Slope Activity 1: Is GBS up to Code?
INTRODUCTION
 Stairs are made up of treads, the horizontal distance that you step on, and risers, the vertical distance that you step up. For both safety and comfort reasons, treads and risers must meet certain requirements by law. These laws allow for small errors in construction and settling of buildings over time (and consequent changes in riser height, for example). The building laws also change over the years--so, houses and buildings built earlier than 1999 might not be in complete compliance of current laws on risers and treads (however, any "gross" violations do have to be changed!!)

 CURRENT BUILDING LAWS: * The ideal ratio of riser to tread is 0.7 * r + t = 17 inches * r . t = 70 to 75 * there can be no more than 3/16 of an inch variation in height of risers in a  set of stairs.
MAIN BODY

I. Do the stairs in and near Aderhold fit these guidelines?

To find out, use a ruler to measure one risers and each set of stairs in INCHES and complete the following table:

 riser height (in) tread length (in) SLOPE = riser/tread sum, r + t product r . t Concrete stairs to 2nd floor Carpeted stairs down to bottom of pit

Conclusions: Answer the question of part I. in at least TWO complete sentences. _________________________

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II. Angles of Inclination...

 The angle of inclination of a staircase is the angle of the ramp you'd make with the floor if you laid a board up the stairs. Compute the angle of inclination for... 1) the IDEAL staircase: ________ 2) the concrete stairs to the second floor: __________ 3) the carpeted stairs down to the bottom of the pit: ___________

Generally, staircases do not have to be changed if the angle of inclination is LESS than the ideal, but they do if the angle is MORE than the ideal.

Do any of the stairs in or near Aderhold need to be changed? ____________

III. Other common slopes in buildings...

4) You could think of a floor as a BIG tread with no riser. What is the slope of the floor? _____________

5) You could think of a wall as a LONG riser with no tread. What is the slope of a wall? _____________

(remember, division by zero is considered "undefined.")

IV. Another rule for risers and treads...

Risers and treads can also meet the following rule to fit the law:

2r + t = 24 inches, where r = riser height and t = tread length

We can rewrite this equation as: t = -2r + 24, or, written in function notation,

t(r) = –2r + 24, where the tread length, t(r), depends on the riser height, r

Now it looks like equations we've already graphed!

Complete the table below and graph the different possibilities for risers and treads on the grid.

 riser (r, or input) tread (t(r), or output) 1 inch ___________ 2 inches ___________ 3 inches ___________ 4 ___________ 5 ___________ 6 ___________ 7 ___________ 8 ___________ 9 ___________ 10 ___________ 11 ___________ 12 ___________ 13 ___________

6) a) What is the y-intercept of this equation? ___________

b) What does the y-intercept mean for lengths of treads and heights of risers?

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7) a) What riser will give you a tread of 12 inches? Show work

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b) What riser will give you a tread of 9 inches? Show work

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8) Can you have a riser of more than 12 inches? Why or why not?

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9) a) What is the slope of this equation? ____________

b) Explain what the slope means in words, including units:

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10) Do any of the staircases in the pit at GBS satisfy this rule? ______________

Show your calculations to SHOW HOW YOU KNOW:

 SUMMARY

Write a 3-paragraph letter to the principle, Mr. Smith, advising him of your calculations and whether or not he needs to have any stairs changed this summer. Your letter should include the following:

-paragraph 1: a short introduction of yourself and the project

-paragraph 2: an explanation of your measurements and calculations for slope and angle of inclination (sections I & part of II)

-paragraph 3: a recommendation of further action by the school and conclusion

****Be sure you sign your letter!

Extra Credit: Add a paragraph to your letter in which you explain section IV. to Dean Castenell, telling him whether any set of stairs fits this guideline.

EVALUATION

 not yet... OK expert! Measurement & Calculations (I & II) 1 2 3 4 5 Completeness of all questions, graphs, etc. (III & IV) 1 2 3 4 5 Letter appropriate address & signature completeness of explanation & recommendation overall corrections 0 1 1 2 1 2 3 4 2 5 3 Extra Credit 1 2 3 TOTAL: _______/20

DUE DATE: _____________________ if hand-written; ______________________ if typed

STAIR LAB LETTER EVALUATION

 not yet... OK expert! address and signature 1 2 3 three paragraphs 1 2 3 through, correct discussion of slope 1 2 3 through, correct discussion of angle of inclination 1 2 3 through, correct recommendation to Mr. Smith 1 2 3 4 5 correct flow, grammar, spelling, etc. 1 2 3 TOTAL: _______/20

EXTRA CREDIT: Are the stairs in your home regulation?

REMEMBER --

 The ideal ratio of riser to tread is 0.7 INV TAN 0.7 = 35° < ---ideal angle of inclination r+t = 17 inches r.t = 70 to 75 there can be no more than 3/16 of an inch variation in height of risers i a set of stairs.

I. Do stairs in your home fit these quidlines?

To find out, use a ruler to measure one riser and one tread from TWO sets of stairs in INCHES and complete the following table:

 Description of stairs riser height (in) tread length (in) SLOPE = riser/tread Angle of Inclination sum, r + t product r . t

Generally, staircases do not have to be changed if the angle of inclination is LESS than the ideal, but they do if the angle is MORE than the ideal.

Do any of the stairs in or near Aderhold need to be changed? ____________

II. RESULTS

Write a full paragraph explaining your results in the chart above. Tell

what you measured, including a description of the two sets of stairs you investigated

what you calculated

whether or not your stairs fit the guidelines of the current laws

whether, according to the angles, the stairs should be changed

III. EVALUATION

 not yet... OK master architect Calculations are accurate 1 2 3 4 5 6 Paragraph is -complete -mathematically correct -checked for correct spelling, grammar, etc. 1 1 1 2 2 2 3 3 3 TOTAL: _______/15

The Department of Mathematics Education
University of Georgia