Each place value column is a power of 10: the thousands column is 10³, the ones column is 10⁰ = 1. So 4052 = 4 × 10³ + 5 × 10¹ + 2 × 10⁰.
The 4 in 4052 is worth 4000, but the 5 is worth only 50. What does the column a digit sits in do to its value?
You already know that the 5 in 4052 is worth 50, not 5. A number is built from its digits, each worth digit × its column. Every column is ten times the one to its right, so every column is a power of 10:
Column
ones
tens
hundreds
thousands
ten thousands
Power of 10
100
101
102
103
104
Value
1
10
100
1000
10 000
The exponent counts the zeros in the value: 103=1000 has three zeros. Writing a number as a sum of digit × power of 10 is called expanded notation.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
How much is 10³?
How many zeros does 106 have?
TryTry it on the board and compare the result with your prediction.
Build 4052
Each column's number is its digit times its power of 10.
Try it yourselfYour edits are saved in this browser.
Build 4052: 4 thousands, no hundreds, 5 tens and 2 ones. Which power of 10 goes with the 5?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Columns and powers
The board shows the columns. Here is 4052 written out.
Worked exampleBefore each step, say what you think comes next.
4052=4000+0+50+2Each digit is worth its digit times its column.
4000=4×1000=4×103The thousands column is 10³.
50=5×10=5×101The tens column is 10¹.
2=2×1=2×100The ones column is 10⁰.
4052=4×103+5×101+2×100The 0 in the hundreds adds nothing, so we leave it out.
Now try one: which power goes with the 5?
Count the columns from the ones, which is 0.
In 4052, the 5 is in the tens column. What is the exponent of its power of 10? (The 5 is worth 5×10?.)
Sam writes 4052 in expanded notation. Click the line where the working first goes wrong.
ModifyChange the problem on purpose and see what happens.
Build, read, convert
Try it yourselfYour edits are saved in this browser.
Build a number with a 6 in the ten thousands column and a 9 in the ones column, and zeros between.
Use the board. This ticks when you get there.
Write 7×104+2×102+5 as a number.
In 380 000, the 8 is worth 8×10?. What is the exponent?
Build 345, then press × 10 twice. What number do you get, and what happened to each digit?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Expanded notation
What is the value of the 2 in 52 300?
Write 6×105+4×102 as a number.
Review: what is 25?
Write 3 000 000 as 3×10?. What is the exponent?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Three terms
Then read your number out in expanded notation.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The top power is 10⁴, so the first digit is in the ten thousands column.
Another hint
Three columns with digits and two with 0.
Build a number with three digits that are not 0, and a top power of 10⁴.
Use the board. This ticks when you get there.
For teachers
Students build numbers on place value columns to see that each column is a power of 10, write numbers in expanded notation with exponents, and convert back (AC9M7N03, nodes 1 and 2). About 40 minutes.
I can say which power of 10 each place value column is.
I can write a number in expanded notation using powers of 10.
I can turn expanded notation back into a number.
Before class
Do the Run board yourself first. Students need exponent notation (Powers lesson) and place value for whole numbers (Year 6).
The one idea: each column is ten times the one to its right, so the exponent is the column's position counting from 0 at the ones. The common mistake is counting from 1 (Sam), so every exponent is one too big. The empty column (a 0 digit) may be left out of expanded notation; writing 0 × 10² is also correct, and this lesson's answers leave it out.
Two nodes in one lesson: the columns and their powers (Run, Investigate), then expanded notation both ways (Modify, Practise).
Running the lesson
Predict: students who chose 30 read 10³ as 10 × 3.
Run: ask where the 0 in 4052 went when we write the expanded notation.
Investigate: have students predict the value of 10⁰ before the question, from the pattern 1000, 100, 10.
Modify: ask what × 10 does to every exponent.
Make: collect numbers and swap: partners write the expanded notation.
The hard part (on purpose)
Starting the exponent count at the ones column as 0, not 1. Have students say 'ones is ten to the nothing' once, and point at the column while they write the exponent.
When students get stuck
A student writes 5 × 10² for the 5 in the tens (Sam's mistake). Ask: What power of 10 is the ones column? Fix: Count from 0 at the ones: ones 0, tens 1, hundreds 2.
A student writes 10³ = 30. Ask: How many 10s are multiplied? Fix: Write 10 × 10 × 10 the long way, then count the zeros.
A student writes 4 × 10³ + 0 × 10² + 5 × 10¹ + 2 × 10⁰ and asks if they can drop the 0 term. Ask: What does 0 × 100 add? Fix: Nothing, so it can be left out. Both forms are correct.
Stretch and support
Stretch: What would the column to the right of the ones be worth? Continue the pattern 1000, 100, 10, 1, … and write the next two columns as powers of 10 with negative exponents.
Support: Keep the long form (4 × 1000 + 5 × 10 + 2) beside the powers until the exponent is automatic.
Where it leads
Next: Rounding (to a given place) uses the same column names.
Where it leads: scientific notation in Year 9 and 10, and decimal place value, where the columns continue as 10⁻¹, 10⁻², ….
What each step is for
how much is 10³?: Committing to a value for 10³ shows whether they read the exponent as a multiplier (30) or as the number of zeros (1000).
build 4052: Raising the thousands digit to 4 and the tens digit to 5 shows each digit multiplied by its column's power of 10, and the empty hundreds column left as 0.
columns and powers: The exponent is the column's position counting from 0 at the ones; Sam counts from 1 and every power is one too big. 10⁰ = 1 is the ones column, one 10 to no power.
build, read, convert: Building a number with gaps, reading the value of a digit, converting expanded notation back, and shifting by ×10 or ×100 connect the exponent to the number of zeros.
expanded notation: The value of a digit, the number from its expanded notation, and why times 100 adds two zeros; the review keeps powers fresh.
three terms: Building any number with exactly three non-zero terms and a top power of 10⁴ shows that the top term decides the size and the zeros are columns.
Now you can
Say which power of 10 each place value column is.
Write a number in expanded notation using powers of 10.