Unit 10 · Measurement and Data

Imagine a kitchen floor covered with square tiles, each tile 1 foot on a side. If you count the tiles, you know how much floor there is. Mathematicians call that count the area. A unit square is a square with sides of 1 unit. Area tells how many unit squares fit inside a shape.

Try a rectangle drawn on grid paper. It is 4 squares tall and 6 squares wide. Count row by row: 6, 12, 18, 24. The rectangle covers 24 unit squares. We write the area as 24 square units. If each square is 1 centimeter wide, the area is 24 square centimeters.

Two rules matter. The squares must cover the whole shape with no gaps. The squares must not overlap. A common mistake is counting only the squares along the edge. Those squares are the border, not the inside. Count every square that fits inside, including the ones in the middle.

Counting squares one at a time is slow for a big garden. Look at the rows instead. A garden bed that is 7 feet wide and 8 feet long has 7 rows of squares. Each row has 8 squares. That is 7 groups of 8, which is a multiplication problem: 7 × 8 = 56. The area is 56 square feet.

This works for every rectangle. Call one side the length and the other the width. The area is length × width. The rule is a formula: a short sentence in symbols that always works. Area = length × width. For a rectangle 9 cm by 6 cm, the area is 9 × 6 = 54 square centimeters.

Check the formula by counting. A 3 by 4 rectangle has rows of 4, 4, 4. Adding gives 12. Multiplying 3 × 4 also gives 12. Multiplication is just fast adding of equal rows. One common mistake is to add the sides instead: 3 + 4 = 7. Adding gives you part of the border, not the area.

A postage stamp and a soccer field both have area, but you would not measure them the same way. Measure small things in square centimeters or square inches. Measure rooms in square feet or square meters. Measure whole cities in square miles. The unit matches the size of the thing.

Bigger units hold many smaller ones. A square foot is 12 inches by 12 inches. So it holds 12 × 12 = 144 square inches. A square yard is 3 feet by 3 feet, so it holds 3 × 3 = 9 square feet. Notice that the numbers get squared, not just multiplied by 12 or 3. That surprises many students.

Try a classroom rug that is 6 feet by 9 feet. Its area is 54 square feet. In square inches, the rug is 72 inches by 108 inches, which is 7,776 square inches. Both numbers describe the same rug. The square foot answer is easier to picture, so it is the better choice here.

Walk the string around the class garden. It is 8 feet, then 5 feet, then 8 feet, then 5 feet. The whole walk is 8 + 5 + 8 + 5 = 26 feet. That distance around is the perimeter. A fence, a picture frame or a ribbon around a box all follow the perimeter.

Perimeter works for any shape with straight sides. A triangle with sides 5 cm, 7 cm and 9 cm has a perimeter of 5 + 7 + 9 = 21 cm. A five-sided park has sides of 30, 40, 20, 40 and 30 meters. Its perimeter is 160 meters. Add every side. Do not skip any.

Perimeter is measured in plain units like feet or centimeters, not square units. That is because it is a length, a line you could walk. A common mistake is to add only the two sides you can see labeled. In a rectangle, the unlabeled sides match the labeled ones. A 6 by 4 rectangle has sides 6, 4, 6 and 4.

A rectangle has two lengths and two widths. So instead of adding four numbers, add length and width once, then double it. For a 12 by 5 rectangle, 12 + 5 = 17, and 17 doubled is 34. The perimeter is 34 units. Check the long way: 12 + 5 + 12 + 5 = 34. Same answer.

Written as a formula: Perimeter = 2 × (length + width). The parentheses mean add first, then multiply. If you multiply first, 2 × 12 + 5 = 29, which is wrong. Another correct way is 2 × length + 2 × width, which is 24 + 10 = 34.

The shortcut helps with costs. The class garden is 10 feet by 7 feet. Fence costs $3 per foot. First find the perimeter: 10 + 7 = 17, doubled is 34 feet. Then find the cost: 34 × $3 = $102. A student who used the area, 70 square feet, would buy far too much fence.

Suppose the class has exactly 12 seed packets, one per square foot. Many rectangles have an area of 12 square feet. A 1 by 12 bed does. So does a 2 by 6 bed and a 3 by 4 bed. Each uses all 12 packets. But how much fence does each need?

Find each perimeter. The 1 by 12 bed: 2 × (1 + 12) = 26 feet. The 2 by 6 bed: 2 × (2 + 6) = 16 feet. The 3 by 4 bed: 2 × (3 + 4) = 14 feet. Same area, three different perimeters. The long skinny bed needs almost twice the fence of the nearly square bed.

The rule goes the other way, too. Two rectangles with the same perimeter can have different areas. A 1 by 7 rectangle and a 4 by 4 square both have a perimeter of 16. Their areas are 7 and 16. Shapes closest to a square give the most area for the least fence. That is why so many gardens and rooms are almost square.

Not every garden is a rectangle. The class wants an L-shaped bed that wraps around a corner. There is no single formula for an L. But an L is just two rectangles stuck together. Cut it into pieces you know, find each area, and add them. This is called finding the area of a composite shape.

Picture an L made of a 10 by 4 rectangle with a 6 by 3 rectangle attached below its left end. The first piece has area 10 × 4 = 40. The second piece has area 6 × 3 = 18. Add them: 40 + 18 = 58 square feet. The line where you cut does not change the total area.

Sometimes it is easier to subtract. A 6 by 6 square with a 2 by 3 notch cut from one corner has area 36 − 6 = 30. Either way, the key idea is decompose: break the hard shape into easy pieces. A common mistake is counting the shared edge twice or forgetting one piece. Label each piece before you add.

Sometimes the answer is known and a side is missing. The class has 45 square feet of bed and one side is 5 feet. How long is the other side? Area = length × width, so 45 = length × 5. Ask: 5 times what makes 45? That is 45 ÷ 5 = 9. The missing side is 9 feet.

Perimeter works backward, too. A bed has a perimeter of 30 feet and a length of 10 feet. Half the perimeter is length plus width: 30 ÷ 2 = 15. Then 15 − 10 = 5. The width is 5 feet. Check: 2 × (10 + 5) = 30. It works.

Squares are special because both sides match. A square with an area of 64 square inches has a side that, times itself, gives 64. Since 8 × 8 = 64, the side is 8 inches. A common mistake is dividing 64 by 2 or by 4. Dividing by 4 gives a side only when you start from the perimeter, not the area.

The raised beds arrive as open boxes. How much soil fills one? Flat tiles cannot answer that, because soil fills space in three directions: length, width and height. Use unit cubes instead. A unit cube is 1 unit long, 1 unit wide and 1 unit tall. The number of cubes that pack a box is its volume.

Picture a box with a floor that is 4 cubes by 3 cubes. One layer of cubes covers the floor: 4 × 3 = 12 cubes. The box is 2 cubes tall, so it holds 2 layers. Count: 12 + 12 = 24 cubes. The volume is 24 cubic units. If each cube is 1 centimeter, the volume is 24 cubic centimeters.

The word cubic tells you it is volume, just as square tells you it is area. A common mistake is counting only the cubes you can see on the outside. Cubes hidden inside still count. Another mistake is counting the top layer twice. Count one full layer, then multiply by the number of layers.

Counting layers led to a shortcut. The floor of a box is a rectangle, so its area is length × width. Each layer holds that many cubes. Multiply by the height to count all the layers. The formula for a rectangular prism, the math name for a box, is Volume = length × width × height.

The class planter is 8 feet long, 2 feet wide and 1 foot deep. Volume = 8 × 2 × 1 = 16 cubic feet of soil. A bigger planter is 6 by 4 by 5. Its volume is 6 × 4 × 5. Multiply in any order. First 6 × 4 = 24, then 24 × 5 = 120 cubic feet. Or do 4 × 5 = 20, then 20 × 6 = 120.

Think of it as base area times height. A base of 24 square feet with a height of 5 feet is 120 cubic feet. A common mistake is adding the three numbers, 6 + 4 + 5 = 15, or multiplying only two of them. Volume needs all three directions. When you are done, check that the label says cubic.

Pepper’s cage is 4 feet long. The pet store lists cages in inches. To convert feet to inches, remember that 1 foot is 12 inches. So 4 feet is 4 groups of 12 inches. That is 4 × 12 = 48 inches. Going from a big unit to a small unit gives more of them. So you multiply.

The same idea works for yards. A yard is 3 feet, which is 36 inches. A hallway that is 2 yards long is 2 × 3 = 6 feet, or 2 × 36 = 72 inches. A mile is 5,280 feet. That is why Chicago blocks are measured in feet and long trips in miles.

Going the other way, from small units to big units, means dividing. A jump rope is 84 inches long. How many feet is that? Since 12 inches make 1 foot, ask how many 12s are in 84. 84 ÷ 12 = 7. The rope is 7 feet. A common mistake is to multiply when you should divide. Ask: should I get more units or fewer?

The bag of guinea pig food says 2 kilograms. The feeding chart says 40 grams per day. To compare them, convert. A kilogram is 1,000 grams. So 2 kilograms is 2 × 1,000 = 2,000 grams. Kilo means one thousand. That prefix works for length, too: a kilometer is 1,000 meters.

Metric length uses centimeters and meters. A meter is 100 centimeters, about the height of a kitchen counter. A ruler that is 3 meters long is 3 × 100 = 300 centimeters. Pepper is 25 centimeters long. Four Peppers in a line would be 100 centimeters, or exactly 1 meter.

Mixed amounts convert piece by piece. A backpack weighs 2 kilograms 300 grams. Convert the kilograms first: 2 × 1,000 = 2,000 grams. Then add the leftover grams: 2,000 + 300 = 2,300 grams. Going backward, 5,000 meters is 5,000 ÷ 1,000 = 5 kilometers. Metric conversions are all tens, so you never need messy numbers like 12 or 36.

Pepper’s water bottle holds 250 milliliters. The class fills it from a 1-liter pitcher. A liter is 1,000 milliliters. So the pitcher fills the bottle 1,000 ÷ 250 = 4 times. A 2-liter soda bottle holds 2 × 1,000 = 2,000 milliliters. Liquid volume measures how much a container holds.

U.S. kitchens use cups, pints, quarts and gallons. Each step doubles or quadruples. A pint is 2 cups. A quart is 2 pints, which is 4 cups. A gallon is 4 quarts. So a gallon is 4 × 4 = 16 cups. A milk jug from the store holds 1 gallon, and a school milk carton holds 1 cup.

To convert, find the chain. How many pints are in 3 gallons? First gallons to quarts: 3 × 4 = 12 quarts. Then quarts to pints: 12 × 2 = 24 pints. A common mistake is skipping a step, like saying 3 gallons is 6 pints. Write each step down so you do not lose one.

Time is a measurement, too, but its units are not tens. An hour is 60 minutes. A day is 24 hours. A week is 7 days. To change hours into minutes, multiply by 60. Pepper’s play time is 3 hours a week. That is 3 × 60 = 180 minutes of running around the classroom.

Going from minutes to hours means dividing by 60, and there is often a remainder. A movie is 150 minutes long. 150 ÷ 60 is 2 with 30 left over, because 2 × 60 = 120 and 150 − 120 = 30. So the movie is 2 hours 30 minutes. Do not write 2 hours 50 minutes; the leftover is 30.

Days and hours work the same way. Two days is 2 × 24 = 48 hours. A week is 7 × 24 = 168 hours. A common mistake is to treat time like money and think 1 hour 30 minutes is 1.3 hours. It is actually 1 and a half hours, because 30 is half of 60.

Math class starts at 9:45 and ends at 11:20. How long is it? That length of time is elapsed time. Do not subtract 9:45 from 11:20 like ordinary numbers, because 60 minutes make an hour, not 100. Instead, jump along a number line of time.

Start at 9:45. Jump 15 minutes to reach 10:00, a nice round time. Jump 1 hour to reach 11:00. Jump 20 minutes to reach 11:20. Now add the jumps: 15 minutes + 1 hour + 20 minutes. That is 1 hour and 35 minutes. Class is 1 hour 35 minutes long.

Elapsed time also works forward and backward. Pepper’s cage cleaning starts at 10:50 and takes 35 minutes. Jump 10 minutes to 11:00, then 25 more to 11:25. It ends at 11:25. If a movie ends at 4:10 and lasted 1 hour 45 minutes, jump back 1 hour to 3:10, then back 45 minutes: 10 minutes to 3:00, then 35 more to 2:25. It started at 2:25.

Room 12 voted on a name for their second guinea pig. Ms. Chen showed the votes on a bar graph. Each name got a bar. The taller the bar, the more votes. But the numbers on the side went 0, 5, 10, 15, 20. That list of numbers is the scale. Each line up meant 5 votes, not 1.

The bar for Biscuit reached the third line. That is 3 × 5 = 15 votes. The bar for Nugget reached the second line, so 2 × 5 = 10 votes. The bar for Oreo stopped halfway between the first and second lines, so it was between 5 and 10, which is 7 or 8. Ms. Chen said 8.

Use the graph to compare. Biscuit beat Nugget by 15 − 10 = 5 votes. All three names together got 15 + 10 + 8 = 33 votes. A common mistake is to count lines instead of reading the scale, saying Biscuit got 3 votes. Always look at the numbers on the side before you read a bar.

A picture graph uses little pictures instead of bars. The class made one for library books read in March. Each row is a student. Each book picture stands for 4 books. That rule is called the key, and it is printed under the graph. Reading a picture graph without the key is like reading a bar graph without the scale.

Maya’s row shows 3 whole book pictures. That is 3 × 4 = 12 books. Leo’s row shows 2 whole books and a half book. The half picture stands for half of 4, which is 2. So Leo read 2 × 4 + 2 = 10 books. Dario’s row shows 5 pictures, so 5 × 4 = 20 books.

Picture graphs answer questions quickly. Who read the most? Dario, with the longest row. How many more than Maya? 20 − 12 = 8. How many in all? 12 + 10 + 20 = 42. A common mistake is counting each picture as 1 book. Then Maya would seem to have read only 3, which the key says is wrong.

Pepper’s weight chart is a line plot. The bottom is a number line marked in quarters: 1 1/2, 1 3/4, 2, 2 1/4, 2 1/2. Each Monday’s weight becomes one X above its number. When the same weight happens again, the new X stacks on top. So a tall stack means that weight happened many times.

Read Pepper’s plot. There is 1 X above 1 1/2, 2 above 1 3/4, 2 above 2, 3 above 2 1/4 and 1 above 2 1/2. Count all the X marks: 1 + 2 + 2 + 3 + 1 = 9. The class weighed Pepper 9 times. The most common weight is 2 1/4, because that stack is tallest.

Line plots also show range, the distance from smallest to largest. Pepper’s smallest weight is 1 1/2 and largest is 2 1/2. The range is 2 1/2 − 1 1/2 = 1 pound. A common mistake is to think the tallest stack is the biggest measurement. It is only the most frequent one. The biggest is the X farthest right.

The class measured 8 pencils to the nearest half inch. The lengths were 5, 4 1/2, 6, 5, 5 1/2, 4 1/2, 5 and 6. To make a line plot, first find the smallest and largest. Those are 4 1/2 and 6. Draw a number line from 4 1/2 to 6. Mark every half inch: 4 1/2, 5, 5 1/2, 6.

Next, place the X marks. Go through the list one pencil at a time. 5 gets an X. 4 1/2 gets an X. 6 gets an X. The next 5 stacks a second X above 5. Keep going until every pencil has one X. Count your X marks at the end: 3 above 5, 2 above 4 1/2, 2 above 6, and 1 above 5 1/2. That is 3 + 2 + 2 + 1 = 8, one for each pencil.

Give the plot a title, like Pencil Lengths in Inches. Add a label for the unit under the number line. A common mistake is to skip a mark when no pencil has that length. Keep every half inch mark, even an empty one. Otherwise the spacing will lie. Another mistake is placing two X marks side by side. They must be stacked.

Data displays answer questions only after you do some math. Pepper’s line plot shows 9 weights. Ms. Chen asked: on how many Mondays did Pepper weigh less than 2 pounds? Find the stacks left of 2: 1 X above 1 1/2 and 2 above 1 3/4. Add them: 1 + 2 = 3 Mondays.

Some questions need two steps. The bar graph of name votes shows Biscuit 15, Nugget 10, Oreo 8 and Waffles 4. How many students voted for a name other than Biscuit? First add the others: 10 + 8 + 4 = 22. Or find the total, 37, and subtract 15 to get 22. Both routes should agree. If they do not, one has a mistake.

Always check the reasonableness of an answer. Suppose a question about 9 Mondays gives an answer of 15 Mondays. Something went wrong. If the pencil plot has 8 pencils, no answer about them can be bigger than 8. A quick estimate before you compute helps you catch a slip.