Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Counting in Base 2
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Counting in Base 2
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Subject: General Mathematics
Class: Junior Secondary 1
Term: 1st Term
Week: 4
Theme: Number And Numeration
This page supports the lesson note with a companion video and a short classroom-ready summary.
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This topic introduces learners to the concept of different number bases, focusing specifically on Base 2 (binary system). Understanding number bases is fundamental for comprehending how numbers are represented and manipulated, particularly in digital technology. While Base 10 is the everyday system, Base 2 is crucial for computer science and electronics, which are increasingly relevant in modern Nigeria, from mobile phones and digital televisions to ATMs and point-of-sale systems in markets. Mastering the basics of Base 2 at this level provides a foundational understanding for future studies in science, technology, engineering, and mathematics (STEM).
into twos again: Take the 1 group of four. Cannot form another group of two from this.
Remaining 'groups of four' left over:
1. This is our $2^2$ place value digit. (1)
7. Read the remainders from the last step upwards: 1, 0,
1. So, 5 in Base 10 is $101_2$ in Base
2. Verification: $101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)$ $= (1 \times 4) + (0 \times 2) + (1 \times 1)$ $= 4 + 0 + 1 = 5_{10}$ Worked Example 2: Counting 7 Palm Fruits in Base 2 Let's say we have 7 palm fruits.
1. Count the total: 7 palm fruits.
2. Group in twos: 7 items $\rightarrow$ 3 groups of 2, remainder 1. (Digit for $2^0$ is 1)
3. Count the groups of two: We have 3 groups of two.
4. Group these 3 'groups of two' into twos again: 3 groups of two $\rightarrow$ 1 group of (two 'groups of two'), remainder 1 group of two. (Digit for $2^1$ is 1)
5. Count the 'groups of four' (or groups of (two 'groups of two')): We have 1 such group.
6. Group this 1 'group of four' into twos again: 1 group of four $\rightarrow$ 0 groups of (two 'groups of four'), remainder 1 group of four. (Digit for $2^2$ is 1)
7. Read the remainders from the last step upwards: 1, 1,
1. So, 7 in Base 10 is $111_2$ in Base
2. Verification:** $111_2 = (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)$ $= (1 \times 4) + (1 \times 2) + (1 \times 1)$ $= 4 + 2 + 1 = 7_{10}$ This method emphasizes the grouping process, which directly addresses the performance objective. 2.
1. Introduction to Number Bases A number base is the number of unique digits, including zero, used in a positional numeral system. It also dictates how quantities are grouped. For example, in Base 10 (decimal system), we use ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and group quantities in tens. When we count to ten, we don't have a new digit; instead, we form a 'group of ten' and move to the next place value, representing it as 10. 2.
2. Base 10 (Decimal System) – A Familiar Context In Base 10, each digit's position (place value) represents a power of
1
0. Example: The number 235 in Base 10 means: $2 \times 10^2 + 3 \times 10^1 + 5 \times 10^0$ $2 \times 100 + 3 \times 10 + 5 \times 1$ $200 + 30 + 5 = 235$ This shows grouping in tens, hundreds, thousands, etc. 2.
3. Base 2 (Binary System) – The Core Concept Base 2 uses only two digits: 0 and
1. This system groups quantities in twos. Just as in Base 10, when we reach a quantity equal to the base (two, in this case), we form a group and move it to the next place value. Each place value in Base 2 represents a power of 2. 2.
4. Counting in Base 2: The Grouping Method The primary objective of this lesson is "counting in groups of twos." This involves physically or pictorially grouping items into sets of two and noting any remainder. Step-by-Step Reasoning for Counting in Base 2 using Grouping:
1. Start with the total quantity of items.
2. Group items in twos: Take two items and make a single group. Repeat this until no more groups of two can be formed from the remaining items.
3. Count the number of single items left over (remainder): This will be the digit for the 'units' place ($2^0$). This remainder will always be either 0 or 1.
4. Count the number of groups of two: Now treat each 'group of two' as a single unit.
5. Group these 'groups of two' into new groups of two: This forms 'groups of four' (since each new group contains two of the 'groups of two').
6. Count the number of 'groups of two' left over from the previous step (remainder): This will be the digit for the 'twos' place ($2^1$). This remainder will always be either 0 or 1.
7. Continue this process: Group the 'groups of four' into new groups of two to form 'groups of eight', and so on.
8. Record the remainders from each step (from bottom up): The sequence of remainders, read from the last remainder to the first, forms the Base 2 representation.
Worked Example 1: Counting 5 Kola Nuts in Base 2 Let's say we have 5 kola nuts.
1. Count the total: 5 kola nuts.
2. Group in twos: Take 5 kola nuts. Form 1st group of 2. (Left with 3) Form 2nd group of 2. (Left with 1)
Remaining single items:
1. This is our $2^0$ place value digit. (1)
3. Count the number of 'groups of two' formed: We have 2 groups of two.
4. Group these 'groups of two' into twos again: Take the 2 groups of two. Form 1 group containing two 'groups of two' (which is effectively 4 kola nuts).
Remaining 'groups of two' left over:
0. This is our $2^1$ place value digit. (0)
5. Count the number of 'groups of four' formed: We have 1 group of four.
6. Group these 'groups of four' into twos again: Take the 1 group of four. Cannot form another group of two from this.
Remaining 'groups of four' left over:
1. This is our $2^2$ place value digit. (1)
7. Read the remainders from the last step upwards: 1, 0,
1. So, 5 in Base 10 is $101_2$ in Base
2. Verification: $101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)$ $= (1 \times 4) + (0 \times 2) + (1 \times 1)$ $= 4 + 0 + 1 = 5_{10}$ *Worked 3.
1. Introduction (5 minutes)
Teacher Activity: Begin by reviewing Base 10 counting. Ask students how they count common items (e.g., pupils in a class, eggs in a crate, naira notes). Emphasise that they naturally group in tens. Introduce the idea that there are other ways to count or group things, and today's lesson will focus on grouping in twos, which is called Base
2. Mention its importance in computers and phones.
Student Activity: Students share how they count various items and confirm their understanding of Base 10 grouping. 3.
2. Exploration of Base 2 (15 minutes)
Teacher Activity: Provide students with small, countable objects (e.g., stones, bottle caps, sticks, beans, groundnuts, coins). Ensure each student or group has a sufficient quantity (e.g., 10-15 items). Instruct students to count a specific number of items (e.g., 3 items). Guide them to physically group the 3 items into twos.
Ask: "How many groups of two did you make? How many items are left over?" (Answer: 1 group of two, 1 left over). Repeat with 4 items. (Answer: 2 groups of two, 0 left over). Repeat with 5 items. (Answer: 2 groups of two, 1 left over).
Introduce the next level of grouping: "Now, let's take the groups of two you made. If you have 2 groups of two, can you put those 2 groups together to form a bigger group of two?" (This forms a group of four). Demonstrate the full process with an example like 5 or 7 items on the board, showing how the remainders are recorded from right to left to form the Base 2 number. Use a table format (e.g., "Divide by 2", "Remainder").
Student Activity: Students count out specified numbers of objects. Students physically group the objects into twos and identify remainders. Students participate in the discussion, answering questions about the number of groups and remainders. Students follow along as the teacher demonstrates the full grouping process, recording the steps. 3.
3. Formalisation of Concepts (10 minutes)
Teacher Activity: Explain that the remainders from the grouping process (read from bottom-up) form the digits of the Base 2 number. Emphasize that only digits 0 and 1 are used in Base
2. Reinforce the place values ($2^0, 2^1, 2^2$, etc.) without going into complex conversions at this stage, focusing on the meaning of "groups of two," "groups of four," etc. Write the number and its base as a subscript (e.g., $5_{10} = 101_2$).
Student Activity: Students listen and ask clarifying questions. They practice writing numbers with the correct base subscript.
Instructions for Teachers: Present these questions one by one. Encourage students to use their physical objects or draw pictures to perform the grouping. Discuss each solution thoroughly after students have attempted it.
Question 1: Count 3 oranges in Base 2 by grouping them in twos.
Solution: Take 3 oranges.
Group them into sets of two: You can form 1 group of two oranges, and 1 orange will be left over.
Number of groups of two formed:
1. Number of single oranges left over:
1. Now consider the 1 'group of two'. Can you form another group of two from this? No.
Remaining 'groups of two':
1. Reading the remainders from the last step (groups of two) to the first step (single items): 1,
1. Therefore, 3 oranges in Base 10 is $11_2$ in Base
2. Commentary: This simple example reinforces the basic grouping concept and the two levels of remainders.
Question 2: Represent the number of students in a small study group of 6 students in Base 2 using the grouping method.
Solution: Total students:
6. Divide 6 by 2: Gives 3 groups of two, with a remainder of 0. (Digit for $2^0$ is 0) Now take the 3 groups of two.
Divide these 3 groups by 2: Gives 1 group of (two 'groups of two'), with a remainder of 1 'group of two'. (Digit for $2^1$ is 1) Now take the 1 group of (two 'groups of two') (which is equivalent to a group of four students).
Divide this 1 group by 2: Gives 0 groups of (two 'groups of four'), with a remainder of 1 group of (two 'groups of two'). (Digit for $2^2$ is 1)
Reading the remainders from bottom to top: 1, 1,
0. Therefore, $6_{10}$ is $110_2$.
Commentary: This introduces a zero remainder, showing that not all digits will be
1. It also progresses to a slightly larger number requiring three place values.
Question 3: A vendor has 9 loaves of Agege bread. Count this quantity in Base
2. Solution: Total loaves:
9. Divide 9 by 2: 4 groups of two, remainder 1. (Digit for $2^0$ is 1) Take the 4 groups of two.
Divide 4 by 2: 2 groups of (two 'groups of two'), remainder 0. (Digit for $2^1$ is 0) Take the 2 groups of (two 'groups of two').
Divide 2 by 2: 1 group of (two 'groups of four'), remainder 0. (Digit for $2^2$ is 0) Take the 1 group of (two 'groups of four').
Divide 1 by 2: 0 groups of (two 'groups of eight'), remainder 1. (Digit for $2^3$ is 1)
Reading the remainders from bottom to top: 1, 0, 0,
1. Therefore, $9_{10}$ is $1001_2$.
Commentary: This example introduces a quantity that requires more place values, further solidifying the repeated grouping process.
Digital Devices and Computers: The most significant application. Computers, mobile phones, tablets, and all digital electronics communicate and process information using Base 2 (binary) because their internal circuits operate on two states: on/off, high/low voltage, represented as 1 and
0. Understanding Base 2 provides a foundational insight into how these ubiquitous Nigerian technologies function, from text messages to online banking.
Coding and Programming: For students interested in technology, Base 2 is the language of computers. Basic understanding of binary is a prerequisite for understanding how instructions are given to computers, leading to software development, which is a rapidly growing sector in Nigeria with a high demand for skilled professionals. Even simple "if-else" logic in programming relates to binary decisions (true/false, 1/0).
Traffic Lights and Automated Systems: Many automated systems, like traffic lights at busy junctions in Lagos or Abuja, rely on binary logic (on/off, open/close) to control their operations. When a traffic light is red (0) or green (1), this can be seen as a binary state. Similar logic applies to automated gates, alarm systems, and even some agricultural irrigation systems, all of which are becoming more common in Nigeria.