I once taught a poem by Wallace Stevens called “Thirteen Ways of Looking at a Blackbird” to a class of adult literacy students. Before I gave out the poem I put the title on the board and asked students what they thought the poem was going to be about. They had all kinds of ideas about looking at blackbirds. Then I asked them, “What about the first part? What does that mean Thirteen ways of looking at a blackbird?”And they said things like:
- “Thirteen ways to understand the bird is better than one… but you have to take time to see the bird.”
- “If you look at the bird you will find all the different things it does, but you have to look closely.”
- “We don’t pay attention to these things and he wants us to focus.”
- “He stopped to pay attention to something so maybe we will too.”
Today’s meeting was led by Andrew Sanfratello, who is a Mathematics Curriculum & Instructional Specialist for the CUNY LINCT College Transition Program. I was reminded of teaching the Wallace Stevens poem when Andrew said he’d been thinking about the difference between what it means to memorize the multiplication table and what it means to know the multiplication table.
Much of what Andrew led us through is based on the work of Benjamin Dickman:
- Dickman, B. (2014). Problem Posing with the Multiplication Table. Journal of Mathematics Education at Teachers College, 5(1).
- Dickman, B. (2014) Conceptions of Creativity in Elementary School Mathematical Problem Posing (Doctoral dissertation)
- a March 2016 colloquium talk at Teachers College, Columbia University called “Designing Cognitively Demanding Multiplication Tasks: Number Theory for Teachers”
- a Spring 2017 Math for America minicourse, “Problem Posing in Number Theory: Task Design by Teachers and Students” (Link includes course materials)
Ben currently teaches at the Hewitt School in NYC.
Andrew started us off by showing us a multiplication table and asked if we had been required to memorize the table in school. Some of us had and some of us learned it as adults.
He explained that this is called an operation table. In this case, the operation is multiplication, but it doesn’t always have to be. It could involve addition, subtraction, division or other operations.
Here’s are some examples of other operation tables Andrew shared:
What do you notice? What do you wonder? How could any of these alternative operations tables be used with students?
Andrew supplied some vocabulary for the components of an operation table. The operation used is placed in the top left corner. The numbers in the grey area represent inputs, across the top and the other vertically on the left side. In the center the white squares are called outputs. So on the multiplication table, the inputs are the numbers you multiply to get the output (or the product).
Andrew pointed out that the horizontal inputs are considered first and the vertical ones are considered second. So, you choose an input from the top, then insert the operation, choose an input from the left side, and finally read an output from the white boxes. Example: 5 x 3 = 15
The Multiplication Table
For the remainder of the meeting we searched for the new in the familiar, focusing on multiplication. Andrew gave us a blank operation table and asked us to fill it in as a multiplication table as quickly as possible. He gave us about 5 minutes and then asked us to share with a partner how we filled out the table. What order did we go in? What sequences did we use? A few of us moved horizontally counting by 2s, 3s, 4s, etc. Some started by writing in all the square numbers going diagonally (1, 4, 9, 16, 25, 36…).
Andrew emphasized that there are many different ways to fill in an operation table. He then modeled his process for filling out the table.
- Write the operation in the top left corner.
- Fill in grey areas by writing the inputs across the top, then the inputs vertically down the left.
- Moving across, fill in the first row of outputs completely.
- Moving down, use the same numbers to fill in the first column of outputs completely.
- Fill in the 2nd row, not including the first box which is already done.
- Using the same numbers going down, fill in the 2nd column, not including the first two boxes which are already done.
- Continue filling in one row horizontally, then one column vertically at a time until the table is complete.
Andrew’s way of filling out the table focuses on the diagonal of square numbers and emphasizes that there is symmetry in a multiplication table. Going horizontally then vertically is efficient because the numbers repeat. This is because multiplication is commutative (the order of the numbers multiplied doesn’t change the product). For example, 5 x 3 produces the same product as 3 x 5. This wouldn’t necessarily be true with other operations. For which operations does order matter?
Andrew also pointed out a difference between how he recommends filling out a multiplication table and an addition table. What do you notice from these excerpts?
Andrew made the point that most operation tables, like the two examples above, start at the identity. Within an operation, the identity is the number that doesn’t change the other number. For multiplication, the identity is 1. For addition, the identity is 0.
- 5 x 1 = 5 and 1 x 5 = 5
- 5 + 0 = 5 and 0 + 5 =5
- 5/1 = 5, but 1/5 does not equal 5. (There is no identity under division.)
- 5 – 0 = 5, but 0 – 5 does not equal 5. (There is no identity under subtraction.)
Andrew posed a question for us:
Look at the white squares in the 10 x 10 multiplication table. How many of those are odd numbers? Don’t count. Give me a quick gut estimate.
Here were our guesses as a group: 50, 25, 20, 49.
Andrew then asked us to do some work to find out how many odd numbers there are. Eventually, we found out that 25 of the 100 squares are odd and came up with the following explanations:
- In the first row, 5 outputs are odd.
- If the row input is odd, then the row outputs include odd and even numbers.
- If the row input is even, then all outputs in the row are even.
- Any number multiplied by an even number is even. (Tangential question: Is 0 even or odd?)
- Odd x even = even
- Even x odd = even
- Even x even = even
- Odd x odd = odd
- 1 out of 4 of these is odd. This is similar to 25 out of 100.
- Look at the first four squares of the multiplication table. One out of 4 squares is odd. This pattern repeats. 1 out of every 4 would equal 25.
- Another way to see Stephanie’s pattern is to look only at the first 4 squares:
Andrew then posed another question:
How many different numbers are there in the 10 x 10 multiplication table?
After some discussion, we understood this question to mean, how many different outputs are there in the white squares? Andrew took some guesses and again let us work individually to come up with an answer.
After some small group work, Mark presented a solution that used the symmetry of the multiplication table.
- You can eliminate all the numbers to the left of the diagonal of perfect squares, since these numbers are repeated. This leaves you with 55 numbers, since 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55.
- There are 13 numbers are repeated that we need to cross off.
- 13 from 55 leaves us with 42.
Outstanding questions: What do the repeated numbers (4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, and 40) have in common? Is there a way to predict how many outputs will be repeated? Can we generalize this solution for a larger square?
Andrew then posed a bunch of questions for us to explore in pairs. He gave us a bunch of questions and let us decide where to start:
- Add up all the outputs on the table. What is the total?
- How many 2-digit numbers are there?
- Which numbers are missing? Why?
- How many one-syllable numbers are there? Two-syllable? Three-syllable? (Is it different in other languages?)
- How many times does the digit 4 appear? What about 5?
- What patterns appear on the diagonals?
After some time working in small groups, we looked at the first question together:
What is the total if you add up all the outputs in the 10 x 10 multiplication table?
Melissa looked for shortcuts to calculate a sum of all the outputs in the table. She started by finding sums of the rows and discovered a few patterns to complete the sums in the table without adding each number:
1st row = 55
2nd row = 110
3rd row = 165
4th row = 220
The sums in the even rows start with multiples of 11 and end with zeros. The sum of the 2nd row is 110, the sum of the 4th row is 220, so the sum of the sum of the 6th row would be 330.
In the sums of odd rows, the tens digit and the hundreds digit are each going up by 1 and the final digit is always 5. The sum of the 1st row is 55 and the sum of the 2nd row is 165, so the sum of the 3rd row would be 275.
Melissa then explained that she realized that 55, 110, 165, 220… are multiples of 55. 55 is one group of 55, 110 is two groups of 55, 165 is three groups of 55, and so on.
So, the sum of each row is 55 x 1, 55 x 2, 55 x 3, etc. Since 55 is being multiplied by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, you can sum all these numbers and multiply by 55 to find the total. So, Melissa reasoned that the sum of the outputs would be 55 * 55 = 3,025
Gregory responded to Melissa’s explanation with a noticing about what happens with multiples in each of the rows of outputs. He used the 7 row as an example:
Each product in the 7 row is a multiple of 7, so therefore you could think of the sum of that row as an equation.
(7 x 1) + (7 x 2) + (7 x 3) + (7 x 4) + (7 x 5) + (7 x 6) + (7 x 7) + (7 x 8) + (7 x 9) + (7 x 10) = 385
This can be rewritten as:
7(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) = 385
The sum of the numbers in parentheses is 55, so the final calculation is
7 x 55 = 385
which is the same multiple Melissa found for the seventh row.
Continued for all the rows, it might look something like this:
- 1(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 2(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 3(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 4(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 5(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 6(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 7(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 8(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 9(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
- 10(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)
(10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1) * (10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)
As a problem-solving strategy, Eric focused on a simpler, similar problem and built his way up. He looked at the sums of the outputs of different sized multiplication tables:
There seems to be a pattern in the sums of the tables. What would be the sum of the 4 x 4 multiplication table? Why? If you continue this method, what do you predict was the total Eric got for all the outputs on a 10 x 10 multiplication table?
In trying to make sense of the pattern Eric noticed, Mark drew the following:
Mark noticed a few of the things:
- The sums of all of the multiplication tables as you build up to the 10 x 10 table all come from the diagonal of perfect squares – 9, 36, 100, 225, 441, 784, 1296, 2025, 3025.
- Those sums are the squares of the triangular numbers! (1, 3, 6, 10, 15, 21, 28, 36, 45, 55…)
Mark posted some pictures from the meeting on Twitter and Benjamin Dickman saw something interesting going on in the “elbows” of the different squares he drew and contributed the following:
My guess?: in the 36 elbow, you have: (6+12+18+24+30)+(30+24+18+12+6)+36. Pair respective numbers in the two sets of parentheses for (36+36+36+36+36)+36 = 36*6 = 6^3. So the ten elbows add up to 1^3 + 2^3 + … + 10^3. Equating @eappleton‘s approach and Melissa’s formula yields: pic.twitter.com/D2xKsup5DN
— Benjamin Dickman (@benjamindickman) January 11, 2018
Which inspired the following attempt to show this connection with colors. (I’m always looking for a chance to open my recently purchased box of 64 crayons.)
Let’s focus on the elbows… we start with 1. Then there are two 4s, three 9s, four 16s, five 25s, etc. If written out, that pattern might look something like this, which shows that the sum of the cubes of the numbers 1-10 is equal to the square of the sum of the numbers 1-10.
Additional Questions to Explore
One of the things that was so great about this meeting was the feeling that we could just keep exploring and finding all of these new and amazing things in something we’ve been seeing but not really looking at for most of our lives.
What made today’s work especially productive was the types of questions Andrew put out there. In that spirit, here are some more paths to follow from Ben’s work linked above:
- Find the only 49 in the multiplication table. Add to it the six numbers above in its column, and the six numbers to the left in its row. What is the total?
- Without thinking too much, are there more multiples or 3 or non-multiples of 3 in a 10 x 10 multiplication table? Once you have an answer, see if you were correct.
- Using a multiplication table, explain what it means for a number to be “prime”?
- Look at the numbers on the south-east diagonal starting from either 2 entry in the 10 x 10 times table: 2, 6, 12, 20, 30, 42, 56, 72, 90. What pattern or patterns do you see?
- Starting at the 3 in row one of the multiplication table, jump like a knight (from chess) to move one square down and two to the right. Doing this repeatedly, you get the following numbers: 3, 10, 21, 36. What pattern or patterns do you see?
- What kinds of symmetry can you find in the 10 x 10 multiplication table?
- In the 10 x 10 multiplication table, the number directly to the right of x is x+8. What is the next number over to the right?
- How would you extend the 10 x 10 multiplication table to cover the negative numbers?
- Which numbers are missing from the 10 x 10 multiplication table and why?
- The numbers in the 2 row are called “even.” What would you call the numbers in the 5 row?
- Can the same number appear more than once in a single column?
- How many one syllable number names are found in a 10 x 10 multiplication table?
- Think about dividing a number in the multiplication table by the one directly to its left. When does this result in a whole number?
- Choose a column or row and write a short story about the numbers in it.
- Which row of the multiplication table is the hardest to remember and why
- If you were trying to fill in a blank multiplication table and got stuck, what would you do?
In Attendance: Deniece, Eric, Melissa, Greg, Stephanie, Mark, Raheem, Andrew
Programs Represented: Metropolitan Detention Center Brooklyn, CUNY Start, the Manhattan EOC, ParentJobNet, CUNY LINCT, CUNY Adult Literacy and HSE Program
Respectfully submitted by Mark and Eric