# Long Multiplication Worksheets

### Contents

This page includes Long Multiplication worksheets for students who have mastered the basic multiplication facts and are learning to multiply 2-, 3-, 4- and more digit numbers. Sometimes referred to as long multiplication or multi-digit multiplication, the questions on these worksheets require students to have mastered the multiplication facts from 0 to 9.

There are a variety of strategies for completing long multiplication including the classic paper and pencil methods, lattice multiplication (which we feature on this page), mental strategies, manipulative use, technology, and various other paper and pencil algorithms. Multi-Digit multiplication can be a frustrating experience for many students. Try to teach multi-digit multiplication using more than one strategy.

## Long Multiplication Worksheets

• Long multiplication practice worksheets including a variety of number sizes and options for different number formats.

• Two-Digit multiplication is a natural place to start after students have mastered their multiplication facts. The concept of multiplying two-digit numbers requires a knowledge of place and place value, especially if students are to fully understand what they are accomplishing with the various strategies they use. A question such as 24 × 5 can be thought of as (20 + 4) × 5. Mentally, this becomes much easier as students multiply 20 by 5 then 4 by 5 and add the two products. A good way to build understanding of place value is with base ten blocks. These manipulatives also translate very well into paper and pencil and mental math strategies.

• An extra digit can throw off some students but add an extra challenge to others. Always ensure that students are ready for three-digit multiplication or both you and your student will be frustrated. Three-digit multiplication worksheets require a mastery of single-digit multiplication facts and a knowledge of a multi-digit multiplication strategy that will enable students to both understand the question and get the correct answer. Four-digit multiplication was invented in 350 B.C. as a way of punishing children who stole bread from the market. Just kidding! It's actually a great challenge for students who have experienced success with their multiplication facts and have a good handle on a long multiplication strategy. What do you give students who have mastered their multiplication facts and long multiplication and who love a challenge? Look no further than five- to eight-digit multiplication. Enjoy!

• There are no thousand separators in the numbers on the first worksheets. It makes it a little more difficult to read the numbers, but sometimes it is better not to have too many things in the way when students are learning long multiplication. The answer keys include answers with the steps shown, so students and teachers can diagnose any problems in the steps they took to answer the questions. The answers use a paper and pencil algorithm that is commonly used in the U.S. and other countries.

• Commas are included as thousands separators for the numbers on the next worksheets. Commas are used in the U.S. and other English-speaking countries as a way of making numbers easier to read. As with the other long multiplication worksheets on this page, the answer keys include the steps.

• Separating thousands with spaces avoids any confusion with commas and periods. Various number formats in different countries and languages use commas and periods for both decimals and thousand separators, but a space is only ever used as a thousands separator. It is more common in some countries such as Canada and France, but it is being adopted more in other parts of the world.

• In some places, periods are used as thousands separators and commas are used as decimals. This is very confusing to people who are used to U.S. formatted numbers.

## Other Long Multiplication Strategies

• Lattice, or sieve, multiplication is a great strategy for students to use to calculate long multiplication problems on pencil and paper. We've made the first step of preparing a lattice easy as the worksheets below have them pre-drawn. With a little practice, students can use graph paper or draw their own lattices freehand. The first factor is separated by place value along the top of the lattice, giving each place value its own column. The second factor is separated in the same way, but along the right side with one place value per row. The single digit column and row numbers are multiplied together and their product is written in the corresponding box, separating the tens and ones places on either side of the diagonal. Finally, the diagonal "rows" are summed and regrouped starting with the diagonal in the lower right hand corner which will only have a singl-digit in it. The answer keys we've provided should give you a good idea of how to accomplish lattice multiplication like a pro. Once students have a little practice, you might find that this is their preferred method for calculating the products of large numbers. This method is highly scalable, which means it is a straight-forward task to multiply a 10-digit by a 10-digit number, etc.

• The distributive property of multiplication allows students to "split" factors into addends to make the multiplication easier. This also helps students learn to complete multiplication mentally rather than using paper and pencil methods. Often times, the numbers are split up by place value, so 123 becomes 100 + 20 + 3. If one wanted to multiply 123 by 4, you could multiply 100 by 4, 20 by 4 and 3 by 4 to get 400, 80 and 12 which sums to 492. If you multiply a multi-digit number by a multi-digit number, you can split both numbers into place value addends. For example, 938 × 74 = (900 + 30 + 4) × (70 + 4) = (900 × 70) + (900 × 4) + (30 × 70) + (30 × 4) + (8 × 70) + (8 × 4) = 63000 + 3600 + 2100 + 120 + 560 + 32 = 69412. All of the worksheets in this section have a model question included. Of course, it might be easier to complete questions with larger numbers using box multiplication.

• Multiplying on graph paper helps students "line up" their numbers when completing long multiplication questions. These worksheets include custom grids that have the right amount of room for one question.

• In case you or your students want to make up your own questions, these blanks should expedite the process.

• The halving and doubling strategy is accomplished very much in the same way as its name. Simply halve one number and double the other then multiply. In many cases, this makes the multiplication of two numbers easier to accomplish mentally. This strategy is not for every multiplication problem, but it certainly works well if certain numbers are involved. For example, doubling a 5 results in a 10 which most people would have an easier time multiplying. Of course, this would rely on the other factor being easily halved. 5 × 72, using the halving and doubling strategy (doubling the first number and halving the second in this case) results in 10 × 36 = 360. Practicing with the worksheets in this section will help students become more familiar with cases in which this strategy would be used.

• Multiplying numbers in number systems other than decimal numbers including binary, quaternary, octal, duodecimal and hexadecimal numbers.