Liberia - Grade 6 - Mathematics - Changing Base Ten to Base Five and Vice Versa

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Subject: Mathematics

Semester: 1

Period: 2

Week: 8

School Name:
Teacher’s Name:
Subject: Mathematics
Grade Level: Grade 6
Date: Week 8
Lesson Duration: 45 minutes
Week & Period: Week 8, Period 2
Topic: Changing Base Ten to Base Five and Vice Versa
Sub-topic: Conversion Between Bases

Learning Objectives
By the end of the lesson, students should be able to:
Define the base five system
Count in base five (digits 0–4)
Convert base ten numbers to base five
Convert base five numbers to base ten

Previous Knowledge
Students already know base ten number system, place value, and basic arithmetic.

Instructional Materials
Mathematics textbook for Grade 6, sticks, counters, chart paper.

Lesson Development – ABC Model
A – Anticipation (Warm-up / Starter)
Time: 5–10 minutes
Teacher asks: “How many groups of 5 can we make from 20 objects?” Introduce the idea of base five counting.

B – Building Knowledge (Main Lesson Body)
Time: 25–30 minutes

Definition

The Base Five Number System, also called the Quinary System, is a positional numeral system that uses five digits (0, 1, 2, 3, 4) to represent numbers. Like the familiar base ten system, the value of each digit depends on its position, but instead of powers of 10, it uses powers of 5.

Each place value represents powers of 5:

5⁰=0 (ones place)
5¹=5 (fives place)
5²=25 (twenty-fives place)
5³=125, and so on.

Because only digits 0 through 4 are used, once a place reaches 4, the next number moves to the next place value, similar to how in base ten, digits go from 0 to 9 before increasing the next place value.

Conversion from Base Ten to Base Five

To convert a base ten number (decimal) to base five:

Divide the decimal number by 5.
Write down the remainder (it will be between 0 and 4).
Divide the quotient by 5 again.
Repeat until the quotient is 0.
The base five number is the remainders read from bottom to top (last remainder is the highest place value).

Example: Convert 23 (base 10) to base 5

23÷5=4 remainder 3
4÷5=0 remainder 4

Read remainders bottom to top: 43 (base 5)
So, 23₁₀=43₅.

Conversion from Base Five to Base Ten

To convert a base five number to base ten:

Multiply each digit by its place value (powers of 5).
Add the results.

Example: Convert 132₅ to base 10

1×5²=1×25=25
3×5¹=3×5=15
2×5⁰=2×1=2

Add them up: 25+15+2=42₁₀.

More Examples

Convert 17 (base 10) to base 5:
17÷5=3 remainder 2
3÷5=0 remainder 3
Reading bottom to top: 32₅
So, 17₁₀=32₅.
Convert 214₅ to base 10:
2×25=50
1×5=5
4×1=4
Total = 50+5+4=59₁₀.
Convert 50 (base 10) to base 5:
50÷5=10 remainder 0
10÷5=2 remainder 0
2÷5=0 remainder 2
Reading remainders bottom to top: 200₅.

Learners’ Activities (Expanded)

Activity 1: Counting in Fives
Use groups of five objects (e.g., sticks, beads) to help learners visualize the base five system. Count from 1 to 20 in base five notation (0 to 4, then 10, 11, 12, …).
Activity 2: Conversion Practice
Provide learners with a list of numbers from 1 to 50 in base ten. Ask them to convert these numbers into base five using the division-remainder method.
Activity 3: Reverse Conversion
Give learners base five numbers (e.g., 43, 132, 214, 100) and have them convert these back to base ten using the multiplication and addition method.
Activity 4: Place Value Expansion
For given base five numbers, write out the expanded form showing powers of 5. Example: 1325=1×25+3×5+2×1132_5 = 1 \times 25 + 3 \times 5 + 2 \times 11325=1×25+3×5+2×1.
Group Work:
Learners work in pairs or groups to create flashcards with base ten numbers on one side and their base five equivalents on the other side. They quiz each other.

Assessment Checks

Oral questions:

“Convert 17 in base ten to base five.” (Answer: 32₅)
“Convert 214₅ to base ten.” (Answer: 59)
“What is the base five place value of the digit 3 in 132₅?” (Answer: 3×5=15)
“Convert 50 from base ten to base five.” (Answer: 200₅)

Quick written quiz:

Convert the following base ten numbers to base five: 7, 12, 25, 44.
Convert the following base five numbers to base ten: 101, 34, 420, 13.

Notes (Expanded & Detailed)

The base five system is an example of a numeral system different from the base ten system we use daily. Understanding it is useful in computer science, digital electronics, and exploring how different cultures use various number systems.

The key skill in base conversions is mastering the division-remainder method for converting from base ten to another base, and the expansion method using powers of the base for converting back.

This strengthens learners' understanding of place value as a concept, and also enhances their number sense and flexibility in thinking about numbers.

Practicing conversions improves not only computational skills but also prepares learners for understanding binary, octal, and hexadecimal systems in future studies.

Additional Assignments

Convert the following base ten numbers to base five: 9, 15, 28, 33, 48.
Convert these base five numbers to base ten: 121, 404, 33, 102, 241.
Write the expanded form of the following base five numbers: 231, 104, 42, 1001.
Using groups of 5 objects, demonstrate how you can count up to 25 in base five notation.

C – Consolidation (Conclusion & Assessment)
Time: 5–10 minutes
Summary: Review base five, counting, and conversions in both directions.

Evaluation Method (Expanded)
Exit slip/quiz: Convert 12₁₀ to base five and 103₅ to base ten. Teacher provides oral feedback.

Assignment (Expanded):
Practice converting 10 numbers from base ten to base five and vice versa.

Follow-up Activity:
Use everyday objects to group in fives and represent numbers in base five.

Differentiation / Inclusive Strategies
Provide visual aids and manipulatives; pair learners for peer practice.

Teacher’s Reflection (After Class)
What worked well? ___________________________________________
What needs improvement? ____________________________________
Students’ engagement level: ☑ High ☐ Medium ☐ Low