Digital Information - Lesson 4: Binary Numbers

Overview

In this lesson, students will practice representing numbers in binary (base 2), transitioning from the circle-square representations they made in the last lesson. Students will create and use a "Flippy Do", a manipulative which helps students convert between binary (base 2) and decimal (base 10) numbers. They will practice converting numbers and explore the concept of place value in the context of binary numbers.

Goals

Students will be able to:

• Represent decimal numbers using combinations of binary (base 2) digits 0 and 1
• Represent binary numbers using combinations of decimal (base 10) digits 0-9
• Explain how the position of each binary digit determines its place value and numeric value

Purpose

This lesson is designed to give students as much time as possible using the Flippy Do to get comfortable with the relationship between binary and decimal numbers and the concept of place value.

Resources

Supplemental Code.org material

For the teachers

• How to Make a Flippy Do in print
• Flippy Do worksheet in braille, Duxbury file
• Flippy Do worksheet in braille, .brf

For the students

• Activity Guide in print
• Activity Guide in braille, Duxbury file
• Activity Guide in braille, .brf

Getting Started (5 minutes)

Discuss: Yesterday, you created your own number system using circles and squares. What can we communicate using only two symbols? Is there a limit?

Students should quietly write an answer, then share with a partner, then discuss with the whole class.

Activity (35 mins)

Lecture: Today's activity is introducing students to the binary number system. As a visual aid, you can use Code.org's presentation slides for Unit 1, Lesson 4: Binary Numbers. These slides use a lot of animations. The lecture notes below describe when to move to the next slide or click through an animation -- if you aren't using the slides, you can ignore these prompts.

Lecture

Flippy Do Instructions

1. In the top-right cell of the Flippy Do, write the first power of 2, 2⁰. In the cell to left, write the next power of 2, 2¹. Do this until you've finished the entire top row. The last value should be 2⁷.
2. In the second row, write the whole number equivalent of each power of two in the cell above. For example, the right-most cell in the second row should read "1", because that is the value of 2⁰ in the cell just above it.
3. In the third row, write a "0" in each cell.
4. Write a "1" on the back of each flap on the bottom -- these will be flipped up over the zeroes in the third row. Make sure the ones aren't upside down when they're flipped!
5. Cut on the dotted lines at the bottom.

Continue with the FlippyDo

Note: It may be necessary to demonstrate how values can be calculated by flipping up a "1" for each value required to arrive at the sum of values equal to the decimal number.

Represent these decimal numbers in binary:

Try: Try Out Your Flippy Do!

Do This: Use your Flippy Do to try out these problems:

For example, To convert the decimal number 10, I would flip up a one in the 8's position, because eight can fit in 10 (The next bit to the left is 16, which is too big). Then I have 2 left. I flip up a one in the 2's position. This gives me the binary number "1010", which means 10 in decimal.

If students are having a difficult time understanding the rules of the system, remind them of the concept of place value and relate to base 10.

Try: Flippy Do Activity Guide

Say: Let's continue to practice with our own two number bases, decimal and binary. After you finish each of the four parts of the Activity Guide, I want you to check your work with your partner. Feel free to use your Flippy Do as you work.

Distribute: Activity Guide

Note: Encourage students to use their Flippy Do as a resource.

• Challenge 1 - All 4-bit Numbers: Students should produce all binary numbers with a length of 4 bits, from 0000 to 1111. They should see that all odd numbers end in 1 and even numbers end in 0. Students may also notice that the binary digits increasingly "roll-over" to 1's (from right to left) as numbers become larger and larger.
• Challenge 2 - Binary Numbers with Exactly One 1: The goal here is for students to systematically find all binary numbers that have all zeros, except for one bit. Students should notice that the resulting decimal values are all powers of 2.
• Challenge 3 - Conversion Practice: This section gives students more practice converting between number bases. The last two decimal numbers, 256 and 513, are too big to represent using the Flippy Do, however students should make the connection that more bits could be added to the left of the Flippy Do using increasing powers of 2.
• Challenge 4 - Putting it all Together: The last three questions on the Activity Guide ask students to apply their understanding to new situations. The first question asks them to consider odd values. The last two questions ask students to think about how many bits are required to represent specific values in binary.

Note: As you circulate, take an opportunity to be a Lead Learner. Help students discover the items below using the suggested questions:

• Reading a number vs. Placing a Number: Do we fill in the places on the Flippy Do starting on the left or right? Does it matter? (Yes. If we have a 5-bit number, we actually use the 5 bits on the far right. If we were to use the bits on the far left, this changes the value of the number. This is similar to adding more zeros to a decimal number.)
• Highest value possible with a given number of bits: What is the largest number we can make with 4 bits? Is the last number we can make always odd? (A meaningful pattern is that we can count as high as one less than the next bit on the left. If we have four bits, we can count up to the number 15, because the next bit has a value of 16.)
• Number of numbers we can make: How many total unique numbers are possible with 4 bits? (This is a base 2 number system. With each new bit, we double the amount of unique numbers we can make. With four bits, we can make the decimal numbers 0 to 15 (0000 to 1111), for a total of 16 unique numbers.)

Wrap up (5 Minutes)

Remarks

• It's important to know the differences between binary and decimal number systems. As a review, the decimal number system is base-10. There are ten different symbols used to represent numbers (0-9). The binary numbers system is base-2. There are two different symbols used to represent numbers (0-1). Using our Flippy Do, we can convert between Binary and Decimal number systems.
• While it is easier for humans to use the decimal number system in our everyday lives, we will see later in this unit how electrical signals inside computers can be best represented by using the the binary number system.

Prompt: Now that we've had a chance to practice, let's find out what we've learned and what we still have questions about. Write down:

• 3 things you learned today
• 2 things you found interesting
• 1 question you still have.

Journal

Add to your journal the vocab definition for bit and byte.

• bit: A contraction of "Binary Digit"; the single unit of information in a computer, typically represented as a 0 or 1
• byte: 8 bits (For example 10010101)

Assessment: Check for Understanding

For Students

Open a word doc or google doc and copy/paste the 3 following question.

Question 1

How many bits would be needed to count all of the students in class today?

Question 2

Each time we add another bit, what happens to the amount of numbers we can make?

Question 3

What are the similarities and differences between the binary and decimal systems?

Standards Alignment

• CSTA K-12 Computer Science Standards (2017): DA - Data & Analysis: 3A-DA-09 - Translate between different bit representations of real-world phenomena, such as characters, numbers, and images.
• CSP2021: DAT-1.A.2, DAT-1.A.3, DAT-1.A.4, DAT-1.C.1, DAT-1.C.2, DAT-1.C.3, DAT-1.C.4, DAT-1.C.5

Next Tutorial

In the next tutorial, we will discuss CSP Digital Information Lesson 5, which describes learn about limitations of binary number systems.