6.NS.B.3: Adding, Subtracting, Multiplying and Dividing Multi-Digit Decimals
In plain English: 6.NS.B.3 is the Common Core grade 6 math standard that asks students to add, subtract, multiply and divide multi-digit decimals fluently with the standard algorithm for each operation. Students line up place values to add and subtract, count decimal places to place the point in a product, and move the decimal point to divide by a decimal. It prepares for rational number operations in grade 7.
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Compute fluently with multi-digit numbers and find common factors and multiples. Also written as 6.NS.3 · Official standard
Students build fluency with the four operations on multi-digit decimals. Fluently means accurately, efficiently and flexibly, without needing a drawing or model each time. A decimal is a number written with a decimal point, such as 12.8, where the digits after the point stand for tenths, hundredths and thousandths. The standard algorithm is the usual written method for an operation: stacking numbers in columns to add or subtract, multiplying digit by digit, and long division.
In grade 5, students worked with decimals to hundredths using models and place-value strategies. In this lesson they move to the written algorithms for longer decimals. To add and subtract, they line up the decimal points so every digit sits under its own place value. To multiply, they multiply as if the numbers were whole numbers and then count decimal places to place the point. To divide by a decimal, they multiply the divisor (the number you divide by) and the dividend (the number being divided) by the same power of 10, such as 10, 100 or 1,000, so the divisor becomes a whole number. Then they use long division. Every problem starts with an estimate, so students can tell if the decimal point landed in a sensible place. All numbers in this lesson are positive, as expected in grade 6.
Learning Objectives
By the end of this lesson, students will be able to:
Add multi-digit decimals by lining up decimal points and annexing zeros where needed
Subtract multi-digit decimals, including regrouping across zeros
Multiply multi-digit decimals and place the decimal point by counting decimal places
Divide by a whole number or a decimal with long division, annexing zeros until the division ends
Estimate first and use the estimate to check where the decimal point belongs
Prior Knowledge Required
Students should already be comfortable with:
Reading and writing decimals to thousandths 5.NBT.A.3
Multiplying and dividing by powers of 10 and moving the decimal point 5.NBT.A.2
Adding, subtracting, multiplying and dividing decimals to hundredths with models 5.NBT.B.7
Multiplying multi-digit whole numbers with the standard algorithm 5.NBT.B.5
Dividing multi-digit whole numbers with long division 6.NS.B.2
Write this work on the board and ask students to find the mistake before anyone computes the answer:
Warm-Up Prompt
"Theo added 4.6 + 2.35 by lining up the last digits on the right, like whole numbers. He got 2.81. Without working it out, how do you know 2.81 is wrong? What is the correct sum?"
Collect answers. An estimate (a quick, rounded answer) settles it: 4.6 is about 5 and 2.35 is about 2, so the sum is about 7, not about 3. Theo lined up the 6 tenths with the 5 hundredths, so he really added 46 + 235 = 281 and then guessed where the point goes. Lining up the decimal points puts tenths under tenths and hundredths under hundredths: 4.60 + 2.35 = 6.95. Introduce the word place value: the value of a digit depends on its place, so the 6 in 4.6 means 6 tenths.
Direct Instruction20-25 minutes
Part 1: Adding and subtracting. Write the numbers in a column with the decimal points in a straight line, so each digit is under its own place value (Diagram 1). A whole number has its decimal point at the end: 16 is 16.0. If the numbers have different lengths, annex zeros, which means writing zeros at the end of the shorter decimals. This does not change the value, because 3.45 and 3.450 both mean 3 and 45 hundredths. Then add or subtract from right to left, just as with whole numbers, and bring the decimal point straight down into the answer. Regrouping (carrying or borrowing) works the same way: 10 thousandths make 1 hundredth, and 10 tenths make 1 one.
Adding three decimals
Priya biked 3.45 km on Monday, 12.8 km on Saturday and 6.075 km on Sunday. How far did she bike in all?
Equation: 3.450 + 12.800 + 6.075 = 22.325, so 22.325 km (estimate: 3 + 13 + 6 = 22)
Subtracting across zeros
A 2-liter bottle of juice is full. Sam pours out 0.375 liter. How much juice is left?
A car used 11.5 gallons of gas to go 331.2 miles. How many miles did it go per gallon?
Equation: 331.2 ÷ 11.5 = 3312 ÷ 115 = 28.8, so 28.8 miles per gallon (estimate: 330 ÷ 11 = 30)
Dividing a decimal by a whole number
A 7.5 km trail is split into 4 equal sections. How long is each section?
Equation: 7.500 ÷ 4 = 1.875, so each section is 1.875 km (check: 1.875 × 4 = 7.5)
Part 2: Multiplying. Multiply the digits as if the numbers were whole numbers. Then count the decimal places (digits after the point) in both factors and give the product that many decimal places. In the third example, 6.40 has 2 decimal places and 3.25 has 2, so the product has 4: 20.8000, which is $20.80. Each partial product (the result of multiplying by one digit) shifts one place left, just as in whole-number multiplication. Always compare with the estimate: 20.80 is close to 18, and 208.00 or 2.08 would not be.
Part 3: Dividing. In a division, the dividend is the number being divided, the divisor is the number you divide by, and the quotient is the answer. When the divisor is a whole number, as in the fifth example, divide and place the decimal point in the quotient straight above the point in the dividend. Annex zeros to the dividend and keep dividing until the remainder is 0. When the divisor is a decimal, as in the fourth example, multiply the divisor and the dividend by the same power of 10 (10, 100 or 1,000) so the divisor becomes a whole number. The quotient does not change, because both numbers grow by the same factor. Diagram 2 shows the full long division.
Guided Practice15 minutes
Pairs work through four problems on grid paper, one digit per square, so the columns stay lined up. Each pair writes an estimate before computing. After each problem, one pair shows its grid.
Guided practice problems and answers
Problem
Estimate
Answer
1. Ms. Lee pays for $23.68 of art supplies with a $50 bill. What is her change?
50 - 24 = 26
$26.32
2. Grapes cost $2.98 per pound. What do 1.5 pounds cost?
3 × 1.5 = 4.5
$4.47
3. A bracelet uses 0.24 m of cord. How many bracelets can be made from 9.12 m?
9 ÷ 0.25 = 36
38 bracelets
4. Add 45.3 + 0.87 + 112.
45 + 1 + 112 = 158
158.17
In Problem 2, 298 × 15 = 4,470, and 2 + 1 = 3 decimal places gives 4.470, or $4.47. In Problem 3, multiply both numbers by 100: 912 ÷ 24 = 38. In Problem 4, write 112 as 112.00. Listen for students who line up the last digits instead of the decimal points, and ask them to check their answer against the estimate.
Independent Practice10-15 minutes
Students solve five problems on their own, estimate first, and check each answer with the inverse operation (addition checks subtraction, multiplication checks division). (1) 305.4 - 78.96 (226.44; check: 226.44 + 78.96 = 305.4). (2) 17.6 × 0.25 (176 × 25 = 4,400 with 3 decimal places, so 4.4). (3) 6.72 ÷ 0.32 (672 ÷ 32 = 21). (4) 8.009 + 14.99 (22.999). (5) 0.03 × 0.7 (3 × 7 = 21 with 3 decimal places, so 0.021: write a zero in the tenths place to make 3 decimal places).
Closure5 minutes
Exit ticket: (1) 72.4 - 9.87 (62.53). (2) 2.8 × 0.05 (0.14). (3) 10.2 ÷ 0.6 (102 ÷ 6 = 17). Then ask one question aloud: "Why can you move the decimal point in both numbers of a division, but not in just one?" (Multiplying both by 10 keeps the quotient the same. Moving only one changes the problem.)
Differentiation Strategies
For Struggling Students
Give place-value grid paper with a bold column for the decimal point, and have students write one digit per square
Have students write the annexed zeros in a different color so every number has the same number of decimal places before adding or subtracting
For multiplication, give a three-step card: multiply as whole numbers, count decimal places in both factors, place the point and compare with the estimate
For Advanced Students
Ask students to write a multiplication problem whose product has fewer decimal places than the count predicts, such as 2.5 × 0.4, and explain where the extra zeros went
Give a division where the divisor has 3 decimal places, such as 4.2 ÷ 0.105, and ask what power of 10 to multiply by
Ask students to find two decimals with a sum of 5 and a product of 6.16, then check their answer with the algorithms
Assessment Guidance
What to Look For
Check that students write an estimate before they compute and compare it with the answer. In addition and subtraction, the decimal points should form a straight column, with annexed zeros where the numbers have different lengths. In multiplication, students should count decimal places in both factors, and they should not line up the decimal points. In division, look for the divisor and the dividend being multiplied by the same power of 10, and for the decimal point in the quotient placed straight above the point in the dividend. Ask students to check a quotient by multiplying it by the divisor.
02
Classroom Activities
3 Activities
1
Grocery Receipt Challenge
15 minPairs
Each pair gets 8 price cards and a budget of $40.00. Pairs add prices with the standard algorithm, then subtract to find the change. A second card set lists the weights of three bags of produce, which have different numbers of decimal places.
Estimate the total of all 8 prices by rounding each to the nearest dollar
Add the 8 prices on grid paper with the decimal points lined up
Subtract the total from $40.00 to find the change
Add the three weights, annexing zeros so each has 3 decimal places
Discussion Questions
Can the pair buy all 8 items with $40.00? (Yes: the total is $39.29, and the change is $0.71)
What is the total weight of the three bags? (4.625 kg)
Why do prices always have 2 decimal places, while weights can have 1, 2 or 3?
Modification for Distance Learning
Share the cards on a slide. Students type their column work in a shared table with one digit per cell, then post their estimate next to the exact total.
2
Decimal Point Detective Card Sort
15 minPairs
Each pair gets 8 product cards. Every card shows a multiplication and its product with the decimal point missing. Pairs estimate, place the decimal point, and check by counting decimal places.
Product Cards (8 cards)
2.4 × 3.15 = 7560 (7.56)
0.8 × 0.07 = 56 (0.056)
12.5 × 0.4 = 500 (5)
3.06 × 2.5 = 7650 (7.65)
0.25 × 16.8 = 4200 (4.2)
45.2 × 0.03 = 1356 (1.356)
1.05 × 1.2 = 1260 (1.26)
7.5 × 0.12 = 900 (0.9)
Procedure
One partner rounds the factors and says an estimate; the other writes the digits of the product with the decimal point placed
Count the decimal places in both factors and confirm, writing zeros in front where needed (56 with 3 decimal places is 0.056)
Sort the cards into two piles: product greater than 1, and product less than 1
Discussion Questions
Which card has a product smaller than both of its factors? (0.8 × 0.07 = 0.056)
Several digit strings end in zeros, such as 500 for 12.5 × 0.4. Why can the zeros at the end be dropped after the point is placed?
How many cards are in the "less than 1" pile? (2: 0.056 and 0.9)
Challenge Variation
Pairs write two new cards of their own with the decimal point missing, one with a product less than 1, and trade with another pair.
3
Measure and Divide Stations
20 minPairs
Pairs work at two stations. At the first, they solve 6 division cards with long division. At the second, they measure a textbook and use division to find how many fit on a shelf.
Station 1: Division Cards (6 cards)
A 3.6 m ribbon is cut into bookmarks 0.15 m long. How many bookmarks? (24)
27.3 ÷ 0.7 (39)
0.84 ÷ 0.012 (70)
56.7 ÷ 8.1 (7)
2.25 ÷ 0.5 (4.5)
19.5 ÷ 1.25 (15.6)
Station 2: Measure and Divide
Measure the thickness of the textbook, from the front cover to the back cover, in centimeters to the nearest tenth. Many textbooks are 2.5-4.5 cm thick
A shelf is 150 cm long. Divide 150 by the thickness to find how many copies of the book fit standing side by side. For example, with a 3.2 cm book: 150 ÷ 3.2 = 1500 ÷ 32 = 46.875, so 46 books fit
Check with multiplication: 46 × 3.2 = 147.2 cm fits, and 47 × 3.2 = 150.4 cm is too long
Discussion Questions
Which division card has the largest quotient? (0.84 ÷ 0.012 = 70)
On the bookmark card, the quotient 24 is larger than the dividend 3.6. Why does that happen?
The shelf answer was 46.875, but the answer to the question is 46 books. Why?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Lining Up Decimal Points to Add and Subtract
Top: Priya's distances, 3.45 + 12.8 + 6.075, written in place-value columns. Annexed zeros make every number end in the thousandths place, and the sum is 22.325 km. Bottom: 2 - 0.375. The 2 is written as 2.000, then regrouped from the ones place down to the thousandths, and the difference is 1.625 liters.
Diagram 2: Long Division With a Decimal Divisor
To divide 331.2 by 11.5, multiply both numbers by 10 so the divisor is the whole number 115. Long division of 3312.0 by 115 gives 28.8, with the decimal point in the quotient straight above the point in the dividend. Multiplying 11.5 × 28.8 gives back 331.2.
04
Homework Assignment
~30 min
6.NS.B.3 Homework: Operations With Decimals
Directions: Write an estimate before each problem. Show your work with the standard algorithm, one digit per square on grid paper if you can. Check every subtraction by adding and every division by multiplying.
Part 1: Adding and Subtracting (Problems 1-2)
Three grade 6 classes collected recycled paper: 14.6 kg, 9.35 kg and 21.075 kg. (a) How much paper did they collect in all? (b) How much more did the class with the most paper collect than the class with the least?
Subtract, then check each answer by adding: (a) 400 - 187.46 (b) 6.2 - 3.875.
Part 2: Multiplying (Problems 3-4)
Gas costs $4.15 per gallon. A family fills its car with 11.4 gallons. How much does the gas cost? Estimate first, then use the standard algorithm.
Multiply: (a) 0.09 × 0.8 (b) 38.4 × 2.05. For each, write how many decimal places the product has and why.
Part 3: Dividing (Problems 5-6)
A pot holds 5.25 liters of soup. Each bowl holds 0.35 liter. How many bowls can be filled? Show how you made the divisor a whole number.
(a) 93.6 ÷ 3.6 (b) Four friends share a $30.60 restaurant bill equally. How much does each friend pay? (c) 5.1 ÷ 0.24. Annex zeros until the remainder is 0.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Adding and Subtracting
Decimal points lined up, zeros annexed, answers correct and checked
One computation error, or no check
Digits lined up on the right, or answers missing
Multiplying
Products correct with the decimal point placed by counting places
Digits correct but the decimal point misplaced once
Products incorrect or missing
Dividing
Divisor made a whole number, quotients correct and checked by multiplying
Correct setup with one computation error
Decimal point moved in only one number, or missing
Estimates
A sensible estimate for every problem, compared with the answer
Estimates for some problems
No estimates
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Estimate before you choose. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What is 5.3 + 2.46?
Answer: B
Line up the decimal points and annex a zero: 5.30 + 2.46 = 7.76. The estimate 5 + 2 = 7 agrees. Choice A lines up the last digits on the right, which adds 53 + 246 = 299, and then places the point to make 2.99. Choice C treats 5.3 as 5.03. Choice D adds 530 + 246 correctly but leaves out the decimal point.
Question 2 of 20 · Multiple Choice
Ana's cat weighed 4.85 kg in May. By June it had gained 0.3 kg. How much did the cat weigh in June?
Answer: D
Write 0.3 as 0.30 and line up the points: 4.85 + 0.30 = 5.15 kg. Choice A lines up the 3 with the hundredths, so it adds 0.03. Choice B adds 8 + 3 = 11 tenths but forgets to carry the 1 to the ones place. Choice C reads 0.3 as 3. A cat that gained weight must weigh more than 4.85 kg, which also rules out choice B.
Question 3 of 20 · Multiple Choice
What is 9.4 - 3.57?
Answer: C
Write 9.4 as 9.40. Regroup 1 tenth as 10 hundredths: 10 - 7 = 3. Then regroup 1 one as 10 tenths: 13 - 5 = 8. Finally 8 - 3 = 5, so the answer is 5.83. Check: 5.83 + 3.57 = 9.40. Choice A subtracts the smaller digit from the larger in each column (7 - 0, 5 - 4, 9 - 3). Choice B forgets that the tenths digit became 3 after the first regrouping. Choice D forgets that the ones digit became 8.
Question 4 of 20 · Multiple Choice
Leo pays for a $13.45 T-shirt with a $20 bill. How much change does he get?
Answer: B
Write $20 as 20.00 and regroup: 20.00 - 13.45 = 6.55, so the change is $6.55. Check: 6.55 + 13.45 = 20.00. Choice A regroups for the cents but forgets that the ones digit dropped from 10 to 9, so it computes 10 - 3 = 7 in the ones place. Choice C adds instead of subtracting. Choice D subtracts the dollars (20 - 13 = 7) and then just copies the 45 cents.
Question 5 of 20 · Multiple Choice
What is 0.9 × 0.5?
Answer: A
Multiply 9 × 5 = 45. Each factor has 1 decimal place, so the product has 2: 0.45. It makes sense that the product is less than 0.9, because you are taking half of 0.9. Choice B gives the product only 1 decimal place. Choice C gives it 3. Choice D adds the numbers instead of multiplying.
Question 6 of 20 · Multiple Choice
What is 4.25 × 1.6?
Answer: C
Multiply 425 × 16 = 6,800. The factors have 2 + 1 = 3 decimal places, so the product is 6.800, which is 6.8. The estimate 4 × 1.6 = 6.4 agrees. Choice A places the point with only 2 decimal places (68.00). Choice B uses 4 decimal places. Choice D forgets to shift the second partial product: it adds 2,550 + 425 = 2,975 instead of 2,550 + 4,250.
Question 7 of 20 · Multiple Choice
Tomatoes cost $2.40 per pound. How much do 3.5 pounds cost?
Answer: A
Multiply 240 × 35 = 8,400. The factors have 2 + 1 = 3 decimal places, so the product is 8.400, or $8.40. Estimate: a little more than 2 × 3.5 = 7. Choice B uses only 2 decimal places instead of 3. Choice C adds the numbers. Choice D multiplies only by the 3 and leaves out the half pound.
Question 8 of 20 · Multiple Choice
Without multiplying, which number is closest to 45.8 × 0.49?
Answer: B
0.49 is close to 0.5, which is one half, and half of 45.8 is about 23. So the product is close to 22. The exact product is 22.442. Choices A and D place the decimal point too far left, and choice C places it too far right. An estimate like this is a fast way to check where the point belongs.
Question 9 of 20 · Multiple Choice
What is 7.2 ÷ 0.08?
Answer: D
Multiply both numbers by 100 so the divisor is a whole number: 720 ÷ 8 = 90. Check: 0.08 × 90 = 7.2. Choice A moves the point only in the divisor and computes 7.2 ÷ 8. Choice B moves the point in the dividend only 1 place (72 ÷ 8). Choice C moves it 3 places (7,200 ÷ 8).
Question 10 of 20 · Multiple Choice
A 2.4 kg bag of trail mix is split into snack bags of 0.12 kg each. How many snack bags can be filled?
Answer: A
Multiply both numbers by 100: 240 ÷ 12 = 20 bags. Check: 20 × 0.12 = 2.4. Choice B multiplies the divisor by 100 but the dividend only by 10 (24 ÷ 12). Choice C multiplies the dividend by 100 but the divisor only by 10 (240 ÷ 1.2). Choice D multiplies 2.4 × 0.12 instead of dividing, and a number of bags cannot be less than 1 here.
Question 11 of 20 · Multiple Choice
What is 12.6 ÷ 5?
Answer: D
Place the decimal point in the quotient above the point in the dividend. 12 ÷ 5 = 2 remainder 2, then 26 tenths ÷ 5 = 5 tenths remainder 1 tenth. Annex a zero: 10 hundredths ÷ 5 = 2 hundredths. The quotient is 2.52. Check: 2.52 × 5 = 12.6. Choice A divides 126 ÷ 5 and never puts the point back. Choice B multiplies instead of dividing. Choice C stops at the remainder instead of annexing a zero.
Question 12 of 20 · Multiple Choice
A car travels 157.5 miles on 4.5 gallons of gas. How many miles per gallon is that?
Answer: C
Multiply both numbers by 10: 1575 ÷ 45 = 35 miles per gallon. Check: 4.5 × 35 = 157.5. Choice A moves the point only in the divisor and computes 157.5 ÷ 45. Choice B multiplies instead of dividing. Choice D divides in the wrong order, 4.5 ÷ 157.5, which gives gallons per mile.
Question 13 of 20 · Multiple Choice
Mia had $25.00. She bought a book for $8.99 and a snack for $2.35. How much money does she have left?
Answer: D
Add what she spent: 8.99 + 2.35 = 11.34. Then subtract: 25.00 - 11.34 = 13.66, so she has $13.66 left. Choice A is the amount she spent, not the amount left. Choice B subtracts the book but adds the snack. Choice C regroups for the cents but forgets that the ones digit dropped from 5 to 4.
Question 14 of 20 · Multiple Choice
Which division has the same quotient as 0.36 ÷ 0.009?
Answer: A
The divisor 0.009 has 3 decimal places, so multiply both numbers by 1,000: 0.36 × 1,000 = 360 and 0.009 × 1,000 = 9. So 0.36 ÷ 0.009 = 360 ÷ 9 = 40. Choice B multiplies the dividend by only 100. Choice C multiplies it by only 10. Choice D multiplies it by 10,000.
Question 15 of 20 · Short Answer
Add 128.4 + 0.935 + 16. Show how you lined up the numbers.
Write 128.400 + 0.935 + 16.000 with the decimal points in one column. The sum is 145.335. The estimate 128 + 1 + 16 = 145 agrees. A common error is to put the 16 under the 4 tenths, as if it were 0.16.
Question 16 of 20 · Short Answer
Subtract 70.05 - 26.8. Then check your answer by adding.
Write 26.8 as 26.80: 70.05 - 26.80 = 43.25. Check: 43.25 + 26.80 = 70.05. The tenths place needs regrouping (0 - 8), and the ones digit is 0, so first regroup 1 ten as 10 ones, then 1 one as 10 tenths.
Question 17 of 20 · Short Answer
One lap of a school's track is 0.4 km. Jo runs 6.5 laps. How far does she run? Explain where the decimal point goes.
Multiply 65 × 4 = 260. The factors have 1 + 1 = 2 decimal places, so the product is 2.60, which is 2.6 km. Estimate: 6.5 laps of a little less than half a kilometer is about 3 km.
Question 18 of 20 · Short Answer
Divide 43.68 ÷ 1.2. Show how you changed the divisor to a whole number.
Multiply both numbers by 10: 436.8 ÷ 12. Long division gives 36.4. Check: 1.2 × 36.4 = 43.68. Estimate: 48 ÷ 1.2 = 40, and 43.68 is a little less than 48, so the quotient is a little less than 40.
Question 19 of 20 · Short Answer
Pencils cost $0.55 each. How many pencils can you buy with $9.90? Check your answer with multiplication.
Jordan wrote 1.4 × 0.2 = 28 and 0.6 ÷ 0.02 = 3. Find the error in each answer and give the correct answers.
In the product, 14 × 2 = 28, but the factors have 1 + 1 = 2 decimal places, so 1.4 × 0.2 = 0.28. Jordan forgot to place the decimal point. In the quotient, multiply both numbers by 100: 60 ÷ 2 = 30, so 0.6 ÷ 0.02 = 30. Jordan multiplied the dividend by only 10 (6 ÷ 2 = 3). Check: 0.02 × 30 = 0.6.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.NS.B.3 mean?
6.NS.B.3 means students can add, subtract, multiply and divide decimals with many digits, quickly and accurately, using the standard written method for each operation. That means stacking numbers with the decimal points lined up to add and subtract, multiplying digit by digit and counting decimal places, and using long division.
Is 6.NS.B.3 a grade 5 or grade 6 standard?
It is a grade 6 standard. In grade 5 (5.NBT.B.7), students work with decimals to hundredths using models, drawings and place-value strategies. In grade 6, the numbers can be longer and students are expected to use the standard algorithms fluently, without a model.
What is the standard algorithm?
The standard algorithm is the usual written method for an operation. For addition and subtraction, you write the numbers in columns and work from right to left with regrouping. For multiplication, you find partial products and add them. For division, you use long division. With decimals, the only new step is deciding where the decimal point goes.
Why do you line up the decimal points when adding decimals?
Lining up the decimal points puts digits of the same place value in the same column, so you add tenths to tenths and hundredths to hundredths. For example, in 0.7 + 1.25, the 7 means 7 tenths, so it goes under the 2, not under the 5. Annexing a zero (0.70) can help keep the columns even.
How do you know where the decimal point goes when you multiply decimals?
Count the decimal places in both factors and give the product the same total. For 1.3 × 0.21, there are 1 + 2 = 3 decimal places, and 13 × 21 = 273, so the product is 0.273. You do not line up the decimal points to multiply. An estimate confirms the answer: 1.3 × 0.21 is about 1 × 0.2 = 0.2.
Why can you move the decimal point when dividing by a decimal?
Multiplying the divisor and the dividend by the same number does not change the quotient, just as 6 ÷ 2 and 60 ÷ 20 both equal 3. So 1.8 ÷ 0.6 has the same answer as 18 ÷ 6, which is 3. You must move the point the same number of places in both numbers.
What are common mistakes with decimal operations?
A common mistake is lining up the last digits instead of the decimal points when adding or subtracting. Others are lining up the decimal points in multiplication, moving the decimal point in only one number when dividing, and forgetting to annex zeros in a division that has a remainder. Asking students to estimate first catches many of these errors.
Should students use calculators for 6.NS.B.3?
The standard asks for fluency with the written algorithms, so students should compute by hand. A calculator is useful for checking an answer after the work is done, or for exploring patterns, such as what happens to a product when one factor is multiplied by 10.
How is 6.NS.B.3 connected to 6.NS.B.2?
6.NS.B.2 asks students to divide multi-digit whole numbers fluently with long division. 6.NS.B.3 uses the same long division for decimals: after the divisor is made a whole number, the steps are the same, and the decimal point in the quotient goes straight above the point in the dividend.
What comes after decimal operations in grade 7?
In grade 7, students extend all four operations to negative numbers and fractions together, the rational numbers (7.NS.A.1, 7.NS.A.2). They also convert fractions to decimals with long division and solve multistep problems with money, measurements and percents (7.NS.A.3, 7.EE.B.3). Decimal fluency from grade 6 is used in every one of these topics.
07
Related Standards
6 standards
These standards connect to 6.NS.B.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NBT.B.7Prerequisite
Add, subtract, multiply and divide decimals to hundredths with models and strategies
Lesson coming soon
5.NBT.A.2Prerequisite
Explain patterns in zeros and decimal point placement with powers of 10
Lesson coming soon
Alongside
6.NS.B.2Parallel
Fluently divide multi-digit numbers using the standard algorithm
