Are you ready for multiplication the MathBait™ way? 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 ones 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 hundreds 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 tens 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.

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.