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]]>Multi-digit multiplication is a difficult concept to teach. Long gone are the days where we teach one method, such as long multiplication, and just *hope* that all of our students catch on and can use that method effectively. Today we know the importance of teaching multi-digit multiplication more strategically. This ensures that every student in your classroom can experience success in some way. It also ensures that students’ knowledge is built on strategically, and that they really UNDERSTAND the process of multi-digit multiplication.
Now, before I get started talking about some of the methods for multi-digit multiplication, I do want to let you know that I have a free webinar on this topic. Throughout the approximately 40-minute webinar I’ll discuss a multitude of different strategies for multi-digit multiplication, as well as a suggested progression that will help to maximize student understanding. You can read more about that webinar and get registered HERE.
Alternatively, if you are looking for a resource where all of the work is done for you, you may be interested in this Multi-Digit Multiplication Station, where students work through a variety of strategies at their own pace, mastering each one as they go. Strategies are integrated in a strategic manner, ensuring that students build on their understanding progressively. See the Multi-Digit Multiplication Station HERE.
So how should you begin teaching multi-digit multiplication?
It’s important to start with strategies that will help students solve multi-digit equations mentally. Rather than jumping right into long multiplication or an effective alternative, begin with the following:
1. The commutative and associative properties. First of all, it’s important for students to remember these properties. The commutative property states that the order of the factors does not change the product. For example, 4×3 and 3×4 both equal 12. The associative property states that the factors can be grouped in different ways. For example, (7×2)x5 gives you the same product as (2×5)x7. These properties help students understand that they can manipulate equations to solve them easier.
2. Using Factors. This is a great way to teach students that numbers can be manipulated in order to make an equation easier to solve. When we teach multi-digit multiplication, our goal is not always to get a correct answer as quickly as possible. Sometimes our goal is to be able to think creatively when it comes to number. This is one of those instances. We could take the equation 4×15, and break the 15 into its factors, 3 and 5. Now we have this equation: 4x3x5. Now we could solve it like this: (4×3)x5 -> 12×5 ->60. This is just to show that there is not only one right way to solve this equation.
3. Multiplying by 10, 100, and 1000, as well as multiples of 10, 100, and 1000. Although I’ve grouped these two concepts together for the purposes of this blog post, this should be taught slowly and carefully, piece by piece. When you teach this concept, it’s important to focus on the place value rules before teaching tricks like the “adding zeros” trick. For example, when students are faced with the equation 45×100, they need to understand that the place values increase by 2 places, to make the product 4500. Similarly, when multiplying an equation like 3×1000, the place values increase by 3 places to make 3000. After students have mastered this concept, we can teach them that when there are 2 zeros in the factors, we add 2 zeros to the product. Keep in mind that these tricks should only be taught AFTER students possess an excellent understanding of the math behind the concept.
4. Breaking Up Numbers. This is one of the most useful mental math strategies out there. It involves breaking up one of the factors, multiplying in groups, and then adding those groups together. Here’s an example: In this example we break up the 12 into a 10 and a 2, and then multiply it in parts. So 12×30 becomes (10×30) + (2×30). This is much easier to solve!
We can also use this strategy to multiply bigger numbers, like 103×9. We can break the 103 up into a 100 and a 3, and then multiply in parts, like this: (100×9) + (3×9).
5. The Box/Window Method. I love the box/window method because it utilizes the expanded form of each factor, making this a great strategy to reinforce number sense concepts. To use this strategy, we draw a box (the number of columns and rows depend on the number of digits in the factors), and then write the expanded forms of the factors along the top and the side. Then we multiply each part, and add the parts together when we are finished. If you’d like a more detailed tutorial for this strategy, please see THIS blog post, which also includes a video tutorial.
6. Partial Products. This is one of the most important strategies to teach as an alternative to long multiplication. In partial products, the equation is set up like in traditional long multiplication, but the way we multiply is different. For example, for the equation 35×3, we first multiply 3×5 to make 15. Then we multiply 3×30 to make 90. Notice that we multiplied by THIRTY, not three. This is because that 3 represents 30. This gives us 90. Now we add the 15 and 90 together to make 105. If you’d like a more detailed tutorial for this strategy, please see THIS blog post, which also includes a video tutorial.
The strategies that I’ve outlined above are the MOST important ones for teaching multi-digit multiplication. All of these strategies place an emphasis on number sense understanding and ensure that students really understand what the numbers in each equation mean. But what about strategies like traditional long multiplication?
This is a controversial topic. Some teachers believe that our teaching should be ONLY number sense focused, so that we don’t teach strategies that don’t focus on number sense understanding. These teachers tend to use strategies like partial products all year long as a very effective alternative to traditional long multiplication. Other teachers believe that we should teach the way that multiplication was taught years ago. It worked then, so why wouldn’t it work now?! These teachers tend to focus more on strategies like traditional long multiplication, and less on more current methods like box/window or partial products.
I’m not here to tell you which way is better This depends on you and your students. However, I will tell you my personal belief. Personally I tend to not go to either extreme. I am a huge believer in strategies that promote number sense understanding. However I also believe that there is a place for traditional methods for SOME of your students. You will have to be the judge here. If you have students who are struggling with multi-digit multiplication, you’ll probably choose to let them focus on partial products and box/window and leave it at that. Why add more confusion? They can be very successful with those strategies. HOWEVER, you might have some students who have an excellent number sense understanding of what you have taught so far, and are ready for more of a challenge! These students might thrive with other, less number-sense focused methods, since they already have a strong grasp of math concepts. For these students, I am going to talk about a couple of other strategies.
These next strategies are less number-sense focused, but they can be a fun way to multiply for those students who are ready for a challenge.
If you would like additional support with these strategies, I recommend registering for the free Multi-Digit Multiplication Webinar, which will allow you to explore these strategies in greater depth.
Have a great day,
Shelley
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]]>Do you struggle with teaching multi-digit multiplication in your classroom? Whether it’s time restraints, figuring out how to differentiate your instruction, or just finding the right materials, multi-digit multiplication is a difficult concept to teach! This is especially true since we know the importance of teaching multiple strategies in order to ensure success for all of our students.
I’d love to help you navigate through some effective strategies for multi-digit multiplication! This will help you develop a plan to target all students in your classroom.
I’d love for you to join me for my Multi-Digit Multiplication Webinar! This webinar will go through a multitude of strategies for multi-digit multiplication that you can implement in your classroom immediately!
To register, just visit THIS PAGE, or click on the link below:
MULTI-DIGIT MULTIPLICATION STRATEGIES WEBINAR – REGISTER HERE
Hope to see you there!
Shelley
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]]>I like this strategy as a follow up to the area model (or box/window method), which you can read more about HERE.
I’ve broken down the steps to this strategy in detail for you, but if you’d rather watch this in video format, just click play below! Otherwise, keep scrolling for the step-by-step pictures.
When we use partial products to solve a multiplication equation, we can set it up like a traditional long multiplication equation, as shown below.
Just like traditional long multiplication, we multiply the ones digit of the second factor first. In this case there is only one digit in the second factor. So first we multiply 4×5 to make 20.
Here’s where this strategy differs from traditional long multiplication. We write the entire 20 rather than using carrying as in traditional long multiplication.
Next, we multiply the 4 by the tens digit of the first factor. However in partial products we use the value of this digit. In this example, the value of the 4 in 45 is 40. So we multiply 4×40 to make 160.
We write the entire 160 beneath the 20.
Now that we’ve taken care of the multiplication, we just need to add our products up, in order to make the final product. In this example, 20+160=180, so we know that 180 is the product of 45 and 4.
When we work with more digits, we just multiply the ones first, and then the tens. The example below shows how we would solve a 2-digit by 2-digit equation. Remember that we always use the VALUE of the digits, not just the digits themselves.
If you’d like to learn about other strategies for multi-digit multiplication, you may be interested in my FREE Multi-Digit Multiplication Webinar which you can read more about HERE, or these blog posts on the box window method, and lattice multiplication.
Looking for more support with multi-digit multiplication in your classroom? Check out my self-paced, student-centered Multi-Digit Multiplication Math Station HERE. It includes many different strategies for multi-digit multiplication so that all of your students can experience success in their own way.
Good luck!
Shelley
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The box/window method is an excellent strategy for multi-digit multiplication. This strategy places an emphasis on number sense understanding, as students use the expanded form of each factor. This helps students to understand what the digits in each factor REALLY mean.
This approach is also referred to as the “area model” by some teachers.
I’ve included detailed, step-by-step instructions for this method down below. However, if you would like to watch the steps in video format instead, simply press play on the video below. Otherwise, keep scrolling down to the step-by-step instructions!
I like the box/window method as an introduction to the partial products strategy. Although this basically is partial products, it is a more visual way to do the multiplication. Place value is emphasized because each factor is broken up into its expanded form, as shown below.
When we use the box/window method, we first draw a box or window! The number of rows and columns will depend on the number of digits in the factor. The one below is drawn for a 2-digit by 2-digit equation; therefore there are two rows and two columns.
Next, we arrange the expanded form of our factors along the top and one side of the box. For example, the box shown below is for the equation 57×25.
Now it’s time to multiply! We multiply the numbers that meet in each space on the box. The picture below shows the numbers that are being multiplied in each box. Once your students are used to using this strategy, they will not need to write the entire equation in each box anymore, but can just write the products instead.
Lastly we add all of the smaller products, to make our final product. In this example, the smaller products are added up to make 1425. This means that the product of 57 and 25 is 1425.
When you use this strategy with larger numbers, you simply increase the size of the box to make room for more rows and columns. This is also an effective strategy for 2-digit by 1-digit multiplication, simply by making a rectangle divided into two separate sections, and finding the “area” of each section.
Looking for more help with multi-digit multiplication in your classroom? Check out my self-paced, student-centered Multi-Digit Multiplication Station HERE. It includes many different strategies for multi-digit multiplication so that each of your students can experience success in his own way.
You may also be interested in my FREE Multi-Digit Multiplication Webinar, which you can read more about HERE.
Good luck!
Shelley
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Lattice multiplication is an alternative to traditional long multiplication. Before I begin explaining this strategy, I do want to take a second to talk about multi-digit multiplication strategies in general.
We know that number sense is an essential component of today’s classrooms. We teach math in a way that enhances number sense understanding, so that students really understand what they are doing, rather than just memorizing a series of steps.
Now let me begin this post on lattice multiplication by saying that this is not necessarily one of those strategies that enhances number sense understanding.
So why would you want to teach this strategy?
Once students have a solid understanding of the place value concepts behind multiplication, some can thrive with traditional methods such as long multiplication, or this alternative – lattice multiplication. I do believe that before you teach this method, you should focus on mental math strategies that DO encourage number sense understanding, such as the partial products strategy, or box/window method (area model). If your students have mastered those mental math based strategies, and are ready for more, this is a fun one to teach!
How do you perform lattice multiplication?
Lattice multiplication utilizes a grid to keep numbers organized. This is especially helpful when it comes to regrouping, as the numbers that are carried are also written within the grid to make the adding easier.
I’m going to explain this strategy step-by-step, with lots of pictures, but if you’d rather watch my Lattice Multiplication video, simply press “play” below! Otherwise keep scrolling for the step-by-step instructions.
Let’s begin with an equation that does not require regrouping: 18 x 31. To solve this equation, we follow the steps below:
Step 1: Draw a grid. The number of rows and columns will depend on the number of digits in the factors. For example, if you are multiplying a 2-digit by 2-digit equation, your grid will have two rows and two columns.
Step 2: Next, we arrange the factors along the top and right side of the grid, as shown below.
Step 3: Now it’s time to multiply. We multiply the numbers that meet in each space on the grid. For example, in the top right corner, we are multiplying 8×3 to make 24. The tens and ones are split on either side of the diagonal line.
Step 4: We continue multiplying for each space on the grid.
Step 5: Lastly, we add! We add using diagonal rows, and write the sum of each diagonal row along the left side and bottom of the grid. So in this example, the final product is 558.
Now what happens if we need to regroup? Let’s take a look below.
When we regroup, we simply carry the tens digit to the next diagonal row. In the example below, we have done all of our multiplying on the grid. Now when we add, let’s see what happens.
First we add the diagonal row in the bottom right, to make 0. (see example above)
Now we add the next diagonal row. The sum is 15, so here we write the ones digit (5), and carry the tens (1) to the next diagonal row (I’ve circled that carried digit in this example so that it stands out). Now, when we add that diagonal row, we simply add that carried digit in there as well.
This is often helpful for students because the carried digits stay in the row that they need to be added in, eliminating the confusion that carrying can often bring in traditional long multiplication.
What if we have more digits in the factors?
Easy!
We simply increase the numbers of rows or columns based on the number of digits in the factors. The example below shows a 2-digit by 3-digit equation, so there are 2 rows and 3 columns.
I do want to reinforce one more time that this is not a strategy that you would teach students who are just learning multi-digit multiplication. Please make sure you are focusing on mental math strategies that reinforce number sense first. THEN strategies like this one can be a fun addition to your multi-digit multiplication, and can really be helpful for some of your students.
Looking for more help with multi-digit multiplication in your classroom? Check out my self-paced, student-centered Multi-Digit Multiplication Station HERE.
You might also be interested in my FREE Multi-Digit Multiplication Webinar! In this webinar I’ll go over many different strategies for multi-digit multiplication and a suggested approach to maximize understanding. See more information HERE.
Good luck!
Shelley
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]]>With greater access to technology than ever before, math apps and websites are widely used in today’s classrooms. I wanted to provide you with a comprehensive list of some of the best ones out there, so I sought advice from our teacher community! Several hundred teachers got back to me with some of their favorite apps and websites, and no surprise – some of the same ones kept appearing over and over again.
Below is a HUGE list of some of the best Math apps and websites out there. Because I know how valuable other teachers’ feedback is, I’ve included some comments from the teachers who use them.
Please note that I am not affiliated with any of these companies – all of these recommendations are coming directly from our teacher community.
This app was mentioned by more teachers than any other one. Here are what a few people had to say about it:
This is another app that many teachers are having great success with! Here are what a few teachers had to say:
My good friend Tessa from Tales from Outside the Classroom has actually written a super informative blog post all about Front Row. You can find that post HERE.
Teachers raved about this app in the replies that they sent to me. Here are just a couple :
This is another really popular one that teachers are using! Here is some feedback:
Although this website is free, the app does cost money. Many teachers that I heard from have this app paid for by their administration. Here’s what a few teachers had to say:
A lot of teachers that I heard from are using apps in order for students to make video showing their thinking. I LOVE this idea! What a fantastic way to assess, find gaps in knowledge, and reinforce important skills! Educreations, Show Me, and Explain Everything are three of the apps that teachers are using to make this happen in their classrooms. Here’s a bit of feedback from a few teachers:
The apps and websites mentioned above are the ones that were mentioned by the MOST teachers. However, there were many, many other apps and websites mentioned as well. I’ve included over 150 other math-related apps and websites below, listed in alphabetical order. Please note that this list includes both free and paid apps/websites.
Also, I received a comment from one of our teacher community members that I thought was important to remember when it comes to timed drills. Obviously all students are motivated differently, but this is some great advice. Here’s what Dawn had to say:
A note about timed math drills: I would like to encourage to avoid any apps that have timed activities. Recent research has shown that math anxiety in children is much more likely to occur in students who participate in timed math activities, especially girls. Deep mathematical thinking takes time and should not be rushed.
Now, here’s that list!
I hope that you found this list useful! If you have any other great math apps or websites that you use in your classroom, please feel free to leave the names of them in the comment section below!
Shelley
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]]>DAY ONE
The Day One (November 23) giveaway is for these Reading Response Booktivities. Already have them or can’t use these? Choose a $5.00 TpT gift card instead! Enter to win HERE.
DAY TWO (Thursday, November 24)
The Day Two (November 24) giveaway is for an Addition Station! Thousands of teachers have used this resource with huge success! You’ll find that your students are begging you to do it! Choose from 1st, 2nd, 3rd, or 4th grade. Already have it or can’t use it? Choose a $10.00 TpT gift card instead! Enter to win HERE.
DAY THREE (Friday, November 25)
DAY FOUR (Saturday, November 26)
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]]>As a teacher, I know that you have dedicated your life to making our world a better place. I know that you have a job that never stops, and although it is the most rewarding job out there, it is difficult.
I also know that you don’t hear “thank-you” nearly enough.
So, THANK-YOU! Thank you for everything that you do to help our children. Thank you for your endless dedication to the teaching profession.
Today I’m expressing my thanks in two different ways! First of all, I’m making one of my best-selling resources FREE on November 20 and 21! These Classroom Expectations and Community Building posters will help you build a positive sense of community in your classroom. Be sure to come back and grab them on the 20th and 21st! Check them out HERE.
Additionally, I’m giving away one $25 TeachersPayTeachers gift card to one lucky winner! To enter right now, just fill out the Rafflecopter form below! There are three ways to enter – choose just one, or all 3 for extra entries!
Now that you’ve entered this giveaway, please visit my friends who are also all giving away a $25 TpT gift card and a free resource this weekend! You’ll find all of their links below. Have a fantastic day!
Shelley