Another Peek Beyond the Point · Lesson 1 of 13
A Quick Recap of Decimals
“Reconnect decimal digits with fractions and place value before multiplying and dividing them.”
• Read the value of each digit in a decimal number. • Write decimals in expanded form and as decimal fractions. • Convert simple gram measurements into kilograms. • Explain whole-number division by 10, 100, and 1000 using place value.
A Quick Recap of Decimals
A decimal is a way to write a quantity using place values smaller than one. The decimal point separates the ones place from the tenths place. Each move one place to the right makes the place value one-tenth of the previous value. So the sequence tens, ones, tenths, hundredths, and thousandths follows one connected pattern, rather than two unrelated systems.
For example, 27.53 means 2 tens, 7 ones, 5 tenths, and 3 hundredths. The digit 5 represents 5/10, not 5 whole units. The digit 3 represents 3/100. Saying the digits after the point individually can help you read the written number, but understanding their values is what lets you calculate with it.
Problem
Write 8.047 as a sum of its place values.
- 1.There are 8 ones, 0 tenths, 4 hundredths, and 7 thousandths.
- 2.So 8.047 = 8 + 0/10 + 4/100 + 7/1000 = 8 + 0.04 + 0.007.
- 3.The zero keeps the tenths place empty. Removing it to make 8.47 would change the number.
Decimals and their fraction forms
A decimal fraction has a denominator such as 10, 100, or 1000. Its denominator tells us which place to use for the final digit. For 254/1000, the 254 counts thousandths. Regrouping those thousandths gives 2 tenths, 5 hundredths, and 4 thousandths, so the decimal is 0.254. If there are too few numerator digits, use zeroes to keep the places correct.
A fraction with a denominator that is a power of ten, such as 10, 100, or 1000.
Problem
Express 254/1000 as a sum of decimal fractions and as a decimal.
- 1.Split the numerator: 254/1000 = 200/1000 + 50/1000 + 4/1000.
- 2.The first two parts simplify to 2/10 and 5/100, so the sum is 2/10 + 5/100 + 4/1000.
- 3.These are 0.2 + 0.05 + 0.004 = 0.254.
Trailing zeroes after the decimal point do not change a number: 0.5 = 0.50 = 0.500. They name the same amount in tenths, hundredths, or thousandths. Zeroes between the point and a non-zero digit do matter. For instance, 0.05 is one-tenth of 0.5. Keep this distinction in mind when later calculations require extra decimal places.
Use place value in measurement and division
One kilogram contains 1000 grams. Thus a gram measurement divided by 1000 gives the same quantity in kilograms. This is a change of unit, not a change in the amount of spice. Similarly, dividing a number by 10, 100, or 1000 makes every digit’s value one-tenth, one-hundredth, or one-thousandth as large.
Problem
Express 50 g of cinnamon, 100 g of cumin, 25 g of cardamom, and 250 g of pepper in kilograms.
- 1.Use 1000 g = 1 kg, so divide each gram measurement by 1000.
- 2.50/1000 = 0.050 = 0.05 kg; 100/1000 = 0.100 = 0.1 kg.
- 3.25/1000 = 0.025 kg; 250/1000 = 0.250 = 0.25 kg.
- 4.Each fraction and decimal describes the same amount using kilograms.
To calculate 123 ÷ 10, think of 123 ones becoming 123 tenths: 12 ones and 3 tenths, or 12.3. In 24 ÷ 100, 24 ones become 24 hundredths, or 0.24. The familiar instruction to move the decimal point left describes this change of digit values. It is a useful writing shortcut once the reason is understood.
Do not delete a zero that holds an empty place. 0.023 and 0.23 are different, even though 0.230 and 0.23 are equal.
Quiz
In 27.53, what is the value of the digit 3?
Which decimal equals 23/1000?
How many kilograms are 68 grams?
Which number has the same value as 0.5?
What is 12 ÷ 1000?
Practice Problems
- Write 46.208 in expanded form and identify the value of each digit.
- Convert 847/10000, 173/100, and 23/1000 into decimals.
- Explain the difference between the zeroes in 0.050 and 0.005.
- Express 125 g and 8 g in kilograms, as both fractions and decimals.
- Find 12345 ÷ 1000 and explain the resulting place values.
- Take turns naming a decimal fraction and its equivalent decimal. Include fractions whose numerator has fewer digits than the denominator has zeroes.
Key Takeaways
• Decimal places continue the same ten-to-one place-value pattern used by whole numbers. • Tenths, hundredths, and thousandths represent fractions of one. • Zero placeholders preserve the positions of other digits. • Trailing decimal zeroes do not change a number’s value. • Dividing grams by 1000 expresses the same quantity in kilograms.
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Decimal Multiplication