Share
Pin it!
Google Plus

TCM Blog

Reflecting on the How Many Squares on a Checkerboard? Problem

 Permanent link

So, how did things go in your classroom with the How Many Squares on a Checkerboard task? I’m told that we’re still getting a good number of visits to the blog, but few visitors are taking the next step and leaving a comment. The hope is that the blog will become an interactive way for members to share thoughts, comments, ideas, and so on. Please realize that a comment need not be long—a simple “Tried it with my class—they really liked it” is great. Maybe your students solved the task in a really interesting way that you hadn’t thought of before; maybe you had to remove roadblocks before students got going on the task. We would love to hear from you!

All right, so let’s have a look at the How Many Squares on a Checkerboard task and some approaches to solving it. A common strategy is to start with a simpler problem:

How many squares on a 1 × 1 square? 1 

How many squares on a 2 × 2 square? 5
(four 1 × 1 squares and one 2 × 2 square)

How many squares on a 3 × 3 square? 14

For this last one, solvers must see that the board has not only squares of different sizes but also overlapping squares, so a 3 × 3 square has 9 (nine) 1 × 1 squares; 4 different 2 × 2 squares (overlapping, as demonstrated in the checkerboard examples below; and 1 (one) 3 × 3 square), and so on, until eventually arriving at the following solution:

CBoard01-200x192.jpg 

 CBoard02-200x601.jpg 

 checkerboard 

How many squares on an 8 × 8 checkerboard? 204 

64  1 × 1 squares    49  2 × 2 squares 
36  3 × 3 squares    25  4 × 4 squares 
16  5 × 5 squares    9  6 × 6 squares 
4  7 × 7 squares    1  8 × 8 square 

Strategies for identifying and extending patterns, drawing diagrams, making a table, and so on soon come into play. Often the final result for the Checkerboard problem is presented in a table like the one below.

Number of Squares on Various Boards 

Students and teachers readily identify patterns that emerge. The problem can then be extended to determine the total number of squares on any size square board, with the corresponding algebra being introduced as appropriate.

In the higher grades, an extension might be to find the total number of rectangles that can be found on an 8× 8 checkerboard. (Warning: This is not at all trivial!)

So what did you think of the Checkerboard task? How did your students respond to the task? What strategies did they use? Did they encounter any stumbling blocks? How did the student resolve the stumbling blocks? You are welcome to share class photos or student work samples. Hope to hear from you!

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

How Many Squares on a Checkerboard?

 Permanent link

 

Now that I’m an official blogger (with two blogs posts under my belt), I found selecting the next problem to be a real dilemma. I have decided to post another “classic” problem.

How many squares are on a standard (8 x 8) checkerboard? 

checkerboard 

As with the Handshake problem, the appeal of this problem (and what probably makes these problems classics) is its accessibility to students across many grade levels, the variety of problem-solving strategies that can be brought to bear in its solution, and the large number of variations/extensions. The simplicity in stating and setting up the problem is also part of its appeal.

A word of caution when introducing this task: Often students see this problem as somewhat trivial, counting just the 64 small squares; some go an extra step and realize that the whole board is also a square, for a total of 65. So, realize that students (or teachers) might need some prompting to recognize that the board also has 2 x 2 squares, 3 x 3 squares, and so on.

So, there you have it. Go ahead and have some fun with this task!

I was gratified to see the response to the launching of the TCM Blog. The site had lots of visits and a few comments. I’m hoping that for this post, we’ll get even more visits, and that more of you who visit the site will take the extra step to post a comment/question/random thought/whatever drawn either from your own experience/reflections or from introducing the problem in your classroom.

As with the first task, I’ll be back in a couple of weeks to post solutions/thoughts/extensions/variations to the task. I hope to hear from you soon and that you’re enjoying “Math Tasks to Talk About.”

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

Reflecting on The Handshake Problem

 Permanent link

Well, I’ve now been officially initiated into the blogosphere (is that actually a word?) I really appreciated those who took the time to comment on the first task, and I am sincerely hoping that this blog entry, the discussion of the task, encourages more discussion/comments.

So, how’d you do with the Handshake task? If you missed it, here’s the link.

As you know, I love this problem! It’s overflowing with the variety of problem-solving strategies that can be brought to bear in its solution—act it out, draw a diagram, look for a pattern, solve a simpler problem, make an organized list, make a table, use logical reasoning, . . . .

Young students (and math-anxious teachers) can use (and combine) the strategies of acting the problem out, solving a simpler problem, and looking for a pattern, as they build up to the given problem. For example, have 2 students act it out—1 handshake

3 students—3 handshakes

4 students—6 handshakes

5 students—10 handshakes

6 students—15 handshakes

I always find it interesting when students act out the problem: They often go from random hand shaking, which they discover is very hard to count, to organized hand shaking (lining themselves up and going down the line), and finally (what we hope for) to the realization that when an extra person joins the group, they don’t have to repeat all the handshakes that came before, but rather just add on how many handshakes the new person has to do. Therefore, the 7th person would have to do 6 handshakes, 15 + 6 = 21, so 7 students—21 handshakes. Often at this point (if not before), the pattern-seeking students will see that as the number of people goes up by 1, the number of handshakes goes up by first 2, then 3, then 4, then 5, and so on. Continuing, they arrive at the solution:

8 students—21 + 7 = 28 handshakes

9 students—28 + 8 = 36 handshakes

10 students—36 + 9 = 45 handshakes 

Older students (and teachers) will tackle the whole problem with a combination of organized listing and looking for a pattern. They mentally or physically line up 10 people and reason that the 10th person will go down the line and shake hands with 9 people; the next person will go down the line and shake hands with 8 people; the next one, 7; the next one, 6, and so on. So the total number of handshakes is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 handshakes.

This usually leads to the discovery of a generalization: for 20 people, add 19 + 18 + 17 + … + 1. The generalization still takes a lot of computation to find the total number of handshakes, but it’s certainly within the capabilities of students in elementary school. A wonderful YouTube video shows two grade 3 girls solving the problem in this manner, using linking cubes as an aid, for the number of handshakes for 35 people! How’s that for “perseverance in problem solving”?

With teachers and older students, if no one has yet suggested this problem-solving strategy, I like to demonstrate using logical reasoning and throw in my incredibly well-reasoned (but incorrect!) hypothesis:

Well, if I’m one of the 10 people at the party, I shake hands with 9 other people. So does every other person at the party. Since there are 10 of us, and we each shake hands with 9 other people, it’s obvious that the answer is 10 × 9 = 90 handshakes. But this doesn’t match the answer arrived at by using the other methods. What’s wrong?

After some pondering, they realize that I’ve broken one of the ground rules and have counted every handshake twice, so the correct answer would be 90 ÷ 2 = 45. This discovery will then lead to the algebraic generalization that for any group of n people, the number of handshakes will be

n(n – 1)/2,

which matches the formula for finding the sum of all the numbers up to (but not including) a given number. The process is a nice, concrete way of showing why that formula makes sense: n people would each shake (n – 1) hands, but that counts every handshake twice, so we have to divide by 2. Thus,  

n(n – 1)/2.

The handshake problem has many variations in presentation. A way of incorporating the problem into a history context is effectively shown on NCTM’s Illuminations website, which discusses the tradition of the Supreme Court Justices all shaking hands with one another before each session. Then follows the Handshake problem, which asks how many handshakes that scenario would take.

The Illuminations website also has an applet that draws a diagram, along with creating a chart, for the number of handshakes for 2 people up to 12 people.

And, for students in grade 6 and beyond, Illuminations has a nice extension/connection between the Handshake problem and triangular numbers.

OK—so there you have it. I hope you’ll agree with me that this is indeed a “Math Task To Talk About.” Maybe you have some other interesting connections to the Handshake problem, great ways that your students thought about it, or thought-provoking activities that build on it. Please share!

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

The Handshake Problem

(Geometry) Permanent link

kidshandshakeI have the honor of being the “inaugural blogger” for the new Teaching Children Mathematics (TCM) blog, “Math Tasks to Talk About.” Now, to be clear, what I know about blogging could fit in a thimble with plenty of room still left for your finger. However, the talented staff at NCTM can take whatever I submit and magically make it blog-worthy, so here goes!

My absolute favorite math task to talk about is a classic known as the Handshake problem. Alternatively, you may know it as the How Do You Do? problem or the Meet and Greet problem or one of more than at least a dozen different names. No matter what you call it, this problem is my favorite because it can be easily made accessible and interesting to students at all levels, from first grade through high school!

All right—here’s the problem:

Ten [or however many you want] people are at a party, and you want everyone to meet (shake hands with) everyone else at the party. How many handshakes will it take?  

For students in the early grades, simply reduce the number of people who are shaking hands; and for students in the upper grades, move to a generalization for a large number of people shaking hands.

The other reason I find the problem appealing is that the conditions are few and easy to understand. In exploring the problem, students will discover two things: (1) you wouldn’t shake hands with yourself, and (2) when two people shake hands, it counts as one handshake, not two.

OK—that’s it! Solve away! Clearly, the only way this will become a robust and interesting blog is if there is interaction. Give a version of the problem to your class, talk about the different ways your students approach solving it, and talk about your own strategies/reasoning as you think about the problem. Afterward, come back here and post a comment about how it went. You are also welcome to share sample student work and photos. We need your help to make “Math Tasks to Talk About” a rich problem-solving resource.

I’ll be back in two weeks as a follow-up to this post with solutions and ruminations about the problem. In the meantime, I look forward to hearing from you!

 

RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

Please note that only logged in NCTM members are able to comment.

Reload Page