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MathBait™ Multiplication

The Algorithm

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Ready to introduce multi-digit multiplication to your students? In this article we breakdown Partial Products providing key information on the mathematics behind the method, when and how to introduce Partial Products to students, and the pros and cons teachers and parents must consider.


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Multiplication Methods







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.

876 times 5
Begin by transcribing the product vertically, ensuring each place is properly aligned.

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

876×5, multiplication of the one's place, 6×5, with 30 transcribed below
Multiply the ones 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 tens place, it really represents 70. Therefore, we multiply 5×70=350 and write this below our 30.

876×5 with 30 and 350 below the product line
Multiply the tens 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 tens place. As we have 35 tens and 0 ones, we document a zero in the ones 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.

876×5 followed by the partial products 30, 350, and 4000, summed to 4380
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:

Multiplying 876 by 5 using the Standard Algorithm

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

  • Quick

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

  • 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:

3 out of 5 stars

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.

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