This Warm-up draws students’ attention to the meaning of zeros in a decimal, in particular, whether and when a zero affects the value of the decimal.
Students first think about zeros in the context of addition. When adding the thousandths in 1.009+0.391, students may write 0.010 or 0.01 for the sum of 0.009 and 0.001. When recording the sum of 1.009 and 0.391, they may write 1.400, 1.40, or 1.4, depending on whether they think of 10 thousandths as 1 hundredth and 10 hundredths as 1 tenth. Then students reason about whether two decimals with different numbers of digits—one with more zeros than the other—could have the same value.
As they reason about the place values of the digits and the meaning of zeros in decimals, students practice looking for and making use of structure (MP7).
Arrange students in groups of 2. Give students 1 minute of quiet time to mentally add the decimals in the first problem and another minute to discuss their answer and strategy with a partner. Then ask students to pause and write down the sum. Poll the class on the decimal that they wrote: 1.4, 1.40, or 1.400. Then ask students to complete the rest of the Warm-up.
Find the value mentally: 1.009+0.391
Decide if each statement is true or false. Be prepared to explain your reasoning.
34.56000=34.56
25=25.0
2.405=2.45
Invite students to share whether they think each statement in the last problem is true or false, and ask for an explanation for each. Students may simply say that we can or cannot just remove the zeros. Encourage them to use what they know about place values or comparison strategies to explain why one number is or is not equal to the other.
If not mentioned in students’ explanations, highlight, for instance, that 34.560 (thirty-four and five hundred sixty thousandths) is equal to 34.56 (thirty-four and 56 hundredths) because in both numbers, there are 3 tens, 4 ones, 5 tenths, 6 hundredths, and no thousandths, ten-thousandths, or hundred-thousandths.
If time permits, use Critique, Correct, Clarify to give students an opportunity to improve a sample written claim about the zeros in a decimal by correcting errors, clarifying meaning, and adding details.
If not illustrated in students’ revised statements, consider presenting an example that can counter the claim in the first draft. For instance, 12.90 is equal to 12.9, but 12.09 is not equal to 12.9.
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This Warm-up draws students’ attention to the meaning of zeros in a decimal, in particular, whether and when a zero affects the value of the decimal.
Students first think about zeros in the context of addition. When adding the thousandths in 1.009+0.391, students may write 0.010 or 0.01 for the sum of 0.009 and 0.001. When recording the sum of 1.009 and 0.391, they may write 1.400, 1.40, or 1.4, depending on whether they think of 10 thousandths as 1 hundredth and 10 hundredths as 1 tenth. Then students reason about whether two decimals with different numbers of digits—one with more zeros than the other—could have the same value.
As they reason about the place values of the digits and the meaning of zeros in decimals, students practice looking for and making use of structure (MP7).
Arrange students in groups of 2. Give students 1 minute of quiet time to mentally add the decimals in the first problem and another minute to discuss their answer and strategy with a partner. Then ask students to pause and write down the sum. Poll the class on the decimal that they wrote: 1.4, 1.40, or 1.400. Then ask students to complete the rest of the Warm-up.
Find the value mentally: 1.009+0.391
Decide if each statement is true or false. Be prepared to explain your reasoning.
34.56000=34.56
25=25.0
2.405=2.45
Invite students to share whether they think each statement in the last problem is true or false, and ask for an explanation for each. Students may simply say that we can or cannot just remove the zeros. Encourage them to use what they know about place values or comparison strategies to explain why one number is or is not equal to the other.
If not mentioned in students’ explanations, highlight, for instance, that 34.560 (thirty-four and five hundred sixty thousandths) is equal to 34.56 (thirty-four and 56 hundredths) because in both numbers, there are 3 tens, 4 ones, 5 tenths, 6 hundredths, and no thousandths, ten-thousandths, or hundred-thousandths.
If time permits, use Critique, Correct, Clarify to give students an opportunity to improve a sample written claim about the zeros in a decimal by correcting errors, clarifying meaning, and adding details.
If not illustrated in students’ revised statements, consider presenting an example that can counter the claim in the first draft. For instance, 12.90 is equal to 12.9, but 12.09 is not equal to 12.9.