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Event: Pentominos: Introducing Algebra Through an Arithmetic Puzzle

Tomorrow we will be joined by Heidi Schuler-Jones, our guest expert and presenter, to discuss very innovative and effective ways to develop algebraic habits of mind through thinking about computations independently of the numbers used.

To get the most out of our upcoming three-day discussion, “Introducing Algebra through an Arithmetic Puzzle,” you are invited to and encouraged to explore the following resources:

  1.  How to use Push & Support Cards since these will be provided during the activity.

    • Scroll down to and read “Before Using Push and Support in Your Classroom” and “How to Use Push and Support in Your Classroom”.

    • This will prepare you to use the Pentomino Push and Support Cards during the activity.

  2. The Fostering Algebraic Thinking Toolkit: A Guide for Staff Development https://files.eric.ed.gov/fulltext/ED476802.pdf. Participants could consider it supporting material to extend their conversations during the three days and as they implement the activity and similar resources into their classrooms.
  3. The following Mathematical Practices will be particularly relevant to the activity:

    1. Make sense of problems and persevere in solving them. (MP.1) .
    2. Reason abstractly and quantitatively. (MP.2)
    3. Construct viable arguments and critique the reasoning of others. (MP.3)
    4. Model with mathematics. (MP.4)
    5. Look for and make use of structure. (MP.7)

This specific, hands-on” discussion will focus on integrating learning through the following objectives that will guide our dialogue:

 1. Analyze and examine activities and lessons that develop algebraic habits of mind

2.  Create and share noteworthy practices and problems for participants to share different strategies and experiences. 

3.  Share suggestions on how to assist learners who might struggle or need more challenging work to complete.

I cordially invite you to begin exploring the resources and invite you to join in our discussion with Heidi, a long-time teacher, and trainer, for an engaging discussion that you will immediately be able to implement in your mathematics classroom.  Please be sure to become a member so you can participate in the discussion.

Brooke Istas
Math and Numeracy Moderator

Comments

Heidi Schuler's picture
Fifty

Before we begin this activity, here are a few things to keep in mind:

  1. Be the student first. Explore this activity just as your students would with a deep curiosity to examine, try out ideas, make mistakes, and try a different approach.
  2. You can use the online version of the Pentomino Puzzle, but feel free to use a 100-grid chart found here, some tracing paper, and a highlighter as you progress through the virtual activity.
  3. Use this LINCS Math and Numeracy Community to collaborate and help one another. We encourage our students to work together with peers in their exploration of new ideas, so why shouldn’t we do the same?

Let’s get started!

For those who are new to pentominoes, these are shapes that can be formed by joining five identical squares by their edges. In this activity, you will work through a series of “pentomino sum” puzzles. The online version will guide you through the puzzles.

  • Go to student.desmos.com and enter this class code: 2SW QTJ.

    • You may use your desmos login if you already have one, create a new account, or simply continue with the activity without signing-in. One advantage to signing-in, however, is that you can return to this activity later and continue with where you left off.
  • Try the first 5 slides and then share some initial thoughts in this forum before continuing with the remaining Pentomino Puzzles.
    • What are you noticing and wondering about the puzzles?
    • Note: Slide 4 (and 6) is a Partner Challenge, so you'll want to share your work with a colleague or friend. You can show them your screen on Slide 5, or feel free to post your sum to this forum for others to test out. Some sample Partner Challenge responses captured from this morning are shown below.
  • We’ll be posting some Push and Support Cards on this forum throughout the day for those who may want some extra guidance or those who are ready for some more challenging activities, so be sure to check back to this discussion thread often.

Have fun exploring the Pentomino Puzzles!

 

Partner Challenge 1    More Partner Challenge 1

Rachel Baron's picture
One hundred

I found that as I went through the different puzzles, I noticed different relationships. At first, I was very focused on the single target and on figuring out easy ways to estimate the sums. Later on, I found myself looking at larger patterns across rows and up and down columns. I'm looking forward to trying slide 5 once some other people join in! :)

Connie Rivera's picture
One hundred

I hope you'll be able to post some of the responses and thinking happening in Desmos because it was really neat to be seeing other participants' responses as I went through the activity!  Thanks for sharing this :-)

Heidi Schuler's picture
Fifty

Here you go, Connie!

There are so many wonderful responses to the Pentomino Puzzles at student.desmos.com, and I encourage everyone to jump on the site and see what's been going on the last couple of days with this activity. Christin and Eric have already identified a few, and the summaries below represent a cross-section of other responses to various puzzles. I focused on the strategies and the notice and wonder reasoning shared throughout.

  • One person first guesstimated with mental math but switched to a different strategy and method when the sum was much larger because guesstimation wasn’t an efficient use of time.
  • (Regarding the shape change) One person had to change methods again but found an interesting inverse pattern between the new shape on the Push Card and Puzzle 6.
  • Another person realized his method wouldn’t work when he got to the L, and that’s when he started treating the pentomino like a function.
  • Here's a different person's reasoning about the L-shaped Pentomino:

Puzzle5 

Puzzle5     Puzzle 5 Response

 

  • When I created my own pentomino, I then thought more about how placing squares in different directions affected the constant in the function. If a square was one space right of my start, I'd add one. If a square was one space below my start, I'd add 10. Two spaces would double the amount, and a combination move (down and right) would add 10+1. The opposite directions would subtract by the same amounts (much like coordinate planes!). The coefficient in the function is always 5, and the constant is the sum of the moves as described. Once a function is found, it can be used to find the sum given a start or find a start given a sum by solving the equation. I was creating equations to simplify the process and discovered more and more generalized ways of doing so.”
  • “I wonder what about those two pentomino shapes makes for that opposite similarity?”
  • “I treated the pentomino like a function, found the rate of change (5) and used a few pairs of inputs and outputs to write a function rule to find the sum. I could then reverse the rule if needed to find the central square given the sum.”
  • “When I created my own pentomino I then thought more about how placing squares in different directions affected the constant in the function.”

 

ecappleton's picture
One hundred

Hi all,

I love how the Partner Challenges are posted in the image above. It's magic!

I noticed that the Desmos activity gives us our challenge number back to us. I think the idea is that we might be sitting with someone who would take the next screen after we post our challenge sum. I wonder how we might simulate that activity here. Maybe we could post images of a pentomino solution to the challenge numbers?

I think I have a solution to Connie's sum of 245:

I used the web site https://postimages.org/ to get an image URL that I used to insert the image (Heidi's suggestion!).

Fun!

Eric

Heidi Schuler's picture
Fifty

Thanks for the workaround option, Eric! Yes, desmos assumes you'd be able to share your screen or device with someone nearby for the partner challenges, but I appreciate your taking the screen-sharing to a whole new level!

For those who want to use the web site https://postimages.org/ for uploading a screenshot, be sure you copy the Direct Link option and then paste it into the URL field found in the Image box found in the toolbar of your LINCS post.

LINCS toolbar

Heidi

Patricia Helmuth's picture
One hundred

I noticed there is always a difference of 11 between the "north" square and the "east" square. There is also a difference of 11 between the "west" square and the "south" square. And then a difference of 9 between the "west" and "north" square and the same difference between the "east" and "south" square. I wonder if those differences are helpful in figuring out an efficient way to figure out where the entire 5-square pentomino shape should be placed for any sum?

Heidi Schuler's picture
Fifty

Several of you have already identified some important discoveries in the desmos Pentomino Puzzle activity, and I encourage all of you to share those in this forum when you get a moment. In the meantime, I thought I'd share out a couple of Push Cards for those who are ready to explore the puzzles a bit more. Take a look at these cards and share your explanations in this new thread.

Push5       Push4    

Patricia Helmuth's picture
One hundred

Heidi,

I really like exploring the different pentomino shapes, that is, after I figured out a more efficient strategy than what I started with.

For the first two puzzles, I guesstimated where the pentomino would be placed based on mental math: adding up all the numbers in the tens place to see if I was in the ballpark of the target sum. With both Puzzles 1 & 2 this worked efficiently enough as it did put me in the ballpark and I only had to move the pentomino one space over to the right (interestingly both times) to get to the target sum.

However, when I got to Puzzle 3 I felt as though guesstimating wouldn't be very efficient and would take much more time because the target sum I was looking for was much larger. So, I came up with a method for finding the value of the middle square which allowed me to find where to place that pentomino shape for any sum.

I enjoyed when the shaped changed, as in your push card above, because as the shape changed I had to adjust my method to reflect the shape. I am intrigued by the shape you posted because the method I used to solve for the top square in the push card has the inverse of the operation I used to solve for the bottom right square for Puzzle 6! Same numbers but I used subtractiton for Puzzle 6 and I used addition for your shape in your push card. Very interesting! 

I wonder what about those two pentomino shapes makes for that opposite similarity? I have to think about that. 

I had to draw a pentomino, like the pentomino in your push card, so I could visualize the relationship between the numbers in the squares as shown below.

-Patricia

Heidi Schuler's picture
Fifty

Only a few folks have responded to Puzzle 6, so I encourage you to check it out if you haven't already. Once you post a response, take a look at the explanations others have provided. Consider if there's more than one way to answer this puzzle and why or why not.

Heidi

Heidi Schuler's picture
Fifty

For those who are just getting started or for those who are wanting to jump back into the puzzles to explore some more, here are a couple of Support Cards to help guide your efforts.

Support 1     Support 6

 

Heidi Schuler's picture
Fifty

Now that you’ve had a chance to explore the puzzles as a student, let’s engage the content from an instructor’s perspective. Share your thoughts on these questions:

  • What discoveries did you make as you worked through the first few puzzles?
  • What strategy did you use when you first got started? Did you change your strategy along the way? Why or why not?
  • Where did you see and/or use algebra in your work, if at all?
  • Patricia Helmuth’s work on Push and Support Cards notes that “Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills....Providing students with differentiated instruction in the form of Push and Support cards is one way to level this imbalance, keeping all students engaged in one high-cognitive task that supports and encourages learners who are stuck, while at the same time, providing extensions for students who move through the initial phase of the task quickly..”
    • How did your experience in using the Pentomino Push and Support Cards (as a student) compare to her statement?
    • Did you try out multiple cards? Why or why not? 

 Finally, go back and review the work of the "class" in each of the Pentomino Puzzles. Take note of the connections between responses, particularly between informal and formal language used to describe strategies.

Patrick Rodenborn's picture
Ten

I discovered for the first pentomino that the central square was always the mean of all of the squares. I'm pretty sure that will be true for any symmetrical pentomino with a central square. That discovery carried me through the first few activities without any algebra. I did notice that some others started to use algebra and were coming to similar conclusions (divide by 5).

Once I got to the L I realized that my strategy would no longer work. I then did turn to some algebra. I treated the pentomino like a function, found the rate of change (5) and used a few pairs of inputs and outputs to write a function rule to find the sum. I could then reverse the rule if needed to find the central square given the sum.

When I created my own pentomino I then thought more about how placing squares in different directions affected the constant in the function. If a square was one space right of my start, I'd add one. If a square was one space below my start, I'd add 10. Two spaces would double the amount, and a combination move (down and right) would add 10+1. The opposite directions would subtract by the same amounts (much like coordinate planes!). The coefficient in the function is always 5, and the constant is the sum of the moves as described. Once a function is found, it can be used to find the sum given a start or find a start given a sum by solving the equation. I was creating equations to simplify the process and discovered more and more generalized ways of doing so.

I think the same approach would work for -ominos (?) that are more or less than 5 squares, but the new coefficient would match the number of squares.

I'm curious if any useful play can come from working with something other than the sum of the squares. Products might become unwieldy, but maybe there is some more interesting math in there somewhere.

Christin Smith's picture
First

Hi everyone!

Heidi has given me back-end access, and I have really enjoyed reading through shared strategies. My favorite part is when I learn different ways of expressing similar things.

What do you notice about the three strategies shared below? What do you wonder?

ecappleton's picture
One hundred

Hi Christin,

Thank you for sharing these strategies. I love this explanation for finding the sum of the L shape:

"I knew that sliding the shape over two spaces would get me to the right place (every time you move right one square, you add 5 because each of the five little squares are adding one)."

I notice:

- The L sums seem to have this pattern: 341, 346, 351, 356, 361, ... The sum is always 1 more than a multiple of 5.

It makes me wonder...

- Are all multiples of 5 plus 1 possible? Are any sums possible that aren't a multiple of 5 plus 1?

- If 5 times n plus 1 describes each sum, where is n in the shape? In the cross pentomino, n could be the middle square. In the L, which square is n? For example, if we write an equation for a sum of 356 as 5n + 1 = 356, then n = 67. However, there is no way to place the L on 67 so that the sum equals 356.

sum of 301

sum of 351

sum of 401

- How much is added to sum of the L shape when you move the pentomino down one square? How much is added if you move the L diagonally?

Eric

 

 

Christin Smith's picture
First

A theme I noticed from the language we used to talk about the first pentomino was “averages”:

  • “spreading a total of 65 across 5 columns evenly”

  • “divided 135 by 5 to get 27”

  • “I tried using the averaging strategy [like on the 1st pentomino], but it didn’t work”

 In the “+”-shape pentomino an average of all the square values produced the “center” square number.

Patrick conjectured earlier today that this would be true for “any symmetrical pentomino with a central square”. I love this! 

  • I wonder... is there a different pent-omino with a “central square”?

  • I wonder… are there any ____-ominos that can’t be symmetrical? 

Eric’s observation about the sums of the L-shaped pentomino is so real: 341, 346, 351, 356, 361, ...WHY can’t we place the shape correctly using 5n+1 somehow?

I played around a bit by choosing what seemed like the most “central” square in our asymmetrical L-shape and creating an algebraic expression to depict the sum. 

5n - 9 is kind of like adding 1…So wait:

The sums ARE one more than a multiple of five, but Eric already showed that the exact number of 5s won’t be in our L. It’ll be off to the side a little. 

  • I wonder… can we figure out where to place the L-shaped pentomino by finding where the “average” square (n when 5n = Sum) will be in relation to the pentomino?

Heidi Schuler's picture
Fifty

Below are some contributions to the Sketch Your Own Puzzle from Screen 11 in desmos. My hope is to give a few days for others to contribute and then make these into a card sorting activity we all can use. In the meantime, see if you can match the Pentomino Puzzles to the algebraic expressions created by your peers. Share your responses and reasoning in this thread.

Pentomino Match

Test your algebraic expression skills! What expression could fit this Pentomino Puzzle?

5 squares horizontal Pentomino

Heidi Schuler's picture
Fifty

Use this if you want to try an online version of the Pentomino Puzzle Match-Up!

Pentomino Puzzle Match-Up

 

Want to make your own set of cards? Feel free to copy and edit the activity from this link.

Heidi

Heidi Schuler's picture
Fifty

The Pentomino Puzzles offer opportunities to examine and explore algebraic reasoning through arithmetic puzzles. Along the way, these activities also give us a chance to experience some of the  CCRS Mathematical Practices (pp. 48-50).

Here are some ways you demonstrated those Practices:

  • Make sense of problems and persevere in solving them. (MP.1)

    • You discovered that your method for solving the different shape from the Push Card was the inverse of your findings for the Puzzle 6 pentomino shape.
  • Reason abstractly and quantitatively. (MP.2) 
    • “When I created my own pentomino, I then thought more about how placing squares in different directions affected the constant in the function. If a square was one space right of my start, I'd add one. If a square was one space below my start, I'd add 10. Two spaces would double the amount, and a combination move (down and right) would add 10+1. The opposite directions would subtract by the same amounts (much like coordinate planes!). The coefficient in the function is always 5, and the constant is the sum of the moves as described. Once a function is found, it can be used to find the sum given a start or find a start given a sum by solving the equation. I was creating equations to simplify the process and discovered more and more generalized ways of doing so.”
  • Construct viable arguments and critique the reasoning of others. (MP.3) 
    • You found the central square was always the mean of all the squares and reasoned this should be true of any symmetrical pentomino with a central square.
    • Your demonstrated your algebraic reasoning about another person'sconjecture on the L-shaped pentomino.
  • Model with mathematics. (MP.4) 
    • Everyone who attempted even one of the pentomino puzzles did this.
    • Everyone who modified their strategy at some point did this.
  • Look for and make use of structure. (MP.7) 
    • (From Puzzle 1)  "I noticed that the numbers in the ones place are the same all the way down each column. First, I thought about which columns I could combine to get a 5 in the ones place of the total. I knew the center column had to have odd numbers because otherwise the total would come out even (odd + odd = even, but odd + odd + odd = odd). That meant I had to use the 3's column. 3x3= 9 and 9+2+4= 15. I tried the first (top) option because I knew that if I already had 15 from the ones, I only needed 50 more from the tens. 10 x 3 = 30 and adding another 20 from the bottom of the pentomino gave me 50. So, my answer is 3+12+13+14+23=65. I checked it on the calculator just to make sure that I didn't overlook something."
    • See Patricia’s Notice and Wonder posts on the LINCS forum
    • Check out this reasoning from Puzzle 1:

Puzzle 1

 

 

 

 

 

 

 

 

  •  

    • "If 5 times n plus 1 describes each sum, where is n in the shape? In the cross pentomino, n could be the middle square. In the L, which square is n? For example, if we write an equation for a sum of 356 as 5n + 1 = 356, then n = 67. However, there is no way to place the L on 67 so that the sum equals 356."

 

Surprised by all you can do with some arithmetic puzzles? Want to know what to do with that newfound knowledge?

Now it’s your turn to try out the activity with your students. Modify this activity for your own classroom by going to the Pentomino Puzzle found in the Activity Builder portion of teacher.desmos.com. You also can follow the lesson plan designed to accompany this activity.

  1. Try out the activity in your classroom.
  2. Download and modify the Pentominoes Push and Support Cards to make a set of cards for your specific student population.
  3. Share with your colleagues (at your local program and in the Math and Numeracy Community) some feedback on how things went with the activity and the cards, discoveries made by students, and any a-ha’s or u-oh’s you discovered while facilitating the activity.
  4. Recall that on Day 2 we talked about Patricia Helmuth’s work on Push and Support Cards and specifically responded to the following quote from her work: “Adult education classrooms are commonly comprised of learners who have widely disparate levels of mathematical problem-solving skills....Providing students with differentiated instruction in the form of Push and Support cards is one way to level this imbalance, keeping all students engaged in one high-cognitive task that supports and encourages learners who are stuck, while at the same time, providing extensions for students who move through the initial phase of the task quickly..”
    • How did your experience in using the Pentomino Push and Support Cards (as a teacher) compare to her statement?
    • Did you distribute multiple cards to the same students? Why or why not?
    • How did you decide which card(s) to give to your students?

 

Want more activities and ideas like this?

Check out these Youcubed tasks for some inspiring, multilevel, and fun explorations.

Scroll about ¾ of the way down this page to see Patricia Helmuth’s list of Tasks for some adult education activities that include Push and Support Cards.

 

Let's keep the conversation going!

Even though the activity may have ended, our discussion about the work we’re doing with our students to connect arithmetic to algebra doesn’t have to stop. Continue to explore and share out your ideas and resources in this forum and challenge one another to dig into the discoveries first as a student. In the end, we’re all lifelong learners.

Heidi

BrookeIstas's picture
One hundred

As we close this activity, but not the discussion, I want to thank Heidi for leading this fun activity that you can take into your classroom and use.  Thank you to everyone who pushed each other to think deeper and broader, and challenged our thinking!  Please continue to share your thoughts, ideas, and pictures.  Thank you, thank you, thank you!  If you would like to lead a special discussion in this group please reach out to me, at brooke.istas@cowley.edu.  I would love to see more participants share their techniques and have fun in our group!

Keep thinking like mathematicians! 

Brooke Istas