# What is Lattice Multiplication

Back to Course Page: Factoring Quadratics Using Lattice Multiplication

## Introduction

What is Lattice Multiplication and where does it come from? Good question! Lattice multiplication is a process that was first founded in the 10th century in India. This method was later adopted by Fibonacci in the 14th century and seems to be becoming the "go-to" method in teaching elementary students how to multiply two numbers in which at least one of them is a two-digit number or greater. This process uses the exact same algorithm you probably learned in your own elementary classes, but organizes it into a box; thus, this is why many people also refer to this method as the "box-method". Through the use of the distributive property, we can use this same process for any type of multiplication problem. This method not only teaches students on how to multiply two larger numbers, but also allows them to work on their organizational skills and practice identifying the place value of a given number. Now that we know what lattice multiplication is and where it comes from, let's look at a specific example.

## Two-Digit Number Multiplied By a One-Digit Number

Before going through an example of our own, let us first watch a short clip that demonstrates this process for us:

- Note: This video will take you to another page. It is suggested that you right click on the link and open it up in a new tab so you do not lose your current page.
- Please watch this video up until you hit 2 minutes and 15 seconds.

Two-Digit Number x One-Digit Number

### Activity #1

After watching through the video, have you seen this process before? If not, that's okay! We are going to go through a step by step analysis here using a different example than in the video.

#### Let's try multiplying 61 x 4.

**Step #1:** In order to determine which number you should place for your rows and which number to place for your columns, it is best to first find the number with the smallest number of place values. The number 61 has 2 place values and the number 4 has only 1 place value. Therefore, the number with the *smallest number of place values* will be the number located on our __row__ and the number 61 will be represented on our columns.

**Step #2:** Next, draw the box so that it has 2 columns for each of the place values in 61 and 1 row for each place value in 4.

We write the 4 on the right hand side of the row so that it lines up with the traditional algorithm that many of us have learned when we were in elementary school. Therefore, if the 4 was on the left hand side, the order of the multiplication would be incorrect.

**Step #3:** In each individual box, draw a diagonal line from the upper right hand corner to the lower left hand corner like this:

- Note: the line can go past the lower left hand corner if you prefer, this may help when determining your end result.

**Step #4:** Going along with our standard algorithm, we are now going to multiply the 4 in the row and the 1 in the far most right column. The reason each box is now divided in half is so that you place the number of tens in your answer in the top left portion and the number of ones in your answer in the bottom right portion. Since 4 x 1 gives us 4, that means there are 0 tens and 4 ones. This is where students will be reminded of their place values and how important it is to understand the definition of place value. Now, write your answer in the box like this:

Continue this same process for 6 x 4. This gives you 24 which has 2 tens and 4 ones.

**Step #5:** Once you have all of your values written, this is where having the longer lines adds a little bit of an advantage. You are going to follow your diagonals and add the numbers that fall along the same portion of the diagonal. Follow the arrows in the picture below:

Under the first diagonal, there is only the value 4. Under the second, there is the value 4 and 0, so 4 + 0 gives us 4. Under the third, the only value there is 2. Therefore, reading from *left to right* this time, you get your final answer of: **244**.

### Activity #2

As many individuals always say, practice makes perfect! Therefore, in this next activity, you are going to practice using lattice multiplication on each of the problems found on the PDF worksheet. All of the directions are written for you on the worksheet, so I definitely suggest printing it out so you can work them out by hand!

- Note: Clicking on this link will directly take you to the worksheet. It would probably be best to right click the link and open a new tab so you do not lose the current page.

After you have completed the worksheet, check your results with the Answer Key found below:

## Two-Digit Number Multiplied by a Two-Digit Number

Now that we know how to use lattice multiplication when multiplying a two-digit number by a one-digit number, we are going to apply these same steps to multiplying two two-digit numbers together!

Before going through an example of our own, let us watch a short clip that demonstrates this process for us:

- Note: This video will take you to another page. It is suggested that you right click on the link and open it up in a new tab so you do not lose your current page.
- Please watch this video starting at 3 minutes until the end of the clip.

Two-Digit Number x Two-Digit Number

### Activity #1

Now that you have seen an example of using lattice multiplication to multiply two two-digit numbers, there are many similarities and differences to the two-digit by one-digit multiplication process.

**Differences:**

- Since both numbers each have two place values, it does not matter which number is placed above the columns and which number is placed to the right of the rows.
- The lattice lines are longer in this particular case since the box has to be a 2 x 2 box for the two place values in each number.

**Similarities:**

- In both cases, you place one number in the columns portion and one number to the right of the rows.
- You still multiply each individual number in the row by each individual number in the column.
- Add along the lattices, even now that they are longer.
- If the lattice adds up to more than 9, carry over whatever is in the "tens" place.

Let us look at an example of our own!

#### What if we multiplied 87 x 24?

Our first step is to develop our "box". Since both 87 and 24 each have 2 place values, our box is going to be a 2 x 2, where either number can be in the column or row portion. I will put 87 on top of the columns and 24 along the rows. In addition to drawing our boxes, draw your lattice by starting from the upper right hand corner of each box and drawing diagonally down to the bottom left hand corner of the box.

Next, multiply each number in the row by each number in the column. For example, 2 x 7 is 14, 2 x 8 is 16, etc. Place all "tens" in the upper left hand portion of the box and place all "ones" in the lower right hand portion of the box, just like in the two-digit by one-digit process.

Add along each of the lattices (diagonal lines). If any lattice is more than 9, carry over any "tens" that occur. For example, if a given diagonal adds up to 13, write the 3 down and carry the 1, just like in our traditional process. Once you finish adding your lattices, you should arrive at this answer:

Reading from left to right we get our answer. Therefore, 87 x 24 = 2,088.

### Activity #2

As many individuals always say, practice makes perfect! Therefore, in this next activity, you are going to practice using lattice multiplication on each of the problems found on the PDF worksheet. All of the directions are written for you on the worksheet, so I definitely suggest printing it out so you can work them out by hand!

- Note: Clicking on this link will directly take you to the worksheet. It would probably be best to right click the link and open a new tab so you do not lose the current page.

After you have completed the worksheet, check your results with the Answer Key found below:

**You have now completed Unit 1 of this course!**

To continue onto Unit 2: Factoring Polynomials where a = 1

To go back to the Course Home Page: Factoring Quadratics Using Lattice Multiplication

## References

(2008, August 14). *Lattice Multiplication*. [video file]. Retrieved from http://www.watchknowlearn.org/Video.aspx?VideoID=17808&CategoryID=4625, November 2012.

CEMSE University of Chicago. *Everyday Mathematics Lattice Algorithm for Multiplication*. [powerpoint]. Retrieved from http://everydaymath.uchicago.edu/teaching-topics/computation/documents/mult-lattice-ex-2.pdf, November 2012.

West, Lynn. (2011, July). *An Introduction to Various Multiplication Strategies*. [Document]. Retrieved from http://scimath.unl.edu/MIM/files/MATExamFiles/WestLynn_Final_070411_LA.pdf, November 2012.