Computer Science - ICT Grade 6 Binary Numbers Converting to Decimal

Grade 6 Computer Science – ICT
Topic: Binary Numbers: Converting to Decimal

1. The Driving Question

"If computers only understand 0s and 1s, how do they turn those weird binary strings—like 1011—into the numbers we actually use, like 11? And why can’t they just count like we do?"

2. The Core Idea – Built, Not Listed

Imagine you’re playing a game of "Power of Two Tag" with your friends at recess. The rules: - Only two people can be "it" at a time (like binary’s 0 and 1).
- Each round, the "it" players double their value: the first round, they’re worth 1 point; the next round, 2 points; then 4, 8, and so on.
- If a player is not "it" in a round, they’re worth 0 points for that round.

Now, let’s say the game ends after 4 rounds, and the "it" players were: - Round 1: Player A (1 point) - Round 2: Player B (2 points) - Round 3: No one (0 points) - Round 4: Player A and B (8 points each)

To find the total score, you’d add up the points where players were "it": 1 + 2 + 0 + 8 = 11. That’s exactly how binary works! The binary number 1011 is just a scorecard for this game, where each digit tells you whether a "power of two" is "it" (1) or not (0). The rightmost digit is Round 1 (1), the next is Round 2 (2), then Round 3 (4), and so on. Add up the "it" rounds, and you’ve converted binary to decimal.

Key Vocabulary:
- Binary: A number system using only 0 and 1, where each digit represents a power of 2.
Example: The binary 1101 is like a light switch panel with 4 switches—only the 1st, 2nd, and 4th switches are flipped "on" (worth 1, 2, and 8 points).
Grade 9–12 note: In computer science, binary expands to represent any data (text, images) by grouping bits into bytes (8 bits). The "powers of two" logic stays the same, but the scale grows exponentially.

• Decimal: The base-10 number system we use daily (digits 0–9).
  Example: The number 13 isn’t just "thirteen"—it’s 1 ten + 3 ones, just like binary is 1 eight + 1 four + 0 twos + 1 one for 1101.

• Bit: A single binary digit (0 or 1).
  Example: The binary 10 has 2 bits. If you think of bits as pixels on a screen, a single bit is like one tiny dot that’s either black (0) or white (1).

• Place value (in binary): The value of a digit based on its position, which doubles as you move left.
  Example: In 101, the leftmost 1 is worth 4 (not 100 like in decimal), because it’s in the "fours place."

3. Assessment Translation

How this appears in class:
- Exit tickets: "Convert the binary number 1101 to decimal. Show your work." - Proficient: Writes 8 + 4 + 0 + 1 = 13 and labels each digit’s place value (e.g., "the 1 on the left is 8").
- Developing: Adds 1 + 1 + 0 + 1 = 3 (ignores place values) or writes 13 without showing work.

• Short constructed response: "Explain why the binary number 1000 equals 8 in decimal. Use the term ‘place value’ in your answer."
• Proficient: "The 1 is in the ‘eights place’ because it’s the 4th digit from the right (2³ = 8). The other digits are 0, so the total is 8 + 0 + 0 + 0 = 8."
• Developing: "Because 1 + 0 + 0 + 0 = 1" or "It’s just 8" (no explanation).

State standardized test framing (e.g., PARCC, SBAC):
- Multiple choice: "What is the decimal value of the binary number 1010?" - Distractors:
- 10 (student adds digits like decimal: 1 + 0 + 1 + 0).
- 5 (student counts the number of 1s).
- 16 (student misplaces the powers: 2⁴ instead of 2³ for the leftmost digit).
- Correct answer: 10 (8 + 0 + 2 + 0).

• Short answer: "A computer stores the number 5 as the binary 101. Show how this binary number equals 5 in decimal."
• Proficient model response:
  > "The binary 101 has three digits. From right to left, they represent 1, 2, and 4. The 1s are in the 1s place and the 4s place, so you add 4 + 0 + 1 = 5."

4. Mistake Taxonomy

Mistake 1: Adding digits like decimal
- Prompt: "Convert 1101 to decimal." - Common wrong response: 1 + 1 + 0 + 1 = 3.
- Why it loses credit: Ignores place values—treats binary like a decimal sum.
- Correct approach: 1. Write the place values above each digit: 8 4 2 1.
2. Circle the digits with 1s: 8 and 4 and 1.
3. Add only the circled values: 8 + 4 + 1 = 13.

Mistake 2: Mislabeling place values
- Prompt: "Explain why 1000 in binary is 8 in decimal." - Common wrong response: "The 1 is in the thousands place, so it’s 1,000." - Why it loses credit: Applies decimal place values (1s, 10s, 100s) to binary.
- Correct approach: 1. Count the digits from the right (starting at 0): 1 is in position 3.
2. Calculate 2³ = 8.
3. All other digits are 0, so the total is 8.

Mistake 3: Skipping the "why" in explanations
- Prompt: "A student says 101 in binary is 6 because 1 + 0 + 1 = 2, and 2 × 3 = 6. Is this correct? Explain." - Common wrong response: "No, it’s 5." (No explanation.) - Why it loses credit: Doesn’t address the student’s flawed reasoning or show understanding.
- Correct approach: 1. Identify the error: The student added digits like decimal and multiplied randomly.
2. Show the correct method: 101 = 4 + 0 + 1 = 5.
3. Explain why place values matter: Each digit’s value doubles, not adds.

5. Connection Layer

• Within CS: Binary → Hexadecimal — Hexadecimal (base-16) is like a "shortcut" for binary. Four binary digits (e.g., 1101) map to one hex digit (D), making long binary strings easier to read. Understanding binary’s powers of two makes hex’s powers of 16 click.

• Across subjects: Binary → Genetics (DNA base pairs) — DNA uses four "digits" (A, T, C, G), just like binary uses two. Scientists convert DNA sequences into binary to store them in computers (e.g., A=00, T=01). Both systems rely on combinations of simple units to represent complex information.

• Outside school: Binary → Barcode scanning — When a cashier scans a barcode, the black and white stripes are like binary: black = 1, white = 0. The scanner reads the pattern, converts it to binary, then to a decimal number (the product ID). Next time you see a barcode, notice how it’s just a "secret binary message" for computers!

6. The Stretch Question

"If you had to invent a number system for aliens whose hands have 7 fingers, how would their ‘base-7’ system work? How would they write the decimal number 10 in their system—and why would it look different from binary?"

Pointer toward the answer: Start by thinking about how we count in base-10: we run out of digits after 9, so we "carry over" to the next place (10 = 1 ten + 0 ones). In base-7, you’d run out of digits after 6, so 7 would be written as 10 (1 seven + 0 ones). To write decimal 10 in base-7, you’d ask: "How many 7s fit into 10?" (1, with 3 left over), so it’d be 13 in base-7. The key is that the place values change—just like binary’s powers of two, base-7 would use powers of seven!