Binary Multiplication Worksheet

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All information in a computer is stored and transmitted as sequences of bits, or binary digits. A bit is a single piece of data which can be thought of as either zero or one. This activity demonstrates how sequences of these two symbols can be used to represent any number. Copyright cC. This lesson and supporting materials may be multiplying binary numbers worksheet for nonprofit multiplying binary numbers worksheet use only.

Part of this lesson was adapted from a freely available sample lesson in Computer Science Unplugged cby Bell, Witten, and Fellows. Script for your reference, for guiding children to discover binary numbers.

Have the students make these as described below. Copy of the Secret Numbers worksheet for each student. Help Cinda Get To School worksheet for each student. Binary Piano craft worksheets as desired.

Binary magic trick handouts as desired. Counting to on Your Fingers worksheets as desired. The first part of this lesson is a discovery exercise which should stimulate students to learn to count in binary, as well as to reinforce their understanding of place value. You should review the questions in the script before leading the discussion with your students, but don't feel like you have to memorize the whole thing.

Keep the script handy! Also note that your discussion will probably not follow the script exactly. It is provided as a guide to help you keep your discussion on track. The dialogue took place between Rick Garlikov and a class of 3rd graders.

Explain the motivation for the lesson, and tell the students that we're now multiplying binary numbers worksheet to play some games which will give us practice in writing binary numbers. Divide students into small groups optional - this lesson multiplying binary numbers worksheet be done by individuals, pairs or small groups.

Distribute flash cards, one set to each student or group. The first time you do this lesson you'll have to have the students make their cards. The set should look something like this example: The large cards are approximately 3in x 4in, and the small squares are 2. Note that the small cards have a zero on one side and a one on the multiplying binary numbers worksheet. Have students sort the cards in descending order so that the largest is on the left and the smallest is on the right.

High school kids should say something like "powers of 2. Another fun thing to point out is that each card is one more than the sum of all the cards lower than it. Have the students turn over the cards so the numbers are hidden. To reinforce their memory of the different place values call out numbers for them to "find. Some students might point out that they don't have 3, but they do have 1 and 2.

Do multiplying binary numbers worksheet couple other sums which involve 2 cards, then move to 3 cards, etc. Now flip the cards back over so that the number is showing.

Call out a number, and have the students place 1s above the cards which sum to that number, multiplying binary numbers worksheet 0s above all other cards. For example, if you say 11, students place 1s above cards 8, 2, and 1, and 0s above 16 and 4. If some students find the answers quickly, challenge them to find another solution they won't be able to do so. Have older kids turn over the flash cards after the first example so they get to practice remembering the values.

Ask if anyone in the class has a system for finding an answer. Upper grades should have done so. Request that a student demonstrate the system to the group quickly. A good method for doing this is to subtract the largest power of two you can from the original number, then subtract the largest power of two you can from that number, then subtract the largest power of 2 you can from that number, etc.

Then, write 1s in the places of the powers of two you subtracted and 0s elsewhere: Older kids should see the binary expansion as a sum of products where the decimal value is equal to the sum of each binary digit multiplied by its corresponding power of 2. Spend a few minutes reemphasizing the connection between binary numbers to decimal numbers.

For example, the decimal value is equal to four s plus five 10s plus three 1s. Similarly, the binary value is equal to multiplying binary numbers worksheet plus one multiplying binary numbers worksheet one 64 plus one 4 plus one 1. You may want to point out that just as the place values in the decimal representation are powers of 10, the place values in the binary representation are powers of 2.

What number is binary ? Try to have the advanced students visualize the cards. Can we do all numbers up to the maximum discussed above?

We won't go all the way to That would take too long. Instead we'll go to Each of these 4 students represents one of the flash cards used in the earlier exercises. Have the remaining students direct the 4 students to show 0s or 1s, and sit or stand multiplying binary numbers worksheet. Start with 0, all 4 students should show 0s, and be seated. Next do 1, students should showand the rightmost person should stand up.

Then 2 should beetc. Try to elicit a system for incrementing the numbers. Point out that this system is like adding 1 each time.

Younger kids may not see a system. Can all numbers be represented using only 0s and 1s if I gave you enough cards? What's a simple proof of this? Ones and zeros are not explicitly written on the hard drive multiplying binary numbers worksheet transmitted over the modem. Multiplying binary numbers worksheet, they are represented by a magnetic orientation of the segments on a hard drive, and by high and low tones in data transmission.

Since bits by themselves don't represent much information, they are commonly stored together in groups of size 8 called bytes. Distribute multiplying binary numbers worksheet Secret Numbers worksheet for students to complete. Each student creates a secret number and gives it to a friend to decode. Then the original student checks the decoding and completes the remainder of the worksheet, which also multiplying binary numbers worksheet thought about what numbers in base 3 would be like.

The second part asks multiplying binary numbers worksheet deeper understanding of the notion of place value. This lesson extends gracefully into a discussion of bases and number systems. Have the students develop a base 7 number system and practice writing numbers in that system. Compare the number of digits used to represent a number in base 7 with the number of digits used to represent a number in base 2. Advanced students may be able to prove that a binary representation is unique.

How high can you count if you use your toes as well? Allow students to discover certain pleasant characteristics of binary numbers. For example, to multiply a binary number by multiplying binary numbers worksheet, simply add on another 0 in the least significant rightmost bit. How can you divide by two? What number is represented by 1?

What is the pattern? What number is represented by ? Which of these characteristics have analogs in other bases? What base would an alien use to contact us initially? Assuming the alien doesn't know that our numeric system is decimal, the alien would use unary just 1s as a tally of the values. Suppose the alien counts in base If the it communicated to us in base 13, we wouldn't be able to recognize the values. Higher grade students should be asked to articulate the difference between numbers and their representations.

Have two students stand apart with 5 chairs between them. Ask one to walk to the other, going left or right around each chair. See the Help Cinda get to School handout associated with this activity.

How many ways to do this are there? The answer will become more clear if you place a tag on the floor reading "0" to the left of each chair, and reading "1" to the right of each chair, and then ask the children to write down the sequence that they spell out during their walk. How many ways to make a pizza multiplying binary numbers worksheet there, if there are 7 different toppings?

This extends nicely into a lesson on elementary combinatorics: How many ways are there to get dressed if you can choose between 3 pairs of pants, 5 shirts, and 4 pairs of shoes? Rick Garlikov's paper on using the Socratic Method for teaching binary numbers Standards:

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The first article discusses binary addition ; the second article discusses binary subtraction ; the third article discusses binary multiplication ; this article discusses binary division. The pencil-and-paper method of binary division is the same as the pencil-and-paper method of decimal division, except that binary numerals are manipulated instead.

As it turns out though, binary division is simpler. There is no need to guess and then check intermediate quotients; they are either 0 are 1, and are easy to determine by sight.

Pencil-and-paper division, also known as long division, is the hardest of the four arithmetic algorithms. Like the other algorithms, it requires you to solve smaller subproblems of the same type. Solving these division subproblems requires estimation, guessing, and checking. In addition to these division subproblems, multiplication and subtraction are required as well. Here is an example:. Does 88 go into 8? Does 88 go into 83? Does 88 go into ? The first step of long division, as commonly practiced, combines several steps and their substeps into one.

Technically, 88 goes into 8 zero times, so we should write down a 0, multiply 88 by 0, subtract 0 from 8, and then bring down the 3.

Next, we should write down a 0 because 88 goes into 83 zero times, multiply 88 by 0, subtract 0 from 83, and bring down the 1. We tried to divide by 88 before — two steps ago. That means we have a two-digit cycle 45 from here on out. The answer is 9. The red digits are the carries that occur during the multiplication substeps the multiplication is done as if the divisor — the bigger number — is on top, by convention.

Each red digit is crossed out before the next multiplication. To avoid clutter, I have chosen not to mark the borrows that occur during subtraction. My example has a multi-digit divisor, and has an answer with a remainder that I wrote as a repeating decimal. I wanted one example that showed long division to its fullest. I could have picked a problem with a single-digit divisor which would require no guessing, assuming you know the multiplication facts , or one that produced an integer quotient, or one that produced a quotient with a fractional part that terminated.

I could have expressed the fractional part as an integer remainder, or in fraction form. Here it is broken down into steps, following the same algorithm I used for decimal numbers:.

Does 11 go into 1? Does 11 go into 10? Does 11 go into ? We stop here, recognizing that we divided by 11 two steps ago. This means we have a two-digit cycle 10 from here on out. The quotient is When the answer has a repeating fractional part, checking it is not as straightforward as it is for the other arithmetic operations.

What we can do is approximate the quotient to a finite number of places and then check that it comes close to the expected answer. You can check the answer in a few ways.

One way is by doing binary multiplication by hand: Another way to check is to convert the operands to decimal , do decimal division, and then convert the approximate decimal answer to binary. Estimating that as 3. That looks like it wants to be You can also check the answer using my binary calculator.

Again, that looks like It gives the decimal answer we expect: You can also use this tool to convert in the opposite direction, verifying that 3. There are also analytical ways to check the answer exactly: Like the other arithmetic algorithms, I described the division algorithm in a base-independent way. I wanted to stress the mechanical procedure, not why it works in either decimal or binary. When you do binary long division, you might find yourself doing some of the substeps in your head in decimal e.

Be thankful my example only had a two-digit repeating cycle! Thank you for posting this series of article and emailing me to let me know it was up.

Can you share which tool is used to produce it? It is very clear. You gives so quick response. Recently I read several of your articles. Very well written and useful. I can post some testing I have done with some of your programs. One thing I find, on Ubuntu 64 v Gay caused dead loop compiled by gcc test. However, it does work fine with gcc -m32 test.

The issue seems due to integer size. BigFloat to convert binary to decimal, do the math operation and convert back to binary bits. Maybe you can email me with details see my contact page or continue this discussion on one of my David Gay articles.

What are they for? Those are the carries during the multiplication see my article on binary multiplication. Example of Binary Division The pencil-and-paper method of binary division is the same as the pencil-and-paper method of decimal division, except that binary numerals are manipulated instead. Get articles by e-mail. Tom, Those are the carries during the multiplication see my article on binary multiplication.

Thank you so much! I used it as model for a microcontroller routine of an electronics project. Please help me with these.