Powers of ten
Patterns when multiplying and dividing by 10, 100 and 1000, and writing powers of ten with exponents
About three lessons of 45 to 60 minutes
Why does multiplying by 10 just add a zero?
Every student learns the trick: multiply a whole number by 10, and you just add a zero on the end. 47 x 10 = 470. It feels like magic, but it is really place value in action -- multiplying by 10 shifts every digit one place to the left, into a spot worth ten times as much.
The same shifting pattern works for multiplying or dividing by 100, by 1000, and by decimals too, and it is exactly what the exponent in 10 to the power of 3 is counting: how many times you shift. Today you will find the pattern, use it to calculate instantly, and connect it to exponent notation.
- 4.7 x 10the decimal point shifts one place right: 47
- 4.7 x 100the decimal point shifts two places right: 470
- 350 divided by 10the decimal point shifts one place left: 35
- 10 x 10 x 10written as 10 to the power of 3, which equals 1,000
What students will be able to do
Students will explain the pattern in the number of zeros in a product when multiplying a whole number by a power of 10, explain the pattern in the placement of the decimal point when multiplying or dividing any number by a power of 10, and use whole-number exponents to write powers of 10.
- I can explain why multiplying by 10, 100 or 1000 shifts the decimal point right.
- I can explain why dividing by 10, 100 or 1000 shifts the decimal point left.
- I can multiply or divide a decimal by a power of ten mentally, without a written algorithm.
- I can write 10 x 10 x 10 as 10 to the power of 3, and know what the exponent counts.
- I can write a power of ten, such as 100,000, using exponent notation.
Standards this unit teaches
- 5.NBT.A.2Common Core (US)Powers of ten
Explain patterns when multiplying or dividing by powers of ten and use whole number exponents to write them.
- AC9M5N06Australian Curriculum v9 (ACARA)Multiply and divide decimals by powers of 10 (Year 5)
Multiply and divide decimals by multiples of powers of ten without a calculator, using place value and multiplication facts.
- AC9M7N03Australian Curriculum v9 (ACARA)Expanded notation with powers of 10 (Year 7)
Represent whole numbers in expanded form using place value and powers of ten. Australia's formal exponent-notation descriptor for powers of ten sits at Year 7, so the exponent-writing half of this unit runs about two years ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Power of ten
- a number made by multiplying 10 by itself a whole number of times, such as 10, 100, or 1,000
- Exponent
- the small raised number that shows how many times the base is multiplied by itself, such as the 3 in 10 to the power of 3
- Base
- the number being multiplied by itself in an exponent expression; the base in 10 to the power of 3 is 10
- Decimal point
- the dot separating the whole-number part of a number from its fraction part
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. Multiplying by 10 shifts every digit left
ConcreteBuild 47 on a place-value chart, then multiply by 10. Every digit moves one place to the left: the 4, which was worth 4 tens, becomes worth 4 hundreds, and the 7, which was worth 7 ones, becomes worth 7 tens. A new 0 fills the empty ones place. That is why 47 x 10 = 470 -- not magic, just every digit becoming worth ten times as much.
The same shifting explains decimals. 4.7 x 10: the 4 shifts from ones to tens, the 7 shifts from tenths to ones, so 4.7 x 10 = 47. The decimal point looks like it 'moved right', but really the digits shifted left past a fixed decimal point.
Find 6.2 x 100.
- Multiplying by 100 is multiplying by 10 twice, so shift every digit two places to the left.
- The 6 shifts from ones to hundreds. The 2 shifts from tenths to tens.
- 6.2 becomes 620.
Answer: 6.2 x 100 = 620.
- When you multiply by 10, does every digit become worth more or less?
- How many places do digits shift when multiplying by 100?
2. Dividing by a power of ten shifts right
PictorialDividing works the same way in reverse: every digit shifts to the right, becoming worth one tenth as much, once for each power of ten you divide by. 350 divided by 10: the 3 shifts from hundreds to tens, the 5 shifts from tens to ones, the 0 shifts off the end, giving 35.
Divide by 1,000 (three powers of ten) and every digit shifts three places right. 350 divided by 1,000: 350 becomes 0.350, or 0.35, because the digits shift all the way past the ones place into the decimal places.
Find 81 divided by 1,000.
- Dividing by 1,000 shifts every digit three places to the right.
- 81 is 8 tens and 1 one. Shifting three places right: the 8 lands in the hundredths place, the 1 in the thousandths place.
- Fill the empty ones and tenths places with 0.
Answer: 81 divided by 1,000 = 0.081.
- When you divide by a power of ten, does every digit become worth more or less?
- Why does 81 divided by 1,000 need two zeros written after the decimal point?
3. Writing powers of ten with exponents
Abstract10 x 10 x 10 is a mouthful to say and write every time, so mathematicians write it as 10 to the power of 3 (10 cubed), where the exponent, 3, counts how many 10s are multiplied together. 10 to the power of 3 equals 1,000 -- and notice the exponent also matches the number of zeros and the number of place-value shifts.
That connection is the whole point of this standard: multiplying by 10 to the power of n shifts every digit n places to the right (bigger), and dividing by 10 to the power of n shifts every digit n places to the left (smaller). Once the exponent is known, the shift is known.
Write 100,000 using an exponent, and find 3.05 x 10 to the power of 2.
- Count the zeros in 100,000: there are 5, so 100,000 = 10 to the power of 5.
- 10 to the power of 2 means shift two places: 3.05 x 10 to the power of 2 shifts the digits two places right.
- 3.05 becomes 305.
Answer: 100,000 = 10 to the power of 5. 3.05 x 10 to the power of 2 = 305.
- What does the exponent in 10 to the power of 4 count?
- How many zeros does 10 to the power of 6 have, and how do you know without multiplying it all out?
Common misconceptions and how to address them
MisconceptionMultiplying by 10 always just means adding a zero on the end, even for decimals.
Why it happens: The whole-number shortcut ('add a zero') is taught early and gets over-applied once decimals are involved.
How to address it: The real rule is 'shift every digit one place left', which happens to look like adding a zero for whole numbers because a new ones-place zero appears. For 4.7 x 10, the shift gives 47, not 4.70 with an added zero on the end -- show both side by side.
MisconceptionDividing by 10 removes a zero from the end, so 81 divided by 10 removes the (nonexistent) zero and stays 81.
Why it happens: Students mirror the 'add a zero' shortcut in reverse without checking whether there actually is a trailing zero to remove.
How to address it: The real rule is 'shift every digit one place right'. 81 divided by 10 shifts to 8.1, whether or not the original number happened to end in zero. Practise on numbers with and without trailing zeros to break the shortcut's false pattern.
Misconception10 to the power of 3 means 10 x 3, not 10 x 10 x 10.
Why it happens: The small raised exponent looks like it should mean 'multiply the base by the exponent', mirroring how multiplication itself is written.
How to address it: An exponent counts repeated multiplication by the same base, not a single multiplication by the exponent. Write out 10 to the power of 3 as 10 x 10 x 10 = 1,000 every time until the notation is trusted, and contrast it with 10 x 3 = 30 to show they are very different.
Guided practice (with answers)
1. Find 6.2 x 100.
Answer: 620. Shift every digit two places left.
2. Find 81 divided by 1,000.
Answer: 0.081. Shift every digit three places right.
3. Write 100,000 using an exponent.
Answer: 10 to the power of 5, since 100,000 has 5 zeros.
4. Find 3.05 x 10 to the power of 2.
Answer: 305. The exponent 2 means shift two places left.
5. Find 940 divided by 10 to the power of 2.
Answer: 9.4. The exponent 2 means shift two places right.
Independent practice worksheets
Set the matching ChalkBee decimals and place-value worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with whole-number powers of ten, then move to decimal shifts.
Differentiation
- Use a physical place-value chart with movable digit cards so the shift can be shown by hand, not just described.
- Keep early practice to whole numbers before introducing decimal shifts, then to shifts that stay within the decimal places (avoid crossing the decimal point) before mixing both.
- Say the shift out loud every time: 'multiply by 10 to the power of 2, shift 2 places left' as a fixed sentence.
- Provide a written list of powers of ten with their exponents (10 to the power of 1 is 10, to the power of 2 is 100, to the power of 3 is 1,000) as a reference.
- Multiply or divide by powers of ten larger than 1,000, such as 10 to the power of 5.
- Explore what happens when you multiply by a power of ten and then divide by a different power of ten in the same problem, predicting the net shift.
- Write a number in scientific-notation style (a single digit times a power of ten) as an early preview of later grades.
- Investigate the pattern in the number of zeros when two powers of ten are multiplied together, such as 10 to the power of 2 times 10 to the power of 3.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling multiplying, dividing, and exponent notation.
1. Find 5.6 x 1,000.
Answer: 5,600. Shift every digit three places left.
2. Find 720 divided by 100.
Answer: 7.2. Shift every digit two places right.
3. Write 10 x 10 x 10 x 10 using an exponent, and find its value.
Answer: 10 to the power of 4, which equals 10,000.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 multiplying and the leftward shift (section 1), Lesson 2 dividing and the rightward shift (section 2), Lesson 3 exponent notation plus the exit ticket (section 3 and assessment).
- Language to keep saying: digits shift, they do not move the decimal point on their own; the exponent counts how many 10s are multiplied together, not a multiplication by the exponent. These target the three main misconceptions.
- This unit assumes secure decimal place-value naming (tenths, hundredths, thousandths) from the Grade 5 decimals unit; if a student cannot yet read 4.7 as 4 ones and 7 tenths, revisit that unit first.
- Curriculum note: ACARA v9 covers multiplying and dividing decimals by powers of ten at Year 5 (AC9M5N06), matching the shifting-pattern half of this unit closely. The formal exponent-notation half is placed later in Australia, at Year 7 (AC9M7N03), so that half of this unit runs about two years ahead of the ACARA placement.
- Present mode and print both work: use Present to shift digit cards on a projected place-value chart, then print for independent shifting and exponent practice.