Ten times and one tenth: place value
Read the value of a digit in places from 1000 to 0.001.
Begin with the activity ↓Does adding a zero always multiply by ten?
Multiplying by ten changes every place value by a factor of ten. Appending a zero to 0.4 gives the same value, 0.40.
Ten hundreds form one thousand, written 1000. In 3050 there are three thousands, no hundreds, five tens and no ones. Zero holds a place with no bundles.
Now look below one whole. One tenth is 0.1. Divide that piece into ten equal parts to get 0.01, and again to get 0.001, one thousandth.
The number 0.4 is four tenths. Ten times it is forty tenths, or 4. The decimal point is not magic: the place value represented by the digit 4 changes.
Ten times 3.05 is ten times 3 plus ten times 0.05: 30 plus 0.5, giving 30.5. Dividing 3.05 by ten gives 0.305. Now the thousandths place has a purpose.
It is tempting to say that 0.4 and 0.40 differ. But four tenths become forty hundredths when subdivided: the value stays fixed. Appending a zero to the whole number 4 is different from appending a trailing decimal zero.
Multiplying by 100 applies a factor of ten twice; 1000 applies it three times. Division reverses the process. Predict whether a positive value will grow or shrink, then check its places.
Listen one paragraph at a time
Hold on to this distinction
The calculation 0.4×10 gives 4, while 0.40 equals 0.4. Distinguish changing an amount by calculation from changing how the same amount is written.
Change it and check
Choose a number and a multiplication or division factor. Read the places before and after the calculation.
3.05 × 10 = 30.5
| Value | 10000 | 1000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 | 0.0001 |
|---|---|---|---|---|---|---|---|---|---|
| 3.05 | 0 | 0 | 0 | 0 | 3 | 0 | 5 | 0 | 0 |
| 30.5 | 0 | 0 | 0 | 3 | 0 | 5 | 0 | 0 | 0 |
Compare which place holds each digit in the two rows. Scaling by 0.1 divides by ten; scaling by 0.01 divides by one hundred.
This activity uses nonnegative numbers. Ten times zero is still zero. Terminating decimals are displayed without rounding.
Try with paper and objects
Draw columns for thousands through thousandths. Place digit cards for 3.05, then explain each card’s value after multiplying by ten.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Connect the answer to its unit
Return to the explanationCompare with a solution
0.204 · number only
The 2 becomes 0.2 and 0.04 becomes 0.004, giving 0.204.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Interpret decimal multiplication as repetition or a fraction of an amount.