Chapter 2 in math has been all about place value--and we are continuing our work on place value with decimals. Our goals for this chapter are to:
- Read, write, and model decimals
- Represent decimals and relate them to fractions
- Explain whether two decimals are equivalent
- Interpret rounded decimals, and round decimals to the nearest tenth or the nearest hundredth
- Compare and order decimals to decimals thousandths
Examples
Goal 1: Read, write, and model decimals
When we read and write numbers, we do not use the word "and" unless we have a decimal in the number. Most people, if they see the number 375, would say "three hundred and seventy-five", but this is incorrect. See a few correct examples below:
a) 123.745 = one hundred twenty-three and seven hundred fourty-five
b) 52.098 = fifty-two and ninety-eight thousandths
c) 0.32 = thirty-two hundredths
When we see these numbers, we should also be able to say, "the 2 in example (a) is in the tens place, and has a value of 20. The 7 in example (a) is in the tenths place and has a value of 0.7."
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Goal 2: Represent decimals and relate them to fractions
When we look at a thousandths grid (as seen below),
When we read and write numbers, we do not use the word "and" unless we have a decimal in the number. Most people, if they see the number 375, would say "three hundred and seventy-five", but this is incorrect. See a few correct examples below:
a) 123.745 = one hundred twenty-three and seven hundred fourty-five
b) 52.098 = fifty-two and ninety-eight thousandths
c) 0.32 = thirty-two hundredths
When we see these numbers, we should also be able to say, "the 2 in example (a) is in the tens place, and has a value of 20. The 7 in example (a) is in the tenths place and has a value of 0.7."
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Goal 2: Represent decimals and relate them to fractions
When we look at a thousandths grid (as seen below),
- Each column/row is worth one tenth, or 1/10, or 0.1, because there are ten equal columns/rows in the grid.
- Each square is worth one hundredth, or 1/100, or 0.01, because there are 100 equal squares in the grid.
- Each rectangle is worth one thousandth, or 1/1000, or 0.001, because there are 1000 equal squares in the grid.
Students should be able to see a decimal, such as 0.398, or 0.4, and be able to model the number on a thousandths grid, and write the decimal as a fraction (and vice versa)
(a) 0.398 = 398/1000
(b) 0.4 = 4/10 = 40/100 = 400/1000
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Goal 3: Explain whether two decimals are equivalent
When we have to explain equivalency, the best way to do it is through modeling our numbers on either a place value chart, or grids. We then check our explanation using the following communication checklist:
Here is a poor explanation:
"I can model 0.5 and 0.50 on a place value chart. They are equivalent."
Here is an improved explanation:
"I can model 0.5 and 0.50 on a place value chart.
(a) 0.398 = 398/1000
(b) 0.4 = 4/10 = 40/100 = 400/1000
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Goal 3: Explain whether two decimals are equivalent
When we have to explain equivalency, the best way to do it is through modeling our numbers on either a place value chart, or grids. We then check our explanation using the following communication checklist:
- Did you use math language?
- Did you include the right amount of detail? (In general, this means a sentence explaining what each picture is, and a conclusion sentence as to how the pictures prove or disprove equivalency)
- Did you include a diagram?
Here is a poor explanation:
"I can model 0.5 and 0.50 on a place value chart. They are equivalent."
Here is an improved explanation:
"I can model 0.5 and 0.50 on a place value chart.
Both numbers have 0 ones, and five tenths. The blank spot in the hundredths box for the first number has a value of 0 hundredths, which is the same as the second number. Therefore 5 tenths = fifty hundreds, or 0.5 = 0.50."
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Goal 4: Interpret rounded decimals, and round decimals to the nearest tenth or the nearest hundredth
When rounding, look to the digit on the right of the place you need to round to. If it is a 5 or higher, round the digit up; if it is below 5 keep the digit the same that it is.
Round the following numbers to the nearest hundredth and tenth:
(a) 0.396 = 0.40 (hundredth) = 0.4 (tenth)
(b) 0.527 = 0.53 (hundredth) = 0.5 (tenth)
(c) = 0.091 = 0.09 (hundredth) = 0.1 (tenth)
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Goal 5: Compare and order decimals to decimals thousandths
A question you might see is:
Place the following numbers in order of value, least to greatest:
0.492, 0.239, 1.239, 0.4
If you do not see the order right away, place the numbers on a place value chart and compare each column, starting on the left. The number with the largest digit in the ones column will be the greatest number. If you have numbers that are equal in a certain column (for example 0.492 and 0.4 both have a four in the tenths column) then look at the column to the right of it and compare the digits. In the example given, 0.492 has a 9 in the hundredths column, and 0.4 has an "imaginary" zero in the hundredths, and therefore 0.492 is the larger number.
The order of the above numbers from least to greatest value is:
0.239, 0.4, 0.492, 1.239.
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Goal 4: Interpret rounded decimals, and round decimals to the nearest tenth or the nearest hundredth
When rounding, look to the digit on the right of the place you need to round to. If it is a 5 or higher, round the digit up; if it is below 5 keep the digit the same that it is.
Round the following numbers to the nearest hundredth and tenth:
(a) 0.396 = 0.40 (hundredth) = 0.4 (tenth)
(b) 0.527 = 0.53 (hundredth) = 0.5 (tenth)
(c) = 0.091 = 0.09 (hundredth) = 0.1 (tenth)
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Goal 5: Compare and order decimals to decimals thousandths
A question you might see is:
Place the following numbers in order of value, least to greatest:
0.492, 0.239, 1.239, 0.4
If you do not see the order right away, place the numbers on a place value chart and compare each column, starting on the left. The number with the largest digit in the ones column will be the greatest number. If you have numbers that are equal in a certain column (for example 0.492 and 0.4 both have a four in the tenths column) then look at the column to the right of it and compare the digits. In the example given, 0.492 has a 9 in the hundredths column, and 0.4 has an "imaginary" zero in the hundredths, and therefore 0.492 is the larger number.
The order of the above numbers from least to greatest value is:
0.239, 0.4, 0.492, 1.239.