Adding and subtracting decimals
Lining up the point, regrouping across it, and checking with money and estimation
About four lessons of 45 to 60 minutes
Adding up the shopping, dollars and cents
You are at the shop with a snack for $2.50 and a drink for $1.25. What do they cost together, and what change do you get from a five dollar note? You do this kind of sum all the time, and the secret to getting it right is simple: keep the dollars with the dollars and the cents with the cents. In decimal language, that means lining up the decimal point.
Money is decimals you can hold. A dollar splits into a hundred cents, so 25 cents is 0.25 of a dollar and 50 cents is 0.50. Today you will add and subtract decimals by lining up the point, trading ten tenths for a whole just as you traded ten ones for a ten, and checking your answer makes sense with a quick estimate.
- A snack for $2.50 and a drink for $1.25line up the point: 2.50 + 1.25 = 3.75, so $3.75 in all
- Change from a $5 note5.00 minus 3.75 = 1.25, so $1.25 change
- A pencil 4.7 cm plus a rubber 2.6 cm4.7 + 2.6 = 7.3 cm, the ten tenths traded for one whole
- A lap of 6.2 s beating one of 3.75 s6.20 minus 3.75 = 2.45 s faster, once you pad 6.2 to 6.20
What students will be able to do
Students will add and subtract decimals to hundredths by lining up the decimal point and working place by place, regroup ten tenths into a whole and ten hundredths into a tenth, place the strategy on a number line, and check that answers are reasonable using money and estimation.
- I can line up the decimal point before I add or subtract.
- I can pad a decimal with a zero so both numbers have the same places.
- I can regroup ten tenths into one whole when adding.
- I can regroup across the point when subtracting.
- I can estimate first to check my answer is reasonable.
Standards this unit teaches
- 5.NBT.B.7Common Core (US)Add, subtract, multiply and divide decimals to hundredths
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
- 5.NBT.A.1Common Core (US)Place value in decimals
Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Lining up the decimal point is this place-value idea put to work.
- AC9M5N04Australian Curriculum v9 (ACARA)Add and subtract decimals (Year 5)
Use place value to add and subtract decimals, with estimation and rounding to check that answers are reasonable.
- AC9M4N01Australian Curriculum v9 (ACARA)Decimals to hundredths (Year 4 foundation)
Extend place value to tenths and hundredths and use decimal notation to name and represent these numbers. Australia introduces tenths and hundredths in Year 4, so this Year 5 unit puts that place-value knowledge to work in addition and subtraction.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Decimals: tenths and hundredthsreading and placing decimals, the unit this builds on
- How to teach decimals0.1 is one tenth, and money is the everyday model
- Place valueeach place is ten times the one to its right, now past the point
- Additioncolumn addition and carrying, extended to the tenths and hundredths
- Money and counting coinscents are hundredths of a dollar, the concrete model for decimals
Words to teach and display
- Decimal
- a number with a decimal point, showing parts smaller than one whole
- Decimal point
- the dot that separates the whole ones from the tenths and hundredths
- Tenth
- one of ten equal parts of a whole, written 0.1
- Hundredth
- one of a hundred equal parts of a whole, written 0.01, the size of one cent in a dollar
- Regroup
- to trade ten of one place for one of the next, such as ten tenths for one whole
- Sum
- the answer to an addition
- Difference
- the answer to a subtraction
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Line up the point
ConcreteEverything in this unit rests on one rule: line up the decimal points, so that tenths sit under tenths and ones under ones. When you add money you would never add the dollars to the cents, and decimals are the same idea. Write one number above the other with the points in a single vertical line, and the columns take care of themselves.
Take 2.6 + 1.7. Line up the points, then add each column just like whole numbers, starting from the right. The tenths are 6 and 7, the ones are 2 and 1. The point in the answer goes straight below the points above it.
A part-whole picture shows what the columns are doing. The whole is the total 4.3, split into the two amounts you are combining, 2.6 and 1.7. Adding is joining the parts to make the whole. Subtracting is starting from the whole and taking one part away to find the other.
- When you add 3.4 and 12.5, which digits line up under the 4?
- Why would adding 2.6 and 1.7 as if they were 26 and 17 give the wrong place value?
- In 4.3, how many whole ones are there and how many tenths?
2. Adding with a trade across the point
PictorialAdding decimals uses the same carry you already know. When a column reaches ten or more, trade ten of that place for one of the next place to the left. The new rule is only where the trade happens: ten tenths make one whole, so a full tenths column carries a 1 into the ones.
Take 4.7 + 2.6. The tenths are 7 + 6 = 13 tenths. Thirteen tenths is more than a whole, so trade: ten tenths become one whole (carried into the ones) and 3 tenths stay. Now the ones are 4 + 2 + 1 carried = 7. The answer is 7.3.
This is exactly the ten ones for one ten trade from earlier grades, just one place further right. Ten tenths for one whole, and later ten hundredths for one tenth, follow the same ten-for-one pattern that runs through the whole number system.
Add 4.7 + 2.6.
- Line up the points and add the tenths: 7 + 6 = 13 tenths.
- Trade: 13 tenths is 1 whole and 3 tenths. Write 3 in the tenths place and carry 1 to the ones.
- Add the ones: 4 + 2 + 1 carried = 7.
Answer: 4.7 + 2.6 = 7.3.
- In 3.8 + 4.5, how many tenths do you get before trading, and what carries?
- Why is 13 tenths written as 1.3 and not left as 13 in the tenths column?
- What trades into the tenths place when the hundredths reach ten?
3. Adding and subtracting on a number line
PictorialA number line marked in tenths shows the same moves as a count you can see. To add, start at the first number and jump forward. To subtract, start at the larger number and jump back. The jumps are lengths along the line, and where you land is the answer.
To work out 0.4 + 0.5, start at 0.4 and jump forward five tenths to land on 0.9. Every tick on this line is one tenth, so a jump of 0.5 is five ticks. Subtracting 0.9 minus 0.4 is the same picture read backward: start at 0.9, jump back four tenths, land on 0.5.
The number line is the reasonableness check for the column method. If your written answer to 0.4 + 0.5 came out as 0.09 or 4.5, a glance at the line shows it should sit between 0 and 1, near 0.9, so those answers cannot be right.
- Where do you land if you start at 0.3 and jump forward 0.6?
- How would you show 0.8 minus 0.5 on this line?
- Between which two whole numbers must the answer to 0.4 + 0.5 sit?
4. Subtracting with a trade and padding zeros
AbstractSubtracting decimals is column subtraction with the point lined up. Two moves keep it safe. First, if the numbers have different numbers of decimal places, pad the shorter one with zeros so every column has a digit. Writing 6.2 as 6.20 does not change its value, it just fills the hundredths column. Second, when a column is too small, borrow ten of it from the place to the left, exactly as with whole numbers.
Take 6.2 minus 3.75. Pad 6.2 to 6.20 so both have hundredths. The hundredths are 0 minus 5, which does not work, so borrow: 6.20 becomes 6.1 and 10 hundredths, and 10 minus 5 = 5 hundredths. The tenths are now 1 minus 7, which does not work, so borrow again from the ones: 6 ones become 5, and 11 tenths minus 7 = 4 tenths. The ones are 5 minus 3 = 2.
Reading it off in order: 2 ones, 4 tenths, 5 hundredths, so 6.2 minus 3.75 = 2.45. Padding with the zero was what made the hundredths column possible.
Subtract 6.2 minus 3.75.
- Pad 6.2 to 6.20 so both numbers reach the hundredths place, points lined up.
- Hundredths: 0 minus 5 needs a borrow, so 20 hundredths becomes 1 tenth and 10 hundredths; 10 minus 5 = 5.
- Tenths: 1 minus 7 needs a borrow from the ones; 11 minus 7 = 4. Ones: 5 minus 3 = 2.
Answer: 6.2 minus 3.75 = 2.45. Check: 3.75 + 2.45 = 6.20.
- How do you write 5 so you can subtract 2.35 from it?
- In 4.3 minus 1.8, which column needs a borrow first?
- How can you use addition to check a decimal subtraction?
5. Money and the estimation check
AbstractMoney is where decimals earn their keep, and it is also the best place to build the habit of estimating first. Before you add or subtract, round each amount to the nearest whole and get a rough answer. Then do the exact calculation and check it lands near your estimate. An answer far from the estimate is a signal to look again.
For $2.50 + $1.25, estimate first: about 3 dollars plus about 1 dollar is about 4 dollars. Now the exact sum, lining up the point, is 3.75, which is close to 4, so it is reasonable. The answer is $3.75.
For change from a five dollar note, subtract: 5.00 minus 3.75. Estimate 5 minus 4 is about 1 dollar. The exact answer is $1.25, close to the estimate, so it checks out. Estimation does not replace the calculation, it guards it.
A snack is $2.50 and a drink is $1.25. Find the total, then the change from $5.00.
- Estimate the total: about $3 plus about $1 is about $4.
- Add exactly, points lined up: 2.50 + 1.25 = 3.75, close to the $4 estimate.
- Change from $5.00: 5.00 minus 3.75 = 1.25.
Answer: The total is $3.75 and the change from $5.00 is $1.25.
- Estimate 3.9 + 5.2 before adding. Is your exact answer near the estimate?
- Why is estimating first a good habit with money?
- If your total for two items over $3 each came to under $3, what would that tell you?
Common misconceptions and how to address them
MisconceptionDecimals are lined up by their last digit instead of by the point, so 2.6 and 1.75 are stacked with the 6 above the 5.
Why it happens: Students carry over the whole-number habit of right-aligning digits, which puts tenths under hundredths.
How to address it: Line up the decimal points first and draw the vertical line through them. Pad with a zero so both numbers have the same places: 2.60 above 1.75, tenths under tenths.
Misconception13 tenths is written straight into the answer, giving 4.7 + 2.6 = 6.13.
Why it happens: The tenths column is treated like a box that holds any number, without the ten-for-one trade.
How to address it: A column can hold only 0 to 9. Ten tenths make one whole, so 13 tenths is 1 whole and 3 tenths: write 3 and carry 1. The bar model shows the traded whole.
MisconceptionA whole number is treated as having no decimal places, so 5 minus 3.75 cannot be started.
Why it happens: Without a point and zeros, the columns to subtract from look empty.
How to address it: Every whole number has a hidden point and endless zeros after it: 5 is 5.00. Write it that way and the hundredths and tenths columns are there to borrow from.
MisconceptionThe decimal point is left out of the answer or floated to a new position.
Why it happens: Students focus on the digits and forget the point must stay in line with the points above.
How to address it: Drop the point straight down from the lined-up points above into the answer before writing any digits. Its column never moves.
MisconceptionLonger looks larger, so 0.45 is thought to be bigger than 0.5 when subtracting.
Why it happens: More digits after the point reads as a bigger number, echoing whole-number length.
How to address it: Pad to the same places: 0.50 against 0.45. Now 50 hundredths clearly beats 45 hundredths, so 0.5 is larger. A tenths number line confirms 0.5 sits further right.
MisconceptionAnswers are not checked, so an unreasonable result slips through.
Why it happens: Once the columns are done, the calculation feels finished, with no habit of standing back.
How to address it: Estimate first by rounding each number to the nearest whole, then compare. If 2.50 + 1.25 came out far from 4, something went wrong. Estimation catches place-value slips.
Guided practice (with answers)
1. Add 3.4 + 2.5.
Answer: 5.9. No trade needed: tenths 4 + 5 = 9, ones 3 + 2 = 5.
2. Add 4.8 + 3.6.
Answer: 8.4. Tenths 8 + 6 = 14, so write 4 and carry 1; ones 4 + 3 + 1 = 8.
3. Subtract 7.3 minus 2.8.
Answer: 4.5. Tenths 3 minus 8 needs a borrow: 13 minus 8 = 5; ones 6 minus 2 = 4.
4. Subtract 5 minus 1.4.
Answer: 3.6. Write 5 as 5.0, then 10 tenths minus 4 = 6 tenths and 4 ones minus 1 = 3.
5. Add the shopping: $3.75 + $2.20.
Answer: $5.95. Hundredths 5 + 0 = 5, tenths 7 + 2 = 9, ones 3 + 2 = 5.
6. Estimate then add 6.1 + 2.9.
Answer: Estimate about 6 + 3 = 9. Exact: 6.1 + 2.9 = 9.0, which matches the estimate.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with adding and subtracting to tenths, then move to hundredths and money problems.
Differentiation
- Give squared paper or a place-value grid so each digit sits in its own column and the point stays in line.
- Keep coins and notes out so a decimal sum is also a money count students can check by hand.
- Start with same-length decimals (both to tenths) before mixing tenths and hundredths that need padding.
- Pre-print the point and the zeros so a student writing 5 as 5.00 does not have to remember to add them.
- Move to three decimal places (thousandths) and to adding three decimals in one column.
- Solve multi-step money problems: buy several items, total them, then find the change.
- Insert a missing addend: 3.6 + ___ = 5.2, reasoning back from the total.
- Explore why 0.3 + 0.3 + 0.4 = 1 exactly, linking tenths back to one whole.
Assessment: exit ticket
A three-question exit ticket for the last five minutes. It samples adding with a trade, subtracting with a borrow and padding, and a money check with estimation.
1. Add 5.6 + 2.7.
Answer: 8.3. Tenths 6 + 7 = 13, write 3 carry 1; ones 5 + 2 + 1 = 8.
2. Subtract 8.4 minus 3.65.
Answer: 4.75. Pad to 8.40: hundredths 0 minus 5 borrows to 10 minus 5 = 5; tenths 3 minus 6 borrows to 13 minus 6 = 7; ones 7 minus 3 = 4.
3. A book is $6.50 and a pen is $1.75. Estimate, then find the exact total.
Answer: Estimate about $7 + $2 = $9. Exact: 6.50 + 1.75 = $8.25, close to the estimate.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 lining up the point (section 1), Lesson 2 adding with a trade (section 2), Lesson 3 the number line (section 3), Lesson 4 subtracting with a borrow and money plus estimation (sections 4 and 5) and the exit ticket.
- Language to keep saying: line up the point, ten tenths make one whole, pad with a zero, estimate first. These four phrases pre-empt most of the misconceptions.
- Keep squared paper and coins out through the pictorial sections. Money is the model students already trust, so cash out an answer whenever a decimal sum feels abstract.
- The number-line diagram is marked in tenths, which Grade 5 students have already met, so the tick labels show every tenth. The bar models use one-decimal values so the part widths stay clear.
- Curriculum note and a US and AU alignment: both frameworks place adding and subtracting decimals at this stage. The US sits it in Grade 5 (5.NBT.B.7) resting on Grade 4 decimal notation, and ACARA places it at Year 5 (AC9M5N04) resting on Year 4 tenths and hundredths (AC9M4N01), so the US and Australian sequences line up closely here with no real divergence.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.