--- title: "How many patterns?" subtitle: "Data representation · Binary numbers · Lesson 6" --- # How many patterns? **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Colour values in a program stop at 255, and old game scores wrapped round to 0. Both come from the same rule about bits. :::teacher-note **In one sentence:** Students count up on binary cards to find that n bits make 2ⁿ patterns and hold 0 to 2ⁿ − 1, see a count overflow, and choose how many bits a number needs (AC9TDI8K04). **Students should be able to say:** I can work out how many patterns some bits can make. I can find the biggest number some bits can hold. I can explain overflow, and choose how many bits a number needs. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **You have four cards, worth 8, 4, 2 and 1. What is the **biggest** number they can show?** ☐ 4 ☐ 15 ☐ 16 ☐ 1111 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 15. All four cards face up: 8 + 4 + 2 + 1 = 15. There are 16 patterns, but one of them is 0, so the biggest is 15. 1111 is 15 written in binary. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Worked example: count the patterns :::keep-together ### Count the patterns 1. $1111 = 8 + 4 + 2 + 1 = 15$ — All cards face up is the biggest number. 2. $0000 = 0$ — All cards face down is the smallest. 3. $2 \times 2 \times 2 \times 2 = 2^4 = 16$ — Each card has 2 choices, so each card doubles the patterns. 4. $0 \text{ to } 15$ — 16 patterns, and one of them is 0, so the biggest is 15. 5. $2^4 - 1 = 15$ — With $n$ bits: $2^n$ patterns, and the biggest number is $2^n - 1$. Now try five bits. **What is the biggest number **five** bits can show?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 31 11111 = 16 + 8 + 4 + 2 + 1 = 31. Five bits make 2⁵ = 32 patterns: 0 to 31. ::: :::keep-together ### Find the mistake **Sam works out how many numbers 4 bits can show. Click the line where the working first goes wrong.** ``` 1 How many numbers can 4 bits show? 2 Each bit is up or down: 2 choices. 3 4 bits: 2 × 2 × 2 × 2 = 16 patterns. 4 So 4 bits show the numbers 1 to 16. 5 The biggest number is 16. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: the 16 patterns are 0000 to 1111, so the numbers are 0 to 15. Sam left out 0 (all cards face down). ::: :::keep-together ### Three bits **How many different patterns can **three** bits make?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 2 × 2 × 2 = 8 patterns: 000, 001, 010, 011, 100, 101, 110, 111. That is 0 to 7. ::: :::typical ## Vary: one change at a time **One past the biggest.** **Add a card.** **Six cards.** **A byte.** **How many bits?.** Keep doubling: 1, 3, 7, 15, 31, 63, 127. Which is the first that reaches 100? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Patterns and limits :::keep-together ### Ten bits **How many different patterns can **ten** bits make?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1024 $2^{10} = 1024$ patterns, so ten bits hold 0 to 1023. ::: :::keep-together ### Score overflow **An old game stores the score in one byte. The score is 255 and the player scores 1 more point. What does the score show?** ☐ 256 ☐ An error message ☐ 255 ☐ 0 ::: :::teacher-note **Answer:** 0 255 is 11111111. Adding 1 overflows: every bit flips to 0 and the score shows 0. Real games have had bugs like this. ::: :::keep-together ### Colour values **In many programs, the amount of red in a colour goes from 0 to 255. How many bits store it?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 0 to 255 is 256 values, and $2^8 = 256$. So each colour amount is one byte. ::: :::keep-together ### Three bits **What is the biggest number **three** bits can show?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7 111 = 4 + 2 + 1 = 7. ::: :::keep-together ### Review: read binary **Review: what number is **10101** in binary?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 21 10101 = 16 + 4 + 1 = 21. ::: ## Make: enough bits Then: how many more students could join before you need a seventh card? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **How many patterns can 6 bits make, and what is the biggest number they can hold?** {{lines-3}} ::: :::teacher-note **Answer:** 2⁶ = 64 patterns, so 0 to 63. The biggest is 63. :::