Place value with decimals in 5th grade

The correct statement is A: the value of the 4 in the hundredths place is $0.01$ times the value of the 4 in the ones place, since $0.01 \times 4 = 0.04$.

1. 1In $4.24$, the first 4 sits in the ones place with value $4$. The second 4 sits in the hundredths place with value $0.04$.
2. 2The hundredths place is two places to the right of the ones place, so its value is $\frac{1}{100}$ as large — that is, $0.01$ times the ones value.
3. 3Choice A says the hundredths value is $0.01$ times the ones value, which matches. Choice B uses $0.1$, which would only account for a 1-place gap, not a 2-place gap. Choose A.

Common mistake. A common mistake is using $0.1$ instead of $0.01$, as if the hundredths place were only one step to the right of the ones place rather than two.

The idea underneath. Moving one place to the right makes a digit's value $\frac{1}{10}$ as large, so a 2-place rightward move makes it $0.01$ as large.

Answer. 3.3

Type a $3$ into the Ones cell. The chart reads $3.3$ (or $X3.3$ if you also fill the Tens cell with a digit $X$ from $1$ to $9$). In every case the second $3$ in Ones is worth $3$, which is $10$ times the first $3$ in Tenths ($0.3$).

1. 1The given 3 sits in the tenths place, so its value is $3 \times 0.1 = 0.3$.
2. 2A digit whose value is $10$ times $0.3$ must have value $3$. Moving one place to the left of the tenths cell takes the place value from $0.1$ to $1$, so a digit there is worth $10$ times what it would be worth in tenths.
3. 3Type a $3$ into the Ones cell. The chart now reads $3.3$ — the $3$ in Ones has value $3$ and the $3$ in Tenths has value $0.3$. The Tens cell can stay blank (chart reads $3.3$) or hold any digit $1$–$9$ you choose; for example, Tens $= 4$ makes the chart read $43.3$. Either way, the second $3$ in Ones is still $10$ times the first $3$ in Tenths.

Common mistake. A common mistake is placing the second $3$ in the Tens cell (and leaving Ones blank), thinking the next column to the left of Tenths is the $10$-times target. Tens is actually $100$ times Tenths, and leaving Ones blank creates a gap so the chart returns no number at all.

The idea underneath. Moving a digit one place to the left multiplies its value by $10$; moving it one place to the right divides its value by $10$.

The true statement is that moving the 7 one place to the left makes its value 10 times as much.

1. 1In 35.72, the 7 is in the tenths place, so its value is 0.7.
2. 2Moving one place left from tenths crosses the decimal point and puts the 7 in the ones place. Then its value is 7.
3. 3Compare the new value to the original value.
4. 4So the true statement is that the 7 would become 10 times as much.

Common mistake. A common mistake is thinking the value stays 7 because the digit stays 7. In 35.72, the 7 is worth 0.7, and moving it to the ones place makes it worth 7.

The idea underneath. A digit's value depends on its place, and each place to the left is worth 10 times the place to its right.

Liam's work is correct because the second 3 is one place to the left, so it is 10 times as much as the first 3.

1. 1Look at the chart. The first 3 is in the tenths place, so it represents 3 tenths, or 0.3.
2. 2The second 3 is in the ones place, so it represents 3 ones, or 3. The ones place is one place to the left of the tenths place.
3. 3Moving one place to the left makes a digit 10 times as much.
4. 4So the second 3 is 10 times as much as the first 3. The chart matches Liam's goal.

Common mistake. A common mistake is reversing the left-right rule and thinking that moving one place to the left makes a digit one-tenth as much. Moving left always multiplies a digit's value by 10.

The idea underneath. A digit's value depends on its place. Each move one place to the left makes it 10 times as much, and each move one place to the right makes it one-tenth as much.