ParaPro Math Study Guide: Decimals Practice

Understanding Decimals

Decimals are a way of representing numbers that are not whole numbers. They use the base-10 system to show parts of a whole. As a paraprofessional, you’ll use decimals when helping students with money, measurements, and data analysis.

What is a Decimal?

A decimal number has two parts separated by a decimal point (.)

Whole number part: The digits to the left of the decimal point
Fractional part: The digits to the right of the decimal point

Example: 42.75

42 is the whole number part
75 is the fractional part
The decimal point (.) separates these two parts

Place Value in Decimals

Understanding place value is essential for working with decimals. Each position in a decimal number represents a power of 10.

Whole Number Part				Fractional Part
Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths	Ten-thousandths
100s	10s	1s	.	0.1s	0.01s	0.001s	0.0001s
1	4	5	.	2	8	7	3

Reading Decimals

When reading a decimal number:

Read the whole number part as you normally would
Say “and” or “point” for the decimal point
Read the fractional part as a whole number
End with the place value of the last digit

Example: 42.75 can be read as “forty-two and seventy-five hundredths” or “forty-two point seven five”

Converting Between Decimals and Fractions

Converting Decimals to Fractions

Write the digits from the decimal as the numerator
Write the place value of the last digit as the denominator (10, 100, 1000, etc.)
Simplify the fraction if possible

Example 1: Convert 0.25 to a fraction

Step 1: The numerator is 25

Step 2: The place value of the last digit is hundredths, so the denominator is 100

Step 3: Simplify the fraction 25/100 by dividing both numbers by their GCF (25)

Result: 0.25 = 25/100 = 1/4

Example 2: Convert 0.8 to a fraction

Step 1: The numerator is 8

Step 2: The place value of the last digit is tenths, so the denominator is 10

Step 3: Simplify the fraction 8/10 by dividing both numbers by their GCF (2)

Result: 0.8 = 8/10 = 4/5

Converting Fractions to Decimals

Divide the numerator by the denominator
Continue the division until the remainder is zero or a pattern repeats

Example 3: Convert 3/4 to a decimal

Divide 3 by 4: 3 ÷ 4 = 0.75

Therefore, 3/4 = 0.75

Example 4: Convert 2/3 to a decimal

Divide 2 by 3: 2 ÷ 3 = 0.6666…

The digits 6 repeat indefinitely, so we write 2/3 = 0.6̅ or 0.6 with a bar over the 6

Therefore, 2/3 = 0.6̅ = 0.666…

Types of Decimal Numbers

Terminating decimal: The digits end after a finite number of places (e.g., 0.75)
Repeating decimal: One or more digits repeat indefinitely (e.g., 0.333… or 0.272727…)

Comparing and Ordering Decimals

Steps for Comparing Decimals:

Compare the whole number parts first
If the whole number parts are equal, compare the tenths
If the tenths are equal, compare the hundredths, and so on
If needed, add zeros to the right of the decimal to make the numbers have the same length

Example 5: Compare 4.38 and 4.29

Step 1: The whole number parts are both 4, so we need to compare the fractional parts

Step 2: Compare the tenths: 3 > 2

Therefore, 4.38 > 4.29

Example 6: Compare 0.5 and 0.50

Step 1: Add a zero to 0.5 to make it 0.50

Step 2: Compare: 0.50 = 0.50

Therefore, 0.5 = 0.50 (they are equal)

Example 7: Order these decimals from least to greatest: 3.45, 3.4, 3.452, 3.5

Add zeros where needed: 3.45, 3.40, 3.452, 3.50

Compare place values and order: 3.40, 3.45, 3.452, 3.50

Final order: 3.4, 3.45, 3.452, 3.5

Rounding Decimals

Steps for Rounding Decimals:

Identify the place value to which you are rounding
Look at the digit to the right of that place
If that digit is less than 5, round down (keep the digit in the rounding place the same)
If that digit is 5 or greater, round up (increase the digit in the rounding place by 1)
Drop all digits to the right of the rounding place

Example 8: Round 3.748 to the nearest tenth

Step 1: The tenth place has a 7

Step 2: The digit to the right is 4

Step 3: Since 4 is less than 5, round down (keep 7)

Step 4: Drop all digits to the right

Result: 3.748 rounded to the nearest tenth is 3.7

Example 9: Round 5.685 to the nearest hundredth

Step 1: The hundredth place has an 8

Step 2: The digit to the right is 5

Step 3: Since 5 is 5 or greater, round up (8 becomes 9)

Step 4: Drop all digits to the right

Result: 5.685 rounded to the nearest hundredth is 5.69

Example 10: Round 25.097 to the nearest whole number

Step 1: The ones place has a 5

Step 2: The digit to the right is 0

Step 3: Since 0 is less than 5, round down (keep 5)

Step 4: Drop all digits to the right of the decimal point

Result: 25.097 rounded to the nearest whole number is 25

Operations with Decimals

Adding and Subtracting Decimals

Align the decimal points vertically
Add zeros if needed to make the numbers have the same length
Add or subtract as with whole numbers
Place the decimal point in the answer directly below the aligned decimal points

Example 11: Add 5.63 + 2.8

  5.63
+ 2.80  (added a zero for alignment)
------
  8.43

Example 12: Subtract 7.25 – 3.68

  7.25
- 3.68
------
  3.57

Multiplying Decimals

Multiply the numbers as if they were whole numbers (ignore the decimal points)
Count the total number of decimal places in both factors
Place the decimal point in the product so that it has that many decimal places
Add leading zeros if necessary

Example 13: Multiply 2.3 × 1.5

Step 1: Multiply 23 × 15 = 345

Step 2: Count decimal places: 2.3 has 1 place and 1.5 has 1 place, for a total of 2 places

Step 3: Place the decimal point in the product to give 2 decimal places: 3.45

Result: 2.3 × 1.5 = 3.45

Example 14: Multiply 0.04 × 0.7

Step 1: Multiply 4 × 7 = 28

Step 2: Count decimal places: 0.04 has 2 places and 0.7 has 1 place, for a total of 3 places

Step 3: Place the decimal point in the product to give 3 decimal places: 0.028

Result: 0.04 × 0.7 = 0.028

Dividing Decimals

Move the decimal point in the divisor to the right to make it a whole number
Move the decimal point in the dividend the same number of places to the right
Divide as with whole numbers
Place the decimal point in the quotient directly above where it is in the dividend

Example 15: Divide 8.4 ÷ 2.1

Step 1: Move the decimal point in 2.1 one place to the right to get 21

Step 2: Move the decimal point in 8.4 one place to the right to get 84

Step 3: Divide 84 ÷ 21 = 4

Result: 8.4 ÷ 2.1 = 4

Example 16: Divide 6.25 ÷ 0.5

Step 1: Move the decimal point in 0.5 one place to the right to get 5

Step 2: Move the decimal point in 6.25 one place to the right to get 62.5

Step 3: Divide 62.5 ÷ 5 = 12.5

Result: 6.25 ÷ 0.5 = 12.5

Decimal Applications in the Classroom

Scenario 1: Working with Money

A class is raising money for a field trip. They’ve raised $127.85 so far, and their goal is $350. How much more money do they need to reach their goal?

Step 1: Set up a subtraction problem: $350.00 – $127.85

Step 2: Subtract: $350.00 – $127.85 = $222.15

The class needs $222.15 more to reach their goal.

Scenario 2: Grade Calculations

A student has test scores of 85.5, 92.0, 78.5, and 89.0. What is the student’s average test score?

Step 1: Add the scores: 85.5 + 92.0 + 78.5 + 89.0 = 345.0

Step 2: Divide by the number of tests: 345.0 ÷ 4 = 86.25

The student’s average test score is 86.25.

Scenario 3: Measurement Conversions

A science experiment calls for 1.25 liters of water, but the measuring cup only shows milliliters. How many milliliters should be measured?

Step 1: Set up a multiplication problem: 1.25 liters × 1000 milliliters per liter

Step 2: Multiply: 1.25 × 1000 = 1250

The experiment requires 1250 milliliters of water.

Tips for Teaching Decimals

Use concrete models: Base-10 blocks, money, and measuring tools help students visualize decimals.
Connect to prior knowledge: Relate decimals to fractions, whole numbers, and real-world contexts.
Emphasize place value: Use place value charts to help students understand the meaning of each digit.
Use real-world examples: Money, measurements, and sports statistics are excellent contexts for decimals.
Focus on conceptual understanding: Help students understand why decimal procedures work rather than just memorizing rules.

Common Decimal Misconceptions and Errors

More digits means larger value: Students may think 0.75 is greater than 0.8 because 75 > 8.
Decimal point alignment: When adding or subtracting, students may fail to align decimal points.
Ignoring the decimal point: In multiplication and division, students may place the decimal point incorrectly or forget it altogether.
Rounding rules: Students may apply rounding rules inconsistently or incorrectly.
Thinking 0.1 is the same as 1/10: While this is true, students may not connect decimal and fraction representations.

Key Points to Remember

Decimals are a way to represent parts of a whole using the base-10 system.
The place value of each digit determines its value in the number.
When adding or subtracting decimals, always align the decimal points.
When multiplying decimals, the number of decimal places in the product equals the sum of the decimal places in the factors.
When dividing by decimals, move the decimal point to make the divisor a whole number.
Rounding decimals depends on the digit to the right of the rounding place.

Decimals