A unit of the course: the story, then chapter by chapter — sections, numbered lessons, a source or the numbers to read, three checks each — a review per chapter, and the wrap-up at the end.
Drawn scene: a 100-meter dash finish line at stadium dusk with runners at the tape, a photo-finish timing board and a stopwatch on a bench
9Unit
Decimals
Number
A second is cut into hundredths on a stopwatch. A dollar is cut into hundredths as cents. A meter is cut into hundredths as centimeters. Decimals are the way we write those pieces. They are fractions in disguise, with 10, 100 or 1,000 as the hidden bottom number. Once you see that, the decimal point stops being scary.
In this unit you will read and write decimals to thousandths. You will compare them, round them and place them on a number line. Then you will add, subtract, multiply and divide them. Money and measurements will be your practice field. You will learn why the decimal point moves when you multiply by 10. You will also see why it never moves in addition.
By the end, a grocery receipt, a race time or a gas price will make full sense. You will be able to catch a mistake at a cash register. You will split a bill fairly. You will tell which of two runners really won. Decimals are small numbers, but they run the world.
How we figured it out
Ancient
Counting boards in China hold digits in columns of ones, tens and hundreds
c. 500 CE
Mathematicians in India write numbers with ten digits and place value, including zero
952
Al-Uqlidisi in Damascus writes fractions in decimal form, an early use of decimal fractions
1202
Fibonacci's Liber Abaci brings the Hindu-Arabic digits to Europe's merchants
1585
Simon Stevin's booklet De Thiende, 'The Tenth,' teaches decimal fractions to everyday users
1617
John Napier uses a decimal point to separate whole numbers from decimal parts
1792
The U.S. Coinage Act makes the dollar 100 cents, a decimal currency
1795
France adopts the metric system, with units built on tens
1875
Seventeen nations sign the Treaty of the Meter in Paris to share one standard meter
1971
Britain switches to decimal money: 100 new pence in a pound
2009
Usain Bolt runs 100 meters in 9.58 seconds, a record timed to the hundredth
18
Chapter
Tenths, Hundredths and Thousandths
Decimals
Big questionHow can a few digits after a dot tell us exactly how big a piece of one whole is?
The story
Four Hundredths of a Second
Eight sprinters cross the line together, and only the digits after the point can say who won.
The starter's pistol fires. Eight sprinters explode from the blocks at a Chicago high school track meet. For a hundred meters they run flat out. Then they lean across the finish line almost together. The crowd cannot tell who won. Neither can the runners. Everyone turns to the scoreboard and waits.
The board lights up. Lane 4: 12.43 seconds. Lane 6: 12.47 seconds. Lane 2: 12.50 seconds. The winner beat second place by four hundredths of a second. That is less time than a blink of an eye. Without decimals, all three runners would just be 'about 12 seconds.' The race would be a tie.
Sprint races are timed to the hundredth of a second. A whole second is cut into 100 equal parts. Each part is 0.01 second. The stopwatch counts them. Two runners can finish in the same whole second and still be ranked first and second. The digits after the decimal point decide it.
The fastest 100 meters ever run took 9.58 seconds, by Usain Bolt in 2009. Say it slowly: nine and fifty-eight hundredths. Before that race, the record was 9.69, also his. The difference, 0.11 second, does not sound like much. In sprinting, it is enormous.
This chapter is about those digits after the point. You will learn to read them, write them, compare them and round them. You will see them on stopwatches, price tags, rulers and gas pumps. Tenths, hundredths and thousandths are small, but they decide who wins.
Talk about itTwo runners both finish in 12 seconds and some hundredths. What extra information does the timer need to say who won?
Section 1
Decimals Are Fractions
18.1
Tenths on a Grid
Main ideaA decimal is a fraction whose bottom number is 10, 100 or 1,000, written with a decimal point instead of a fraction bar.
Cut a chocolate bar into 10 equal strips. Each strip is one tenth of the bar. As a fraction, that is 1/10. As a , we write it 0.1. The dot is the . Digits to its left count whole things. Digits to its right count parts of one whole. So 0.1 means no whole bars and 1 strip.
Now eat 3 strips. You ate 3/10 of the bar, or 0.3. Eat all 10 strips and you ate 10/10, which is one whole bar, 1.0. If you had 2 whole bars and 7 strips, you write 2.7. The 2 sits left of the point. The 7 sits in the place, the first spot to the right of the point.
A common mistake is reading 0.3 as ’three.’ The point tells you the 3 is a part, not a whole. Another mistake is writing 3/10 as 0.03. Ask: how many equal parts? Ten parts means one digit after the point. Say the fraction out loud, ’three tenths,’ and the decimal writes itself: 0.3.
Words to know
decimal
a number that uses a decimal point to show parts of a whole, like 0.3 or 2.7
decimal point
the dot that separates whole numbers on the left from parts on the right
tenths
the first place to the right of the decimal point; one tenth is 1/10, or 0.1
Check yourself
1. A bar is cut into 10 equal strips. You have 7 strips. Which decimal shows that?
Why: 7 out of 10 equal parts is 7/10. Tenths use one digit after the point, so 7/10 = 0.7.
2. What does the digit 5 mean in 2.5?
Why: The 5 sits in the first place right of the point, the tenths place, so it means 5/10.
3. Which fraction equals 0.9?
Why: 0.9 has one digit after the point, so it is 9 tenths, which is 9/10.
18.2
Hundredths and Money
Main ideaSplitting a whole into 100 equal parts gives hundredths, the second place after the decimal point, just like cents in a dollar.
A dollar is 100 cents. One is 1/100 of a dollar. We write it $0.01. The 1 sits two places right of the point, in the place. A dime is 10 cents, which is 10/100 of a dollar, or $0.10. A quarter is 25 cents: 25/100, written $0.25. Money already uses decimals every day.
Picture a square cut into a 10 by 10 . It has 100 small squares. Shade 37 of them. You shaded 37/100, or 0.37. Read it ’thirty-seven hundredths.’ The shaded part also fills 3 full columns plus 7 extra squares. So 0.37 is 3 tenths and 7 hundredths. The 3 is in the tenths place. The 7 is in the hundredths place.
Try 6 hundredths. That is 6/100. A common mistake is writing 0.6, but 0.6 is 6 tenths, which is 60 squares. Six hundredths is only 6 squares. The 6 must sit in the second place: 0.06. The zero in the tenths place holds the spot. Without it, the 6 would slide into the wrong place.
Words to know
hundredths
the second place to the right of the decimal point; one hundredth is 1/100, or 0.01
cent
one hundredth of a dollar, written $0.01
grid
a square cut into rows and columns of equal small squares; a 10 by 10 grid has 100
Check yourself
1. How do you write 42 hundredths as a decimal?
Why: Hundredths need two places after the point. 42/100 = 0.42.
2. On a 10 by 10 grid, 0.7 of the squares are shaded. How many small squares is that?
Why: 0.7 is 7 tenths, and each tenth is a column of 10 squares. 7 × 10 = 70 squares.
3. Which amount of money is 8 hundredths of a dollar?
Why: 8 hundredths is 8/100. The 8 goes in the second place after the point: $0.08, or 8 cents.
18.3
Thousandths and the Stopwatch
Main ideaThousandths are the third place after the decimal point; one thousandth, 1/1,000, is written 0.001.
Cut one hundredth into 10 equal pieces. Each piece is one , 1/1,000, written 0.001. The 1 sits three places right of the point. A thousandths grid would need 1,000 tiny squares. That is 10 hundredths grids stacked together. So 0.001 is tiny: a thousand of them make one whole.
Some stopwatches count in thousandths of a second. A time of 10.347 seconds means 10 whole seconds, 3 tenths, 4 hundredths and 7 thousandths. As a fraction, the part after the point is 347/1,000. Read it ’ten and three hundred forty-seven thousandths.’ The word ’and’ marks the decimal point.
Watch the names. Each place to the right is 10 times smaller. 0.5 is 5 tenths. 0.05 is 5 hundredths. 0.005 is 5 thousandths. A common mistake is thinking 0.005 is bigger because it has more digits. It is the smallest of the three. More digits after the point can mean smaller pieces, not more.
Words to know
thousandths
the third place to the right of the decimal point; one thousandth is 1/1,000, or 0.001
place
the position of a digit in a number, which tells its size, like tenths or hundredths
Check yourself
1. Which decimal is 8 thousandths?
Why: Thousandths need three places after the point. 8/1,000 = 0.008.
2. In 4.219, which digit is in the thousandths place?
Why: Count from the point: 2 is tenths, 1 is hundredths, 9 is thousandths.
3. Which number is the smallest?
Why: 0.003 is 3 thousandths. 0.03 is 3 hundredths, ten times bigger. 0.3 and 0.33 are bigger still.
Section 2
Place Value Past the Point
18.4
Reading and Writing Decimals
Main ideaRead the whole number, say 'and' for the point, then read the digits after the point with the name of the last place.
Take 3.25. Read the : three. Say ’and’ for the decimal point. Then read the digits after the point as a whole number: twenty-five. Finish with the name of the last place, hundredths. So 3.25 in is ’three and twenty-five hundredths.’ The last place tells how many equal parts the whole was cut into.
Now go the other way. Write ’six and four tenths’ as a decimal. Whole part: 6. Point. One digit for tenths: 4. Answer: 6.4. Write ’twelve thousandths.’ There is no whole part, so start with 0. Thousandths need three places. Twelve fills two of them, so add a zero in front: 0.012. Always count places from the point.
Out loud, ’three point two five’ is fine. But ’three and twenty-five hundredths’ shows the value. A common mistake is writing ’seven hundredths’ as 0.7. Say the place name, then count the places: hundredths means two places, so 0.07. Another mistake is writing 0.012 as 0.12. Thousandths need three places.
Words to know
whole number part
the digits to the left of the decimal point
word form
a number written in words, like 'three and twenty-five hundredths' for 3.25
Check yourself
1. How do you read 0.35?
Why: The last digit, 5, is in the hundredths place, so 0.35 is thirty-five hundredths.
2. Write 'nine and four hundredths' as a decimal.
Why: Whole part 9, then two places for hundredths with a 4 in the second: 9.04.
3. Which decimal is 'fifty-two thousandths'?
Why: Thousandths need three places. 52 fills two, so a zero goes first: 0.052.
18.5
Expanded Form
Main ideaExpanded form writes a decimal as a sum of each digit's value, like 4.36 = 4 + 0.3 + 0.06.
Break 4.36 into its parts. The 4 is 4 ones. The 3 is 3 tenths, or 0.3. The 6 is 6 hundredths, or 0.06. Written as a sum, 4.36 = 4 + 0.3 + 0.06. This is . It shows exactly what each is worth. You can also write it with fractions: 4 + 3/10 + 6/100.
Try 20.507. The 2 is in the tens place, so it is worth 20. The 0 in the ones place adds nothing. The 5 is 5 tenths: 0.5. The 0 in the hundredths place adds nothing. The 7 is 7 thousandths: 0.007. So 20.507 = 20 + 0.5 + 0.007. Skip the zeros in the sum, but keep them in the number. They hold the other digits in place.
Go backwards too. What number is 3 + 0.02 + 0.009? Line up the places: 3 ones, 0 tenths, 2 hundredths, 9 thousandths. That is 3.029. A common mistake is writing 3.29, dropping the zero in the tenths place. Nothing was in tenths, so a 0 must sit there. Every place needs a digit.
Words to know
expanded form
a number written as the sum of the values of its digits, like 4 + 0.3 + 0.06
digit
one of the ten symbols 0 through 9 used to write numbers
Check yourself
1. Which is the expanded form of 5.62?
Why: The 6 is in the tenths place (0.6) and the 2 is in the hundredths place (0.02).
2. What number is 8 + 0.04 + 0.001?
Why: 8 ones, 0 tenths, 4 hundredths, 1 thousandth. The empty tenths place needs a 0: 8.041.
3. In 60.35, what is the 3 worth?
Why: The 3 is the first digit after the point, the tenths place, so it is worth 0.3.
18.6
Zeros That Matter
Main ideaA zero added to the right end of a decimal does not change its value, but a zero between the point and a digit does.
Is 0.5 the same as 0.50? Picture a hundredths grid. 0.5 is 5 tenths, or 5 full . That is 50 small squares. 0.50 is 50 hundredths, also 50 small squares. Same amount. So 0.5 = 0.50 = 0.500. These are . Zeros added on the right end, after the last digit, do not change the value.
A zero in the middle is different. 0.5 and 0.05 are not equal. 0.05 is 5 hundredths, only 5 small squares. The zero pushed the 5 into a smaller place. So look at where the zero sits. Right end, after the digits: no change. Between the point and a digit: the value shrinks by 10 times.
Equivalent decimals help you compare. To compare 0.7 and 0.68, write 0.7 as 0.70. Now both have two places: 70 hundredths and 68 hundredths. 0.70 is more. A common mistake is thinking 0.68 is bigger because 68 is bigger than 7. Matching the places first stops that mistake.
Words to know
equivalent decimals
decimals that name the same amount, like 0.5 and 0.50
column
one up-and-down strip of a grid; on a hundredths grid, one column is one tenth
Check yourself
1. Which decimal is equivalent to 0.8?
Why: 8 tenths is 80 hundredths. The zero on the right end does not change the value.
2. Which is NOT equal to 0.30?
Why: 0.03 is 3 hundredths, but 0.30 is 30 hundredths, which is 3 tenths. The others all equal 3/10.
3. Why is 0.4 equal to 0.40?
Why: Four full columns of a hundredths grid are 40 small squares, so 0.4 and 0.40 cover the same amount.
Section 3
Comparing and Rounding
18.7
Comparing Decimals
Main ideaTo compare decimals, line up the decimal points and compare place by place, starting from the left.
Which is more, 3.46 or 3.5? Line up the points. the ones: 3 and 3, tied. Compare the tenths: 4 and 5. Five is bigger, so 3.5 is more. Stop there. You never need the hundredths once a place breaks the tie. Write it 3.5 > 3.46. The open side of the symbol faces the bigger number.
Now compare 0.209 and 0.21. Add a zero so both have three places: 0.209 and 0.210. Ones: 0 and 0. Tenths: 2 and 2. Hundredths: 0 and 1. The second number wins. So 0.21 > 0.209. Fewer digits does not mean smaller. Only the places matter.
Put 1.8, 1.08 and 1.78 in order from . Write them with two places: 1.80, 1.08, 1.78. The tenths are 8, 0 and 7. So 1.08 is least, then 1.78, then 1.80. Order: 1.08 < 1.78 < 1.8. A common mistake is calling 1.08 the biggest because it ’has more digits.’ Compare places, not lengths.
Words to know
compare
to decide which of two numbers is bigger, or whether they are equal
greater than
the symbol >, which opens toward the bigger number, as in 3.5 > 3.46
least to greatest
an order that starts with the smallest number and ends with the biggest
Check yourself
1. Which statement is true?
Why: Write 0.7 as 0.70. Then 62 hundredths is less than 70 hundredths, so 0.62 < 0.7.
Why: Write 0.5 as 0.50. Then compare hundredths: 45 < 50, so 0.45 < 0.5.
18.8
Rounding Decimals
Main ideaTo round a decimal, look at the digit one place right of the rounding place: 5 or more rounds up, less than 5 keeps it.
A gas pump shows $3.479 per gallon. to the nearest cent, the hundredths place. Find the hundredths digit: 7. Look one place right: 9. Nine is 5 or more, so round up. The 7 becomes 8 and everything after it drops. So $3.479 rounds to $3.48. Rounding gives a shorter number that is close to the real one.
Round 6.23 to the . The tenths digit is 2. The next digit is 3. Three is less than 5, so the 2 stays. Drop the rest: 6.2. Round 6.25 to the nearest tenth: the next digit is 5, so round up to 6.3. Round 6.97 to the nearest tenth: the 9 must go up to 10, so it carries. 6.97 rounds to 7.0.
A number line helps. 6.23 sits between 6.2 and 6.3, closer to 6.2. A common mistake is looking at all the digits instead of just the next one. To round 4.851 to the nearest tenth, only the 5 matters: 4.9. Another mistake is rounding twice, turning 4.849 into 4.85 and then 4.9. Round once. 4.849 is closer to 4.8.
Words to know
round
to replace a number with a nearby number that is shorter and easier to use
nearest tenth
rounding so the number ends in the tenths place, like 6.23 to 6.2
Check yourself
1. Round 2.746 to the nearest hundredth.
Why: The hundredths digit is 4. The next digit is 6, which is 5 or more, so 4 becomes 5: 2.75.
2. Round 8.35 to the nearest tenth.
Why: The tenths digit is 3. The next digit is 5, so round up: 8.4.
3. Which number rounds to 5.0 when rounded to the nearest tenth?
Why: 4.96: the tenths digit 9 goes up because 6 is 5 or more, so 4.96 rounds to 5.0. The others round to 4.9, 5.1 and 5.2.
Section 4
Decimals in the World
18.9
Decimals on the Number Line
Main ideaBetween any two whole numbers, ten equal jumps mark the tenths, and each tenth splits into ten hundredths.
Draw a from 2 to 3. Cut the space into 10 equal jumps. Each jump is one tenth. The are 2.1, 2.2, 2.3 and so on up to 2.9, then 3. To place 2.6, count six jumps past 2. To place 2.65, zoom in between 2.6 and 2.7. Cut that gap into 10 tiny jumps. Each is a hundredth. Go five jumps, and you are at 2.65.
The number line makes comparing easy. Numbers farther right are bigger. So 2.65 is right of 2.6 and left of 2.7. It also shows rounding. 2.65 is exactly halfway between 2.6 and 2.7, so by the rule it rounds up to 2.7. But 2.63 is closer to 2.6, so it rounds down.
A ruler is a number line. A meter stick marked in centimeters has 100 marks. A point at 37 centimeters is at 0.37 of a meter. A common mistake is placing 2.35 between 2.4 and 2.5. Say it slowly: two and thirty-five hundredths. That is more than 3 tenths but less than 4 tenths. So it sits between 2.3 and 2.4.
Words to know
number line
a line with numbers marked in order, where bigger numbers sit farther to the right
tick mark
a short line on a number line or ruler that shows where a number sits
Check yourself
1. On a number line from 4 to 5 with ten equal jumps, what is the mark after the seventh jump?
Why: Each jump is one tenth. Seven jumps past 4 is 4 and 7 tenths: 4.7.
2. Which number sits between 1.3 and 1.4 on a number line?
Why: 1.35 is 1.30 plus 5 hundredths, past 1.3 but before 1.40.
3. Which decimal is exactly halfway between 0.8 and 0.9?
Why: 0.8 is 80 hundredths and 0.9 is 90 hundredths. Halfway is 85 hundredths, 0.85.
18.10
Metric Units as Decimals
Main ideaMetric units go by tens, so 1 centimeter is 0.01 meter and 1 meter is 0.001 kilometer.
The is built on tens, just like decimals. One is 100 centimeters. So one centimeter is 1/100 of a meter, or 0.01 m. A desk 75 centimeters tall is 0.75 m tall. A student 132 centimeters tall is 1.32 m. Write centimeters as hundredths of a meter, and the decimal appears.
One is 1,000 meters. So one meter is 1/1,000 of a kilometer, 0.001 km. A run of 400 meters is 0.4 km. A walk of 1,250 meters is 1.25 km. A 5K race, common in Chicago parks, is 5 kilometers, or 5,000 meters. Half of it is 2.5 km, which is 2,500 m.
Money and mass work the same way. A dollar is 100 cents, so 8 cents is $0.08. A kilogram is 1,000 grams, so 350 grams is 0.35 kg. A common mistake is writing 5 centimeters as 0.5 m. Five hundredths of a meter is 0.05 m. Count the zeros in the unit: 100 means two decimal places, 1,000 means three.
Words to know
metric system
the measuring system built on tens: meters, liters, grams and their parts
meter
the basic metric unit of length, equal to 100 centimeters
kilometer
1,000 meters, a unit for longer distances like a race or a drive
Check yourself
1. A pencil is 18 centimeters long. How long is it in meters?
Why: One centimeter is 0.01 m, so 18 cm is 18 hundredths of a meter: 0.18 m.
2. A trail is 3.6 km long. How many meters is that?
Why: One kilometer is 1,000 m. 3 km is 3,000 m and 0.6 km is 600 m, so 3,600 m.
3. Which is the same as 7 cm?
Why: 7 cm is 7/100 of a meter. Hundredths need two places: 0.07 m.
Chapter review
Tenths, Hundredths and Thousandths
0 / 8
1. Which fraction equals 0.06?
Why: 0.06 has its 6 in the hundredths place, so it is 6/100.
Why: Write all with three places: 0.710, 0.700, 0.709, 0.170. The biggest is 0.710.
4. Round 12.485 to the nearest tenth.
Why: The tenths digit is 4. The next digit is 8, so round up: 12.5.
5. A sprinter runs 100 meters in 11.09 seconds. How do you read that time?
Why: The 9 is in the hundredths place, so 11.09 is eleven and nine hundredths.
6. Which decimal is equivalent to 2.3?
Why: 3 tenths is 30 hundredths. A zero on the right end does not change the value, so 2.3 = 2.30.
7. A ribbon is 45 centimeters long. How many meters is that?
Why: 45 cm is 45 hundredths of a meter, 0.45 m.
8. Where does 6.72 sit on a number line?
Why: 6.72 is 6.70 plus 2 hundredths, so it is just past 6.7 and before 6.8.
Send it to your teacher
19
Chapter
Operations With Decimals
Decimals
Big questionHow do we add, subtract, multiply and divide decimals without losing track of the decimal point?
The story
The Receipt That Did Not Add Up
A girl who can add decimals catches a cash register in a $3.98 mistake.
Rosa and her dad shop at a grocery store on Chicago's west side every Saturday. This week the list is short: milk, bread, apples and cereal. Rosa's job is to keep a running total in her head so they do not go over $15. She rounds each price and adds as they go. She gets about $14. Close, but under.
At the register the cashier scans the items and the screen says $18.49. Rosa frowns. She is sure she added right. Her dad pays, but Rosa keeps the receipt. In the car she adds the four prices carefully, lining up the dollars and the cents: $3.49, $2.79, $4.25 and $3.98.
The sum is $14.51. That is $3.98 less than the receipt. Rosa knows that number. It is the price of the cereal. She looks at the receipt again. There it is: cereal, scanned twice. The store owes them $3.98. They go back in, and the manager returns the money with an apology.
Rosa caught the mistake because she could add decimals and check with an estimate. Money is decimals: dollars to the left of the point, cents to the right. So are meters, liters and kilograms. Every day, people add, subtract, multiply and divide these numbers, and machines sometimes get it wrong.
This chapter teaches those operations. You will line up decimal points and regroup across the point. You will move the point when you multiply or divide by ten. You will count decimal places when you multiply. Most of all, you will learn to check your work. Then a bad receipt will never fool you.
Talk about itRosa's estimate was about $14, and the receipt said $18.49. How did the estimate alone tell her something was wrong?
Section 1
Adding and Subtracting
19.1
Lining Up the Point
Main ideaTo add decimals, line up the decimal points so each place sits under the same place, then add like whole numbers.
Add 3.49 + 2.75. Write one under the other with the decimal points in a straight line. Hundredths under hundredths, tenths under tenths, ones under ones. Add the hundredths: 9 + 5 = 14. Write 4 and 1 to the tenths. Tenths: 4 + 7 + 1 = 12. Write 2, regroup 1. Ones: 3 + 2 + 1 = 6. Bring the point straight down. The is 6.24.
Now add 5.8 + 2.36. The numbers have different lengths. Give 5.8 a zero: 5.80. Now line up 5.80 + 2.36. Hundredths: 0 + 6 = 6. Tenths: 8 + 3 = 11, write 1, regroup 1. Ones: 5 + 2 + 1 = 8. Answer: 8.16. The zero did not change 5.8, but it kept the places lined up.
The big mistake is lining up the right ends instead of the points. Someone adds 5.8 + 2.36 that way and gets 2.94, treating 5.8 as 0.58. Always line up the points first. Then check with an : 5.8 is about 6, and 2.36 is about 2, so the sum should be near 8. And 8.16 is near 8. Good.
Words to know
sum
the answer to an addition problem
regroup
to carry 10 of one place into 1 of the next bigger place, like 14 hundredths into 1 tenth and 4 hundredths
estimate
a quick, rough answer made with rounded numbers, used to check the exact answer
3. Why do you line up the decimal points when adding?
Why: Only equal places can be added together. Lining up the points puts each place under the same place.
19.2
Subtracting With Regrouping
Main ideaSubtract decimals the way you add them: line up the points, fill empty places with zeros, and regroup when a digit is too small.
Subtract 6.25 − 2.8. Line up the points and add a zero: 6.25 − 2.80. Hundredths: 5 − 0 = 5. Tenths: 2 − 8 will not work. Regroup: take 1 from the 6 ones, making it 5, and give 10 tenths to the 2, making 12. Now 12 − 8 = 4. Ones: 5 − 2 = 3. The is 3.45. Check by adding back: 3.45 + 2.8 = 6.25. Yes.
Subtract 5 − 1.37. The 5 has no decimal places, so write it as 5.00. Hundredths: 0 − 7 needs regrouping, but the tenths are 0 too. Regroup twice: 5.00 becomes 4 ones, 9 tenths, 10 hundredths. Now 10 − 7 = 3, then 9 − 3 = 6, then 4 − 1 = 3. Answer: 3.63. Add back to check: 3.63 + 1.37 = 5.00.
Making is subtraction. You pay $10 for a $7.45 item. Change is 10.00 − 7.45 = 2.55. A common mistake is subtracting the smaller digit from the bigger one no matter which is on top, giving 3.45. Always regroup instead. Another mistake is forgetting to write 5 as 5.00. Empty places are zeros.
Words to know
difference
the answer to a subtraction problem
change
the money you get back when you pay more than the price
Main ideaRound each decimal to a friendly number first; the estimate tells you whether your exact answer makes sense.
A shopper picks up items priced $4.89, $2.15 and $6.95. Is $20 enough? Round each to a , the nearest dollar: 5 + 2 + 7 = 14. That is under 20, so yes. The is 4.89 + 2.15 + 6.95 = 13.99, close to the estimate of 14. Estimating is fast and catches big errors.
Estimate 15.62 − 8.9. Round to whole numbers: 16 − 9 = 7. Exact: 15.62 − 8.90 = 6.72. That is close to 7, so the exact answer is likely right. If your exact answer were 14.73, the estimate would warn you. Something went wrong, probably lining up the right ends instead of the points.
Sometimes round to tenths for a closer estimate. 3.48 + 2.71 rounds to 3.5 + 2.7 = 6.2. Exact: 6.19. A common mistake is rounding after adding, which is not estimating at all. Round first, then add. Another mistake is always rounding down. Use the rounding rule: 4.89 goes to 5, not 4.
Words to know
friendly number
a rounded number that is easy to add or subtract in your head, like 5 instead of 4.89
exact answer
the true answer found by computing with the real numbers, not rounded ones
Check yourself
1. Which estimate is best for 7.92 + 3.18?
Why: 7.92 rounds to 8 and 3.18 rounds to 3. So 8 + 3 = 11. The exact sum, 11.10, is close.
2. Estimate 24.6 − 9.85 by rounding to whole numbers.
Why: 24.6 rounds to 25 and 9.85 rounds to 10. So 25 − 10 = 15.
3. A student adds 6.3 + 2.48 and gets 3.11. What does an estimate show?
Why: 6 + 2 = 8, so the sum must be near 8. The student lined up the right ends. The true sum is 8.78.
Section 2
Multiplying Decimals
19.4
Multiplying by 10, 100 and 1,000
Main ideaMultiplying by 10 makes every digit worth 10 times more, so the decimal point shifts one place to the right; by 100, two places; by 1,000, three.
One pencil costs $0.35. Ten pencils cost 0.35 × 10. Think in cents: 35 cents times 10 is 350 cents, or $3.50. Look at the digits: 0.35 became 3.50. The point moved one place to the right. That always happens when you multiply by 10, because every digit is worth 10 times more. Tenths become ones. Hundredths become tenths.
Multiply by 100 and the point moves two places. So 0.35 × 100 = 35. Multiply by 1,000 and it moves three: 0.35 × 1,000 = 350. When you run out of digits, add zeros. Try 2.7 × 1,000. Move three places. 2.7 has only one digit after the point, so write 2.700 first, then move: 2,700. Each moves the point one more place.
A common mistake is adding a zero to the end, like 2.7 × 10 = 2.70. That is the same number, not ten times more. Move the point instead: 27. Check with sense: ten things at $2.70 each should cost more than $2.70. Another check: 2.7 × 10 is about 3 × 10 = 30, and the 27 is near 30.
Words to know
power of ten
10, 100, 1,000 and so on; each one is ten times the one before
product
the answer to a multiplication problem
Check yourself
1. What is 4.06 × 10?
Why: Multiplying by 10 moves the point one place right: 4.06 becomes 40.6.
2. What is 0.9 × 100?
Why: Move the point two places right. 0.9 becomes 0.90, then 90.
3. What is 1.25 × 1,000?
Why: Move the point three places right. 1.25 becomes 1.250, then 1,250.
19.5
Multiplying a Decimal by a Whole Number
Main ideaMultiply as if there were no decimal point, then place the point so the product has as many decimal places as the decimal factor.
Three friends each run 2.4 kilometers. Total: 2.4 × 3. One way: add 2.4 + 2.4 + 2.4 = 7.2. Another way: ignore the point, 24 × 3 = 72. The 2.4 has one of the , so the product needs one too: 7.2. Both ways agree. Picture it: 2 whole kilometers three times is 6, plus 4 tenths three times is 12 tenths, or 1.2. Total 7.2.
Try 1.35 × 4. Ignore the point: 135 × 4 = 540. Count decimal places in 1.35: two. Put the point two places from the right: 5.40, which is 5.4. Check with an estimate: 1.35 is a bit more than 1, so 4 of them is a bit more than 4. Yes, 5.4 fits.
A common mistake is dropping the point and answering 540. Estimating catches it: 1.35 × 4 cannot be 540. Another mistake is counting the whole number’s digits. The whole number has no decimal places. Only the decimal factor counts. With money, think in cents: $1.35 is 135 cents, times 4 is 540 cents, which is $5.40.
Words to know
factor
a number being multiplied; in 2.4 × 3, both 2.4 and 3 are factors
decimal places
the number of digits to the right of the decimal point; 1.35 has two
Check yourself
1. What is 3.2 × 6?
Why: 32 × 6 = 192. One decimal place in 3.2, so the product is 19.2. Check: 3 × 6 = 18, close.
2. What is 0.75 × 8?
Why: 75 × 8 = 600. Two decimal places: 6.00, which is 6. Check: 0.75 is three quarters, and 8 quarters make 2 dollars, so 6.
3. Each notebook costs $2.45. What do 5 notebooks cost?
Main ideaWhen both factors are decimals, count the decimal places in both, add the counts, and give the product that many places.
A tile is 0.5 meters wide and 0.3 meters tall. Its is 0.5 × 0.3. Picture a hundredths grid: 5 columns wide and 3 rows tall covers 15 small squares. That is 15 hundredths, 0.15. Now check with the rule. Ignore the points: 5 × 3 = 15. Count decimal places: 0.5 has one, 0.3 has one, total two. So 0.15.
Try 1.2 × 0.4. Ignore the points: 12 × 4 = 48. Decimal places: 1.2 has one, 0.4 has one, total two. Product: 0.48. Estimate to check: 1.2 × 0.4 is a bit more than 1 × 0.4 = 0.4. Yes. Try 2.5 × 1.5: 25 × 15 = 375, two places, 3.75. That is a bit more than 2.5 × 1 = 2.5. Yes.
Multiplying by a decimal less than 1 makes a number smaller. So 0.5 × 0.3 = 0.15 is less than both factors. Students who expect a bigger answer sometimes write 1.5. Count the places. A common mistake is placing the point by lining it up, as in addition. Multiplication does not line up points. It counts them.
Words to know
area
the amount of flat space a shape covers, found for a rectangle by length × width
decimal places
the number of digits to the right of the decimal point; 0.48 has two
Check yourself
1. What is 0.6 × 0.7?
Why: 6 × 7 = 42. One place plus one place is two places: 0.42.
2. What is 2.5 × 0.4?
Why: 25 × 4 = 100. Two decimal places: 1.00, which is 1. Check: 0.4 of 2.5 is less than 2.5.
3. Which is true about 0.9 × 0.9?
Why: 9 × 9 = 81, two decimal places, 0.81. Multiplying by a number less than 1 makes the result smaller.
Section 3
Dividing Decimals
19.7
Dividing by 10, 100 and 1,000
Main ideaDividing by 10 makes every digit worth 10 times less, so the decimal point shifts one place to the left; by 100, two; by 1,000, three.
Ten friends split a $24.50 bill evenly. Each pays 24.50 ÷ 10. To by 10, make every digit 10 times smaller. The point moves one place left: 2.45. Each pays $2.45. Check: 2.45 × 10 = 24.50. Dividing by 10 undoes multiplying by 10. The answer, 2.45, is called the .
Divide by 100 and the point moves two places left: 24.5 ÷ 100 = 0.245. Divide by 1,000 and it moves three: 24.5 ÷ 1,000 = 0.0245. Add zeros when you need them. Try 7 ÷ 100: write 7 as 7.0, move two places, 0.07. Seven cents is 7 ÷ 100 of a dollar.
This is how metric units convert. 350 centimeters ÷ 100 = 3.5 meters. 1,750 meters ÷ 1,000 = 1.75 kilometers. A common mistake is moving the point the wrong way. Ask: should the answer be bigger or smaller? Dividing by 10 makes it smaller, so the point goes left. So 6.4 ÷ 10 is 0.64, not 64.
Words to know
divide
to split a number into equal groups, or to find how many times one number fits in another
quotient
the answer to a division problem
Check yourself
1. What is 53.8 ÷ 10?
Why: Dividing by 10 moves the point one place left: 53.8 becomes 5.38.
2. What is 6 ÷ 100?
Why: Write 6 as 6.0 and move the point two places left: 0.06.
3. How many kilometers is 4,250 meters?
Why: Divide by 1,000: move the point three places left. 4,250 becomes 4.25 km.
19.8
Dividing a Decimal by a Whole Number
Main ideaDivide as with whole numbers, and put the decimal point in the quotient straight above the point in the number being divided.
Four friends share 9.6 pounds of apples. Each gets 9.6 ÷ 4. The 9.6 is the . Place the point in the answer right above the point in 9.6. Divide the ones: 9 ÷ 4 is 2, with a of 1. That 1 one is 10 tenths. Add the 6 tenths: 16 tenths. Then 16 ÷ 4 = 4. Answer: 2.4 pounds each. Check: 2.4 × 4 = 9.6.
Sometimes you must add a zero. Divide 7.5 ÷ 6. Ones: 7 ÷ 6 = 1, remainder 1. Tenths: 15 ÷ 6 = 2, remainder 3. Write 7.5 as 7.50 and keep going. Hundredths: 30 ÷ 6 = 5. Answer: 1.25. Check: 1.25 × 6 = 7.50. Adding zeros on the right never changes the number, so you may keep dividing.
A common mistake is dropping the point and writing 24 instead of 2.4. Estimate: 9.6 ÷ 4 is about 10 ÷ 4, between 2 and 3. So 24 cannot be right. Another mistake is forgetting the leftover ones. In 9 ÷ 4 the remainder 1 must join the tenths. Every leftover moves to the next smaller place.
Words to know
dividend
the number being divided; in 9.6 ÷ 4, the dividend is 9.6
remainder
what is left over after dividing one place, which moves to the next smaller place
Check yourself
1. What is 6.4 ÷ 8?
Why: 6 ÷ 8 is 0 with 6 left over. That is 60 tenths, plus 4 is 64 tenths. 64 ÷ 8 = 8 tenths: 0.8. Check: 0.8 × 8 = 6.4.
2. Three friends split a $13.50 bill evenly. How much does each pay?
Main ideaRead the story, decide which operation it asks for, line up the dollars and cents, and check the answer with an estimate.
Dev has $15.00. He buys a book for $8.75 and a pen for $1.60. How much is left? First the spent: 8.75 + 1.60 = 10.35. Then the money left: 15.00 − 10.35 = 4.65. Two steps, two operations. Estimate: 9 + 2 = 11, and 15 − 11 = 4. The exact $4.65 is close to 4, so it fits.
A pack of 6 juice boxes costs $4.50. What does one cost? Divide: 4.50 ÷ 6. Think in cents: 450 ÷ 6 = 75. So $0.75 each, or 75 cents box. What do 4 packs cost? Multiply: 4.50 × 4 = 18.00. The words tell the . ’Each’ with a total means divide. ’Each’ with a count means multiply.
A common mistake is doing only one step. Dev’s problem asks what is left, not what he spent. Answering $10.35 stops too early. Another mistake is mixing units, adding $1.60 to 875 cents. Keep everything in dollars or everything in cents. Write the final answer with a dollar sign and two decimal places.
Words to know
operation
one of the four ways to combine numbers: add, subtract, multiply or divide
total
the whole amount when parts are put together
per
for each one, as in 75 cents per box
Check yourself
1. Mia has $20. She spends $6.45 and $3.80. How much is left?
2. Five tickets cost $31.25 in all. What does one ticket cost?
Why: 31.25 ÷ 5: 3,125 cents ÷ 5 = 625 cents, which is $6.25. Check: 6.25 × 5 = 31.25.
3. Which operation finds the cost of 7 items at $2.30 each?
Why: 'Each' with a count of items means multiply. 2.30 × 7 = 16.10.
19.10
Measurement Word Problems
Main ideaMeasurement problems with decimals work like money problems: match the units, line up the places, then add, subtract, multiply or divide.
A board is 2.4 meters long. A carpenter cuts off 0.85 meters. How much is left? Subtract: 2.40 − 0.85 = 1.55 meters. Then she cuts the rest into 5 equal shelves. Divide: 1.55 ÷ 5. In hundredths, 155 ÷ 5 = 31, so 0.31 meters each. Check: 0.31 × 5 = 1.55.
A recipe needs 0.75 of milk. You make 3 batches. Multiply: 0.75 × 3 = 2.25 liters. Your carton holds 2 liters. Is that enough? 2.25 > 2, so no. You are short by 2.25 − 2.00 = 0.25 liters. Compare the first: both are liters, so the numbers can be compared directly.
When units differ, first. A path is 1.2 km and a second path is 850 m. Total? Change 850 m to 0.85 km. Then 1.2 + 0.85 = 2.05 km. A common mistake is adding 1.2 + 850 to get 851.2. The units did not match. Another mistake is writing 850 m as 8.5 km. Divide by 1,000: 0.85 km.
Words to know
liter
the basic metric unit for liquids; a large soda bottle holds about 2 liters
unit
the thing being counted or measured, like meters, liters or dollars
convert
to change a measurement from one unit to another, like 850 m to 0.85 km
Check yourself
1. A rope is 5.5 m long. You cut off 1.75 m. How much is left?
1. (Unit 8) What is 13/5 written as a mixed number?
Why: Two groups of 5 use 10 fifths, and 13 − 10 = 3 fifths are left. So 13/5 = 2 3/5.
2. (Unit 7) What is 600,000 − 248,317?
Why: Check by adding: 248,317 + 351,683 = 600,000, because 317 + 683 = 1,000 and 248,000 + 351,000 + 1,000 = 600,000.
3. (Unit 6) What number goes in the box? ? × 7 = 63
Why: 7 × 9 = 63, so the missing factor is 9. Subtracting 63 − 7 = 56 is the mistake.
4. (Unit 8) Which is greater, 2/3 or 2/7, and why?
Why: Both have 2 pieces, but a third is bigger than a seventh. Two big pieces beat two small ones, so 2/3 > 2/7.
5. (Unit 7) What is 275,846 + 186,379?
Why: 275,000 + 186,000 = 461,000 and 846 + 379 = 1,225. The sum is 462,225.
Send it to your teacher
Write it
A store sells pencils for $0.35 each. Jaden buys 12 pencils and pays with a $5 bill. Find his change. Explain each step: how you multiplied, where you put the decimal point and why, and how you subtracted. Finish with an estimate that shows your answer makes sense.
State the final answer first, with a dollar sign and two decimal places.
Show the multiplication without the point, then explain how many decimal places the product needs.
Line up the decimal points when you subtract, and show any regrouping.
Check by estimating: 12 pencils at about $0.35 is a little more than $4.
Check again by adding your change back to the cost. It should equal $5.00.
0 wordsSaved on this device as you type.
Practice rooms
Rooms already on the site that belong to this unit — cards, quizzes, a lab.
Every lesson keeps its own three checks; a lesson is ticked when all three are right. Chapter reviews, the unit test and its spiral review (five questions from earlier units in this band) score on the page. When the site is connected to your sheet, or the link carries ?dest=, each one also has a Send box: the first-try score, the standards, the supports used, the attempt number and the minutes go to your sheet as an IEP data point.
Print this page for a paper copy of the readings, the sources, the words and the questions; the answers print as dashed boxes under each question.
Fact-check notes for this course live in the handoff: quotes marked (paraphrased) were set that way on purpose.