Decimal Numbers: Understanding and Teaching

Decimal Numbers: Introduction and Challenges

• Decimal numbers are written with a decimal point.
• Children often struggle with understanding decimals, leading to mistakes.
  • Example: A 15-year-old incorrectly identified 0.5808 as the smallest among 0.05, 5, 0.5808, and 0.056, due to a misunderstanding of the rule about the number of digits after the decimal point.
  • Example: Children between 12 and 15 years of age may incorrectly multiply $40 \cdot 0.8$ to get 5 or $5$.

Terminology

• Decimal numbers are also called decimals or decimal fractions.

Objectives of Decimal Education

• Use familiar materials to teach decimals.
• Help children understand place value in decimal numbers.
• Develop methods for understanding the four operations with decimals.
• Design activities to improve proficiency in calculating with decimal numbers.

Understanding Decimal Numbers

• Question: Why are 1.5, 1.50, and 1.500 considered the same but different from 1.05?
• Goal: Help children learn decimals and avoid misunderstandings.

Example 1: Teaching Decimals Through Division

• Teacher introduces decimals through division by 10:
  • $\frac{7}{10} = 0.7$
  • $\frac{27}{10} = 2.7$
  • $\frac{125}{10} = 12.5$
• Rule: Division by 10 is equivalent to placing a decimal point before the last digit.
• Application to fractions with denominators 100 and 1000:
  • $\frac{27}{100} = 0.27$
  • $\frac{205}{100} = 2.05$
  • $\frac{127}{1000} = 0.127$
• Practice exercises include converting decimal numbers into fractions and vice versa.
• Confusion Example: Writing $\frac{3}{100}$ as a decimal.
  • Children may struggle to find the second digit.
  • Teacher explains starting from the number and moving left, adding zero where there is no digit.
  • Clarification of the difference between 0.03 and 3.

Explaining Place Value

• It is necessary to have knowledge of the concept of local value to fully understand the decimal system.

• Explaining Place Value of Decimal Numbers

  • The teacher initiates this after students practice converting fractions to decimals and vice versa.
  • Students write integers like 324, 271, and 450 as multiples of 100, 10, and 1:
    • $324 = 3 \times 100 + 2 \times 10 + 4 \times 1$
    • $271 = 2 \times 100 + 7 \times 10 + 1 \times 1$
    • $450 = 4 \times 100 + 5 \times 10 + 0 \times 1$
  • The teacher explains the relationship between adjacent place values by dividing by 10 as one moves to the right.
  • To show the end of the ‘unit’, a dot is placed in front of the unit numbers.
    • For example: $2.5 = 2 \times 1 + 5 \times \frac{1}{10} = 2 + 0.5$
    • $11.47 = 1 \times 10 + 1 \times 1 + 4 \times \frac{1}{10} + 7 \times \frac{1}{100}$
    • The decimal point separates integers from fractions.

• Practice Writing Decimal Numbers as Local Value

  • $51.6 = 5 \times 10 + 1 \times 1 + 6 \times \frac{1}{10}$

• Practice writing the number presented as local value in decimal form.

  • $2 \times 100 + 0 \times 10 + 7 \times 1 + 8 \times \frac{1}{10} + 3 \times \frac{1}{100} = 207.83$

• Exercises also include word problems, such as converting the length of a table from centimeters to meters.

• The teacher prefers giving such questions only after students have learned to present numbers in the form of local value.

Example 2: Hands-On Measurement

• Materials: 5 pencils, 5 erasers (pencil length ≈ 10 times eraser length), and threads of different lengths.

• Students are divided into five groups, each receiving a pencil, an eraser, and a piece of thread.

• Task: Measure the length of the thread using the pencil and eraser.

• Thread’s length is measured in pencils (units) and erasers (tenths).

• If ten erasers = one pencil, then one eraser = $\frac{1}{10}$ of a pencil.

• Teacher draws a line the length of the pencil and divides it into 10 parts representing the eraser length.

• A table is used to record the lengths in terms of pencils (units) and erasers (tenths).

• The teacher introduces decimal notation, such as 3.6, to represent 3 units (pencils) and 6 tenths (erasers).

• Teams measure their strings and express the lengths in decimal form (e.g., 2.5, 1.3, 4.7, 5.1).

• A decimal point is introduced as a way of writing units and tenths together.

• Students practice converting between decimal and fraction forms, such as writing 1.4 as $1 \frac{4}{10}$.

Decimal Operations

• Goal: Focus on understanding the concepts involved in operations rather than quick application of rules.
• Emphasize testing understanding and asking questions about the correctness of answers.

Addition/Subtraction of Decimals

• Challenge: Difficulty in adding and subtracting decimals with different numbers of digits after the decimal point.
• Example: $0.123 + 1.1 + 1.101$

Example 3: Teaching Addition of Decimals

• The teacher first teaches addition of decimals with the same number of decimal places (e.g., 2.7 + 1.8 and 18.75 + 20.65).
• Next step: Adding decimals with different numbers of decimal places (e.g., 18.7 + 20.65).
• Teacher’s Method: Add zeros to make the number of digits after the decimal point the same in both numbers.
  • Example: 18.7 is rewritten as 18.70, then added to 20.65.
  • $18.70 + 20.65 = 39.35$
• Mistake Example: 0.05 + 0.101 + 0.003 is incorrectly written and aligned.

Understanding the Rule of Addition

• Students need to understand why 18.7, 18.70, and 18.700 are the same, while 18.07 is different.

• The understanding behind the addition rule must be clear.

• Activities that teach the local value of each digit after the decimal point can help students perform mathematical operations better.

Subtraction of Decimals

• Mistakes are more common when numbers are written in a line or have different numbers of digits after the decimal point.
• Examples of incorrect subtraction:
  • $3.4 - 1.53$

Multiplication/Division of Decimals

• Multiplication is generally easier for children than division.
• Textbook Rule for Multiplying Decimal Numbers:
  1. Multiply as with integers.
  2. Place the decimal point so that the total number of digits after the decimal point in the product equals the sum of digits after the decimal in the original numbers.
• Children often forget the rules and apply them incorrectly.

Example 4: Multiplying Decimals

• A child is asked to multiply 2.14 by 10 and is unsure whether the answer is 214 or 21.4.
• First, assess the student's understanding of the decimal system.
• If the nephew understands the decimal system without problems, they moved forward.
• The nephew is asked to calculate $0.1 \times 5$.
  • Nephew: $0.1 = \frac{1}{10}$
  • Then he wrote $\frac{1}{10} \times 5 = \frac{5}{10}$
  • Then you ask, "How will you write this in decimal form?"
• Nephew: 0.5
• Nephew calculates 0.01 x 5 and 0.001 x 5.
  • $0.01 \times 5 = 0.05$
• $0.6 \times 0.3 = \frac{6}{10} \times \frac{3}{10} = \frac{18}{100}$
  • The nephew then says: 0.18
• Nephew calculates $0.2 \times 0.14$
  • $0.2 \times 0.14 = \frac{2}{10} \times \frac{14}{100} = \frac{28}{1000}$
• He then writes $0.028$
• Ask nephew to find the total number of digits on the right and left side of the “equals” sign to show any relationship with the digits to the right of the decimal point.)
• The nephew discovers the rule: The total number of digits on the right and left side of the equal sign is equal.
• Nephew: The product will be 0.42.
• Nephew checks to find out how to write 2.14 multiplied by 10. Yesterday he was unsure whether it would be 0.214 or 21.4.

Division of Decimal Numbers

• The root cause of the problem is the lack of understanding of the basic concepts involved in the division process.
• Children make more mistakes when the denominator is a decimal number.

Example 5: Dividing Decimal Numbers

• A teacher explains how to divide decimal numbers using the fractional way.

• $24 \div 0.4$

  • $0.4 = \frac{4}{10}$

• $24 \div \frac{4}{10}$

  • $24 \times \frac{10}{4} = 60$

• Some children who have understood how to divide a fraction by another fraction have no problem using this method, but other children find this method difficult.

• A similar method might be better:

  • $\frac{0.36}{0.2} = \frac{0.36 \times 10}{0.2 \times 10} = \frac{3.6}{2}$

• Children often make mistakes when they are unable to decide where to put the decimal point in this type of division.

• Students did not realize that her answer should be close to 7.

Steps to help children determine when to put a decimal point in division.

• It is very important for us to explain to children when to put a decimal point in division. Emphasize that the point is to separate the integer from the fraction part.

• Step 1: How to divide, such as 4.16 by 8

• By doing this we will get a number less than 1 which is a proper fraction. This way there is no integer in the quotient (see Figure 4). We start the quotient from the decimal point to show that there is no integer and the fraction part begins.

• In the next step we can give them questions like $54.16 \div 8$.

• Step 2: Note that first we have to divide 54 by 8 (see Figure 5) then we get 6 as remainder. Then we have to divide 6 by 8. At this step the integer part is less than 8. This situation is same as Step 1 and we have to proceed exactly like this (see Figure 5).

• $8 \overline{)54.16}$

• Step 3: Division of numbers that require adding zeros to the right of the numerator to make it exactly divisible by the denominator. For example, dividing 54.12 by 8.

• To understand this process, we need to understand what is actually happening here. $3.12 = 3 \times 1 + 1 \times \frac{1}{10} + 2 \times \frac{1}{100}$

• In fact, the operation of synthetic division involves many sub-operations. Understanding these sub-operations is necessary to understand division. It is possible that if the student is not ready to understand the logic of these sub-operations, then the sub-operations may confuse him.

• 3.12 = 8 = (8)3.9 since 3 < 8.

• You will agree that these sub-operations can only be understood by children who are well acquainted with the concept of place value.

Key Points

• Children learn from experiences.
• They should be encouraged to think by understanding logic instead of memorizing.
• They are more interested in everyday objects and tools than in specially designed teaching materials.
• Opportunities in their daily lives facilitate development of the concept and skill of decimal numbers.

Summary of Decimal Number Concepts

• Some of the basic problems kids have with decimals.

• Considered some methods through which children can better understand the decimal system of numbers.

• Considered ways to introduce children to operations with decimal numbers.

• Concrete object/experience based learning

• Providing a variety of activities to children

• Helping them understand the logic behind a rule rather than memorizing it.

• Throughout the lesson we stressed that it is very important that children understand exactly why and how numbers are written in the decimal system. Once this is done, children usually have no difficulty in operating on decimal numbers.