- “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

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:

- Deneise:
- 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.

- Stephanie:
- 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

- Eric:
- 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)

or

(10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)** *** (10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)

or

55^{2}

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.

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

]]>During our introductions, we each shared our names, our program and something beautiful we’ve seen recently.

- Cynthia: An image at the end of a trilogy of books she just completed.
- Eric: Building a bike wheel – getting in all the separate spokes and fitting them into this one unified object.
- Michael: Members of his community together and finding connection in each other.
- Mark: Playing SET with my 5 year old daughter last night and watching her see a combination that I didn’t see and having this moment of appreciation that she is her own person, with her own perspective.

Since this activity is in part inspired by him, we looked at a Pier Mondrian painting and shared some of the things we noticed:

- Michael inferred a sense of flow and movement when you look at this – your mind tries to find a pattern and to make sense of the ambiguity
- Eric saw the lines inside the larger rectangle as “roads” and noticed that there are 9 instances where 4 roads intersect. Also that there are some intersections with 3 roads (a lot more than there are of 4)
- Cynthia noticed that the yellow rectangles are always adjacent to the white rectangles.
- Linda noticed that the rectangles around the edge/perimeter of the whole thing were open figures.

The first rule of this puzzle is that you have to cover a square with rectangles. Below are a few possible ways of covering a 10 by 10 square with rectangles:

Between the two arrangements above, the one on the right is the winner. Which is to say, the goal of this puzzle is to find the arrangement with the lowest possible score. Can you see how the scores are determined? Where do the 35 and 23 come from?

.

.

.

.

Scores are determined by the subtracting the area of the smallest rectangle in your arrangement from the area of the largest rectangle? On the left, the score is 35 because 42-7=35.

What would the score of this arrangement be?

Finally, we have an arrangement with an even lower score. But unfortunately, this arrangement is disqualified. The final rule is that no rectangles can appear in an arrangement more than once, so the 3 by 4 rectangle and the 4 by 3 rectangle are no good. NB: the two 10s in the arrangement above are fine because the dimensions of the rectangles are not the same.

Here are the collected rules of the Mondrian Art Puzzle:

- You must cover the square with rectangles.
- Every rectangle on the canvas must be different… so you cannot have both a 4×5 and a 5×4 rectangle.
- Your goal is to minimize your score. To calculate the score of any design subtract the area of the smallest rectangle from the area of the largest rectangle.
- When coloring, use as few colors as possible. The same color cannot touch along edges or corners.

Everyone had some time on their own to play around with different squares. Most of us used trial and error, covering the square with rectangle by rectangle until it was completely covered. This was a useful time for practicing the criteria – in particular, not repeating the dimensions of one of the rectangles took some time to get used to. Then as folks start to come up with arrangements with different scores, some potential strategies begin to emerge.

Below are Eric’s initial explorations with the 4×4, 5×5, 6×6 and 8×8 squares.

Here’s some of my work on the 6×6. In addition to the trial and error method, I tried to list all the possible rectangles that could fit and then use the total area of the square to try and find some :

After exploring on their own and then working together, teachers started putting up and coloring their lowest scoring arrangements for each square. (P.S. You don’t really know a person until you start to get acquainted with their preferred color pallets).

- As you create different designs for a given square, what patterns do you notice?
- As you find a low score for any of the squares, how could you know if it’s the lowest score?
- Did you find the lowest possible score for any of the squares? Find another arrangement rectangles that gives you the same score.
- Is there a way to figure out the lowest possible score for a given square
*before*covering it with rectangles? - Is it possible to predict the lowest score for a
*n*by*n*square? - Can you find the design for the lowest possible score of a square
*and*make your picture at least 40% blue? Try it with other percentages and other colors. - Are there any patterns you see in the number of colors needed for your designs?
- <<SPOILER ALERT>> Here are the lowest scores for the squares 4 by 4 through 32 by 32. Pick one and see if you can figure out the dimensions and arrangement of the rectangles that result in that score. Also, since they are just the lowest scores that have been submitted to Math Pickle so far, can any of them be beat?

If you create your own designs, color them in and send them to CAMI through email, linked in the comments below, or post them on Twitter at #nyccami!

Happy New Year!

In attendance: Cynthia, Michael, Eric, Maggie, Linda, Spencer, Mark

Programs represented: Literacy Assistance Center, York College’s Adult Learning Center, LaGuardia Community College, Pathways to Graduation (District 79), and the CUNY Adult Literacy/HSE PD team.

]]>

- Lesson Plan: Mark’s Metrocard – A 3-Act Math Task
- Act One Video: http://bit.ly/marksmetrocard
- MTA fare information – March 2017 (abridged)
- Act Three – pre-valued MTA cards
- Act Three Video: http://bit.ly/marksmetrocardact3

We started by watching a short video. We didn’t have access to a projector so we huddled around a tablet. The small size of the video made it difficult to see some of the numbers, which created some challenges later when the group found itself in disagreement about the value on the Metrocard. In the end, we watched the video three times.

Video: Mark’s Metrocard

*Share your question with a partner. See if it is the same question or a different question.*

- What is the valuable info? What’s the catch?
**How much was left on the card after all the swipes?****How many swipes were there? How many trips for $20?**- Was it coming or going?
- Was he actually traveling?
- Why was he charged $21 when he chose $20?
- Was it the same person?
- Does he get a bonus for $20? What is it?
- Is he really getting $21 on the card?
- How is he getting the extra dollar?
- Does he get 6 or 7 rides?
- Does he get a bonus ride?

Participants noticed that Mark chose $20 in value to put on the card, was charged $21 because of a new card fee, but got $21 in value on his card, which was proved by the fact that he had $18.25 on the card after one swipe. However, one group explained this by saying that the first trip cost only $1.75. There was a lot of discussion about how much value Mark had on his card when he left the MetroCard machine and went to the turnstile the first time.

*I asked the group to focus on the questions in bold to start. Before trying to answer them, though, I asked what information they wanted from me.*

- What is the ratio for the bonus? (I answered this by saying that there is a 5% bonus for all amounts put on the card over $5.50.)
- What dollar amount bonus do I get for $19? (I didn’t answer this question.)
- Is the bonus growing linearly or exponentially? (I didn’t answer this question either.)

I made a decision not to hand out the MTA fare info sheet yet because it seemed that the group had all the information they needed to answer the current questions.

*I then gave the group 7 minutes of individual problem-solving time. *

*When we came back together, Emerald explained her method of answering the questions.*

Emerald knew she was starting with $21 because 5% of $20 is $1, which is added to the value of the card. She could also confirm that it was $21 at the start since after taking away the first 2.75, there was 18.25 left over. If you subtract 2.75 from 21 seven times, you have 1.75 left over. She then counted the number of times she subtracted 2.75 to find there were 7 trips.

I think this could be the end of our work on this problem, but I had a concern I wanted to share with the group. I asked the group what other questions they had. And shared my concern. This led to a new…

*What about that $1.75? That doesn’t bother you?*

- How much are you actually paying per swipe? With the bonus, with the new card fee…
- Are we really getting a bonus? Since they give 5% on $20, then take it away with the new card fee…
- Is it possible to break even? There was skepticism about whether it is even possible to add money so that you don’t have value left over on the card.

I asked the group to focus on these two questions:

*What amount should Mark have put on his card originally?*

*OR*

*What amount should Mark add to $1.75 on his card?*

I gave out an abridged MTA fare information sheet that explains the 5% bonus, new card fee, and allowable increments of new value ($.05).

*I asked the group to work individually for 10 minutes on these questions, then to work in pairs.*

We were starting to run out of time, but the group did get to a couple solutions:

Q: How much should Mark have put on the card originally? A: $55. (Looking for when the 5% bonus gives us exactly one extra ride. 55*.05 = 2.75. 55/2.75 = 20. $55 plus bonus results in exactly 21 trips.)

— Eric Appleton (@eappleton) November 9, 2017

Q: What amount should Mark add to the $1.75 on his card? A: $1.00 (Forget the bonus. Who cares about losing a nickel?) I’m not sure if I’m satisfied by this…

— Eric Appleton (@eappleton) November 9, 2017

Open questions:

- Is there an amount smaller than $55 that can be put on a MetroCard so that it doesn’t result in leftover money after all the swipes?
- Is there an amount larger than $1 that can be added to Mark’s MetroCard value of $1.75 so that there wouldn’t be leftover money the next time?

At the end I showed a short video that demonstrates the MTA’s MetroCard Calculator. I also shared cards with information about the MTA’s pre-valued cards with value amounts that swipe to zero, accounting for the 5% bonus.

- Act 3 video – Mark’s Metrocard (http://bit.ly/marksmetrocardact3)
- MTA’s pre-valued card info

In preparation for this meeting, I spent the last few weeks asking for feedback and solutions from friends outside of CAMI. Open the MetroCard Problem Google Doc to see my attempted solution of the MetroCard Problem and interesting approaches from other people, including use of graphing calculators and spreadsheets. Feel free to comment with your own solution strategies.

- Extensions to the MetroCard Problem
- What if you had a card with ___ in value on it? How much money could you add so that none would be wasted?
- Imagine this situation: When Mark gets down to $1.75 on his card, he adds another $20 in value. How much is on the card now? If I swipe, swipe, swipe until all trips are gone, is there a remainder? If so, add another $20 in value. How long will it take to get to an empty card?
- My friend David adds $40 each time he sees “insufficient funds” when he swipes at the turnstile. If he starts by spending $40 on a new card and adds $40 every time he runs out, how long (how much money, how many refills) will it take him to get to 0 value on the card?
- Why do you think the MTA created a system with such messy remainders? Can you think of a better system to collect money and give bonus for larger purchases?

In attendance: Maggie, Maryam, David, Deneise, Emerald, Eric

Programs represented: CUNY Adult Literacy/HSE Program, Queens EOC, RiseBoro, York College

]]>Here’s a framing question I posed: *Which of these puzzles and problems do you think could work (with modification possibly) in an adult education math class?*

I started with a concrete area activity inspired by Connie Rivera and Amy Vickers’ webinar on area models. I gave out 36 tiles to each table and asked the following question:

We came up with the following rectangles along with a question about whether orientation matters when deciding if a rectangle is unique. For example, is 3 x 12 a different rectangle than 12 x 3?

I then asked the group to look at the following and write down what they noticed.

After a few minutes, I asked them to share at their tables and then as a group.

The bottom left “stacked division” or “cake division” was new to everyone. The group noticed that it worked best when primes were used as the divisor.

The group noticed that the tree diagrams and cake division produced the same primes, though different non-prime factors showed up in the different methods.

Next, I introduced the area model puzzle:

- Keep this next part secret. Pick a pair of two digit whole numbers and create an area model for multiplying them. Label the four partial products but not the sides.

- Exchange your creation with a classmate and see if you each can figure out the two-digit numbers the other person multiplied to get those partial products.

We briefly considered the following questions, which might be good for classroom discussion:

- What strategies are useful for solving an area model puzzle?
- What strategies are useful for making hard area model puzzles?

Benjamin Dickman, who thought of the puzzle, shared some extensions on Twitter:

& find all sets of part’l prods with 3 two digit solutions (the max). Solved 1st by @MathFireworks! See Fig 5 in https://t.co/vgXwWZCPVd 2/2

— Benjamin (@benjamindickman) October 14, 2017

We then moved on to the main problem I brought:

*Consider the operation of counting the factors of a whole number (including 1 and the number itself). You might think of this as a function that counts factors. For example, the number 6 has the factors 1, 2, 3, and 6. If 6 is the input, 4 is the output. The function *d* of 6 might be written as *d*(6) = 4.*

I asked the group to start by thinking of questions and/or conjectures individually, then asked them to share at their tables. Here are some of the early conjectures and questions:

- Primes always have two factors.
- Squares have an odd number of factors.
- Which numbers have the most factors? What are the properties of these numbers?
- Is there a function? What is it?
- Can we use combinations or permutations to find the number of factors from a prime factor tree?
- Which numbers have the most factors? Multiples of 12 have lots of factors.
- What does the graph of
*d*look like? What patterns do you see in the graph? - What is
*d*(192)? Is it 12?

Tables worked on different questions for about 30 minutes, then I started to share the following table from a handout I prepared (available for download at the link above). My hope was that the table would point teachers in the right direction to derive the function, but wouldn’t tell them what it was.

*Complete the prime factorization of at least 4 integers. Add the exponents of the prime factors to the table below. (Add more columns if necessary.) What do you notice?*

integer | prime factorization |
exponent of 1st prime factor |
exponent of 2nd prime factor (if there is one) |
exponent of 3rd prime factor (if there is one) |
number of factors |

36 |
2^{2}3^{2} |
2 |
2 |
9 |

Here are few entries from Mark’s table, with additions from the group:

integer |
prime factorization |
exponent of 1st prime factor |
exponent of 2nd prime factor (if there is one) |
exponent of 3rd prime factor (if there is one) |
number of factors |

36 |
2^{2}3^{2} |
2 |
2 |
– | 9 |

24 | 233 |
3 | 1 | – | 8 |

18 | 3^{2}2 |
2 | 1 | – | 6 |

12 | 2^{2}3 |
2 | 1 | – | 6 |

We noticed that 18 and 12 both have 6 factors. They also both have 2 and 1 as exponents in their prime factorization: *3 ^{2}*

One big outstanding question: What is rule for function d? Can it be derived from the table?

In attendance: Ramon, Mauricio, Linda, Maggie, Solange, Maritza, Jeremy, Mark, Lionel, Eric

Programs represented: BMCC’s Adult Basic Education Program, NYC College of Technology’s Adult Learning Center, LaGuardia Community College’s Adult Basic Skills Program and CCPI, York College’s Learning Center, Literacy Partners, and the CUNY Adult Literacy/HSE PD team

]]>We started with a little estimation to warm-up, watching the following video:

We shared some noticings and wonderings, and then we took about 45 seconds to come up with our best estimate as to how many boxes of girl scout cookies were loaded into the back of this minivan.

The range was 144 to 1200, with Solange coming the closest with 840.

Then as a way to draw out background knowledge and misconceptions, we looked at the following figure and in pairs, took turns sharing observations about it:

Some of the things that came out:

- You can see it as 7 two-dimensional shapes (2 right triangles, 1 rectangle and 4 trapezoids)
- As a three-dimensional shape is has: six sides (two square sides, 4 rectangle sides), volume (length x width x height)
- the sides we refer to as the length , width and height might be different if we rotate the figure

Then we looked a situation involving rectangular prisms.

The available space in the container measures 7’8” wide by 7’10” tall by 39’6” long. The boxes are all the same and measure 24” by 18” by 36”. You can arrange the boxes any way you want in the shipping container.

Teachers used different methods to get to the maximum numbers of boxes if all of the boxes are oriented in the same direction. Some people figured out the 6 different ways to position the box and then tried filling the measurements of the shipping container with each. Doing we found there are two ways to fit 195 boxes into the container.

As teachers tried the different box orientations, they noticed that there was missing/unused space for each of them. Ramon (and for a while Jane) both focused on calculating which orientation results in the least unused space.

Mark gave out a few push questions to extend the problem into using more than one positioning of the boxes.

Can we fit more than 240 boxes into the shipping container? What is the maximum number of boxes and how would we know that we had the maximum?

This lesson is part of the CUNY Careerkit Project, a set of teaching resources exploring ten different industries. This problem comes from the soon to be available manufacturing careerkit.

In attendance: Solange, Jane, Deneise, Ramon, Jeremy, Eric, Mark

Programs represented: BMCC Adult Basic Education Program, PCACP, CUNY Adult Literacy/HSE PD Team

]]>Steve explained that we should imagine this as a spinning wheel with equal odds of landing on any one of the numbers. We talked together to get some understanding of how the game would be played. We also counted the different sections and noted the number in each category with a total of 54 sections:

Category |
Count |

1 | 23 |

2 | 15 |

5 | 8 |

10 | 4 |

20 | 2 |

Joker | 1 |

Tropicana | 1 |

Steve then shared the following payoff information:

Here’s another representation of the board and the payoff table (available for download above):

Eventually, we understood how the game is played:

- The player makes a bet by putting some money on one of the categories (one, two, five, ten, twenty, Tropicana or Joker).
- The wheel is spun.
- If the wheel stops on the category you chose, you win.
- The amount of money you collect depends on the payoff. If you chose one, you get the same amount you bet. If you chose two, you get double what you bet. If you chose five, you get five times what you bet, and so on.

After some more discussion about how we might start to calculate the odds and consider how payoff would affect our decision, we worked to answer some of these questions:

- What is the best (least bad) bet on the wheel?
- What is the house edge for each category?

Location: Turning Point, 423 39th St., Brooklyn, NY

Attended: Steve, Esther, Eric, Raheem, Stephanie, Stephen, Mark, Leo

Programs represented: CUNY, Metropolitan Detention Center, ParentJobNet, Touro, Turning Point

For full lesson plans on how to teach both a scaffolded and more open approach to developing algebraic thinking through visual patterns, see Unit 8 of the CUNY HSE Math Curriculum Framework.

]]>Usha (in italics):* So, the word, “consecutive.” What does it mean?*

One after the other. 1, 2, 3.

*Can you give me another three consecutive numbers?*

2, 4, 6.

*Ah, these are consecutive even numbers, but for our purposes here, these are not consecutive numbers.*

*Another three?*

10, 11, 12.

*Can someone give us five consecutive numbers?*

14, 15, 16, 17, 18.

Usha handed out the first page of the worksheet on consecutive numbers (attached above) and asked us to work individually for a while. The worksheet contains some samples to get you started. One and 2 are two consecutive numbers that sum to 3. Three consecutive numbers that add to 6 are 1, 2 and 3.

A few questions/observations came up:

- Are negative numbers allowed?
*I’m not going to answer that right now…* - Do they have to be whole numbers?
- Most of the spaces stay blank.

Usha stopped us after about 5 minutes. *Are there any patterns that you’re noticing? Don’t share them yet, but note what they are.*

Usha handed out the 2nd page and then the 3rd page of the worksheet as we continued to work independently.

Usha then gave us a few minutes to see if we could express the patterns we saw as a rule.

*I want to interrupt with a story from the history of mathematics. Do you remember one of those students who always had the answer and kept raising their hands? This is a story about a young boy named Carl Gauss who lived in Germany. His teacher got exasperated with him and wanted to give him something to keep him busy, so he asked him to add up the numbers 1 through 100. Gauss lined up the numbers and saw pairs of numbers that added to 101 (1 + 100, 2 + 99, etc.) and there were 50 pairs.*

*End commercial break.*

(There are many versions of this Gauss story. See link, link, link, link)

Usha*: Does Gauss’ idea help you add the numbers 1 through 6 more quickly? *

Some no’s & some yes’s*.*

Usha: *Just a side note: I am protective of quiet students. I think it’s important to include quiet, reflective time for everyone to try their own ideas. We also know that exciting ideas can come out of collaboration. How do we balance this as teachers?*

*I think what I’ll do is ask, how many people are ready to work with a partner? Who wants to keep working on their own? If you’re ready to work with a partner, move over to this side of the room. Everyone else can stay where they are and keep working on your own.*

As people were looking for patterns on their own or in groups, Usha kept putting out questions, like:

*Look at the sums of three consecutive numbers. How can you predict if three consecutive numbers would add up to a number?**63. Are three consecutive numbers possible?*No’s & yes’s.*Think about it for a minute and we’ll come back together.**396?*- 84?
*123?*

Usha asked the rest of the participants to find a partner and start working together.

Question from the group: In a series of consecutive numbers, is the middle number always the average?

*What about the number 63? How many consecutive sums can you work out for 63?*

- Two consecutive numbers: 31, 32
- Three consecutive numbers: 20, 21, 22

Betty said to see if 63 could be written as the sum of three consecutive numbers she drew three lines:

___ + ___ + ___ = 63

She ignored the three and divided the 60 into three 20s. then she saw that she could write the sum of 63 as 20+21+22.

*What about 84? What would Betty’s method look like if we tried it with 84?*

Usha*: Let’s look at 63 together. Can it be made of 4, 5, 6 or 7 consecutive numbers?*

Stephanie’s explanation for how she found 7 consecutive numbers adding up to 63:

First method: I’m looking for the base number. The total is 63. 63 divided by 7 equals 9. Line up the numbers. Take 2 away from 9. Add 2 to 9. Take 1 away from 9. Add 1 to 9. Line up the numbers. 6, 7, 8, **9**, 10, 11, 12.

Second method:

n + (n +1) + (n +2) + (n + 3) + (n + 4) + (n + 5) + (n + 6)= 63

Add all the n’s to get 7n.

Add all the numbers to get 10.

7n + 21 = 63

7n = 42

n = 6 (So, 6 is the first number in the series: 6, 7, 8, 9, 10, 11, 12)

Stephanie’s second method added some clarity as to where the triangular numbers came from in Mark’s observations.

Mark used a series of equations to determine if 63 could be written as the sum of different numbers of consecutive numbers.

n + n +1 = 63

2n = 62

n = 31, n + 1 = 32

He followed a similar procedure for 3, 4 and 5 consecutive numbers. Then he noticed a pattern that allowed him to create the following expressions:

2 consecutive numbers –> 2n + 1

3 consecutive numbers –> 3n + 3

4 consecutive numbers –> 4n + 6

5 consecutive numbers –> 5n + 10

6 consecutive numbers –> 6n + 15

7 consecutive numbers –> 7n + 21

8 consecutive numbers –> 8n + 28

9 consecutive numbers –> 9n + 36

Mark pointed out that the constant in each expression is always a triangular number, one behind the number of consecutive numbers. For example, in the expression 4n + 6, 6 is the 3rd triangular number (1, 3, **6**, 10…). In the expression 6n + 15, 15 is the 5th triangular number (1, 3, 6, 10, **15**, 21…).

To figure out (1) if a number could be could be written as the sum of consecutive numbers and (2) what those numbers are, Mark used the following method:

Let’s say you wanted to know if there was a way to write 63 as the sum of 4 consecutive numbers. Create the expression as described above – so for 4 consecutive numbers, 4n + 6. Set that equal to 63 and solve.

4n + 6 = 63

4n = 57

Since 57 is not evenly divisible by 4, 63 can not be written as the sum of 4 consecutive numbers.

Let’s try 6.

6n + 15 = 63

6n = 48

48 can be divided evenly by 6. It gives us 8. 63 can be written as the sum of the 6 consecutive numbers beginning with 8 (8+9+10+11+12+13 = 63)

Solange noticed a pattern when it came to which numbers could be produced by the sums of consecutive numbers. For example, consider the numbers that can be written as the sum of two consecutive numbers: 1, 3, 5, 7… every second number. Now consider the numbers that can be written as the sum of three numbers: 3, 6, 9… every third number. Now, look at the numbers that can be written as the sum of five consecutive numbers: 5, 10, 15… every fifth number. So for all odd numbers, Solange realized she could determine if a number can be written as a sum of n consecutive numbers by testing to see if the desired number can be evenly divided by n. So for example, 63 can be written as the sum of seven consecutive numbers because 63 can be divided evenly by 7.

Solange noticed a slightly different pattern when it came to even numbers of consecutive numbers. It partly followed the same pattern as the odds, which is to say that the numbers that can be written as the sums of four consecutive numbers are 2, 6, 10, 14 (i.e. every 4th number). Similarly, the numbers that can be written as the sums of six consecutive numbers are 3, 9, 15… (i.e. every 6th number). The difference from the odd is the starting point. The first number that can be written as the sum of three consecutive numbers is 3. The first number that can be written as the sum of five consecutive numbers is 5. Not so with the evens. The first number that can be written as the sum of two consecutive numbers is 1. The first number that can be written as the sum of four consecutive numbers is 2. The first number that can be written as the sum of six consecutive numbers is 3. Solange saw that, for evens, she needed to adjust her method to take into account the “starting number”. She set up the following expressions to test if you want to know if a particular number can be written as the sum of an even number of consecutive numbers:

- To see if a number can be written as the sum of two consecutive numbers, use x+1 over 2.
- To see if a number can be written as the sum of four consecutive numbers, use x+2 over 4.
- To see if a number can be written as the sum of six consecutive numbers, use x+3 over 6.

When x = the number you are testing

Throughout the meeting, Usha added extension questions:

*Can you find 27 consecutive numbers that add to 63?**Can you think of 4 consecutive numbers that add to 2? (*This forced us to use negative numbers.)*Can you think of a general formula that would work for any sum and any number of consecutive numbers?*

In closing the meeting, Usha asked if this problem would work in our classrooms:

- In order to get started, students don’t need to know any algebra. They just have to add numbers and look for patterns.
- This problem is accessible to students at any level and students can really engage with the math of the problem at their own pace.
- To simplify the task, Usha suggested that the worksheet could be modified by removing some of the columns on the right if we think it would be too intimidating for some of our classes.

In Attendance: Lucinda, Eric, Solange, Mark, Linda, Stephanie, Stephen, Betty,

Programs Represented: BMCC, CUNY LINCT to Success, CUNY Adult Literacy PD Team, LaGuardia Community College, ParentJobNet, YALP

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We started by looking at different rectangles…

*Imagine a rectangle with an area of 20 sq. cm. **What could its length and width be? List at least five different combinations.*

The instructions threw us a bit. The whole number factors of 20 are *1 x 20*, *2 x 10*, and *4 x 5*. That’s only three rectangles. Did they consider 1 x 20 different than *20 x 1*? That didn’t seem right. One is just a rotation of the other. Then they must be including fractional factors (is that the right terminology?).

Solange and Linda found *2 1/2 x 8*, *3 1/3 x 6* and *1 1/4 x 16*. We cut the rectangles out of graph paper and put them on the board.

Solange noticed that if you start with 2 factors (for example, *4 x 5*), multiply the first factor by 2 and divide the second factor by 2, you will get the same product. So, *4 x 2 = 8* and *5/2 = 2.5*, so *8 x 2.5 = 20*. We used this method to find other factors of 20, including fractions.

This brought us to a question: How does Solange’s pattern work in this series of multiplication problems? What do you think? Is it the same as the change from *4 x 5* to *8 x 2 1/2* to *16 x 1 1/4*?

4 x 5 = 20 6 x 3 1/3 = 20 8 x 2 1/2 = 20

*If you enlarge each of your rectangles by a scale factor of 2, what would their new dimensions be? What would their areas be? What do you notice?*

We talked a little about what it means to enlarge a rectangle by a scale factor of 2. We decided it meant to double the height and the width. We discovered that the area of the second rectangle was always four times the are of the original rectangle.

*What happens when you enlarge rectangles with different areas by a scale factor of 2? What if you enlarge them by a scale factor of 3? Or 4? Or 5 …? Or k? What if k is a fraction?
*

We found that tripling the height and width of a rectangle increased the area by 9 times the original area. Davida wrote a conjecture based on the scale factors of 2 and 3.

Davida’s conjecture: The amount of the increase of the dimensions correlates with the increase in size with relation to the original area. The increase amount is squared and multiplied by the original area to obtain the area of the new rectangle.

A scale factor of 2 resulted in an increase of 4 times the original area. A scale factor of 3 resulted in an increase of 9 times the original area. What about scale factor of 1.5?

Solange and Linda’s general rule: The scale factor squared times the original area gives you the new area.

*Do your conclusions apply to plane shapes other than rectangles?*

It looked like the general rule applied to triangles and circles.

Solange and Linda made a new conjecture: The general rule above applies to rectangles, triangles, circles, parallelograms, etc.

This brought up some questions:

- Will it work for irregular shapes?
- Will it work for a cross, an X, a T-shaped figure?

*Now explore what happens to the surface area and volume of different cuboids when they are enlarged by different scale factors. Do your conclusions apply to solids other than cuboids?*

We had time to explore the volume of rectangular prisms, but didn’t get to surface area or what would happen with other 3-dimensional figures (We really wished we had manipulatives such as snap cubes at this point in the meeting. Actually, tiles would have been really helpful when exploring area as well.)

In summary, we talk about how this lesson would be useful to help students understand conversions of square yards to square feet or square feet to square inches. For example, how many square inches are in a square foot? Our first instinct is to say, 12, of course. However, if you draw a square foot and then break it into square inches, you will count 144 square inches. In the terminology we used today, we might say that the scale factor was 12, so the increase of a 2-dimensional object would be 12^{2} times the original number. In truth, the figure is staying the same size, but the numbers used in the dimensions are increase by 12, so the calculation is the same. How many cubic inches are there in a cubic foot?

——-

Location: NYCCT, 25 Chapel St.

Attended: Eric, Davida. Linda, Solange

Programs represented: BMCC, LAGCC, NYCCT

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At the heart of both workshops was the Pyramid of Pennies, which was the first three-act math task we explored in CAMI back in May 2015.

- We started both sessions by asking participants to get into small groups and talk about what real-world math meant to them.
- Then we shared an answer that came out of that May meeting, before diving into the problem.

“Real-world math is the creative process – the creation of the problem is the math, rather than relying on what others have told you”

After we did the problem in 3-acts (the bulk of both workshops), we put another version of the problem up and asked participants how their experience would have been different if we’d simply given out a handout that said:

A pyramid is made out of layers of stacks of pennies. Each stack contains 13 pennies. The base layer is a square with 40 stacks on each side. The next layer has 39 stacks on each side. The top layer has 1 stack. How many pennies are in the pyramid?

Finally we shared a section from Mathematical Mindsets by Jo Boaler

According to Conrad Wolfram, working on mathematics has four stages:

- Posing a question.
- Going from the real world to a mathematical model.
- Performing a calculation.
- Going from the model back to the real world, to see if the original question was answered.

Students spend most of their time on step 3. What if we changed that?

The url link above will take you to a folder with all the materials for the workshops, including out presentation slides and facilitator notes.

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