MEASUREMENT

Introduction **%, Measurement is the assignment of a numerical value to an attribute of an object or to a characteristic of a situation. Measurement experiences in elementary school are examples of applied mathematics. These experiences provide the opportunity to use the ideas of aritmetic, algebra, geometry, and data analysis in a practical and concrete situation. Measurement activities involve understanding what objects and characteristics are measurable, understanding the units and processes for measuring them, and applying techniques, tools, and formulas to perform the measurements.

This packet emphasizes the measurement of length, area,volume, and angle measurement. However, children should also have experiences with weight, time, and temperature. No matter what the students are learning to measure, the steps they go through are similar. First they should learn how to compare or order. Which is longer? Which is heavier? Which is hotter?, etc. Balance scales are a wonderful tool to focus on comparison of weights. If young children are not familiar with units, they should work with non-standard units in order to see the need for standard units. Once they are familiar with standard units, students should have lots of practice estimating and measuring. Lastly, they should learn any appropriate formulas related to the measurements.

PK-K 1. The main attribute the children should spend time on is length. It is an attribute that is very

concrete and easy to see. • The children should be comparing lengths. Which is longer, the pencil or the crayon? -~**\

Who is taller, Sam or Mary? Is it farther to walk from our classroom to the lunchroom or the playground?

• Children should begin to measure using non-standard units. How many crayons long is your desk? How many of Karen's feet will it take to go across the room? If you line up a bunch of pennies, how many pennies will it take to be as long as your shoe?

• The Cuisenaire Rods are a good tool to practice measurement with. Put the rods in order from the shortest to the longest. How many white rods does it take to make an orange rod? How many red rods does it take to make an orange rod? How many orange rods long is your desk?

• There is no need to introduce rulers at this age. 2. Children should have some experience with weight, especially using balance scales. They

also should have some informal experience with time, probably related to the school day.

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1. Length should continue to be the main attribute that the children experience. • They should continue to work with non-standard units. A good question to ask

children to determine if they understand the measurement process is, "If we measured how long the room is with the teacher's foot and with Susie's (student) foot, which would get the higher number?"

• The fact that non-standard units lead to differing answers for the length of an object should lead to children understanding the need for standard units. A cute story that you can tell the children is about a country where they invented a foot as a way to measure how long things were. In fact, the King's foot was the unit that everyone used. So if you went to the store and bought 10 feet of material, it meant material that was as long as ten of the king's feet. However, eventually the king grew old and died. The new king had a much shorter foot. The people were very upset to see that when they ordered the material, they were not getting as much as before. Tales such as this will make it clear that everyone should be using the same unit in order to make measurements useful.

• When you introduce standard units, you need to decide when to introduce the USA system, and when to introduce the metric system. I believe that it is too confusing to introduce both at the same time. My recommendation is to introduce the USA system (inches, feet, yards) at this age level because it is more likely to be what they have heard about. I would delay the introduction of the metric system until third grade. Some people would disagree with this, especially since the rods are calibrated

/#»s according to the metric system. It is a choice you will have to make. The choice may 1 be made for you if your school follows a specified curriculum.

• Children should be learning how to use a ruler at this age. You need to show them where to start and how to read the measurements. I recommend using rulers that are graduated in inches, with only a half-inch mark in between. The children should give the answer to the closest inch. Almost all activities involving the use of a ruler should involve estimating first, and then measuring. Some children will be ready to measure the length of objects longer than the ruler because they can add. Some may not yet be ready for this activity. You should pay careful attention to how the children are working with the rulers and how they are talking about it. Some children have difficulty understanding what it means to say that an object is 7 inches long.

2. Children should continue to have informal experiences with weight. The pound may be introduced at this age level.

3. You should also continue to work with time, including beginning how to tell time and understanding elapsed time.

4. You can expose the students to the concept of area with free play on the geoboards. 5. You can expose the students to the concept of volume with free play with containers and

liquids.

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M 1. Length

• By this age children should be mastering how to use rulers and tape measures. As they learn about fractions and decimals, they will be able to make measurements that are more accurate.

• However, students should start to realize that measurement is only an approximation. It is impossible to measure exactly. You can demonstrate this by having several children use the same ruler to measure how tall you are. Their answers will almost certainly vary. They should then have a discussion about why the answers vary.

• Choosing appropriate units should also be a goal for this age student. For example, if they wanted to measure how long the math book was, they should know to choose inches as a unit. If they wanted to measure how long a football field was, inches would be very inefficient. They should use yards.

• Children should learn how to use benchmarks. For example, if they know that a ruler is 12 inches long, and they look at a pencil, they should be able to estimate its length fairly accurately. If they know that their teacher is 6 feet tall, they should be able to look at their father and be able to say," He is a little shorter than my teacher, so he probably is 5 feet 10 inches tall."

• Children should know: 12 inches=l foot 3 feet=l yard 36 inches=l yard They should have some exposure to converting from one unit to the other. This is a difficult task for many children, and it should not be overemphasized.

• You should introduce the metric system at this grade level. Just as in the USA system, students should have lots of experience estimating and measuring. Most experience with the metric system should be limited to the more commonly used units such as, centimeters, meters, and kilometers.

• When recording the measurement of an object, it is important to state the unit of measurement as part of the answer.

• You should not emphasize conversions between the metric system and the USA system. However, some approximate conversions will help children get a better sense for the metric system. Two useful approximate conversions are: A centimeter is a bit less than one-half an inch, and a meter is a bit more than a yard.

• Perimeter is the length around the edge of a two-dimensional shape. Therefore, the units are units of length, and the concept of perimeter is merely an extension of previous work with length. Students should have experience finding the perimeter of a variety of shapes, including circles and irregular shapes with straight edges. When they work with circles, the students should discuss options to avoid having to measure its perimeter with a straightedge ruler. (Strings or tape measures are preferable.) As they work with rectangles, they should learn the formula for the perimeter of a rectangle, which is P=(2xlength)+(2xwidth).

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2. Area Area involves a completely different attribute than length. Informally, it means the space inside the shape. The most common way to measure area is with square units. Students should have many experiences where they place a transparent overlay covered with square inches or square centimeters over a shape and simply count how many squares are in the shape. When they record the answer, they should be careful to indicate the correct unit. As always, they should be encourage to estimate before they actually perform the measurement.

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c Students should measure areas of shapes where the overlay does not fit evenly. Some like the shape on the left below involve work with fractions; some like the circle involve a lot of approximation.

Only after the students have had many experiences with measuring area with a plastic overlay, should the special case of a rectangle be introduced. It can be clearly demonstrated why the area of a rectangle can be measured by using the formula A= height x base. Students should understand that the formula works for rectangles, not for all shapes.

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3. Angles Students should learn that an angle is made up of two rays with the same vertex (end point). They should learn that the more "opened-up" the rays are, the bigger the angle. They should become familiar with a 90 angle or a right angle. An angle whose measure is less than 90 is called acute; an angle whose measure is more than 90 is called obtuse. Students should learn how to use a protractor. A common difficulty that children experience is figuring out which number to use on the protractor. If they use the 90 angle as a benchmark, and first estimate the measure of the angle, they should be less likely to make a mistake.

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4. Volume Students should continue to experiment with liquid volume. They should be aware that a liter is slightly larger than a quart. A challenging long-term activity for children of this age is to have them figure out how many marbles (or whatever you choose) it would take to fill up something very large (for example, a file drawer, a cabinet, or even a room. This activity involves a great deal of problem solving and arithmetic.

\ 5. An activity for grades 3 and 4 (Drawings on the following page)

If you follow the lines of the graph paper, draw all the rectangles you can that have a perimeter of 20 cm. Find the area of each rectangle. Which one has the greatest area? Will that shape always have the greatest area. Experiment with other shapes.

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6. Another activity for grades 3 and 4 (Drawings on the following page)

a. If you follow all the lines on a graph paper, draw all the rectangles you can with an area of 12 square centimeters.

b. If you do not have to follow the lines, can you draw any more rectangles? (There are an infinite number of rectangles if you do not have to follow the lines.) Number one is 1 V2 cm by 8 cm. Number two is Vi cm by 24 cm. The children should realize that you could have rectangles of 1/3 by 36, Vt by 48,1/5 by 60, etc. Numbers three and four have sides whose lengths are irrational numbers. If the children discover either of them, you should tell the students that the measurements of the sides can not be determined until they learn about a special kind of number. They will learn about those numbers in middle school.

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72 GRAPH PAPER MASTERS • Copyright © by Dale Seymour Publications ©

5 ^ 1. Length

Students should be able to convert within the USA system and within the metric system. They should gain an appreciation of the elegance of the metric system: 4.2km = 42hm = 420dkm = 4,200m = 42,000dm = 420,000cm = 4,200,000mm They should know the rough conversions between the metric and the USA systems: A yard is a little shorter than a meter; an inch is about 2.5 cm, and a foot is about 30 cm.

2. Area of shapes with straight edges • Students should continue to explore the relationship between area and perimeter. • Students may still use overlays as needed, but they also should be learning the

formulas for the areas of the following shapes with straight edges, and the explanation of how each formula is derived.

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Students should be able to apply the above to determine the area of irregular shapes, such as pictured below. This knowledge can then be applied to find the area of the classroom floor, or the area of the school building, or even the area of the school grounds.

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3_. Circles Although the formulas for the circumference and area of a circle involve TT , which is an irrational number, I feel that children are curious about circles and so should be learning about them. The following explanations of how to figure out the circumference and area of a circle can be understood by students of this age. TT can be explained as a special kind of number that is different than any other numbers they have worked with before. It is a decimal that goes on forever, but with no pattern. It is approximately equal to 3.14, and for our purposes of measurement, that is accurate enough.

- Students should use tape measures or string to find the circumferences of many different size circles. For each circle, they should measure the diameter, and then calculate the circumference divided by the diameter. All answers probably will be in the range of 2.8 to 3.5. Explain that the Greeks actually proved that the ratio of the circumference to the diameter of any circle was always exactly the same, if you could measure perfectly. The Greeks were not sure exactly what the number was, but they gave it the name TT • They thought that M was approximately 3, but now we are able to be much more accurate. Mathematicians have calculated TT to more than 200,000,000,000 places. For our purposes, we will approximately to be 3.14. Once the students know that Area/diameter =Tt", then they should be able to realize that the formula for the area of a circle is Area = diameter x TT , Since it is usually easy to measure the diameter of a circle, they can calculate the area.

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- Students should explore the area of different size circles using a square cm or sq inch overlay. They can then try to hypothesize what the formula for area is. If you feel that some of your students can understand the following explanation, you should show it to them.

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4. Three dimensional shapes • Volume of prisms and cylinders

Students can experiment to discover the formula for the volume of a rectangular prism. If you give them an empty cereal box and have them fill the bottom of the box with white Cuisenaire Rods (each is a cubic centimeter), they should realize that the number of cubic cm is equal to the area of the base of the box. Then all you have to do is figure out how tall the box is. That will tell you how many tiers of white rods there will be. Therefore, the formula for the volume of a rectangular prism is Volume = (height of prism) x (area of base)

The formula can be generalized to any prism or cylinder. Volume = (height) x (area of base)

Surface area of prisms and cylinders Finding the surface area of a prism is simply finding the total of the areas of each of the faces. As long as each face is a shape for which the students know the formula, it should not be difficult to find the surface area of a prism. Finding the surface area of a cylinder is much more difficult. Except for your most advanced students, it is probably best left for 7th or 8th grade. An explanation is on the following page.

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/P* 5. Some problems for your class • See page 10 of the Algebra packet for a problem that relates measurement and

algebra. • Below is a roll of toilet paper. The height of the roll is 5 inches. The diameter of the

roll is 5 inches. The diameter of the hole is 2 inches. Find the volume of the roll.

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The problems on the following two pages were taken from Exemplars.

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fences Cor grazing.

Create a fenced in space with the maximum area for your cow Bessie given 100 feet of fencing. How many poles would you have for this area if you need one every 5 feet? 1. How do you know it is the maximum area? Explain 2. What is that area? Explain Now, instead of having the grazing area in the middle of a field you decide to use a side of your barn. With the same amount of fence and the side of the barn being 50 feet, find the maximum area of this alternative grazing pasture. 1. How do you know this is the maximum area? Explain 2. What is the area? Explain 3. Compare the two answers and reasoning behind them. What do they suggest? What conjectures can you make? Extend your thoughts.

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Dog H°use

Given a piece of plywood 150 cm x 300 cm, design a doghouse that can be made from the piece. Make your doghouse as large as possible. Show what your finished product looks like and give good enough information so that someone else could use your plans to build a house for her dog.

©Exemplars r>o3H«use -Pagei- 100 Best Tasks 1997

MEASUREMENT Choose a grade level.

PK-K Develop a series of activities that allow students to explore measuring objects in the room using non-standard measurement units.

1-2 Develop a series of activities that allow students to measure objects in the room using a ruler. Include a discussion of the results.

3-4 Develop a lesson or series of lessons that involve the concepts of both area and perimeter.

OR Develop a lesson or series of lessons that involve solving a volume problem such as, "How many marbles will it take to fill up the classroom?"

5-6 Develop a lesson or series of lessons that lead to the measurement of the area of an irregular plot of land or the floor space of an irregularly shaped building.

OR Develop a lesson that involves an integration of algebra and measurement.