Laborers’ Union Curriculum

Related Instructor’s Working Copy

Unit: Scaffolding

Objectives:

Student Laborers’ Skills: Define a scaffold, list four trades that work on scaffolds, list five unsafe scaffold conditions, list how the following scaffold systems: frame, tube and clamp, system, adjustable masonry, suspension, shoring, rolling tower, and Elevated work platform.

OSHA Standards: 1926.461, 1926.417, 1926.450, 1926.451, 1926.452, 1926.453, and 1926.454

SCANS: Resources, Interpersonal Skills, Information, Systems, Technology, Basic Skills, Thinking Skills, and Personal Qualities

National Technology Foundation Standards: Technology productivity tools, Technology communications tools, Technology research tools, and Technology problem-solving and decision-making tools ELA: Reading 1 & 2, Writing 1,2, & 3, Oral Communications 2, Critical Thinking 1 & 2History: ________________________________________________________________Math: __________________________________________________________________Science: ________________________________________________________________Bloom’s Taxonomy: Knowledge, Comprehension, Application, Analysis, Synthesis, and EvaluationInstructional Material Needed: Scaffold User Manual, Computers, Microsoft Excel software, printers, overhead projector, DVD player, and calculators.Resources: Scaffold User Manual Multimedia: ____________________________________________________________

Assessment:

OSHA Standards: Manual Activities, section 2 Assignment sheet and additional manual section worksheetsSCANS: Resources, Interpersonal skills, Information, Systems, Technology, Basic Skills, and Thinking skillsNational Technology Standards: Pythagorean Theorem Project Microsoft SpreadsheetELA: Notebook activities 1-10 and Portfolio activities 1-4 and OSHA self-generated signMath: Pythagorean Theorem Project answers 1-8Science: Pythagorean Theorem Project visuals 1-8 from template

Instructor’s Notes: _______________________________________________________________________________________________________________________________________________

1

Unit: Scaffolding

Lessons: 12 total

1. C/W “Introduction to Scaffolds”Have students refer to page 1-1 in manual and record the “Trainee Objectives/Introduction to Scaffolds” in their notebooks. Label “Notebook Activity 1” in notebook.Have students read to themselves pages 1-3 “introduction” Present “Safety” video (11 minutes) “Working at Heights” (18 minutes)Present students with a quiz for a test grade reviewing key information presented in videos and have them self-correct and retain quiz in their portfolio. Label “Portfolio Activity 1.” Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Assign page 1-13 questions 1- 4 in their manual. Inform students that the test will be 1-4 “Trainee Objectives” and to record this information in their notebook as they come across it. Label “Notebook Activity 2.”Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. C/W Have students refer to page 1-4 “Types of Scaffolds” in manual and read to themselves.Present PowerPoint presentation on scaffolding. Instructor will show overhead transparency of figure 1-4 on page 1-7 of “Adjustable masonry scaffold” and have students identify the correct items on the blank activity sheet provided in this unit. Label “Portfolio Activity 2” Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Referring to page 1-14, have students write a brief explanation in their notebook, on how each type of scaffolding is used. (Bloom’s Tax. Comprehension SCANS technology) Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

2

OSHA Capacity3. C/W Have students refer to page 2-1 “Summary of OSHA Standards” and list the “Trainee Objectives” in their notebook labeled “Notebook Activity 3”. Instructor will review reading material on pages 2-3 & 4 “Capacity” and provide math examples of what the maximum weight allotment is for a scaffolding system. Weight limits for stall loads, suspension ropes, and adjustable scaffold systems. (SCANS basic skills) Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________ H/W Students will read pages 2-5 through 2-8. Students will be responsible for identifying the items in figure 2-3 on worksheet provided and retain in their portfolio labeled “Portfolio Activity 3.” Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

OSHA Scaffold Platform Construction 4. C/W Referring to Homework (H/W) the instructor will utilize the overhead transparency for figure 2-3 and have students review the elements necessary for a pump jack scaffold. Students will read pages 2-9 through 2-16 to themselves. Referring to page 2-16 through 17 students will be asked to write a brief process essay in their notebook labeled “Notebook Activity 3”explaning the necessary steps to properly install stairway units. MCAS process writing, Bloom’s Tax. Procedure and synthesis, SCANS technology”. As an independent project, the instructor will assign the Pythagorean Theorem project; requiring students to ultimately answer eight problems utilizing this new mathematical skill to problem solve the necessary areas for stairs and ladders. (The instructor will assign a due date they believe to be reasonable for the grade level they have allocated it to, most likely to an upper-classmen grade.) Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Students will read pages 2-18 through 22 and in their notebook record Table 2-2 and label it “Notebook Activity 4”. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

3

Scaffolding

OSHA Fall Protection5. C/W Students will read pages 2-22 through 28 to themselves. Referring to the overhead transparency provided, the instructor will show a visual representation of the following: float scaffold, needle beam scaffold, boatswain chair, and catenary scaffold. Students will be asked as a class to identify each of these types of scaffolding. Students will be asked to read pages 2-29 through 30. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

OSHA Competent Person H/W Students will be asked to read pages 2-30 through 32 and in their notebook paraphrase the list of what is necessary to scaffold assembly listed on pages 2-31-32. Label this “Notebook Activity 5”.Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

Section 2 Assignment Sheet6. C/W Referring to pages 2-33 through 39, students will independently answer questions 1-10 using their manual. Instructor will review the correct answers for 1-10 and discuss any pertinent questions within this set of questions. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Referring to the reading on page 3-6 students will answer questions 1-4 on page 3-7 independently. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

7. C/W Instructor will review Section 3 page 3-7 with the class and draw a double cleat ladder on an overhead/board as a SCANS activity “Thinking Skills.” Then refer to page 4-1 “Trainee Objectives” and have students record them in their notebook as “Notebook Activity 6”.Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

4

Fall Protection for General ConstructionH/W Students will read pages 4-13 Section 4 Assignment Sheet and read the applicable pages 4-3 through 4-11 and answer accordingly.Scaffolding

Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

8. C/W Instructor will divide the class up into each subsection of this reading, have the students paraphrase the manual into their own words, and present the condensed paraphrasing to the remaining groups within the class. Each group will be graded for presentation and graded according to a rubric to be left in their portfolio. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Referring to their notes and readings from the day, students will answer questions 1-6 on page 4-13 in their manual.Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

Electrical 9. C/W Instructor will have students refer to page 5-1 “Trainee Objectives” and have students record them in their notebook as Notebook Activity 7”. Referring to Section 5 Assignment Sheet pages 5-11 through 5-12, students will independently answer these questions referring to the readings on 5-3 through pages 5-9. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Instructor will inform students of their H/W assignment and divide the class up accordingly referring to table 5-2 “Electrical citations”. There are ten specific citations, so the assign each student or a team of two a citation and require the students to create a visual representation of each citation, with an attractive and obvious title, visual representation of the citation, OSHA code related to it, and informative and brief essay indicating what the possible hazard is if this code is violated. These representations will act as visual reminders to the class of these hazards and be graded by the applicable rubric. Should be labeled “Portfolio Activity 4”.Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

5

Scaffolding

Scaffold User10. C/W Instructor will have students refer to page 6-1 “Trainee Objectives” and have students record them in their notebook as “Notebook Activity 8.” Referring to Section 6 pages 6-1 through 6-17, students will independently read these pages and then, as a class, refer to page 6-14, review the exercise in figure 6-6 & 6-7. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Students will refer to page 6-15 and answer the activities in “Review Exercise for Calculating Deflection” problems 1-12 and “Review Exercises” on page 6-17 1-12 in their notebooks and label it “Notebook Activity 9.” Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

11. C/W Instructor will review the homework exercises and address any confusion the students may have encountered. Then, as a class, read pages 6-18 6-32 aloud and review on the board what “lateral forces” may effect a scaffold system. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Students will refer to 6-33 through 6-35; Section 6 Assignment sheet and answer questions 1-12. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

12. C/W Instructor will review questions 1-12 from the previous assignment and address any questions or concerns that may arise during the lesson. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________

H/W Students will refer to the 6-1 Trainee Objectives they have previously recorded in their notebook as “Notebook Activity 10” and answer/address each topic. Instructor’s Notes: ________________________________________________________________________________________________________________________________________________________________________________________________________________________***Instructor may ask for the Pythagorean Theorem project to be due at this time as well.

6

The History of Pythagoras and his Theorem

In this lesson you will learn about the life of Pythagoras and how it is that the theorem is known as the Pythagorean Theorem.

Pythagoras was born in the island of Samos in ancient Greece1. There is no certainty regarding the exact year when he was born, but it is believed that it was around 570 BC That is about 2,570 years ago! Those were times when a person believed in superstitions and had strong beliefs in gods, spirits, and the mysterious. Religious cults were very popular in those times.

Pythagoras of Samos

Pythagoras' father's name was Mnesarchus and may have been a Phoenician. His mother's name was Pythais. Mnesarchus made sure that his son would get the best possible education. His first teacher was Pherecydes, and Pythagoras stayed in touch with him until Pherecydes' death. When Pythagoras was about 18 years old he went to the island of Lesbos where he worked and learned from Anaximander, an astronomer and philosopher, and Thales of Miletus, a very wise philosopher and mathematician.

7

Thales had visited Egypt and recommended that Pythagoras go to Egypt. Pythagoras arrived in Egypt around 547 BC when he was 23 years old. He stayed in Egypt for 21 years learning a variety of things including geometry from Egyptian priests. It was probably in Egypt where he learned the theorem that is now called by his name.

By the time he was about fifty-five years old he returned to his native land and started a school on the island of Samos. However, because of the lack of students he decided to move to Croton in the south of Italy.

In Croton he started a school that concentrated in the teaching and learning of Mathematics, Music, Philosophy, and Astronomy and their relationship with Religion. It is said that as many as 600 of the worthiest people in the city attended the school, including Theana whom he married when he was sixty. The school reached its highest splendor around the year 490 BC. He taught the young to respect their elders and to develop their mind through learning. He emphasized justice based on equality. Calmness and gentleness were principles encouraged at the school. Pythagoreans became known for their close friendships and devotion to each other. More than anyone before him Pythagoras combined the spiritual teachings with the pursuit of knowledge and science.

Pythagoras also headed a cult known as the secret brotherhood that worshiped numbers and numerical relationships. They attempted to find mathematical explanations for music, the gods, the cosmos, etc. Pythagoras believed that all relations could be reduced to number relations.

At some point Pythagoras was exiled from Croton and had to move to Tarentum. After 16 years he had to move again, this time to Metapontus where he lived four years before he died at the age of 99.

Here we have a picture of a statue of Phytagoras in the island of Samos. If you click on the figure you'll be able to see a larger picture. On the bottom of the statue the text is "

". The literal translation is "Pythagoras the

8

Samosan", but the preferred translation is "Pythagoras of Samos".

Now let's talk a bit about the theorem that bears his name. The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90o angle. As a matter of fact, they had a rope with 12 evenly spaced knots like this one:

that they used to build perfect corners in their buildings and pyramids. It is believed that they only knew about the 3, 4, 5 triangle and not the general theorem that applies to all right triangles.

The Chinese also knew this theorem. It is attributed to Tschou-Gun who lived in 1100 BC. He knew the characteristics of the right angle. The theorem was also known to the Caldeans and the Babylonians more than a thousand years before

9

Pythagoras. A clay tablet of Babylonian origin was found with the following inscription: "4 is the length and 5 the diagonal. What is the breadth?"

So why is it called the Pythagorean Theorem? Even though the theorem was known long before his time, Pythagoras certainly generalized it and made it popular. It was Pythagoras who is attributed with its first geometrical demonstration. That is why it is known as the Pythagorean Theorem. There are hundreds of purely geometric demonstrations as well as an unlimited (that is right -- an infinite number) of algebraic proofs.

The Pythagorean Theorem is one of the most important theorems in the whole realm of geometry. We will conclude this section by stating the theorem in words:

The square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides.

Another way of saying the same thing is:

When the two shorter sides in a right triangle are squared and then added, the sum equals the square of the longest side or hypotenuse.

You can now move on to the last section and work on some interesting problems. 

The objective of this lesson is for you to figure out the Pythagorean Theorem -- an important relationship between the three sides of a right triangle.

10

The lesson starts with a couple of pages that review all the essentials you need to know. This review is followed by five pages that give you hints to be able to discover the relationship. Some of the hints are based on applets1 that you will use. Others use geometrical figures. On each of the hints' pages you will be able to click on an icon to check if you derived the formula correctly relating the three sides of the right triangle. A template page with the figures used in three of the hints' is provided. You can print it out, cut out the figures, and follow the instructions with physical figures.

Towards the end of the lesson (and hopefully once you've figured out the right answer), you'll be able to compare your results against all the other students that have gone through the lesson. The lesson concludes with a section that gives some historical information and examples about this theorem, one of the most important and fundamental theorems in all of mathematics.You should now be ready to go on with the quick review of some basic concepts.

Review of the Necessary Basic Concepts

You may be familiar with some of the concepts presented in this and the next page, allowing you to go through them very quickly. However, I hope that you will find some interesting concepts.

Angles and Triangles --

Let's first define an angle. When two lines intersect in a point, called a "vertex", the circular span between the lines is called an angle. The following figure shows the angle n between the lines A and B:

Lesson Objective

11

The size of the angle n describes how open or closed the lines are in a circular way. The unit measuring the size of an angle is called a degree, and the symbol ° is used to indicate degrees. How big are the degrees? A full circle of rotation has 360° (as defined by the ancient Greeks). Half a circle has 180°, and half of that, or a quarter of a circle has 90°. The following figure shows a perfect 90° angle between the lines A and B:

Note the small black square between the lines. That special symbol is used to indicate a perfect 90° degree angle, also called a right angle.

Now let's define a triangle. A triangle is a geometric figure consisting of three points or vertices which are connected with straight line segments called sides. When one of the angles of a triangle is a right angle, we call that a right triangle, as shown in this figure:

The letters a, b, and c are normally used to refer to the three sides of a triangle. The letter a stands for altitude and b for base. The c is typically used for the third side, called the hypotenuse. Both sides a and b can be switched; however, the hypotenuse is always the longest side -- the side opposite the right angle, as shown in the figure.

What is a relationship? --

Since you need to understand and state the relationship between the three sides of a right triangle (that is, between a, b, and c in the previous figure), let's look at an example of a

12

relationship. In this example we will be calculating the perimeter or distance around the following rectangle:

To obtain the perimeter p, you need to add the lengths of the four sides or a + b + a + b. This can be expressed in a formula as:

p = 2a + 2b

This formula or equation is a relationship, in this case, between a, b, and p. If we know the values of a and b, we can easily calculate the perimeter p. If we know the size of the perimeter and any one of the sides, we can calculate the size of the other side.

Similarly, for the triangle in which we are interested, you need to define a formula which when given the values of a and b, you can calculate the value of c.

Units --

What units will we use throughout this lesson?  We could use inches or centimeters; however, since we can't really draw things to an exact scale through these computer pages, we will refer to the units simply as "units".

Square Areas --

What is an area?  It is the number of square units needed to cover a given surface.  Suppose that you have a line and that you want to come up with a square that measures exactly the same as the line in both dimensions. In other words, if you have a line that measures 4 units, you want a square that measures 4 units in length and 4 units in width. This is illustrated with the following figure:

13

You could say that the square on the right corresponds to or was derived from the line on the left. Notice that the line measures 4 units and that the corresponding square measures 4 x 4 = 16 square units (count the small squares on the red square). We can say that when you square a 4 you get 16. The mathematical notation for this is:

42 = 16

The 2 right next to and above the 4 indicates that we must multiply the 4 by itself 2 times such that we have a total of 2 fours, or 4 x 4. If we had a 5 instead of the 2, that would imply multiplying 4 by itself 5 times, or 4 x 4 x 4 x 4 x 4.

We can also do the inverse operation. Suppose that you start with a square that measures 25 square units, and you want to get the line that corresponds to either of its sides (they are both the same). Let's illustrate this with the next figure:

As you can see, we can extract the corresponding line that measures 5 units. In this case we can say that when you UN-square a 25 you get a 5. UN-square is not the correct word, however. The correct word is to take the square root. So the square root of 25 is 5. This is represented in mathematical notation as:

which is the inverse operation of the previous one (you can leave the 2 out and it still indicates a square root).

14

To summarize this concept:

72 = 49 The square of  7 is 49, andThe square root of 49 is 7

Since getting the square root of a number is the inverse of squaring a number, both operations cancel each other as in the following two examples:

(Please note that this does not work with negative numbers!)

Let's look at one more example to generalize the concepts. In this case, let's start with a line that is made out of two segments, one measuring a and the other measuring b. The following figure shows the corresponding square derived from the line or the line extracted from the square:

You can see that there are four different areas in the square: two perfect squares of different sizes and two identical rectangles, except for their orientation. A formula to express the size of the square would be:

(a+b) x (a+b)  =  a2 + b2 + axb + axb

Review of a few more Concepts 15

Triangles and Rectangles --

Observe the following figure of a rectangle. What can you tell about the two triangles formed by the diagonal?

Are they the same size?  Are they right triangles? Can you explain it? The answer should be pretty obvious from what we've covered. And a final question, what is the area of any of the triangles with respect to the area of the rectangle? I hope that it is clear to you that it is exactly half, as we have divided the rectangle in exactly two halves. The area of the rectangle is calculated as

a x b

so the area of each triangle would be:

Squaring Patterns --

We will now work a little with the very interesting patterns from the sequence of the squares of whole numbers, i.e. the sequence of the squares of 1, 2, 3, 4, 5. . . .  Two things that you should get out of this exercise are:

becoming familiar with the squares sequence (i.e., 1, 4, 9, 16, etc.), and

an ability to quickly relate a number with its square or a square number with its square root.

We start with a square with a side that measures 1 unit and has an area of 1 x 1 = 1 square units:

16

The square for the next size needs a side that measures 2 units. So we add to the original one, one pink square to the right, one pink one below, and a purple one on the lower right corner to get:

This one has an area of 2 x 2 = 4 square units. The next size up needs a side that measures 3 units. So we add to the previous one, two pink squares to the right, two pink ones below, and again one purple one on the lower right corner. Here is the next square that has an area of 9 square units:

By following the same pattern we can get the next square in the sequence that has an area of 16 square units:

Now let's put a square growth table together:

(A)Pink

Squares to add

(B)Total

Squares to Add

(C)Size of Side

(D)Size of

Area

(E)Area Increase vs.

Previous-- -- 1 1 --2 3 2 4 34 5 3 9 56 7 4 16 7

Let's make sure that you understand the table. We start with a square with a side of size 1. To go to the next one, i.e., to the

17

line with the square with a side of size 2, we add two pink squares plus the one in the corner for a total of 3 squares. The area of this square is 4. The area increased by (4 - 1) = 3 square units as compared to the first square.

I am sure that you have no trouble seeing the pattern in columns A, B and E.  But can you see how to get the number in column B based on the number in the C column of the previous row?  That is, how can you get the 5 from the 2, or the 7 from the 3? Try to figure that out. I suggest that you get a piece of paper (or you can do it in a computer program) and complete this table until you get to 20 in column (C). 

==>==>==>==>==>==>==>

You should now be ready to go through the tips section and try your best to deduce or figure out the formula or relationship between the three sides, a, b, and c of a right triangle, the famous Pythagorean Theorem.

Don't be afraid of making wild guesses.

Be creative and good luck   !!!

Once you discover the Pythagorean Theorem you should revisit all of the tip pages to make sure that you understand why they make sense.

 

Pythagorean Theorem ProblemsIn this last section we present a few problems that require the use of the Pythagorean Theorem. Try to resolve as many problems as you can.

1) The Pythagoras Proof:

18

This is a hands-on exercise for you to convince yourself that the Pythagorean theorem works. It is based on the actual proof that is attributed to Pythagoras. On the following figure we have a right triangle with a square associated with each of its sides:

Using the dimensions associated with the three sides, calculate the area of each of the squares. Then make sure that the area of the hypotenuse's square (brown) equals the areas of the other two squares together.

Now for the hands on part. Draw an equivalent picture on a piece of paper. You can use any size triangle as long as it is a right triangle. Cut up and reassemble the two small squares to form a square identical to the larger one.

2) The classical ladder problem:

There is a building with a 12 ft high window. You want to use a ladder to go up to the window, and you decide to keep the ladder 5 ft away from the building to have a good slant. How long should the ladder be?

Answer: _____________________________________

3) Baseball diamond:

On a baseball diamond the bases are 90 ft apart. What is the distance from home plate to second base in a straight line?

19

Answer: _____________________________________

4) Equilateral triangle:

An equilateral triangle has vertices at (0,0) and (6,0) in a coordinate plane. What are the coordinates of the third vertex? You may want to sketch it out.Note: The sides of an equilateral triangle are identical in length.

Answer: ____________________________________

5) An algebraic problem:

Find out the length of sides a and b on the following triangle:

Answer: ___________________________________

6) An iterative problem:Look at the following figure. Start by finding the value for X1, then for X2, then X3, and so on until you get the value for X6. Write the lengths as square roots, as that makes it simpler.

What is the value of X6?

Answer: __________________________________

7) A 3D problem:

20

We have a wooden box that measures 4 ft. by 3 ft. by 2 ft.:

Figure out: What is the longest straight pole, like the red one, that you can have inside the box?

Answer: ________________________________________

8) Pythagorean Triples:

The Pythagorean Triples were described with Tip number 1. Here we will describe a method to generate all of the Pythagorean Triples. There is a simple formula that gives all the Pythagorean triples. If m and n are two positive integers and m < n, then the triples can be generated with the following equations:

a = n2 - m2

b = 2mnc = n2 + m2

It's easy to check algebraically that the sum of the squares of a and b is the same as the square of c.

Answer: ______________________________________

21

EXTRA CREDIT: Now try it out and produce as many triples as you can by substituting any positive integers for m and n (as long as m < n). If you know how to use a Microsoft Spread-sheet program (like Excel), you can very quickly do a table that generates tons of triples.

Templates for TipsTemplates for Tip # 2:

 

Templates for Tip # 3:

22

Templates for Tip # 4:

23