The Staircase Continues Below One
What makes a tenth different from a hundredth?
The story unfolds
The mathematics beneath the story
Extend the tenfold relationship through decimals to thousandths. Compare using common place-value units; rename trailing zeros without changing value. Multiplying by a power of ten changes the value of each digit relative to the fixed ones place.
A learning breakpoint to notice
Longer numerals are assumed to be larger, or decimals are read as unrelated whole-number strings. Fix the whole, name the units, and connect fractions with place-value notation. A trailing zero differs from a zero inserted between digits.
See the structure
Teacher investigation
Compare 0.6 and 0.58 with a common one-meter whole. Rename 0.6 as 60 hundredths and locate both on a number line.
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
0.6 = 6/10 = 60/100 = 0.60. Therefore 0.60 > 0.58, since sixty hundredths exceeds fifty-eight hundredths. Also 3.47 × 10 = 34.7: three ones become three tens, four tenths become four ones, and seven hundredths become seven tenths.
Compare approaches
A hundred-grid shows equal areas of one fixed whole; a number line shows magnitude; expanded form names the units. These should agree. “Move the decimal” describes a notation shortcut but does not explain the scaling.
Analyze an incorrect response
“0.58 is greater than 0.6 because 58 is greater than 6.” The digits count different units. Compare 58 hundredths with 60 hundredths instead.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
2.46 = 2 ones + 4 __ + 6 __. Dividing by 10 gives __.
Check after trying
tenths; hundredths; 0.246. Each digit’s value becomes one tenth as large.
Transfer independently
A ribbon is 0.75 meter long. Another is 0.705 meter. Which is longer, by how much, and how do you know using a common unit?
Compare your reasoning
0.750 > 0.705; the first is longer by 0.045 meter, or 45 thousandths of a meter.
Pause and reflect
Which unit must students keep track of? What would an accurate answer conceal? Choose a different context or representation to test whether the idea transfers.
Bring it to your classroom
Grade 5 · Printable student practice and a separate teacher key. Use the developmental guidance in the key to select tasks.