The Tenfold Staircase
How does the same digit change value from place to place?
The story unfolds
The mathematics beneath the story
Extend whole-number place value, compare values by the highest differing place, and use benchmarks for rounding. Grade 4 makes the tenfold relationship explicit. Expanded forms connect a numeral with the quantities it represents.
A learning breakpoint to notice
A digit and its value can be confused, and zero may be treated as dispensable. Ask what unit each digit counts. Rounding is a nearest-benchmark question, not simply a last-digit rule.
See the structure
| Thousands | Hundreds | Tens | Ones |
|---|---|---|---|
| 4 | 4 | 4 | 4 |
| 4,000 | 400 | 40 | 4 |
Each value is ten times the next value to its right.
Teacher investigation
Compare the values of 4 in 4,444. How many times as large is each occurrence as the next to its right? Build an expanded form.
Try it before revealing the solution. Name every quantity and unit. Accept several valid representations.
Study a completed solution
4,444 = 4,000 + 400 + 40 + 4. The 4 hundreds represent ten times the amount of the 4 tens: 400 = 10 × 40. In 4,040, the zero hundreds and zero ones preserve the positions of the other digits.
Compare approaches
A place-value chart organizes units; expanded form records their sum. To compare 4,305 and 4,350, equal thousands and hundreds shift attention to the tens: zero tens is less than five tens.
Analyze an incorrect response
“The 7 in 7,060 has value 7.” Its digit is 7, but its place value is 7,000. “Removing the zero makes the same number” also fails: 7,060 is not 706.
Identify the first invalid inference. Ask what evidence would help the learner revise it.
Complete the reasoning
In 6,060, the left 6 has value __ and the other 6 has value __. The first value is __ times the second.
Check after trying
6,000; 60; 100. There are two tenfold steps between their places.
Transfer independently
A count is 3,498 items. Locate the nearest hundreds, then round to the nearest hundred. Explain the decision using distance.
Compare your reasoning
Between 3,400 and 3,500; distances 98 and 2. It rounds to 3,500.
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
Grades 3–4 · Printable student practice and a separate teacher key. Use the developmental guidance in the key to select tasks.