MathBait™ Big Book of Multi-Digit Multiplication

Welcome to Part 4 of MathBait™ Multiplication. In a change of pace, this article does not contain lessons, activities, and games. In Parts 1-3, we focused on helping students develop a strong conceptual understanding of multiplication and provided over a dozen digital games for students to amp up their fact fluency through play. Parts 5-7 will continue to build single-digit fluency while also helping students to tackle multi-digit multiplication.

But how do you teach multi-digit multiplication? There are an astonishing number of approaches and methods. Which methods are grounded in solid pedagogy? Which are not worth our time? Should students learn more than one method? Before we dive head first into more fun and exciting resources, we thought it was important to take a moment to address the many methods of computing multi-digit multiplication and provide our best advice, techniques, and hacks to support your students on this journey. This article is for the parents, the teachers, the aids, and the tutors to help you sift through the noise and find a solid way to help your students attack multiplying larger values.

Understanding New Math

Before we discuss the many methods for multiplication, let's take a look at why so many methods have been popping up in the classroom.

In the last 50 years, our collective view of mathematics education has shifted. This is a good thing. With so many changes in our way of life, advances in technology, and better understanding of the science of learning, we want our educational system to adjust with the times and provide students the knowledge they need for tomorrow's landscape.

This has been dubbed "new math" and it drives everyone crazy. First and foremost, there is no new math, math is basic truth and beauty, and while we have made advances in our understanding, the math we learned 50 years ago isn't different than the math students are learning today. However, how students are learning has changed.

Decades ago calculators in your pockets or access to Google, Bing, and AI from anywhere in the world didn't exist. Thus, teaching how to compute has become less important. In order to prepare our students to be the problem solvers of tomorrow, we don't need to train human-computers, we need to help students build the skills they need to program the computers and this has been the catalyst to update our curricula.

We won't dive deep into this topic. MathBait™ founder Shayla Heavner published an article on the topic in the Journal of Computers in Mathematics and Science Teaching titled "Building Conceptual Understanding: Improving Mathematics Education in Online Courses". Her research was one of the big drivers for the creation of MathBait™ as she found evidence to support a better way of learning. But we do want to name the struggle – understanding and dealing with "new math" is a thing – and help parents and educators navigate this updated way of approaching mathematics. It isn't about the answer, but about the process. In understanding how things work and why they work, students are building the logical skills they will need to be successful.

Problem solving is key for today's students. However, that doesn't mean students need to master every method. By providing multiple perspectives, students gain a better understanding of why and how something works and then pick the method they like best or a method that makes most sense in the problem they are solving. This means many parents find their child's homework nonsensical, confusing, and riddled with steps and procedures they never learned. If this sounds familiar, this article is for you. We are lifting the veil and explaining the most common methods step-by-step while also providing our best insights on if the method should be taught, if so when, and how best to introduce each method to help your students soar.

What is Multiplication (and Multi-Digit Multiplication)?

We certainly can't talk about how to multiply if we don't have any understanding of what multiplication really is. Math is funny that way. Many of us have been multiplying our entire life yet still have a very limited view of what we are actually doing.

To make matters worse, there isn't just one thing multiplication is. This makes the operation both powerful and confusing. The most common views on multiplication are:

• Repeated Addition

• Area

• Scaling

Each viewpoint can provide us with valuable insight on how to multiply.

Repeated Addition

This is the best way to first introduce multiplication for students. As they already know how to add, it builds on previous knowledge and it is easy for students to understand the resourcefulness of this tool.

If I need to add many of the same thing, it's easy to see why multiplication is helpful. For instance, a box of taco shells contains 9 items. How many shells will I have if I purchase ten boxes? I could write out,

9+9+9+9+9+9+9+9+9+9

but that is time consuming and tedious. It is much easier to consider (and write) 9×10. With an understanding of commutativity, students now have a strategy allowing them to pick the easier and quicker approach. Counting by 9's is tedious,

9 18 27 35 45 54 63 72 81 90

but counting by 10's is simple,

10 20 30 40 50 60 70 80 90.

Further, students with an understanding of place value can ascertain that 9 tens is nothing more than a 9 in the ten's place and zero ones, which quickly becomes 90.

The other value of the repeated addition viewpoint is that it can be extended beyond multiplication and into exponents. Exponents can be viewed as repeated multiplication and as multiplication is repeated addition, we have summed up PEMDAS as nothing more than adding. How cool is that? The main operations we use are all just adding. This perspective of math, as nothing more than numbers and adding, can be powerful and take mathematics from what seems like an overwhelming subject back down to child's play. But we can't stop there. Only viewing multiplication as repeated addition is limiting.

The Area Model

An understanding of repeated addition leads perfectly into the Area Model. If I have three squares, and I continue to stack identical 3-square blocks, my total number of blocks can be found by repeatedly adding 3.

This view not only helps students transition into two-dimensional thinking, it will support larger products, partitioning, distribution, and binomial multiplication. It's a must for students to learn how area relates to multiplication.

The Area Model not only supports a better understanding of multiplication and products, but it is also the best way to help students understand commutativity. From the image above, they can quickly see if we count the stacks rather than the rows, we have 3 stacks of 4 blocks each. The way we count doesn't change the number of items. Thus, we can conclude 3×4=4×3.

Area is all around us. Planting a garden, building a house, painting a room. The Area interpretation gives students an understanding grounded in the world around them rather than an abstract idea.

Scaling

Allowing students to play with scaling is not only fun, but supports their foundation for working with ratios and fractions. Further, scaling is the most powerful view of multiplication as it will allow students to later see that with only multiplication can we create any value.

At MathBait™ we present scaling as a shrink ray or growth laser to better understand algebraic manipulations. For instance, if I need to make Dino 6 times his size, I must multiply him by 6. But just like in Honey, I Shrunk the Kids we cannot pick and choose what scales. If we set off our laser, everything in the path is enlarged or shrunk.

As students progress, this view of multiplication will greatly benefit their algebraic endeavors. For instance, in solving the equation 2x+4=6, students can multiply by 1/2, casting their shrink ray on everything: (1/2)(2x+4=6). Making everything half its size, the 2x becomes x, the 4 becomes 2, and the 6 becomes 3 to find x+2=3. This multiplicative view can greatly enhance a student's ability in algebra. They don't need to guess between operations, "Multiply? Divide? Add? Subtract?" as we have thrown out the need for division. With scaling, we can reach any value and make any number larger or smaller as we please.

Each of these views of multiplication is powerful. And while students should eventually have an understanding of each one, it is ideal to spread them out over time. Skip counting is a great introduction to multiplication as repeated addition. Once students are confident in their skip counting, they can begin to explore multiplication through shapes and blocks leading to the area model. Finally, as students build fluency, introduce scaling as growth. This will provide a solid foundation to re-examine scaling in upper-elementary and middle school.

Method 1: The Area Model

This model of teaching multiplications is particularly useful as a first step in multiplying larger values. Clearly leaning on the third interpretation of multiplication, the Area Model can be completed in three steps. For our example we will examine multiplying 38×46.

Step 1: Draw a rectangle

Students begin by drawing a rectangle in which each side represents one factor they are multiplying. In this case, we can assign one side to represent 38 and the second to represent 46.

Step 2: Partition each side

This is the step that matters. Students practice manipulating numbers, which is a big theme in Marco the Great and the History of Numberville. Seeing numbers as malleable is an incredible power when it comes to math. Students may turn 46 into 40+6 or 20+26 or 10+36 or even 10+10+10+10+6. We see this as the great equalizer in mathematics. It allows students to play to their strengths as they can select values they are comfortable with multiplying. As a rule of thumb, it is generally easiest to partition the rectangle into tens and ones, so we'll follow this convention.

Step 3: Find the area of each partition

Next, students find the area of their partitions. Instead of having to multiply large values like 46 and 38, the problem is reduced to smaller and easier computations. Here, 30×40 can be computed as (3×10)×(4×10)=(3×4)×100=1200. Similarly, 6×30 can be quickly determined as 180 and 40×8 as 320. The final computation gives us 8×6=48.

Step 4: Sum the smaller areas

To find the product of 46×38, students sum the areas of the smaller rectangles to determine the total area of the shape. In this case we have 1200+320+180+48=1748.

While the Area Model has many benefits and is a great choice for introducing the multiplication of larger products, it has some downfalls as well. However, the math behind the area model makes for an excellent extension to the Distributive Property. Students will find themselves reliant on this property well into college math.

We introduce the Distributive Property in MathBait™ Multiplication Part 3.

Not only is the Area Model and the Distributive Property the great equalizer, it continues to build students' ability to manipulate numbers which is one of the most impactful skills a student can have. This will support their work in prime factorization which in turn contributes to a multitude of topics in Prealgebra, Algebra, and Geometry. For this reason, we highly recommend students learn the area model.

Overall Assessment of the Area Model

PROS

• Helps students develop a strong number sense and uncover hidden patterns in numbers

• Allows for flexible thinking as students can build their fences in whatever way they'd like

• Excellent introduction to the Distributive Property

• Introduces students to the Area Addition Postulate, beneficial in application problems and Geometry

• Connects to multiplying binomials in Algebra and having a result of 4 terms helps avoid common errors

CONS

• Drawing out squares and rectangles is tedious and time consuming

• If students do not understand why the model works it is not beneficial in strengthening their foundation

Our Rating:

Our Recommendation

The Area Model earns an A+ with us. If your students are just learning to multiply multi-digit values, this model is a great place to start. It allows students to lean on their existing knowledge and make sense of place value. If your students are struggling with the standard algorithm, this is the method you want to pivot to. It allows students to better organize their work and minimize errors.

We recommend all students learn the area model. Not only is it beneficial for basic multiplication, it can also support students in Algebra, as they identify patterns (such as finding perfect squares and completing the square) and in multiplying binomials or expressions with variables.

To transition your student to the area model, try this mini-lesson.

Transitioning to the Area Model

Revisit the activity "What's That Called?" from MathBait™ Multiplication Part 3. Next, provide students with an empty multiplication table and allow them to select a product (preferably not a square). Instruct students to draw their rectangle on the chart.

Example of the product 3×4

Previously, they noticed the number of 1×1 squares matched their product (here we have 12 1×1 squares in our 3×4 rectangle).

Now ask students to draw along one of the grid lines inside their rectangle from top to bottom and another from left to right. Students may pick any grid line within their rectangle.

Invite students to share their thoughts. Why would we do this? How is this helpful? Highlight that we have now created 4 smaller rectangles. Ask students how many 1×1 squares are in the large rectangle (this value hasn't changed with our markings, in our drawing we still have 12).

Next, have students count how many small 1×1 squares are in each of the four smaller rectangles. Here we have 4, 4, 2, and 2. Allow students to sum these values to confirm together they still have their total.

Finally, ask students to write a multiplication statement for each smaller rectangle they have created. In our example we have 2×2, 2×2, 1×2, and 1×2. Compute the products, what do they notice?

Ask students what in their life they might connect to the rectangles. Modeling can be helpful. For instance, if I start with 12 LEGO and break them into four smaller piles, I still have the same number of blocks because I haven't added or taken any away, just rearranged them. After sharing, explain that we can use this idea to make more complex questions easier.

Provide students with graph paper and the expression 22×13. Allow students to work in pairs as they attempt to solve the product using rectangles. Encourage students to draw out the large rectangle and break it into smaller pieces as they previously did.

After having time to generate their solution, ask different groups to share their findings. Highlight particularly useful strategies such as breaking the rectangle into 20+2 and 10+3. This allows us to lean on the work completed in the Distribution section of MathBait™ Multiplication Part 3 and carry over our knowledge of one-digit products to multiples of 10.

Conclude by explaining to students they are now able to complete multiplication of larger values! They can change any multiplication question into smaller questions they already know how to find. Allow students time to practice this new skill.

Method 2: The Lattice Method

We are going to come right out and say it: this method is mind-boggling, complicated, and does not support student understanding. It is unfortunate as at its core, the Lattice Method relies on Napier's Bones, and we are a huge proponent of learning with Napier's Bones. (You are in luck! MathBait™ Multiplication Part 5 is all about Dem Bones. We will provide you with some amazing resources to amp up fluency, understanding, and give students a major confidence booster using Napier's Bones.) However, the Lattice Method strips away all discovery and exploration and distills multiplication down into a set of memorized steps, leaving students with a procedural view of mathematics and little understanding.

That being said, we know many teachers rely on the Lattice Method so let's take a look at this approach to multiplication to better understand its origins and process.

Step 1: Create a Lattice

Construct a lattice based on the values you are multiplying. For instance, if multiplying a 3-digit number by a 2-digit number, your lattice should have 3 columns and 2 rows. For our example, we will multiply 871×57.

Begin by creating a lattice based on the values you are multiplying.

Step 2: Calculate the Smaller Products

Next, calculate the product of the digits in each row and column. Place the tens value above the diagonal and the ones value below. For instance, since we are multiplying 871, we start with 8 and find the product of 8 and each digit in 57. As 8×5=40, our first entry will be a 4 above the diagonal and a 0 below. And as 8×7=56, our next entry will be a 5 above the diagonal and a 6 below. We continue for each column.

Determine the intermediate products of each row by column. Split the products by place value.

Step 3: Sum the Diagonals

Starting in the bottom right corner, sum the diagonals. In the case your sum is more than 9, we must carry the overage to the next diagonal.

Sum the diagonals, carrying any values greater than 9.

Step 4: Transcribe your Solution

The bottom right corner represents the ones digit, as it is the product of the ones digit of 871 and the ones digit of 57. This means, to read this product you must start in the top left corner and move down, followed by moving along the bottom from left to right.

Read your product.

Thus, the product 871×57 is 49,647. If you are not familiar with the Lattice Method, this might look like magic. That feeling of awe is what makes the idea behind this method so outstanding and also why most students who learn the method have no understanding at all why it worked!

The Lattice Method is akin to using Napier's Bones, the first practical western calculator and an amazing tool for students. However, it is vital students develop an understanding of why this method works if introducing it. While we highly encourage playing with Napier's Bones and having seen wonderfully positive effects from rod multiplication, the Lattice Method falls flat.

Why it Works

This method is simply a different way of organizing the Standard Algorithm and implores the power of the Area Model.

We can consider 871×57 as (800+70+1)×(50+7). Distributing provides us with exactly the six intermediate products shown above.

871×57

=(800+70+1)×(50+7)

=(800×50)+(800×7)+(70×50)+(70×7)+(1×50)+(1×7)

Because of the lattice organization, students need not worry about place value. Thus, this product is calculated as,

(8×5), (8×7), (7×5), (7×7), (1×5), and (1×7).

Finally, students sum the amount in each place to reach the final computation.

Overall Assessment of the Lattice Method

PROS

• Focuses on lower multiplies (1-digit products)

• Excellent organization helping to avoid place value confusion

• Based on a strong mathematical concept

CONS

• Drawing out a lattice is tedious and time consuming

• Too many steps and procedures to remember

• It is so far removed from a mathematical principle, it is almost impossible to develop an understanding of the why behind the process

Our Rating:

This method stinks and it is not on our recommendation list. However, the Lattice Method stems from Napier's Bones which is a powerful tool and an excellent way for students to explore and build number sense.

Our Recommendation

Skip this method entirely. The cons far outweigh the possible benefits. That being said, the mathematics this method is based on is powerful. Explore MathBait™ Multiplication Part 5, where students will have the opportunity to play and explore Napier's Bones, developing a strong number sense and graduated levels of understanding.

If your district or curriculum insists on the Lattice Method (or you just love it), there is hope! While we maintain this method is not a viable long-term strategy as it is time consuming and easy to make an error, before introducing your students to the Lattice Method, take the time to explore Napier's Bones. This will help ensure students understand the method and why it works, helping them to make fewer errors and improve effectiveness.

Remember, in this century we aren't teaching multiplication for our students to do complex calculations, we have a tool for that! We are teaching them to recognize and understand patterns, to solve problems, and to build a foundation for higher-order thinking. This method alone doesn't do that.

Method 3: Partial Products and the Standard Algorithm

Most likely the method parents learned in school, the standard algorithm is a tried and true way to multiply values. That being said, it isn't without its problems. Pure memorization of the method doesn't give students what they need beyond a handful of elementary test questions, and it can be easy to make errors due to forgetting a step, not carrying correctly, or simple arithmetic mistakes.

Partial products are a variation on the standard algorithm that helps build student understanding of the process. It also supports a better understanding of place value and easily connects to the Distributive Property.

Partial Products

Forget about carrying! Partial products is a multiplication strategy that focuses on place value. Let's take a look at multiplying 876×5.

Just like in the Standard Algorithm, we write the problem vertically.

Begin by transcribing the product vertically, ensuring each place is properly aligned.

Next, we multiply the one's place. We find 5×6=30, and rather than carrying we simply write 30 below the line.

Multiply the one's place and transcribe the full product below.

Continuing to move to the left, we now come across the 7. This is the great part about partial products. In other algorithms we treat this value as if it is 7, but in partial products we understand that since this 7 is in the ten's place, it really represents 70. Therefore, we multiply 5×70=350 and write this below our 30.

Multiply the ten's place, making sure to understand the 7 represents 7 tens or 70.

Note: We highly recommend avoiding telling students to simply "bring down a zero", or to "add a zero to the end". This causes confusion that carries over all the way to high school. We cannot simply add zeros to numbers or bring down anything. The correct reasoning above is, since 70=7×10, we can find 5×70 as (5×7)×10. Or, in other words, 5×70 represents 35 tens. As such, we must place our value in the ten's place. As we have 35 tens and 0 ones, we document a zero in the one's place.   

The final step in this problem is to find 5×800 (or (5×8)×100)). We sum each partial product and have the solution.

Continue to multiply and then sum the partial products.

The Standard Algorithm

The most common, this algorithm is analogous to partial products, the only difference is we ask students to process multiple bits of information at once. Rather than multiplying and then summing, we ask them to multiply, carry, and sum in one big burst.

The Standard Algorithm for computing 876×5 would look like this:

Now, we won't negate the usefulness of this algorithm. It has stood the test of time and if we really needed to multiply something by hand, most of us would probably jump to this method. However, that doesn't mean it is the best method to teach students, particularly when first being introduced to multiplication. Because we group steps, it is easy to make a mistake and harder to find the error. It also jumps over understanding multiplication and distribution for a quick fix. Lastly, it is a horrible method for struggling students as it forces them to complete multiple steps all at once and often mentally.

Overall Assessment of Partial Products and the Standard Algorithm

PROS

• Quick

• Partial Products focus on place value and the Distributive Property, both key ideas important for student understanding

CONS

• The Standard Algorithm avoids key understandings students will need later on

• Students oft succumb to memorization tricks like "bring down the zero", which hurts their understanding

• Both require neat handwriting if students have any hope of success

• Partial Products can be confusing for students, especially without a strong understanding of place value

Our Rating:

Our Recommendation

Our rating might surprise you. The Standard Algorithm is "standard" for a reason. We do believe every child should know this algorithm so long as it is taught in a way that supports a conceptual understanding and scaffolding.

We recommend when students first begin multiplication of larger values they start with a different model such as the Area Model. This will help them to easily transfer their understanding of 1-digit products to larger values and builds a strong base to build on. When transitioning to this algorithm, introduce partial products first. This gives students time to connect their understanding of place value. Just as in MathBait™ Multiplication Part 2 we provided students with ample time and resources to begin processing multiple bits of information at once, the same should be done here. Partial Products allows students to understand the steps they are taking as a bridge between other methods and the standard algorithm.

When practicing with Partial Products, encourage students to speak their steps. In the above example, we have our students say "70 times 5" to help them process all the information that we store in simply writing a number. In fact, as a first step we often have students jot down 70×5 and 800×5 to help them keep track of their reasoning.

Once students are confident with Partial Products, rather than introducing the Standard Algorithm, try to have students "discover" it on their own. Remind them of the Symbols and Scales lesson from MathBait™ Multiplication Part 2. There, we discussed how much of math follows the idea of using symbols to allow us to minimize how much we must write. Is there a way we can minimize our steps here?

The Art of Problem Solving's Beast Academy does an excellent job of this in their comics. The students of Beast Academy, after learning partial products, explore how they can complete their multiplication with less writing.

To replicate this in your classroom or homeschool, provide students with 3-5 multiplication problems to be completed using Partial Products. Ask them to look for patterns. Where does the final value in each place come from? Can they make a rule that the products follow? Help and encourage as needed. When students feel like they have a part in their learning, they aren't simply given rules to follow, they enjoy the process more, develop a higher level of understanding, and build confidence.

Method 4: Chinese Sticks

This method is quite interesting. It has surged across the internet as a smart and interesting way to multiply. As we dug deeper into its origins, we came up blank. It is referenced as the Chinese Stick Method, the Japanese Stick Method, Mayan Multiplication, and more. Some articles claim it originated in China in 11BCE, however we were not able to confirm this as these articles lacked any citation. Despite its name and official origin, we see this method as a fun exploration but nothing more.

The method connects to the geometric interpretation of multiplication using a compass and a straightedge.

To allow you to compare this method to the Standard Algorithm, we will consider 876×5. If interested in completing this method as an interactive, you'll want to gather a lot of sticks. However, students can explore the method by simply representing sticks as lines on a piece of paper.

Step 1: Gather and place your sticks

Each place value is represented by a stick. Thus, each place can have up to 9 sticks. Already we can see why this method can become a bit cumbersome...

For our example, when multiplying 876×5, we will need to gather 8 sticks to represent 8 hundreds, 7 sticks to represent 7 tens, 6 sticks to represent 6 ones, and finally another 5 sticks to represent our 5 ones in the second factor.

Line up each group of sticks in the first factor, 876, in parallel lines. Make sure to provide some space between the place values to organize your work.

Now, gather the sticks for the second factor, in our case 5, and lay them on top of your existing sticks. The sticks for the second factor should be perpendicular (or at a contrasting angle) to be able to "read" the product.

It's time to read our sticks! We are looking for the intersections our crossing sticks form. Begin with the one's place. We must count every intersection our sticks make, in this case, that's 30!

If you have seen this method across the internet you might be wondering – it didn't seem this complicated in the video I saw! This is because many of the "math trick" videos are made to deceive you. It's unfortunate, but luckily if you scroll through the comments you will find at least someone who has pointed out the deception. These videos show "tricks" using carefully selected values. The problem with tricks in mathematics is they never always work and they detract from understanding. Thus, students end up worse off and more confused than ever because of this.

To finish off the method, continue to count the intersections of the remaining places.

We are left with 40 intersections for the hundreds, 35 intersections for the tens, and our initial 30 intersections for the ones. Just like in any other method, we must sum, and in this case carry. It is essentially 4000+350+30, leaving us with a product of 4380.

Overall Assessment of Chinese Stick Method

PROS

• Interesting for small or select products

• A great option if completing a unit on the different ways to multiply as it allows students to make connections between different methods and their existing understanding

CONS

• Too much to write out

• Can easily become very confusing

• Time consuming

Our Rating:

Our Recommendation

We cannot see any practical purpose to this method. That being said, it is a great exploration for select values.

This is a fun activity even before students are introduced to the Area Model. Provide students with a basic multiplication problem like 13×2 that they can already compute (in this case by adding 13+13 rather than completing any 2-digit multiplication). Allow them to play with actual sticks, such as Popsicle sticks, and walk them through the steps.

Allow students to guess at how to use the sticks to "read" the product. Encourage them to look for patterns. Since they can find the product, they have a direction or destination to help guide them.

Follow up with a discussion on why this method works to multiply two values. Ask students to think about the pros and cons of such a method. If time permits, provide them will additional problems such as 13×6 (to consider carrying) and even 14×23 to push their reasoning.

For a cross-curricular connection, consider asking students why this method may have been beneficial long ago. When people did not have calculators, this method provides a great way to organize place value and can be used and afforded by anyone, they simply have to gather enough sticks!

Method 5: Vedic Multiplication

Vedic mathematics is a system of computation discovered by Indian mathematician Tirthaji. The word Veda derives from the Sanskrit for "knowledge". A benefit of Vedic mathematics is that it is said to decrease the speed of calculations. However, this also brings about the problem with this form of math: it does not rely on understanding.

The first volume on this method was written in 1957. Students being taught at this time certainly needed a quick way to compute. Today, if a computation must be done quickly, we have calculators, phones, computers, and tablets that can handle this much better than a human ever could. This makes Vedic mathematics less useful now than at the time of its conception.

However, we should not discount Tirthaji's work; it is in fact exactly what MathBait™ encourages from its students. He looked at a problem and asked "is there an easier way?". In order develop new methods that could be taught to millions, he had to possess a deep understanding of what mathematics was. How amazing! This is our goal. We work to help our students develop a deep conceptual understanding so they can create and discover their own methods and strategies. With this strong foundation, there is no limit on what they are able to do.

Let's take a look at the Vedic method of multiplication. If you have read the previous sections, you probably won't be surprised to know it is all the same, the only difference is the viewpoint.

Step 1: Write the Product Vertically

To fully understand this method, we will use an example of 1362×298. We start by writing the product vertically, however, for this method it can be helpful to leave a bit of space between the factors.

As with other methods, we begin by multiplying the one's place and transcribing the result below the line. In this case we have a value larger than 9, and yes, we still have to carry! Rather than place the value atop the first factor, we place the tens digit below and one place to the left.

Multiply the one's place.

Next, we multiply in the form of a cross and sum the results. Again, we place the one's digit on the first line and any value carried below and to the left.

Form an X between the tens and one's places.

For our next digit we must form a star. Again, we find each product and the sum, transcribe the ones digit on the first line and any value carried below and to the left.

Create a star centered on the ten's place.

Our next step is to create a star in the center of our product. It is important to note Vedic Multiplication seems to be based on equal digit multiplication. To help account for this, students can add leading zeros as needed.

Create a star in the center of the product. Add a leading zero if needed for organization.

The previous step was our "middle" step. Much like Pascal's triangle there is a symmetry to our crosses and stars. For this problem we will have 1 2 3 4 3 2 1. Thus, now we will make our way back down the ladder, creating another star of three products, this time centered on the hundred's place.

Create a star centered on the hundred's place.

Continuing down, we are again at 2, meaning we create a cross.

Create a cross between the thousand's and hundred's place.

Our final step is to multiply our line straight down. In this case, as we have a 4-digit number times a 3-digit number, our value has a leading zero and thus equates to zero, leaving us only with our value to sum.

Multiply the thousand's place of each factor.

This step-by-step may have seemed overwhelming. In all fairness, we selected a more difficult example to avoid underestimating the complexity of this method. Here is a view of the line, cross, star..., cross, star with only arrows, which may help better understand the pattern present.

Vedic Multiplication Pattern

Depending on the number of digits you are multiplying, the pattern will change. Thus, this very procedural method can very easily become nonsensical to students.

Overall Assessment of Vedic Multiplication

PROS

• A beautiful result of a deep understanding of algebra

• May be easy for students with smaller values, such as 2-digit by 2-digit multiplication problems

CONS

• Very procedural

• Can easily become confusing

Our Rating:

Our Recommendation

This is really a beautiful display of a deep understanding of mathematics and algebra. Unfortunately, students learning multiplication are n where near equipped to truly understand this method. For this reason, our rating is a 2. However, when students hit Algebra 1 or Algebra 2, this is a great exercise!

In case you are interested, here is the proof we created in order to help our writers make sense of the Vedic method.

As you can see, this is much too complex for a young student to understand and it is why we recommend to skip this method.

However, this method of using algebra to help teach basics is quite in line with our MathBait™ model. The reasoning behind this is because it sets students up to already understand the basic structure of algebraic statements from a young age, making more complex topics easier to pick up by building on prior knowledge. Thus, we conclude MathBait™ Multiplication Part 4 with the MathBait™ way...

Method 6: The MathBait™ Way

We've saved what we think is the best for last! The method we have developed for advancing students to multiple-digit multiplication combines the best of all the previous methods while also building a strong foundation for algebra. In fact, in our next installment of Marco the Great, we tackle algebra and thus introduce this method to help students better understand how to multiply polynomials.

Like Vedic Multiplication, our method relies heavily on algebra. Unlike the Vedic method, students do not need to have any understanding of algebra to be able to pick up multiplication and understand the steps they are taking.

Deviating from our structure a bit, in this section, we will provide a lesson with step-by-step instructions on how to teach your students to multiply larger values, the MathBait™ way!

Warm Up

Students should ideally have completed MathBait™ Multiplication Part 3 and been given ample time to play and build fluency before beginning this lesson.

Present students with a multiplication problem such as 53×5. Ask them to use what they learned from the Distributive Property games to solve.

Students should find we can write 53 as 50+3. This allows us to compute 53×5 as (50+3)×5. Distributing, we have (50×5)+(3×5). Again, leaning on their prior knowledge from our previous activities, we understand 50×5=5×10×5=5×5×10=25×10=250 and 3×5=15 to conclude 53×5=250+15=265.

Next, display the problem vertically for students. Ask how we might use the same method to solve if the question is expressed in this way.

Guide students as needed to recognize we can still write 53 as 50+3. As we have to distribute the 5 to each term, we can multiply 5×3 and 5×50 and complete by adding.

Allow students 5-10 minutes to play with the idea in pairs or small groups. They can make up problems, trade, and work together to try to agree on the product.

Conclude by explaining to students they will now begin working on multiplying larger numbers! The wonderful thing about place value is we will never need to find a product outside our existing table. Soon they will be able to multiply very large values by only using the products they are familiar with.

Activity 1: The Area Model

To provide students with a solid base, we next recommend they complete our Area Model activity. You can find this activity in the Area Model section above, under Our Recommendation.

Activity 2: Handshakes

Previously, students explored the Distributive Property when multiplying by a single term. In this activity, students will develop an understanding of what is required to apply the Distributive Property with many terms.

Rules and Objective

This is a multi-team game. The idea is simple. All teams will play simultaneously and the team with the highest score wins.

The objective of the game is for every member of a team to "shake hands" with every member of their opposing team. Teams will be given time to strategize and determine how they will keep track of their "handshakes".

Part of the fun of this game is both the strategy of the playing team and the strategy of the opposing team.

Set Up

Split your class into groups. If homeschooling, we recommend using stuffed animals, action figures, or anything lying around that will fit the activity.

Provide students with a name-tag to identify teams. For instance, you can provide everyone in the first group a name-tag sticker with the number 1 or the letter A (or have students make up their own team names!) and everyone in the second group with a similar name-tag or sticker to identify them, and so on. The only purpose of this is for students to easily identify if a peer is or is not on their team.

One good way to structure the game is to gather an index card for each student. Use a hole punch to place a hole in the top-left and top-right corners of the card and tie a piece of string to each hole. This creates a sort of necklace for each student which can be used to identify their team (consider using colored cards giving each team their own color) as well as a tool students can use when they determine their strategy.

The number of teams will depend on your class size. We recommend 3-4 teams and staggering the objective. During game-play, students will need to make contact with every member of their target team. Thus, if using 4 teams, set it up so that Team A targets Team B, Team B targets Team C, Team C targets Team D, and of course Team D targets Team A.

You should also determine what is most appropriate for your class in terms of handshakes/contact. Students can handshake, high-five, or you can create your own method of making contact such as a tap on the shoulder, adding a sticker, or marking a card. If your class is rowdy, things can get out of hand without clear rules. For a rowdy class we recommend clarifying that if a student from another team approaches you, you must allow them to make contact in the method determined. This avoids students refusing to high-five or trying to run away when a member of the other team approaches them. Some alternatives to a handshake or high-five are:

• Each student can be given a unique sticker or marker color, when they approach a member of the opposite team they can mark their index card with their sticker or marker.

• Each student can have a list and have the other student write their name on their list when approached.

Getting Ready

Once each team has their identifier and understands the rules, allow about 5 minutes for teams to strategize. They should determine a method to (1) keep track of each student they have made contact with and (2) how they will make sure everyone on their team makes contact with everyone on the other team.

This activity works best when students are given space to make their own strategy. Hopefully, some groups come up with a systematic approach, making it easier. For instance, they could write the name of each student on their opposing team down during strategizing and create a schedule.

Once teams have determined their strategy and gathered any supplies they need (i.e. paper and pencil), let the game begin!

Play

Set a timer depending on your class size. The timer should be challenging but not impossible. A well organized team should be able to accomplish their task in the time given while a team without a solid strategy should have trouble completing the task in the set time.

Begin! Start the timer and allow students to wander the room and employ their strategy. When the timer is up, all students should return to their seats and calculate the scores together. Any member of a team who successfully made contact with every member of their opposing team earns a point for their team.

Allow each team to share their strategy and if it was effective. After a brief discussion, give teams another 5 minutes to make changes to their strategy and run the simulation again. Tally the points and declare the winner.

Have students share their thoughts on the activity. What is the best strategy? It can be helpful to draw a diagram for teams to see.

Explain that it is helpful to stay organized. In the example above, if Mandy goes in order shaking each students hand: John, Sasha, Ria, then Zander, it is easy for her to ensure she reached every student. Tell students they have just completed the Distributive Property. They needed to "distribute" their handshakes (or whatever was used) with everyone in the opposing group.

Before this, students used the Distributive Property with one term such as 5×(50+3). In this game, they needed to distribute much more! In the example above, we distributed (Mandy+Sujay+Charles+Lin)×(John+Sasha+Ria+Zander). By using the Distributive Property, we can multiply very large values!

Activity 3: Let's Eat (Again)

Bouncing off the previous activity, students will cycle back to the Let's Eat activity in MathBait™ Multiplication Part 3.

We recommend placing students in small groups to work together.

Provide students with a combo meal (or alternatively have them create their own). For this exercise it is best to begin with two items in the meal, you can expand to more items as students get the hand of things.

Set up the scenario. Two people are in a car and approach the drive-thru. The driver orders 3 combo meals while the passenger orders 2. Ask students to determine (1) how much of each item each person should receive, (2) how many items all together each person should receive, and (3) how many items the car should receive.

For instance, if our combo meal contains 1 drink and 5 churros, the driver, having ordered 3 combo meals, should receive 3 drinks and 15 churros, and the passenger, having ordered 2 combo meals, should receive 2 drinks and 10 churros. All together, the driver has 3+15=18 items, while the passenger has a total of 2+10=12 items. Finally, the car should expect to be handed a total of 18+12=30 items.

After providing students with time to solve their setup, allow a few students to share their findings. Write out their examples for everyone to see.

In the case above we have (2+3), which represents each driver and the number of combo meals they ordered and (1+5) which represents the combo deal of 1 drink and 5 churros. We can multiply this (2+3)×(1+5) just like in the Handshake activity. We need to make sure that everyone in the first group is distributed to everyone in the second. This gives us (2×1)+(2×5) and (3×1)+(3×5). Notice 2×1 is exactly how may drinks our passenger had, and 2×5 is exactly how many churros they had. Similarly, 3×1 is the number of our driver's drinks and 3×5 is the number of our driver's churros.

Ask students to find their values for questions (2) and (3) in the expression. They should notice that 2 drinks and 10 churros (our passenger's meal) is 12 items, and 3 drinks and 15 churros (our driver's meal) is 18 items, matching their answers in part 2. Finally,12+18=30 is the total number of items.

Allow students time to practice with other values and meals. Many teachers report difficulties with students and word problems or with students transferring their understanding from an activity to practice. The key to success here is to work both sides simultaneously. Students begin with a word problem of their own creating that has context they can understand. At first, their goal is not to put it into a mathematical expression but instead to use their logic and intuition to determine how many items each person should receive and the total number of items. Next, they connect this to an expression. This is important because they already have the answers. It allows them to work backwards towards the goal as needed until they strengthen their understanding. Finally, they compare their logical steps with their mathematical expression. Note, the example here is 5×6, which students are able to compute. Make sure to provide simple problems at first which will allow students to check their answer against their existing understanding.

Activity 4: Expand It!

We are ready to put everything together! Previously, students strengthened their fluency of 1-digit multiplication, developed a strong foundation of the Distributive Property, and applied the Distributive Property to basic 2-digit by 1-digit multiplication. In the activities provided here in MathBait™ Multiplication Part 4, students built an intuitive understanding of the Distributive Property with multiple terms and practiced applying the Distributive Property to compute known multiplication facts.

In this activity, students will continue to build their fluency with both 1-digit multiplication through practice and multi-digit multiplication.

The MathBait™ Method

We have essentially taught very young students to FOIL by leaning on their existing knowledge and intuition, without the use of a confusing acronym. They will now use this binomial expansion to multiply 2-digit by 2-digit values and then increase the intricacy to 3-digit by 2-digit and beyond.

We will provide a 3-digit by 2-digit example: 871×56. Students begin by writing each value in expanded form in the traditional vertical arrangement. This leans on the Standard Algorithm and partial products, but takes away a student's need to remember the place value as it is written.

Leaning on their knowledge of the Distributive Property, the value in the one's place of the second factor must "shake hands" with everyone on the other team. In this case, it means our 6 must multiply by 1, 70, and 800. Continue to encourage students to use decomposition and their 1-digit knowledge to solve. This gives us 6×1, 6×7×10, and 6×8×100. Again, we highly recommend avoiding telling students to "add a zero" or other similar terminology. Instead, note that 6×8×100 simply means we need to place 6×8 in the hundred's place, giving us 48_ _. Since we have no tens and no ones, this is why we have zeros in these places to result in 4800.

Now, our second digit, our driver, must also shake hands with everyone in the opposing group. This gives us 50×1, 50×70, and 50×800. Help students as needed to organize their work. We have 50, 50×70=5×10×7×10=5×7×10×10=35×100=3500, and 50×800=5×10×8×100=5×8×10×100=40×1000=40000.

There are many ways for students to organize their work. We prefer to encourage students to place their values from the second line starting in the ten's place, as we are multiplying 5 tens by 1, 5 tens by 7 tens, 5 tens by 80 tens.

If students are struggling with the zeros, consider using words or symbols. For example, leaning on the previous lesson Symbols and Scales, we could assign a new symbol to represent tens, hundreds, thousands, etc. In the image below, we have assigned # to tens, & to hundreds, and @ to thousands. This allows students to focus on the lower-digit addition they are more familiar with. It breaks up their processing into smaller steps that are more easily understood.

Replacing the zeros with symbols can help students to more easily process larger numbers.

If using zeros or symbols, students should now sum their results. They may sum horizontally first, then vertically, or vertically first then horizontally. Here we now have 40 thousands, 83 hundreds, 47 tens, and 6 ones. Students who are not yet strong in place value may benefit from regrouping in this form. As 47 tens is 4 hundreds and 7 tens they can write this as 7 tens and move the 4 hundreds to the next column. Similarly, 83 hundreds becomes 87 hundreds with our added 4, and we can give our 8 to the thousands, leaving only 7 hundreds. Finally, we have 40 thousands plus another 8 to result in 48 thousands. All together, students have 48776. Students who are strong in place value and addition can add directly.

As you can see, the MathBait™ Method follows the Standard Algorithm but uses a modified notation to help students organize the place value information. This will also greatly help students when they reach algebra and learn how to multiply and divide polynomials. They will see a great parallel between polynomials and place value that will allow them to connect to previous understanding.

A modified version of the MathBait™ Method uses the same steps described above, but allows student to organize their work by place value. This saves us from having to write the zeros. If introducing this methods to students, we recommend first using a grid to help keep place value. As students continue practice they will no longer need to write the grid. This begins to look much more like the Standard Algorithm but still allows student to lower how much they must process and forms a solid understanding of mathematical structure for algebra in which the Standard Algorithm generally lacks.

Our Recommendation

The MathBait™ way leans on the strong methods described above and focuses on developing an algebraic understanding as well as strengthening student understanding of place value and distribution. It further allows students to see place value as objects or as a hierarchy which can be particularly helpful if they struggle with trailing zeros. But, the best part about the MathBait™ Method is that it will allow students to more easily transition to complex algebraic products much more easily. Below is an image showing how we can use the MathBait™ multiplication methods, with polynomials.

Conclusion

It is vital that our curricular material moves away from calculation. Students will need to develop problem-solving skills to be competitive in college and the workplace. This is the "new math" that many fear and misunderstand. The many many methods of multiplying larger values can be mind boggling. In this article we broke down each method to help parents and teachers better understand the origins and mathematics behind the steps and procedures. So many of us were taught "how to" and never saw behind the curtain to understand why.

We find the Area Model to be a strong method to introduce students to multi-digit multiplication. While the Standard Algorithm is useful, the MathBait™ Method leans on some of the benefits of the Standard Algorithm while also helping students to develop a stronger understanding of place value, as well as an understanding of structure that will greatly benefit them in later years and as they begin to study algebra.

Next week, we are providing an amazing tool that will continue to help students develop fluency while also allowing them to play with place value and multiply large numbers with ease (when a 6-year old can find 146789×567 in only a few minutes you know it's good!).

Take your students on an adventure through math with Marco the Great! Our unique storytelling approach is sure to engage your students and provide a new perspective of mathematics which is both enjoyable and easier to understand.

Try it today!

NOTE: The re-posting of materials (in part or whole) from this site is a copyright violation! We encourage you to use these activities with your students. You can not take any part of these activities and post them as your own or crediting MathBait™ without written permission. This includes making derivatives for Teacher Pay Teacher or other websites. The material here is not considered "fair use". Thank you for respecting the author's rights.