How to explain multiplying decimals: whole numbers first, decimal point last
By 15Q20M Editorial · Last updated
Explain multiplying decimals as whole-number multiplication plus a size check. For 0.6 × 7, think 6 tenths × 7 = 42 tenths, which is 4.2. Then ask whether the answer makes sense: 0.6 is less than 1, so the product must be less than 7. Teach the count-the-decimal-places rule last, as a shortcut.
How do you explain a decimal times a whole number?
Start where your child is already strong: whole numbers. The trick is to say the decimal as a count of tenths or hundredths. "Zero point six" is just a name; "six tenths" is an amount your child can multiply. Day 17, Question 10 of our Grade 5 workbook asks which equation is correct: 0.6 × 7 = 0.42, 42, 4.8, or 4.2. Read the left column out loud and let your child answer.
| You say | Child does or says |
|---|---|
| "Say 0.6 the long way." | "Six tenths." |
| "So 0.6 × 7 is six tenths, seven times. How many tenths is that?" | "42 tenths." |
| "How many tenths make one whole?" | "Ten." |
| "So how many wholes are in 42 tenths, and what is left over?" | "4 wholes and 2 tenths. That's 4.2." |
| "Is 0.6 more or less than 1?" | "Less." |
| "So if you have seven of them, should the total be more or less than 7?" | "Less than 7." |
| "Does 4.2 make sense?" | "Yes." |
That last question rules out 42 right away. 0.42 is smaller than a single 0.6, so it fails too. 4.8 is the right size but comes from a times-table slip, so the size check works alongside the multiplication: it catches a misplaced point, and the multiplication itself still has to be right.
Money works well for hundredths. Day 18, Question 7 says a student wrote 4 × 0.35 = 14. Ask your child to say 0.35 as money: 35 cents. Four times 35 cents is 140 cents, which is $1.40, not $14. The student multiplied 4 × 35 = 140 correctly, then put the decimal point in the wrong place: 140 cents is $1.40, not $14.
Standard 5.NBT.B.7 asks 5th graders to "Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used." Counting tenths and counting cents are both place value strategies of this kind.
How do you explain multiplying decimals by 10 and 100?
Draw a place value chart with columns for hundreds, tens, ones, tenths, hundredths, and thousandths, and a dot between the ones and the tenths. Write 3.5 in it. Multiplying by 10 makes every digit worth 10 times as much, so each digit moves one column to the left: the 3 moves to the tens and the 5 moves to the ones. That gives 35. Multiplying by 100 moves each digit two columns, giving 350, with a 0 holding the ones place. These are Day 4, Questions 1 and 2 of our Grade 5 workbook.
Many classrooms, and our workbook, describe the same move by watching the decimal point: multiplying by 10 moves the decimal point one place to the right, and multiplying by 100 moves it two places. Both descriptions give the same answer, so use whichever words your child's teacher uses. To check the direction, ask whether the number should get bigger or smaller. For the positive numbers in these problems, times 10 makes it bigger.
Then turn it around. Day 3, Question 9 asks: 0.034 × ___ = 34. Your child can count the moves on the chart: the 3 goes from the hundredths to the tens, which is three columns, so the missing number is 1,000.
Watch for one leftover habit. With whole numbers, times 10 means "add a zero": 35 × 10 = 350. A child who carries that over to decimals writes 3.5 × 10 = 3.50, which is still 3.5. Our answer explanation for Day 3, Question 10 makes this point: adding zeros at the end only works for whole numbers.
Standard 5.NBT.A.2 asks students to "Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10."
How do you explain a decimal times a decimal?
Money gets awkward here. 0.3 × 0.3 as money would be three tenths of 30 cents, which works out to 9 cents, but few children can picture it. A 10-by-10 grid shows it more clearly. Graph paper works, or draw a square and divide it into 10 rows and 10 columns. The whole square is 1, each column or row is one tenth, and each small box is one hundredth.
Day 19, Question 13 of our Grade 5 workbook asks for the area of a square tile 0.3 m on each side. The workbook gives only the words, so draw it together:
- "Shade 3 columns from top to bottom. That's 0.3 of the square."
- "Now draw stripes across 3 rows, all the way across. That's the other 0.3."
- "Count the boxes that are both shaded and striped." Your child counts 9.
- "Each box is one hundredth of the square. So what is 0.3 × 0.3?" Your child answers: nine hundredths, 0.09.
One of the wrong choices in that question is 0.6, which is what you get by adding 0.3 + 0.3 or by doubling. The grid shows why 0.6 can't be right: 0.6 would be 60 boxes, and only 9 boxes sit where the shaded columns and striped rows overlap.
The grid also explains something the rule never says out loud: a tenth of a tenth is a hundredth. That is why tenths times tenths lands in the hundredths place, and why 0.3 × 0.3 has two decimal places even though each factor has only one.
Where does the count-the-decimal-places rule come from?
Once the grid makes sense, show your child that the rule is a shortcut for what they already did. Day 18, Question 6 gives a head start: if 14 × 23 = 322, what is 14 × 2.3?
| You say | Child does or says |
|---|---|
| "2.3 is 23 tenths. So what is 14 × 23 tenths?" | "322 tenths." |
| "How do you write 322 tenths as a decimal?" | "32.2." |
| "14 × 2 is 28, and 14 × 3 is 42. Does 32.2 make sense?" | "Yes, it's in between." |
The wrong choices in that question are 322 (no decimal point), 3.22 (the point one place too far), and 3,220. All of them fail the between-28-and-42 test.
Now state the rule: multiply as if there were no decimal points, then give the answer as many decimal places as the two factors have together. For 2.3 × 0.7 (Day 19, Question 5): 23 × 7 = 161. Each factor has one decimal place, so the answer has two: 1.61. Check the size: 2 × 0.7 is 1.4, so 1.61 fits.
One detail trips children up at the end. Place the point first, then drop any zeros at the far right. For 8 × 1.25 (Day 18, Question 5): 8 × 125 = 1,000, and 1.25 has two decimal places, so the answer is 10.00, which is 10. Dropping the zeros before counting loses track of where the point goes.
The Grade 5 overview says students use "the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense." In plain words: turn the decimals into whole numbers, multiply, then undo the change, which is what counting the decimal places does.
How can my child check where the decimal point goes?
Round each factor to a nearby whole number and multiply. Day 17, Question 14 asks for the best estimate of 3.7 × 2.1. Rounding to 4 × 2 gives about 8, and the exact answer is 7.77. One wrong choice rounds both numbers down to 3 × 2 = 6, but 3.7 is closer to 4 than to 3.
Two more quick questions help. Is the decimal less than 1? Then the answer is smaller than the other number (for a positive number; Day 18, Question 15). Are both decimals close to 1? Then the answer is close to 1 too: Day 19, Question 10 asks which product is closest to 1, and the answer is 1.2 × 0.8 = 0.96.
What mistakes do kids make multiplying decimals?
| Mistake | What it looks like | How to catch it |
|---|---|---|
| Misplacing the decimal point | 4 × 0.35 = 14 (Day 18, Q7) | Say it as money: 4 × 35 cents is 140 cents, or $1.40. |
| Too few decimal places | 2.3 × 0.7 = 16.1, the slip our Day 19, Q5 answer note warns about | 2.3 × 0.7 is about 2 × 0.7 = 1.4, so 16.1 is ten times too big. Two factors with one place each give two places: 1.61. |
| Adding instead of multiplying | 0.3 × 0.3 = 0.6 (a wrong choice in Day 19, Q13) | Draw the 10-by-10 grid: only 9 boxes sit in the overlap, so 0.09. |
| Adding a zero when multiplying by 10 | 3.5 × 10 = 3.50 | 3.50 is the same as 3.5, and times 10 must make the number bigger. The answer is 35 (Day 4, Q1). |
| Moving the point too far | 14 × 2.3 = 3.22 (a wrong choice in Day 18, Q6) | 14 × 2 is 28, so the answer is more than 28: 32.2. |
Most of these are caught by a size check before the exact answer. When a mistake keeps coming back after the size check, the trouble is often decimal place value itself rather than multiplication. Our guide to why decimals are hard in 5th grade has a five-minute check to find where.
Why decimals are so hard in 5th grade, and a 5-minute check to find the spot
Should a 5th grader use the count-the-places shortcut?
Yes, once your child can explain why it works. Common Core asks 5th graders to multiply decimals to hundredths using a concrete model or drawing and a strategy they can explain, such as one based on place value, and to relate it to a written method. The standard algorithm for decimals, done fluently, is named in 6th grade. If your child can tell you why 0.3 × 0.3 has two decimal places, the shortcut is safe to use.
Standard 6.NS.B.3: "Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation."
Next: how to explain multiplying fractions with coins and folded paper
Frequently asked questions
Do you line up the decimal points when multiplying decimals?
No. Lining up the decimal points is for adding and subtracting. To multiply, write the numbers right-aligned like whole numbers, multiply, and place the decimal point in the answer at the end. For 2.3 × 0.7, multiply 23 × 7 = 161, then give the answer two decimal places: 1.61.
Can I use money to teach multiplying decimals?
Yes, especially for a decimal times a whole number. 4 × $0.35 is 140 cents, or $1.40. When both numbers are decimals, money gets awkward: 0.3 × 0.3 becomes three tenths of 30 cents. It works, but a 10-by-10 grid is easier to picture, and ordinary prices stop at hundredths.
How do I explain why 0.1 × 0.1 = 0.01?
Read it as "one tenth of one tenth." On a 10-by-10 grid, shade one column (a tenth) and stripe one row (another tenth). Only one small box is covered by both, and that box is one hundredth of the square. A tenth of a tenth is a hundredth, so 0.1 × 0.1 = 0.01.
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Sources
- Common Core State Standards for Mathematics (combined PDF), Grade 5 critical area 2, 5.NBT.A.2, 5.NBT.B.7, 6.NS.B.3 — Grade 5 critical area 2 (p.33): "Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately." 5.NBT.A.2 (p.35): "Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10." 5.NBT.B.7 (p.35): "Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used." 6.NS.B.3 (p.42): "Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation." The phrase "decimal places" does not appear in the document (full-text search); the count-the-places rule is a classroom shortcut, not standard text. (verified 2026-10-06)
- 15Q20M question bank (Grade 5 workbook) — Day 3 Q9: 0.034 × 1,000 = 34. Day 3 Q10 explanation: "Adding zeros at the end only works for whole numbers." Day 4 Q1 and Q2: 3.5 × 10 = 35, 3.5 × 100 = 350; the workbook's wording is that the decimal point moves right. Day 17 Q10: correct equation 0.6 × 7 = 4.2 (wrong choices 0.42, 42, 4.8). Day 17 Q14: best estimate for 3.7 × 2.1 is about 8 (4 × 2); exact 7.77; wrong choice rounds both down to 6. Day 18 Q5: 8 × 1.25 = 10. Day 18 Q6: 14 × 23 = 322 so 14 × 2.3 = 32.2 (wrong choices 3.22, 3,220, 322). Day 18 Q7: a student says 4 × 0.35 = 14; correct 1.4. Day 18 Q15: a whole number greater than 0 times a decimal less than 1 is always less than the whole number. Day 19 Q5: 2.3 × 0.7 = 1.61; the bank's mistake note names 16.1 as the common slip. Day 19 Q10: 1.2 × 0.8 = 0.96 is closest to 1 (others 0.09, 0.25, 1.25). Day 19 Q13 (text only, no picture): a 0.3 m square tile has area 0.09 sq m (wrong choices include 0.6). The tenths script, the money walkthrough for Day 18 Q7, the 10-by-10 grid drawing steps, the 3.5 × 10 = 3.50 example, the 30-cent framing of 0.3 × 0.3, and the 14 × 2.3 tenths script are illustrative adaptations, not workbook problems or pictures. Days 17 to 19 practice decimal operations (5.NBT.B.7, with whole-number fact warm-ups), 15 problems a day. All answers independently recalculated. (verified 2026-10-06)
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