a side view of your proposed Challenge students to design staircase. plans for the steps to submit for review. Suggest they sketch a Next, tell the students that they are going out of town with drawing. their family, and cannot make the next drama club meeting. Suggest the students answer Write a letter noting you will be out of town, but would like the following questions: your proposal considered. In the letter, explain how you * How many steps will be created the steps. Include why you feel this is the best set needed? of stairs for the project. Also include in your letter all * What will be the riser height calculations and work so that the drama club members can and tread width? better see how you reached your final product. * Sketch the steps and label measurements. (See Appendix for the scoring rubric.) * Identify the slope of your staircase. Key Vocabulary: Tread - the part of the step you actually step on. Riser - The part of the step that is perpendicular to the treads. Slope - The steepness of a line. The rate of change of a line; Rise over run - Vertical change divided by horizontal change Steps for Instruction: 1. Organize students into small groups; pass out needed materials. 2. Ask students what they look for when they choose a place to go sledding. (Students should respond that they look for "steep" hills that will cause the sled to go faster and make the ride more exciting.). 3. Have students discuss how to describe the level of steepness of a hill to a friend. Students make a sketch of their favorite sledding hill. 4. Talk about "steepness" as a concept with the class. 5. Have students brainstorm a definition of "steepness". (I.e. what makes one hill "steeper" and more fun than another?) 6. Ask students to cite other places where they can observe the "steepness" of an object. (Students might cite roads, steps, ramps for wheelchairs, rooftops, ...). 7. Have students apply the class generated definition of "steepness" to steps in general. 8. Ask students to think about what causes some steps to be "steeper" than others. 9. Next, discuss with students the fact that steps have two distinct parts - the tread and the riser. 10. Sketch one set of steps on the board and label "tread" and "riser" for each step. 11. Cut out and copy the pictures included with this lesson. Distribute one set of pictures to each group of students. 12. Give each group of students two of the pictures you cut out and have them discuss which picture has steeper steps, and why? 13. Give each group the remaining pictures, and instruct them to organize the pictures according to their perceived steepness of the stairs from least steep to most steep. 14. Direct students to discuss the following questions in their groups: * "What is different about the pictures?" * "Why did you order them the way you did?" * "How did you make your decisions as to how to place the pictures in this order?" 15. Have a representative of each group report a synopsis of their group's discussion. 16. Instruct students to recall the discussion about selecting a hill for sledding. Have two volunteers try to convince the rest of the class why their individual hill is better for sledding by demonstrating/sketching the steepness of the hill on the board or overhead. 17. Ask the class, "How can we measure the steepness of each hill in order to compare the hills and determine which hill would be more fun to sled down?" 18. Suggest that students draw a segment representing the height of the hill on the diagram and ask how they should do this. 19. Students should draw a segment from where the hill meets the ground to where the base of the hill ends. Discuss why this would be the correct line to draw. 20. Distribute a copy of Table 1 to each group. Here they have the actual measurements of the stairs shown in the pictures in customary measurement similar to what a carpenter would use to actually make the steps from wood or concrete. 21. Groups are to calculate the ratio of the riser to the tread width for each step. 22. Have each group record their ratios in Table 1 then check them for accuracy against those listed on an overhead slide shown to the full class. 23. Instruct students to refer to the actual measurements listed in Table 1 to answer the following questions within their group: * Which stairs are the steepest? * Who thinks they found stairs that are the least steep? * What is the difference between these two stairs? * Which stairs are very similar in steepness? What do they have in common? * Which stairs would be the easiest for people to climb? Why? Commentary: * If you have an opportunity to take the class on a field trip outside of school, bring the students to downtown Cleveland. Give each student a blank form for Table 1, and have pairs of students physically measure a step at each given location, filling in the table using measurements in inches. For this activity, one student should measure the parts of each step using customary units, while the other student records the measurements in a chart. * If you live in another city, identify steps at buildings in your business or downtown area, measure them and take the students there for the field trip. 24. Invite each group to report out their observations, for a full class discussion. Point out to students that they will now have the opportunity to "draw" stairs on graph paper so that they can explore the concept of "slope". 25. Define "slope" for students; illustrate some examples. 26. Next students next should be directed to use graph paper to "draw" four different sets of steps following these instructions: * Begin at the origin (0,0), marking this point as "A". * Draw a line segment AB to represent the riser height. * From the end point of this riser segment, point "B", the students are to draw another line segment, BC, to represent the tread width. * Connect the points at the "bottom back" of each step, points A to C. * Use a different colored pencil for each graph you sketch. 27. Direct students to discuss the following questions in their groups, then facilitate a class discussion so that groups can explain their answers and/or ask questions: * Which graph of a step is the steepest? * Which is the least steep? * What is the difference between these stairs that might account for the change in steepness? * Which step is easier for people to climb and why? 28. Next, direct students to "draw" a set of steps with a riser height of 9 inches and a tread width of 9 inches. Invite answers from the class for these questions: * What do you think these stairs will look like? * Will they be more or less steep than the others? * Will they be easy or hard to climb? * What made previous steps steeper than these? What made other steps not as steep as these? 29. Instruct students to go in the hallway to measure the rise and tread of stairs in several locations inside and around the school building (sites you select). (NOTE: Tell students to round their measurement to the nearest ¼ inch for ease in finding the ratio.) 30. They are to record these measurements in Table 2, then compute the final column. 31. Discuss with the class the relationship between riser height, tread width, and the steepness of the steps they selected and which steps are easier to climb and why? 32. Connect this discussion with the concept of the slope of a line. Discuss with the class "rise" and "run" and how each relates to the steps that have been investigated. Differentiated Instruction: Intervention: * Help students interpret customary measurements when expressed as a decimal. * Have students use metric measurements only for the exercises in an effort to reduce confusion. Then switch to using only customary measurements. * Have students measure other, smaller items (desk top, book cover, box top?) to strengthen their understanding of how to read the measurements. Enrichment: * Have students create their own scenarios similar to the one illustrated in the post assessment. * Encourage students to think of other examples found in real life that the "step" concept can mirror, and design a lesson for their classmates to work through (example: escalators, pyramids, auditorium risers, bleachers, ...). Extensions: * On a tour of any business district or downtown area, students can see a variety of buildings with several sizes of steps outside and inside the buildings. This is a good way to allow students to find a variety of sizes of steps to measure and climb. * Students can take step measurements and hypothesize the "slope" for graph of each step. * Students can calculate slope using kite measurements; charting the height of a kite and the distance from the starting point along the ground to the maximum height. Homework Suggestions and Home Connections: * Students can measure steps in their homes, gardens, yards, the park or other places they visit. They should graph the measurements and sketch the slopes. * Find the slope of two sets of stairs at home -one set inside the house, one set outside of the house. Are there any noticeable differences? Why would there be differences instead of uniformity? * Have students sketch their "ideal" sledding, snowboarding, or skateboarding hill and find the slope. Interdisciplinary Connection: Health/Fitness Connection - have students devise problems related to a stair-stepper. Technology Connection: Use a graphing calculator to plot the measurements for accurate slope measurements General Tips: Appendix: Tables for step by step 11 18 03.doc APPENDIX for Step by step 11 18 03.doc Step by Step ST.pdf Step by Step TE.pdf