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

]]>

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

]]>

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.

]]>We started with a brief conversation about Cuisenaire rods. Solange has used them to teach fractions and first encountered them in an ESOL class. The rods were invented by Georges Cuisenaire, popularized by Caleb Gattegno, and have been used since the 50’s. Fortunately, BMCC has multiple sets we were able to experiment with.

Then we looked at the Cuisenaire train rod problem:

We first brainstormed a few questions we were interested to answer:

*As we go up different Cuisenaire lengths, is there a pattern in how many ways you can make a train?**What is the most efficient way to arrange the “cars” in a “train”?*We defined the train as the full length (3 in the example above) and car as the rods (that may be different lengths) that make up the train. For example, the green train above has 1 car and the white train has 3 cars.*What if the shape were 2-dimensional: 2×2, 3×3, 4×4, etc? What if the shape were 3-dimensional: 2x2x2, 3x3x3, 4x4x4, etc?*This assumes that we are still building the shapes out of Cuisenaire rods.

We then spent about 45 minutes working on our own. Most people worked on the first two questions. A couple people explored the 2D and 3D questions. While people were working, I individually shared some push and support questions that I wrote, depending where people were in their investigations. I cut out questions as I needed them. (Later in the meeting, we revised some of these questions.)

Solange explains to Kevin her solution for this question: How many trains of length 10 can you make with Cuisenaire rods *without *using length 1 rods?

Andrew works on the 2D question.

We came together for the end of the meeting and explored Gregory’s organization of the possible trains of length 5. He described it as an alphabetical or dictionary approach. He organized the possible trains by the number of cars in the train. There is 1 train of length 5 (the yellow rod). Then there are 4 trains made of 2 cars. The first car can be 1, 2, 3 or 4. The second car in each train will be whatever is needed to add up to 5. The third row shows the 6 trains made of 3 separate cars. Ruben came up to use Gregory’s method to show the 4 possible trains made of 4 cars. Finally, there is only 1 train that can be made of 5 cars (white, white, white, white, white).

Then Solange and Kevin explained their solution to how many trains are possible of length 10 if you don’t use the white pieces. The three check marks above identify the 3 trains of length 5 that don’t have white pieces. I’ll leave out their solution in case you want to find it for yourself. It’s really surprising!

Finally, Kevin showed us a connection to Pascal’s triangle in the train combinations. I’ll leave that for you to find as well. Check the links for classroom resources and articles about the train problem.

——-

Location: BMCC, 25 Broadway

Attended: Andrew, Eric, Gregory, Kevin, Linda, Ruben, Solange

Programs represented: BMCC, CUNY LINCT, CUNY Start, LAGCC, Hostos Community College

]]>

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.

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.*

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 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 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)

I asked the group what they thought of when they heard the word *algorithm*^{2}. 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.

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.*

Each group shared their chart paper demonstrations. We discussed the various methods in this order (A, B, C, D, F and G).

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

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.

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.

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, 18
*0*, 10*0*and 6*00*). - 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).

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 (2
^{n}).

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

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*A dizzy sailor is standing on a 15×15 square tiled board. From their initial square they are able to move to any square sharing a common side. Due to the sailor’s dizziness, after every move they immediately make a left or right turn before repeating this process (that is, they are never able to enter and exit a square in a straight line). What is the largest number of squares the dizzy sailor can walk on if they are not allowed to repeat squares and the last step of their path must end at the square they started at?*

Before we started working on the problem, Solange had us talk about what we noticed. This way she made sure everyone understood the situation being described and all of the conditions.

Here’s what we came up with:

After working on the problem, we shared some questions of what we were wondering:

- Can the dizzy sailor walk in a perfect square?
- What is the math beyond trial and error that can help me here?
- Can the dizzy sailor walk in a rectangle? (if so, can we find the area and perimeter)
- What is the fewest number of squares the dizzy sailor can walk on?
- What if we started with a smaller grid?
- What if we all started with the same starting point?
- What if the tiled board was smaller?
- What if the tiled board had even dimensions?

Both groups came up with a similar question about using smaller boards to look for a pattern. This problem really illustrates the problem-solving strategy of solving a similar and simpler problem. This was the approach Solange used and she gave us a sheet with several grids from 2 by 2 squares to 15 by 15 to use to support our further exploration.

Solange gave each group a different arrangement of squares (see docx files above) and asked the following:

*Can you add squares to this figure to make a new figure with a perimeter of 18? (Each square must share at least one complete side with another square. Trace or draw the shape that you make.)**Consider any figure made of squares where each square must share at least one complete side with another square, what is the minimum number of squares required to build a figure of perimeter of 18?**Under the same conditions, what is the maximum number of squares possible to build a figure of perimeter of 18?**What if a square didn’t have to share a complete side with another square? Would your minimums and maximums change?*

The word docx above includes:

- The Dizzy Sailor Problem
- Blank grids (2×2 to 15×15)
- Two versions of the perimeter of 18 problem

The Supplemental Readings include:

- When Halving is Not Halving: Exploring the relationship between area and perimeter
- Maximum Area of a Rectangle with Fixed Perimeter (from Ask Dr. Math)

At the end of the meeting Solange told us that she had been thinking a lot about the kinds of math we do that challenge us as problem-solvers and the kind of problems that engage our students in productive struggle. Sometimes the same math problems can achieve both but often at CAMI we work on one kind or the other. Her goal in selecting the problems we worked on was to try and do both, so that we were all engaged in a problem and we also had some time to work on and talk about a great, open perimeter/area problem that we could use with students.

Attendance: Stephanie, Meghan, Lionel, Eric, Brian, Solange, Bree

Programs Represented: Literacy Partners, BMCC, Lehman College, CUNY Central

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