Question

solve the problem. use what you learned from the example.

stefan was finding the quotients of decimals and powers of ten.

here is what stefan wrote for two problems:

(0.2 \div 100 = 20) \t (0.02 \div 10 = 0.2)

what did stefan do correctly? what did he do wrong? what can to help him divide correctly with powers of ten the next time?

Explanation:

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<reasoning>

Analyze Stefan's calculations

Using the Dividing Decimals by Powers of Ten knowledge point
\[

$$\begin{aligned} &0.2 \div 100 = 0.002 \quad (\text{Stefan wrote } 20) \\ &0.02 \div 10 = 0.002 \quad (\text{Stefan wrote } 0.2) \end{aligned}$$

\]

Identify what Stefan did correctly

Stefan correctly recognized that dividing by powers of ten changes the place value of the digits by shifting positions. In both of his answers, the non-zero digit remains a \(2\), showing he kept the correct base digit.

Identify what Stefan did wrong

Stefan shifted the decimal point in the wrong direction. Instead of moving the decimal point to the left to make the numbers smaller, he moved it to the right, which multiplied the numbers by powers of ten instead of dividing them.

Formulate advice for next time

When dividing a decimal by a power of ten, the quotient must be smaller than the dividend. He should move the decimal point to the left by the same number of places as there are zeros in the power of ten.
</reasoning>

<answer>

• What Stefan did correctly: He kept the correct non-zero digit (\(2\)) in both of his answers, showing he understood that the digits themselves do not change when dividing by powers of ten.
• What Stefan did wrong: He shifted the decimal point to the right instead of to the left. This resulted in multiplication rather than division, making his answers larger instead of smaller.
• Advice for next time: When dividing by a power of ten, the value must get smaller. Count the number of zeros in the divisor (e.g., two zeros in \(100\), one zero in \(10\)) and move the decimal point that many places to the left, using placeholder zeros as needed.

</answer>

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Answer:

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<reasoning>

Analyze Stefan's calculations

Using the Dividing Decimals by Powers of Ten knowledge point
\[

$$\begin{aligned} &0.2 \div 100 = 0.002 \quad (\text{Stefan wrote } 20) \\ &0.02 \div 10 = 0.002 \quad (\text{Stefan wrote } 0.2) \end{aligned}$$

\]

Identify what Stefan did correctly

Stefan correctly recognized that dividing by powers of ten changes the place value of the digits by shifting positions. In both of his answers, the non-zero digit remains a \(2\), showing he kept the correct base digit.

Identify what Stefan did wrong

Stefan shifted the decimal point in the wrong direction. Instead of moving the decimal point to the left to make the numbers smaller, he moved it to the right, which multiplied the numbers by powers of ten instead of dividing them.

Formulate advice for next time

When dividing a decimal by a power of ten, the quotient must be smaller than the dividend. He should move the decimal point to the left by the same number of places as there are zeros in the power of ten.
</reasoning>

<answer>

• What Stefan did correctly: He kept the correct non-zero digit (\(2\)) in both of his answers, showing he understood that the digits themselves do not change when dividing by powers of ten.
• What Stefan did wrong: He shifted the decimal point to the right instead of to the left. This resulted in multiplication rather than division, making his answers larger instead of smaller.
• Advice for next time: When dividing by a power of ten, the value must get smaller. Count the number of zeros in the divisor (e.g., two zeros in \(100\), one zero in \(10\)) and move the decimal point that many places to the left, using placeholder zeros as needed.

</answer>

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