Before the meeting, Davida showed Rachel and me a multiplication method a student had showed her earlier in the day. The student said that she only knew how to do multiplication using the method on the right and wanted to learn the method on the left. What a coincidence! This is exactly what I was planning to explore today.
I introduced the topic of the meeting by talking about the book, Math Matters by Chapin and Johnson. Written for K-6 teachers, this book is useful for adult educators since it helps us understand basic mathematics and think about ways to incorporate it into our instruction. Chapter 4 is on computation and includes a few examples of alternative algorithms (procedures) and invented student strategies. This chapter inspired today’s meeting.
Number Talk1
I started the meeting with a number talk on 25 x 16.
- Without using paper or pencil to calculate, write down an answer to 25 x 16.
- When you have an answer, give me a thumbs up. Putting your thumb right by your chest lets me know that you’re ready, but doesn’t disturb people who are still thinking. If you have two ways of getting the answer, put out another finger. You might find even more while we’re talking about it.
- Turn to a partner and explain how you got your answer.
- So, what’s the answer? 400 and 230 were possible answers.
- Ask a volunteer to explain how they got the answer. Write their name above the calculations on the board.
Rachel’s method
In talking with Mark, Rachel realized that she had gotten the wrong answer (230) and was willing to explain her strategy in order to find out why it didn’t work.
After Davida demonstrated the area model method at the bottom, Rachel realized that she had forgotten to multiply some of the numbers (5 x 10 & 20 x 6).
Mark’s method
Mark realized that there are four 4’s in 16, so he could multiply 4 by 25 to get 100, then multiply 100 by 4 to get 400.
Davida’s method
Davida knew that 16 is the same as 10 + 6, so she multiplied 25 by 10 and 25 by 6 and added the two products together.
During the number talk, there was some discussion of the distributive and associative properties of multiplication, but we didn’t connect them explicitly to each strategy. After the number talk, I talked about how I tried to take notes on the strategies in a way that would show the mathematical properties that people used to make the calculations easier. For example, Mark’s strategy made use of the associative property of multiplication, which allowed him to separate 16 into the factors 4 and 4, then group and multiply all the factors in a different order than is in the original multiplication problem. Davida made use of the distributive property of multiplication, which allowed her to separate 16 into two addends (10 and 6) and distribute the multiplication into 25 x 10 and 25 x 6. My guess is that Mark and Davida didn’t consciously decide to use the associative and distributive properties, but instead chose what was easiest or most efficient when multiplying mentally. If we’re teaching these properties, a number talk would be a way to show students that they already use the strategies without realizing it. (Find more information on these properties in Chapter 2 of Math Matters, by Chapin & Johnson)
Examining Multiplication Algorithms
I asked the group what they thought of when they heard the word algorithm2. We talked about some possible definitions:
- A recipe, series of steps that work with different ingredients. Order sometimes matters.
- An accepted process or set of rules to be followed in calculations.
We used the second definition to think about multiplication. We quickly looked at standard algorithm for multiplying when using paper and pencil, then broke up into pairs to look at 8 alternative algorithms for multiplication.
Task
Each handout showed a sample calculation, but didn’t name the method or explain how it worked.
| Group 1 | Group 2 | Group 3 | Group 4 |
| B Partial sum A Area model D Chinese |
F Egyptian E Mental B Partial sum |
C Lattice A Area Model H Distribution |
G Peasant D Chinese E Mental |
Small groups responded to the following instructions:
- Use each method to solve 26 x 35. Practice the method with a few other multiplication problems.
- Why does it work?
- What are its advantages? Disadvantages?
- Can it be used for other kinds of multiplication (decimals, fractions, binomials)?
- On chart paper: Demonstrate each of your methods with the multiplication problem 26 x 35.
Presentations
Each group shared their chart paper demonstrations. We discussed the various methods in this order (A, B, C, D, F and G).
The Area Model (A)
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| Linda and Rachel | Mark and Andrew |
Our discussion:
- Mark and Andrew’s method shows the order it was constructed: red, then black, then blue, then pink. Their version is more concrete, since the squares are in proportion to the quantities represented.
- Linda and Rachel’s method is more abstract. The squares are the same size and don’t directly represent the quantities. Mark and Andrew’s method is proportional to the quantities being multiplied.
- If we were to introduce this to students, we might start with concrete representations of multiplication using tiles or grid paper, where students count squares to determine the product, then move towards area model on blank paper (Mark and Andrew’s) and eventually an abstracted area model (Linda and Rachel’s).
- Phil talked about how his son is learning multiplication and how it has been difficult at times to follow the techniques he is taught in school. For information on a progression of multiplication in elementary school, watch this fantastic video: Graham Fletcher’s Progression on Multiplication
The Partial Products Method (B)
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| Davida and Phil | Linda and Rachel |
Our discussion:
- The partial products method shows the same four products as the area model.
- This method forces you to remember the place value of the numbers you calculate. (20 + 6) x (30 + 5) is another way to think about what is happening in this method. For example, the second calculation is 20 x 5, not 2 x 5.
Lattice Multiplication (C)
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| Mark and Andrew |
Our discussion:
- The colors show the order in which the calculation is constructed. First the square, diagonals and numbers being multiplied, then the partial products and, finally, the totals that give an answer.
- It isn’t necessary to remember place value if you know how to follow these steps. The diagonals organize ones, tens, thousands, etc. for adding to the final product.
Chinese Multiplication (D)
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| Singh, Kevin and Greg |
Our discussion:
- Each number is represented by lines. The drawing above represents 35 x 26.
- The intersections represent the partial products (30, 180, 100 and 600).
- Similar to lattice multiplication, place value in mental multiplication is lost, but retain in the structure of the algorithm. Vertical columns represent place value (30 is in the ones column, 10 and 18 are in the tens column, and 6 is in the hundreds column).
Egyptian Multiplication (F)
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| Davida and Phil |
Our discussion:
- To calculate the product of 26 and 35:
- Start with 1 x 35.
- Double both quantities to get 2 x 70.
- Double both again to get 4 x 140.
- Double again to get 8 x 280.
- Finally, double to get 16 x 560. You don’t have to double a third time because that would give you 32 x 560, which is more than 26.
- Look on the left side and choose numbers that you can add to get 26. 2 + 8 + 16 = 26. Now, add the corresponding numbers on the right side (70 + 280 + 560) to get 910.
- This method is nice because it only involves doubling, which can be easier than other kinds of multiplication.
- 1, 2, 4, 8, 16 are binary numbers (2n).
Russian Peasant Multiplication (G)
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| Singh, Kevin and Greg |
Our discussion:
- To calculate the product of 26 and 35:
- Start with 26 x 35.
- Halve left side and double right side to get 13 and 70.
- Halve left side (and discard fraction of .5) and double right side to get 6 and 140.
- Halve left side and double right side to get 3 and 280.
- Halve left side (and discard fraction of .5) and double right side to get 1 and 560.
- Cross out rows that start with an even number.
- Add the right side for rows that start with an odd number (13, 3 and 1).
- This method is similar to Egyptian multiplication in that it uses doubling and binary numbers.
1. For a great description of number talks with classroom examples, read Parrish’s article Number Talks Build Numerical Reasoning. ↩
2. After the meeting, I was reminded that the word algorithm comes from the name of the Islamic mathematician, Al-Khwarizmi, who is often referred to as the father of algebra (another Islamic word). This podcast discussion of Maths in the Early Islamic World is fascinating.↩
Attendance: Linda, Rachel, Mark, Andrew, Greg, Eric, Singh, Phil, Kevin, Davida
Programs Represented: CUNY PD Team, CUNY Start, NYC DOE, District 79, CUNY LINCT to Success, NYCCT
RE: https://groups.google.com/forum/#!topic/cami/VISXuPTzNRw
It seems to me that this problem has something to do with multiples of 9; beyond this fact I haven’t a clue.
This is terrific! Thanks so much for sharing…I think my students will be fascinated by having the ways opened up for them. My ultimate question, though, which I will ask them, is do any of these methods help us do mental math?
I think a few of these (maybe all of them, in some way?) help with mental math. The area model certain helps me. I have used a mental image of it to multiply two digit numbers. Also, the method labeled E above is a representation of a mental way to multiply two-digit numbers. I created the image after reading a description of this way of multiplying.
I do think Amy’s comment below is important for us to think about in the classroom. Which of these models are most useful for our students? I agree that arrays and the area model is the place to start, especially since it can help us lead into a discussion of the distributive property, eventually multiplying polynomials.
Great connection to mental math! I agree that all of the methods could help develop that. Specifically, I think challenging yourself to multiply with any method that is not your most familiar method could be a way to strengthen mental math.
Thank you for sharing all of these ways to multiply–what a rich discussion! As I continue to read more and practice more with all of the methods, I am amazed at all of the math that can be uncovered. I am also learning more about the value of thinking flexibly with numbers.
I want to share how multiplication with area models is proving to be a solid building block as one student works through various math topics. I forget how we actually started using area models. We have used them as a way to explore percentages/fractions. (How do you shade 1/3 of 100 squares?) We have looked at division through the lens of area models. We are using them as we play Go Fish with Jo Boaler’s math cards that are an alternative to multiplication memorization flash cards. I recently used area models to show the “why” of order of operations like 2 x 5 + 3. Finally, I imagine this student will be surprised/happy when we start talking about area and she just already understands it.
In conclusion, explore all models of multiplying, but in your teaching, consider emphasizing the ones that can build knowledge and skills across topics. Which other multiplication methods do this? #MP7 🙂
I think you’re absolutely right. I was thinking we would have time to talk about which algorithms we would bring to the classroom and how, but there was so much to talk about in just understanding how they all work. I agree that area models are the place to start. I loved how you led us through a progression from arrays to a concrete gridded area model to an abstract area model at COABE ’14 (?). Like you said, there are so many ideas that can be woven in (order of operations, area, the distributive and associative properties, multiplying polynomials…).
CAMI folks, I highly recommend the webinar Amy and Connie Rivera did last year on area models for LINCS:
Uncovering Coherence in Area Models: https://www.youtube.com/watch?v=bYvmMcBSDlk&feature=youtu.be
And I learned that an MP7 is a German machine gun, along with being the abbreviation for…
CCR.Math.Practice.MP7 Look for and make use of structure.
In terms of this practice, I would emphasize the area model and the partial product method. I’m not really sure how to bring the others in, though I think the Egyptian and Russian Peasant methods are fascinating and certainly relate to binary numbers, distribution, in/out tables…
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“Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.”
Thanks for the clarification, Eric. I was referring to my favorite mathematical practice, #7. 🙂