Level D Lesson 23
Measurement Equivalence
In lesson 23 the objective is, the student will compare and apply relative sizes of measurement units within one measurement system.
The skills students should have in order to help them in this lesson include arithmetic patterns and basic metric units – kilometer, meter, and centimeter.
We will have three essential questions that will be guiding our lesson. Number 1, what does equivalence mean when referring to measurement? Number 2, how can I show equivalence between kilometers and meters? And number 3, how can I show equivalence between meters and centimeters?
The SOLVE problem for this lesson is, Erika signed up to run the three thousand meter race for the end-of-year track meet. She and her friend, Heather, train by running five days a week after school. What is the length of the race in kilometers?
We will begin by Studying the Problem. First we want to identify where the question is located within the problem and we will underline the question. What is the length of the race in kilometers? Now that we have identified the question, we want to put this question in our own words in the form of a statement. This problem is asking me to find, the race’s length in kilometers.
During this lesson we will learn how to convert meters to kilometers. We will use this knowledge to complete this SOLVE problem at the end of the lesson.
Throughout this lesson students will be working together in cooperative pairs. All students should know their role as either Partner A or Partner B before beginning this lesson.
We will begin this lesson by talking about Measurement with Metric Length. Each pair of students should have several objects to measure and each student should have their own ruler.
We have three basic units of metric measurements, centimeters, meters, and kilometers. Determine the unit of measure on the ruler. The unit of measure on the ruler is centimeters. How many centimeters are shown on the ruler? On the ruler you see here there are fifteen centimeters. Let’s look at how the ruler in the book compares to the ruler your teacher has given you. How are they similar? They both can measure centimeters. How are they different? One has inches and centimeters rather than just centimeters. Also, one of the rulers is longer than the other. Can you identify some items in the classroom that can be measured using centimeters? We can measure pencils, books, the length of your hand or a notebook using centimeters. In order to have the most uniform answers for measurement, we will all measure the same items. We will round to the nearest centimeter. Let’s measure the length of an envelope. We’re going to place the end of the ruler where zero centimeters is located, at the end of the envelope and mark the nearest centimeter. Discuss as a class the length of the envelope.
Now looking at the ruler that your teacher gave you, how many centimeters are there? There are about thirty centimeters. The largest whole centimeter on the ruler is thirty. You are using a ruler that is one foot in length. Can you measure the math book with a ruler? Yes. What is the length? The length of the math book is twenty eight centimeters. What are some objects that would be challenging to measure using centimeters? The chalkboard, the width of the room, the height of the room, and the height of a student are some objects that would be challenging to measure using centimeters. We will be measuring these items in the chart provided. Let’s take a look at the height of a student. Can you measure someone’s height using one ruler? No, but we can use all the rulers in the group, and sometimes more than once. When we need to use the ruler more than once in order to take a measurement we can use a sticky note to mark the height of the ruler and then re-position the ruler above the mark on the sticky note to continue measuring. We can continue to do this until we have measured the entire object or item.
Another tool we have to measure is a meter stick. Use the meter stick to measure one of the following distances, the height of a student, the length of a row of desks, the distance from the teacher’s desk to the door, and the width of the room. Which measurement was easier to use, centimeters or meters? Meters. Why? The meter stick is larger than the centimeter ruler so it took less time and effort to measure. If we know the measurement of a length in meters, we can use that information to find the equivalent length in centimeters. What do you think is the meaning of “equivalent”? Two lengths are equivalent if they have the same value. Discuss with your partner, what was the distance you measured in the room in meters? When you measured the same distance in centimeters, what was the distance? You measured the same distance both times, so you know that they are equal. What is different about the measurements? The only difference is the name or unit of measurement. For example, if a person is one meter tall, what would this person’s height be in centimeters? One meter is equals to one hundred centimeters. So, the person’s height would be one hundred centimeters.
What is the length of the hallway in meters? The hallway is eight meters. How can you find the measurement in centimeters? Multiply the number of meters by one hundred. One meter equals one hundred centimeters. For every meter we have, there are one hundred centimeters. Eight meters equals eight times one hundred to give us eight hundred centimeters. Eight meters equals eight hundred centimeters.
Sometimes, you may need to find a distance that is too large to measure with a meter stick. Let’s name some examples of these distances, the distance from home to school or the distance between towns. Here’s an example. During gym class the students ran around the perimeter of the school property. When they completed the distance around the school, shown below as an oval, students had run a distance of one kilometer. A kilometer is larger than a meter. If we placed one thousand meters end to end, they would be equivalent to a kilometer. Therefore, one kilometer equals one thousand meters. Let’s take a look at the length of a train with one hundred twenty cars. How long is this distance in kilometers? The train with one hundred twenty cars equals two kilometers. What is the equivalent relationship between kilometers and meters? One kilometer equals one thousand meters. So how can we determine the value of two kilometers in meters? We multiply two by one thousand. What is the length in meters of a train with one hundred twenty cars? Two times one thousand equals two thousand. So the length in meters of a train with one hundred twenty cars is two thousand meters.
We are now going to use a table to help us to find equivalent relationships. What is the measurement of the unit in the first column of Table A, Meters. Second column? Centimeters. How many centimeters are in one meter? We said that there are one hundred centimeters in one meter. Let’s record one hundred centimeters as being equivalent to one meter in Table A. How many meters equal two hundred centimeters? Two meters. Let’s record that two meters is equivalent to, two hundred centimeters in Table A. How did you determine the number of centimeters in one meter? Multiply the number of meters by one hundred. How did you determine the number of meters in two hundred centimeters? Divide the number of centimeters by one hundred.
Now let’s look at Table B. What is the difference between Table A and Table B? Table A compares meters to centimeters and Table B compares kilometers to meters. On Table B let’s talk about how many meters are in one kilometer? There are one thousand meters in one kilometer. Let’s record one thousand meters equivalent to one kilometer in Table B. How many kilometers equal two thousand meters? Two kilometers. Let’s record two kilometers is equivalent to two thousand meters in Table B. Why is two kilometers equivalent to two thousand meters? Well, one thousand meters is equivalent to one kilometer. Therefore, one thousand times two equals two thousand meters.
The same prefixes such as centi and kilo and relationships can be used for all basic metric measurements. If one kilometer is equivalent to one thousand meters, then one kilogram is equivalent to one thousand grams. How many grams are equivalent to one kilogram? One thousand grams. If I have two thousand grams, what is the equivalent value in kilograms? Two kilograms. Why? One thousand multiplied by two equals two thousand. Now complete the Table finding equivalent kilograms and grams. Now that you have completed the table let’s summarize. How do you find the equivalent value in grams if you know the measurement in kilograms? Multiply by one thousand. The kilogram is the larger unit of measure, so to find the smaller unit, you must multiply. How do you find the equivalent value in kilograms if you know the measurement if grams? Divide by one thousand. The gram is the smaller unit of measure, so to find the larger unit, you must divide.
We are now going to go back to the SOLVE problem from the beginning of the lesson. Erika signed up to run a three thousand meter race for the end-of-year track meet. She and her friend, Heather, train by running five days a week after school. What is the length of the race in kilometers?
At the beginning we Studied the Problem We underlined the question, what is the length of the race in kilometers? And put this question in our own words in the form of a statement. This problem is asking me to find the race’s length in kilometers.
In Step O, we will organize the facts. First we will identify the facts. Erika signed up to fin the three thousand meter race for the end-of-year track meet, fact. She and her friend, Heather, train by running five days a week after school, fact. What is the length of the race in kilometers? Now that we have identified the facts, let’s eliminate the unnecessary facts. These are the facts that will not help us find how many kilometers the race is. Erika signed up to run the three thousand meter race for the end-of-year track meet, Knowing the length of the race in meters is going to help us to find the length of the race in kilometers. So we will keep this fact. She and her friend, Heather, train by running five days a week after school. Knowing when her and her friend run, is not going to help us find the length of the race in kilometers. So we will eliminate this fact. Now that we have eliminated the unnecessary facts, let’s list the necessary facts. Three thousand meter race, need length in kilometers.
In Step L, we will Line Up a Plan. First we wan to choose an operation or operations to help us to solve the problem. We know the length of the race in meters. We need to find the length of the race in kilometers so we will divide. Now let’s write in words what your plan of action will be. Divide the length of the race by the number of meters in a kilometer.
In Step V, we Verify Your Plan with Action. First we will estimate your answer. We can estimate that the race is about three kilometers. Now let’s carry out your plan. We said that we wanted to divide the length of the race by the number of meters in a kilometer. One thousand meters equals one kilometer. So we will take three thousand and divide it by one thousand. Three thousand divided by one thousand equals three. The length of the race is three kilometers.
Now let’s Examine Your Results. Does your answer make sense? Here compare your answer to the question. Yes, because I was looking for the length of the race in kilometers. Is your answer reasonable? Here compare your answer to the estimate. Yes, because it matches my estimate of three kilometers. And is your answer accurate? Here you want to check your work. Yes, my answer is accurate. We are now ready to write your answer in a complete sentence. The race is three kilometers long.
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, what does equivalence mean when referring to measurement? The unit of measure has the same value but a different name. For example: If the length of the hall is seven meters, it is equivalent to seven hundred centimeters.
Our second question was, how can I show equivalence between kilometers and meters? Create a table to show that every kilometer is equivalent to one thousand meters. To find the value of a distance in meters when given in kilometers, multiply by one thousand.
And our third question was, how can I show equivalence between meters and centimeters? Create a table to show that every meter is equivalent to one hundred centimeters. To find the value of a length in centimeters when given in meters, multiply by one hundred.