SCALE DRAWINGS
INTRODUCTION
The objective for this lesson on Scale Drawings is, the student will use scaling and scale drawings to solve mathematical and real-world problems.
The skills students should have in order to help them in this lesson include, writing ratios, solving proportions and measurement conversions.
We will have three essential questions that will be guiding our lesson. Number one, explain how to use a scale factor to find dimensions in a scale drawing. Number two, describe two different methods to find the perimeter of a scale drawing. Number three, describe two different methods to find the area of a scale drawing.
Begin by completing the warm-up on finding equivalent ratios, solving proportions and determining unit rate to prepare for the lesson on scale drawings.
SOLVE PROBLEM – INTRODUCTION
The SOLVE problem for this lesson is, Nigel is making a drawing of a couch for a new advertisement. The actual measurements of the couch are eight feet long by three point five feet tall. He is using a scale of three centimeters equals two feet. What should the dimensions of the couch be in the advertisement?
S, Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, what should the dimensions of the couch be in the advertisement?
Now that we have identified the question we need to put this question in our own words in the form of a statement. This problem is asking me to find the length and width of the couch in the advertisement.
During this lesson we will learn about scale drawings in order to complete the SOLVE problem at the end of the lesson.
INTRODUCTION TO SCALE DRAWINGS
Mr. Taley is working on a geometry project with his class. He wants students to create a drawing of the classroom.
Take a look around the classroom. What things must we consider if we wanted to complete a drawing of the classroom?
Mr. Taley has his students measure the four sides of the room. The measurements are shown in the table below. What information is contained in the table? The measurements of each side of the room.
If you were one of Mr. Taley’s students, would you want to try and create a drawing that has the measurements in the table above? No, because the drawing would be so large that it would be the actual size of the room.
Discuss with your partner strategies that you could possibly use to create a drawing of Mr. Taley’s room. Some of the strategies maybe to use proportions, or make a smaller drawing. Are there other ideas?
Have you ever seen a drawing of a real object that is proportionally smaller or larger than the real object? In other words, the dimensions of the drawing are in proportion to the dimensions of the real object.
Let’s imagine that the rectangle we see on the top of the page is actually a representation of the local park.
Could that possibly be the actual size of the park? No
Explain your thinking. It is a very small rectangle, which can represent the park.
How can we use the idea of proportional numbers to create a drawing of the park? We can use a proportional relationship.
When we worked with proportional relationships with ratios, there were several ways we could determine whether or not there was a proportional relationship. What were some of those ways? Some examples were if the data could be graphed in a straight line, and we also could use a table.
On way to know if a relationship is proportional is if the two ratios have equal cross products.
Another way we can determine if the relationship is proportional is if the two ratios have equivalent unit rates.
When we are working with a proportional relationship of an object or a drawing, we have to determine the proportion we will use so that each measurement will be consistent. Why is the consistent measurement important? If you change the units it will change the proportion values. Do you have any other ideas?
What are some possible proportions to use for the park drawing? Let’s use a proportion of one centimeter is equal to one hundred feet for our park drawing.
What is the first dimension of our drawing? The width
Describe the relationship that you see between the map and the actual size of the park. There are two ratios set up as a proportion.
What does the x represent in the proportion? The width of the park on the map.
What does the second ratio represent in the proportion? Every one centimeter on the map is one hundred feet in the real world.
Take a look back at the two strategies that we identified to use with proportional relationships.
What strategy can we use with these equivalent ratios? We can use equal cross products.
What does equal cross products mean? When we multiply the numerator of the first ratio times the denominator of the second ratio, the product will be equal to the product of the first denominator and the second numerator.
What is the first operation I need to use? Multiplication
Let’s identify the numbers I have to multiply. Five hundred and one, and one hundred and x, which are indicated by the arrows.
What is the next operation I use? Division
What do I divide by on both sides? One hundred
Explain your answer. I divide by one hundred so that I can isolate the variable.
What is the solution? x is equal to five
What is the width of the park on the map? Five centimeters
Draw a line of five centimeters on the grid paper to represent the width of the park.
Let’s use the same scale to find the length of the park.
Remember that our actual values are represented by the denominator of the ratios, and the map values are represented by the numerators of the ratios.
How is this proportion similar to the proportion for the width? It has one over one hundred which represents the relationship between the map and the actual measurements and it has x over the actual dimensions.
What is the first operation I need to use? Multiplication
How can I find the missing number in the proportion? I can multiply the cross products.
Identify the numbers I have to multiply. One thousand and one, and one hundred and x.
What is the next operation I need to use? Division
What value do I divide by on both sides? One hundred
Explain your answer. I divide by one hundred to isolate my variable.
What is the solution to the equation? x is equal to ten
What is the length on the map? Ten centimeters
Draw a line of ten centimeters on the grid paper to represent the length of the park. Create the rest of the rectangle.
Complete the chart at the bottom of the page. Determine the dimensions of a drawing of the same park with a different scale.
What do you notice is the same for each of the boxes under the width of the park? The ratio that represents the distance on the map, x, over the actual width of the park, which is five hundred.
What do you notice is the same for each of the boxes under the length of the park? The ratio that represents the distance on the map, x, over the actual length of the park, which is one thousand.
Conclusion:
When we use this strategy to scale down each dimension of an object or drawing, we create a representation that is called a scale drawing.
SCALE FACTOR
When we create scale drawings, we multiply by a value that will create the shape or distance in a size that we are able to use in a drawing.
Jarrod is making a scale drawing of his room. He wants to use the representation of one inch is equal to two feet.
Discuss how to write the relationship as a ratio. One inch over two feet.
Jarrod wants to explain the relationship between the actual size of his room and the size of the representation of the room. Share some examples of ways that we compare the size of two objects.
How can we use the ratio we have to find an equivalent ratio that has only numbers and no units? We can multiply by the conversion factor of one foot equals twelve inches written as a ratio one foot over twelve inches.
Why did we choose that conversion factor? Because those are the units contained in the scale.
How can we multiply by that conversion factor and not change the value of the answer? The conversion of one foot over twelve inches is equivalent to the value of one, so we can multiply by one and not change the value.
How can we multiply by that conversion factor and not change the value of the answer? The conversion of one foot to twelve inches is equivalent to the value of one, so we can multiply by one and not change the value.
We multiply our scale by our conversion factor. We then cancel out the units in the numerator and denominator and we’re left with our answer of one over twenty four.
The scale written as a ratio in simplest form with the same units is called the scale factor.
What is the scale factor of Jarrod’s drawing compared to the dimensions of his actual room? One twenty fourth
Explain the meaning of the scale factor. It means that each dimension of the drawing is one twenty fourth of the size of the dimensions of the actual room.
How do we use the scale factor to find the measurements on the drawing? We multiply the actual dimensions times the scale factor.
If the length is twelve feet, we multiply that by our scale factor and we see that the length is actually six inches.
The length of the room is six inches on the drawing.
We use the same process to determine the width of the drawing. We take the actual length sixteen feet and multiply it by our scale factor of one over twenty-four and when we simplify we have an answer of two thirds of a foot which is eight inches.
The width of the room on the drawing is eight inches.
PERIMETER OF A SCALE DRAWING
A door is ninety-six inches tall and forty-eight inches wide. The scale factor of one twenty fourth is used to find the dimensions of the door in a scale drawing.
How do you find the perimeter of a rectangle? Add all four sides.
Let’s take a look at the door that we’re using in our word problem, which is ninety six inches tall and forty eight inches wide.
What is the perimeter of the actual door? Ninety-six plus ninety-six plus forty-eight plus forty-eight is equal to two hundred eighty eight inches.
How can we find the perimeter of the scale drawing of the door? We can use the dimensions of the scale drawing to add together and find the perimeter.
What is the length of the drawing of the door? Four inches
What is the width of the drawing of the door? Two inches
What is the perimeter of the drawing of the door? Four plus four plus two plus two, which equal twelve inches.
What is the scale factor of the drawing? One twenty fourth
What do you get when you multiply the original perimeter by the scale factor? Twelve
What do you notice about the perimeter when you add the dimensions from the scale drawing and when you multiply the original perimeter by the scale factor? When you multiply the actual perimeter by the scale factor, you get the perimeter of the scale drawing.
AREA OF A SCALE DRAWING
A door is ninety-six inches tall and forty-eight inches wide. The scale factor of one twenty fourth is used to find the dimensions of the door in a scale drawing.
How do you find the area of a rectangle? Multiply length by width
What is the area of the actual door? To find the area we multiply the length times the width and our area of our actual door is four thousand six hundred eight inches squared.
What is the height of the drawing of the door? Four inches
What is the width of the drawing of the door? Two inches
What is the area of the drawing of the door? Four times two is equal to eight inches squared.
What is the scale factor of the drawing? One twenty fourth
How did we find the perimeter of the drawing? We could find the measure of each dimension using the scale factor and add to find the perimeter, or we multiplied the perimeter by the scale factor of one twenty fourth.
Will this same strategy work to find the area of the drawing? Why? No, area is a two-dimensional measurement, so we have to multiply it by the scale factor twice or the scale factor squared.
What is the scale factor squared? We multiply the scale factor times itself and our new scale factor is one over five hundred seventy six.
We multiply the actual area by the scale factor squared. Our product is eight.
Is this the same area that we found in Problem Four? Yes
SOLVE PROBLEM – COMPLETION
We’re now going to go back to the SOLVE problem from the beginning of the lesson. The question was, Nigel is making a drawing of a couch for a new advertisement. The actual measurements of the couch are eight feet long by three point five feet tall. He is using a scale of three centimeters equals two feet. What should the dimensions of the couch be in the advertisement?
S, we Study the Problem. Underline the question and completed the statement. This problem is asking me to find the length and width of the couch in the advertisement.
O, Organize the Facts. Identify the facts. We go back to our original problem and we make a vertical strike mark at the end of each fact. Nigel is making a drawing of a couch for a new advertisement./ The actual measurements of the couch are eight feet long/ by three point five feet tall./ He is using a scale of three centimeters equals two feet./
After we identify the facts we eliminate any unnecessary facts. We eliminate any fact that does not help us determine the dimensions of the couch in the advertisement.
Then we list the necessary facts. The actual length of the couch is eight feet, the actual height is three point five feet, and our scale is three centimeters equals two feet.
L, Line Up a Plan. Write in words what your plan of action will be. Set up a proportion comparing the actual length to the scale for each measurement of the couch. Use cross products to solve.
Choose an operation or operations. Multiplication and Division
V, Verify Your Plan with Action. Let’s begin by estimating our answer. Our estimate here is about ten centimeters for the length, and about four centimeters for the height.
Now carry out your plan. We make a proportion with the scale factor and the measurements of the couch. x over eight is equal to three over two; we multiply using cross products, two x equals twenty four and x is equal to twelve. We do the same thing with the width and determine our value of x is five point two five.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, I found the measurement for the drawing.
Is your answer reasonable? Compare your answer to the estimate. Yes, because twelve is more than ten, and five point two five is close to four.
Is your answer accurate? Check your work. Yes
The couch should be twelve centimeters long and five point two five centimeters tall in the advertisement.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, explain how to use a scale factor to find dimensions in a scale drawing. Set up a proportion using the scale factor as one of the ratios. The other ratio is the drawing dimensions over the actual dimensions. Solve using cross products.
Number two, describe two different methods to find the perimeter of a scale drawing. One way is to use proportions to find the dimensions of the scale drawing and then add all sides to find the perimeter. A second way is to multiply the actual perimeter by the scale factor.
And number three, describe two different methods to find the area of a scale drawing. One way is to use proportions to find the dimensions of the scale drawing and then multiply to find the area. A second way is to multiply the actual area by the scale factor squared.