Developing and Using Models and Putting them to Scale
Objective:
Hypothetically, lets say we needed to run an experiment that required 1000 feet, and I need to know if the hallway is long enough to run it. But the problem is, I only have 1 ruler, 1 meter stick and a satellite image. How do we go about doing it?
Cross Cutting Concepts:
Science & Engineering Practice:
Task 1:
Hypothetically, lets say we needed to run an experiment that required 1000 feet, and I need to know if the hallway is long enough to run it. But the problem is, I only have 1 ruler, 1 meter stick and a satellite image. How do we go about doing it?
Cross Cutting Concepts:
- Scale & Proportion
Science & Engineering Practice:
- Making and Using Models
Task 1:
- Create a model of the school by tracing the satellite image onto tracing paper.
- Color it if you can, but it's not necessary.
- Next, cut and paste it into your notebook where there are clearly two clear pages next to each other.
Task 2: Identifying key elements of a model
Today, we identified that this was a model, or a representation of the school, and that it is NOT the school in an of its self. We then later discussed what elements do we need in order to make this model / representation meaningful to you as the scientist and for those who may be examining your notes / work?
Key elements needed for this model *(aka; map) were brought up by students:
During discussion, we collectively title the image, and decide what parts to label a few parts of the image, develop a key, discuss any observations they make along the way, like the fact that the trailers are missing - which leads to what year the photo is (2013), which then leads to modifying the title. The conversation also leads to which direction is North, and how can we tell - which then leads to us all going out side for about 4 minutes to discuss this, then place a compass on our "model", or map.
I then lead the discussion towards scale and ask them how do we find that? After a discussion of how they can do it, I say - "okay, do it! Go and come back in 10 minutes and find your scale."
I then have the students put their newfound scale on the board and have a short discussion. For most periods, I've ended here. For some periods, I have not gotten here yet - but instead we had a discussion about where the hallway is in the school so that we can better measure it. It all depends on the direction of the conversation and how it relates to the overall challenge.
Task 3: Figure out how you can calculate how long the hallway is.
To do that, you'll need to create a scale. To do that, you'll need to:
Note to self:
Today, we identified that this was a model, or a representation of the school, and that it is NOT the school in an of its self. We then later discussed what elements do we need in order to make this model / representation meaningful to you as the scientist and for those who may be examining your notes / work?
Key elements needed for this model *(aka; map) were brought up by students:
- A key or legend with symbols
- Title (Vista Heights Middle School - 2013)
- Labels of parts
- Compass (Arrow pointing North)
- Scale
During discussion, we collectively title the image, and decide what parts to label a few parts of the image, develop a key, discuss any observations they make along the way, like the fact that the trailers are missing - which leads to what year the photo is (2013), which then leads to modifying the title. The conversation also leads to which direction is North, and how can we tell - which then leads to us all going out side for about 4 minutes to discuss this, then place a compass on our "model", or map.
I then lead the discussion towards scale and ask them how do we find that? After a discussion of how they can do it, I say - "okay, do it! Go and come back in 10 minutes and find your scale."
I then have the students put their newfound scale on the board and have a short discussion. For most periods, I've ended here. For some periods, I have not gotten here yet - but instead we had a discussion about where the hallway is in the school so that we can better measure it. It all depends on the direction of the conversation and how it relates to the overall challenge.
Task 3: Figure out how you can calculate how long the hallway is.
To do that, you'll need to create a scale. To do that, you'll need to:
- measure a part of something "out there" that is visible on the map / image.
- measure that same part on the image.
- create a notable scale.
- then measure the hallway on the image.
- use your scale that you generated and calculate how long the hallway is in feet.
Note to self:
- I have all the students post all of their calculations and notes on the opposite page of their maps.
- When I had the students do this activity, I had them go outside to make a few measurements, then return back inside.
- I later had students post their numbers on the image projected in the front of the room.
- I then had students post their scale on the side of the board.
- When I had them calculate, I had them post their numbers in feet on the other side of the board.
- A small discussion ensued as to why the numbers were so vastly different. Some comments were made about accuracy and rounding. I also posted about students may not know how to do the math.
- We then measured using a walking measuring stick to see how close they were.
- This may lead to further discussion as to why numbers were way off, and we may need to spend about a half of a day to learn how to calculate scale, and do this activity over as a class, and recalculate for another item on the map.
- Further discussion on the following items:
- Importance of accuracy of the model
- Importance of measuring 2-3 times for accuracy.
- Where you measured is of utmost importance. I have a feeling that some students didn't know where to measure the beginning and end.
- Understanding how math is used to calculate
Day 3: Calculate & Test
Today, the students posted their scales they came up with on the board. Next, they calculated how long the hallway was and posted their answers on the other board.
We then compared their calculated answers to the real hallway.
The hallway is measured at 479 ft.
Afterwords, I had students discuss why their numbers were off. Some answers included:
Below is how to convert:
To convert from either one unit to another or using the scale to either scale up or down, the process is the same. It's all a matter of ratio's. There are three methods to do this - and I'll discuss all three here.
Say I run a kennel and it just seems to work out that there has to be 3 dogs for every 2 cats. There is better peace and harmony when it's set up like this. So... here's the deal... I've got 18 dogs, how many cats must I have in order for there to be peace and harmony?
Below are the three ways to approach this problem.
Straight Ratio:
With the straight ratio approach, to find out how many cats I would need, you would first divide the total number of dogs by the amont of dogs you would need for the base ratio. In this case, it turns out that 18 dogs divided by 3 dogs gives us a multiple of 6.
Next, I simply multiply the top number by the same multiple. In other words, I would multiply 2 cats by 6 to find out how many cats I'd need. In this case, I'd need 12 cats if I had 18 dogs.
See how I did it below:
Fish Method:
The fish method is a shortcut of the long hand algebra. The basic idea is to cross multiply and divide to find the final answer.
In the example below, you begin with the ratio that has the missing piece. In this case, it's the cats once again. So since we have the total number of dogs, but not cats, we start off with the dogs.
Classic Conversion:
With classic conversions, you're converting based on a rate that is equal to each other. Kind of hard to explain - so bare with me.
Since we have 18 dogs, I know that for every 3 dogs I need 2 cats. So if I multiply the 18 dogs by the ratio of cats to dogs, then I can find out how many cats I need. However, it must be set up correctly.
I want to set it up so that the unit of "dogs" gets canceled out of the mathematical process. To do that, I need to set up the ratio where dogs is on the bottom. By setting up 2 cats per 3 dogs, the unit "dogs" get's canceled out, and we're left with cats.
Next, simply multiply everything in the numerator together and multiply everything in the denominator together, then finish off the calculation.
Today, the students posted their scales they came up with on the board. Next, they calculated how long the hallway was and posted their answers on the other board.
We then compared their calculated answers to the real hallway.
The hallway is measured at 479 ft.
Afterwords, I had students discuss why their numbers were off. Some answers included:
- Accuracy
- Measurement
- Mathematics and Calculation Difficulties
Below is how to convert:
To convert from either one unit to another or using the scale to either scale up or down, the process is the same. It's all a matter of ratio's. There are three methods to do this - and I'll discuss all three here.
Say I run a kennel and it just seems to work out that there has to be 3 dogs for every 2 cats. There is better peace and harmony when it's set up like this. So... here's the deal... I've got 18 dogs, how many cats must I have in order for there to be peace and harmony?
Below are the three ways to approach this problem.
Straight Ratio:
With the straight ratio approach, to find out how many cats I would need, you would first divide the total number of dogs by the amont of dogs you would need for the base ratio. In this case, it turns out that 18 dogs divided by 3 dogs gives us a multiple of 6.
Next, I simply multiply the top number by the same multiple. In other words, I would multiply 2 cats by 6 to find out how many cats I'd need. In this case, I'd need 12 cats if I had 18 dogs.
See how I did it below:
Fish Method:
The fish method is a shortcut of the long hand algebra. The basic idea is to cross multiply and divide to find the final answer.
In the example below, you begin with the ratio that has the missing piece. In this case, it's the cats once again. So since we have the total number of dogs, but not cats, we start off with the dogs.
- Multiply the total no. of dogs with the ratio of the cats.
- Divide that answer with the ratio of the dogs
- Wala, you get a final answer of total cats.
Classic Conversion:
With classic conversions, you're converting based on a rate that is equal to each other. Kind of hard to explain - so bare with me.
Since we have 18 dogs, I know that for every 3 dogs I need 2 cats. So if I multiply the 18 dogs by the ratio of cats to dogs, then I can find out how many cats I need. However, it must be set up correctly.
I want to set it up so that the unit of "dogs" gets canceled out of the mathematical process. To do that, I need to set up the ratio where dogs is on the bottom. By setting up 2 cats per 3 dogs, the unit "dogs" get's canceled out, and we're left with cats.
Next, simply multiply everything in the numerator together and multiply everything in the denominator together, then finish off the calculation.