Reflective notes summary
Exploring Multiplication: Three-in-a-Row Lucky Numbers
Author(s): James A. Russo
Source: Teaching Children Mathematics , Vol. 24, No. 6 (April 2018), pp. 378-383
Published by: National Council of Teachers of Mathematics
Stable URL: https://www.jstor.org/stable/10.5951/teacchilmath.24.6.0378
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms
National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Teaching Children Mathematics
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
378 April 2018 • teaching children mathematics | Vol. 24, No. 6 www.nctm.org
Lucky
This game-based activity prompts students to explore the structure of multiplication, experiment with the distributive property, and begin investigating prime numbers.
Multiplication: Exploring
Numbers
THREE-
ROW IN-A-
Copyright © 2018 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
www.nctm.org Vol. 24, No. 6 | teaching children mathematics • April 2018 379
This game-based activity prompts students to explore the structure of multiplication, experiment with the distributive property, and begin investigating prime numbers.
James A. Russo Launching the game Before playing the game for the first time, our second- grade class spent some time reviewing the idea that multiplication is about adding equal groups. I then introduced the two dice that students were to use in the game: one eight-sided die with digits representing the number of groups, and one six-sided die with dots representing the number in each group. I told students we also need a 120 chart (or a 100 chart) as our game board, whiteboard markers to record our numbers (colored counters can be used as an alternative), and whiteboards to model our multiplication facts.
Game play I explained the basic game play to students: The objec- tive is to be the first player to score three different three-in-a-rows.
• A three-in-a-row is scored when a player records three consecutive numbers—horizontally, vertically, or diagonally (e.g., 2, 3, 4; 6, 16, 26; 21, 32, 43—see the heading Game example).
• The first player rolls the two dice and constructs a multiplication fact reflecting the roll, using the “groups of” representation. For example, if a student rolls a four and four dots, he or she would draw four groups, each showing four dots, and use multiplication or skip counting to work out the total number of dots (i.e., sixteen). The player would then place one counter over the number 16.
• The next player repeats the process, and the game continues until one player records three different three-in-a-rows.
• The strategic (and most engaging) aspect of the game comes into play through the notion of lucky num- bers. If a player lands on a number that is already occupied by another counter (i.e., the number came up earlier in the game), this is said to be a lucky num- ber. The player is allowed to place his or her counter on any other number on the board that is not already occupied. Through this mechanism, otherwise impossible numbers (e.g., prime numbers larger than 7) can be incorporated into a three-in-a-row.
T hree-in-a-Row Lucky Numbers is an engag- ing, enjoyable, mathematically meaning- ful, game-based activity involving dice and a hundred chart, which can be used to introduce students to multiplication. The
game provides a mechanism for students to explore the structure of multiplication, experiment with the distributive property, and begin to investigate prime numbers. When students first encounter multiplica- tion, they benefit from lots of exposure to both con- crete and pictorial representations to build an image of the multiplication operation (Reys et al. 2012). As a classroom teacher who is passionate about mathe- matics instruction in the early years of schooling, I am always experimenting with different representations to assist students in building a conception of multi- plication. I am certainly an advocate for meaningful mathematical investigations, such as a modified ver- sion of the Eggsactly How Many? problem (Enns 2015). However, for building fluency with multiplication pro- cedures and simultaneously engaging students, I love game-based activities.
Meaningful practice through games can help build fluency with operations and can be highly engaging for students (Van de Walle, Karp, and Bay-Williams 2013). Given that a preoccupation with winning a game, as Gough (1999) highlights, can sometimes detract from students paying attention to the key mathematical ideas, the best games are those in which effective game strategy requires students to really focus on the under- lying mathematics.
The game Three-in-a-Row Lucky Numbers is an enjoyable, game-based activity that offers students a chance to explore the structure of multiplication using both “groups of” and array representations. As students become familiar with the game, it can also be used as the basis for examining the distributive property of multiplication and inquiring about composite and prime numbers. Three-in-a-Row Lucky Numbers is designed for two to four players.
SA M
74 10
0 (S
TU D
EN T)
; I LL
YC H
( D
IC E)
/T H
IN KS
TO C
K
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
380 April 2018 • teaching children mathematics | Vol. 24, No. 6 www.nctm.org
Gypsy: Well [gesturing toward the groups of four], if I had four groups of four and I chopped my groups of four in half, they’d turn into groups of two—and instead of having four of them, they’d be eight [groups]. So, they’re the same.
Russo: And how can you be sure that eight groups of two is sixteen?
Gypsy: Well, I know that it has to be an even number, because all the dots have buddies. And I know how to count by twos: two, four, six, eight, ten, twelve, fourteen, sixteen.
Despite the number 16 already contributing toward a three-in-a-row, getting 16 again was highly advantageous for Gypsy. As this square was already occupied, 16 became a “lucky number.” Gypsy was able to color in any square on the board, and she promptly chose the 43 square, completing her third three-in-a-row (21, 32, 43) and winning the game.
Mikayla: Gypsy, you’re lucky you got a lucky number!
Russo: Mikayla, do you think the number 43 would have ever come up as a multiplication fact if Gypsy hadn’t gotten a lucky number?
Mikayla: I don’t think so. We’ve been playing for ages, and none of the numbers that end with a 3 seem to come up very much. No one made 13, 23, 33, or 43 so far.
Russo: Perhaps you’d like to share your game with the class and talk a little bit about some of the patterns you noticed? I wonder if any of the other groups managed to find a multiplication fact that ended with the number 3 (other than 3, of course).
Postactivity discussion At the conclusion of the first session (and sub- sequent sessions), I facilitated a whole-class dis- cussion. This allowed students to discuss game play and to share their strategies. I found the fol- lowing prompting questions particularly useful.
• Were any numbers in the game particularly lucky (i.e., were there any numbers that players kept landing on)? What were these numbers? Answering this question gives students a chance to informally discuss composite num- bers, such as the number 16, which came up multiple times in Gypsy and Mikayla’s game.
We then played a quick demonstration of Three-in-a-Row Lucky Numbers in which the first team to get one three-in-a-row was the winner. After the demonstration (see fig. 1), I allowed students to group themselves into mixed-ability pairs and begin their games.
Game example: Exploring the “groups of” representation I walked over and observed a game between Gypsy and Mikayla. Gypsy (the player using the red marker) had already recorded two three-in- a-rows (2, 3, 4 and 6, 16, 26) to Mikayla’s single three-in-a-row (14, 24, 34), when she rolled two fours and created the corresponding multipli- cation fact of four groups of four. Through skip counting by fours (4, 8, 12, 16), Gypsy was able to calculate a total of sixteen.
Mr. Russo: How can you be sure that four groups of four is sixteen?
Gypsy: Well, I got sixteen a few turns ago when I rolled eight groups of two, and I know that four groups of four is kind of the same as eight groups of two.
Russo: How is it the same?
F IG
U R
E 1 After a quick demonstration of the Three-in-a-Row
Lucky Numbers game, students grouped themselves into mixed-ability pairs and began to explore the “groups of” representation.
JA M
ES A
. R U
SS O
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
www.nctm.org Vol. 24, No. 6 | teaching children mathematics • April 2018 381
distributive property, which is central to grasp- ing multiplication (Kinzer and Standford 2014). In particular, many students, such as Fred, were finding counting by sixes very challeng- ing, and were pursuing alternative strategies.
Russo: I wonder if anyone would like to share any of their strategies for counting by sixes when playing the game?
Fred: Rolling six dots is annoying; it’s hard to count by sixes.
Russo: So, what do you do, Fred?
Fred: I try and count by threes. Three, six, nine, twelve, um [pausing], fifteen [pausing longer]. I’m not sure what comes next.
Jillian: I don’t count by threes or sixes; I count by twos. It’s faster and easier.
Russo: That’s interesting. It’s interesting how lots of people break up the groups in different ways to make the skip counting easier. When we break up groups like this when working with multiplication, it’s called using the distributive property of multiplication. Can anyone tell me if they know another fast and easy way of breaking up groups of six to make the calculation easier?
Mikayla: I do something different. When I rolled five groups of six, I knew that groups of six are kind of like groups of five, but one more. So, first I counted by fives: five, ten, fifteen, twenty, twenty-five. Then I needed to add on the leftover dots, one in each group, so that is five more. Twenty-five and five equals thirty; so five groups of six must equal thirty.
As a class, we spent considerable time discussing the merits of Mikayla’s strategy. Most students endorsed it because count- ing by fives is “easy.” Once students began to grasp Mikayla’s approach to counting groups of six, I suggested that they break the groups of six up into other component parts (e.g., four and two; or three and three, as Fred was doing) and calculate the total. I encour- aged students to decide on the two a p p r o a c h e s to counting groups of six
• Do some numbers never seem to come up as answers to multiplication facts? Did you notice anything interesting about these numbers? This provides an opportunity to informally discuss prime numbers and those composite numbers that, because of the limited range of the dice, never come up in the game (e.g., 22).
• When you landed on a lucky number, where did you choose to place your counter? What was your reason for placing it on this num- ber? Often, but not always, placing your counter on an unlucky number, such as Gypsy did in her game, is advantageous.
Exploring the mathematics During a two-week mini-unit of work on multi- plication, we played the game for five 50-minute sessions altogether. From the second session onward, I encouraged students to predict their answer after they rolled the two dice, and then use the “groups of” representation to check if they were correct. The idea behind this prompt is that as students become familiar with playing the game, this exercise encourages them to move from skip-counting sequences to fluently recall- ing multiplication facts. It also invites students to notice patterns and identify related multipli- cation facts, as Gypsy did in her example.
However, during this second session, I continued to require students to create the “groups of” representation on every turn, even if they thought they already knew the answer. This allows students to build fluency by con- structing multiple representations of similar problems and encourages them to make con- nections among these different representa- tions. Consequently, in addition to reinforcing appropriate back-up strategies, this approach has the benefit of informally exposing students to the distributive property of multiplication, thus helping to foster a conceptual under- standing of multiplication (as demonstrated by Gypsy’s example).
At the beginning of the third session, I encouraged students to begin recording the number sentence next to their “groups of” representation. Students were invited to either record the number sentence as repeated addi- tion (e.g., 2 + 2 + 2 + 2 = 8) or as multiplication (4 × 2 = 8). During this session, I also took the opportunity to more explicitly introduce the
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
382 April 2018 • teaching children mathematics | Vol. 24, No. 6 www.nctm.org
two ten-sided dice) (see fig. 2). In this example, Keaton, the player with the black marker, rolled a 5 and a 2 and constructed an array consti- tuting two rows of five. As the number 10 was already occupied, he was able to select any unoccupied number on the board. Keaton chose the number 13, which simultaneously gave him two additional three-in-a-rows (2, 13, 24 and 13, 14, 15), providing him with three dif- ferent three-in-a-rows overall, allowing him to win the game.
After the game, I spoke to Keaton and his partner, Rylan, about their game.
Keaton: I really needed to get a lucky number in that game, or I was in trouble. I don’t reckon I would have ever been able to roll the dice and make a rectangle with thirteen squares.
Russo: What about if we forget about the dice for a moment: Is it even possible to create a rectangle with exactly thirteen squares?
Keaton: [Thinking and muttering followed by a twenty-second pause] Well, I think the only rectangle that we could make would be a long, skinny rectangle, thirteen across and one up. Does that even count? It wouldn’t even fit on the graph paper!
Russo: It might be long and skinny and not fit into the graph paper, but it definitely still counts as a rectangle.
Rylan: [Interjecting] What about a rectangle that was thirteen up and one across?
Keaton: [Thinking and then pausing for five seconds] Oh, yeah, there’s that one too.
Russo: Is that the same rectangle, or a different rectangle? [I then proceeded to draw a rectangle thirteen across and one up, and another rect- angle thirteen up and one across on different mini-whiteboards.]
Keaton: [Thinking and then pausing for five seconds] Well, it’s facing a different way.
Rylan: [Interjecting] Maybe it’s the same.
Russo: Can you prove that they’re the same?
Keaton: Um, not sure.
Rylan: [Excitedly] What if we turned it over [i.e., rotate one of the rectangles 90 degrees]? Then they’d be the same.
F IG
U R
E 2 Arrays can help transition students from additive thinking
toward multiplicative thinking, and they are useful for illustrating the commutative and distributive properties of multiplication.
PH O
TO B
Y JA
M ES
A . R
U SS
O
that they believed were fastest and easiest. We then discussed, and I modeled, some other uses of the distributive property. For example, if students are finding counting by fours chal- lenging, we explored how they could instead count by twos twice or count by fives and then count back by ones. Mikayla modeled how she could use this latter strategy to calculate eight groups of four: “First I will count by fives: five, ten, fifteen, twenty, twenty-five, thirty, thirty- five, forty. Then I need to take away one dot from each group, because I’ve counted too many. Forty minus eight is thirty-two; so that means eight groups of four equals thirty-two.”
Extending the game: Arrays and prime numbers On the fourth and fifth occasions when we played the game as a class, I modeled the array representation as an alternative to creating “groups of.” Using arrays can help transi- tion students away from additive thinking and toward multiplicative thinking (Day and Hurrell 2015). Arrays also have the advantage of being useful for illustrating both the com- mutative property (e.g., 5 × 2 is revealed to be equivalent to 2 × 5 through rotating the array) and the distributive property of multiplication. During the fifth and final session, some groups of students who were more confident with mul- tiplication also began playing the game with larger dice (e.g., two eight-sided dice or even
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms
www.nctm.org Vol. 24, No. 6 | teaching children mathematics • April 2018 383
This conversation emphasizes how the com- mutative property of multiplication can emerge out of an exploration of arrays. Rylan’s decision to rotate one of the arrays to create two identi- cal rectangles was a powerful opportunity for him and Keaton to begin to grasp that, like addition, multiplication is commutative.
As a class, we then revisited Keaton and Rylan’s game during the postactivity discus- sion. After exploring what Keaton and Rylan had found, I used the opportunity to state that mathematicians have a special name for any number that can be represented only by “long, skinny rectangles”: prime numbers.
The remainder of the lesson (about 10–15 minutes) evolved into an investigation, as stu- dents went about identifying as many prime numbers as possible by experimenting with the array representation and attempting to “squash . . . long, skinny rectangles” into “fat- ter” rectangles. For example, the class initially hypothesized that all single-digit odd numbers must be prime numbers until an astute stu- dent, Liam, realized that nine is not, in fact, a prime number because a 9 × 1 rectangle can be “squashed” into a 3 × 3 square. The example above demonstrates how a discussion of prime and composite numbers can arise relatively organically out of a game of Three-in-a-Row Lucky Numbers, particularly when the array representation is used.
Concluding thoughts As I noted at the beginning of the article, to build an understanding of the multiplication operation, students benefi t from exposure to multiple representations (Reys et al. 2012). The Three-in-a-Row Lucky Numbers game is designed to engage students with ample hands-on experience with “groups of” and array representations, while encouraging stu- dents to link these representations to the mul- tiplication operator as expressed in a number sentence. The game also presents a chance to begin a conversation with students about prime and composite numbers through their exploration of the initially puzzling phenom- ena as to why some numbers in the game are “lucky,” whereas other numbers are “unlucky.”
My students and I have really enjoyed play- ing Three-in-a-Row Lucky Numbers in our classroom, and I hope other teachers fi nd it
a useful resource. If you fi nd Three-in-a-Row Lucky Numbers of value for exploring multipli- cation, you may want to check out other ver- sions of Three-in-a-Row that focus on mental computation skills for addition and subtraction (see Russo 2015).
Common Core Connections
2.0A 3.OA
REFERENCES Day, Lorraine, and Derek Hurrell. 2015. “An
Explanation for the Use of Arrays to Promote the Understanding of Mental Strategies for Multiplication.” Australian Primary Math- ematics Classroom 20 (1): 20–23.
Enns, Ed. 2015. “Problem Solvers: Problem: Eggsactly How Many? Teaching Children Mathematics 21, no. 9 (May): 521–23.
Gough, John. 1999. “Playing Mathematical Games: When Is a Game Not a Game? Australian Primary Mathematics Classroom 4 (2): 12–17.
Kinzer, Cathy J., and Ted Stanford. 2013/2014. “The Distributive Property: The Core of Mul- tiplication.” Teaching Children Mathematics 20, no. 5 (December/January): 302–9.
Reys, Robert E., Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Judith Falle, Sandra Frid, and Sue Bennett. 2012. Helping Children Learn Mathematics. 1st Australian ed. Milton, Queensland, Australia: Wiley.
Russo, James A. 2015. “Get Your Game On: Three in a Row.” Prime Number 30 (4): 16–18.
Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally. 8th ed. New York: Pearson.
James A. Russo teaches at Belgrave South Primary School in Victoria, Australia. His passion is developing engaging games and activities that extend student thinking. Additional
resources developed by Russo are available at http://www.surfmaths.com.
This content downloaded from �����������24.181.66.100 on Sat, 13 Apr 2024 02:59:44 +00:00������������
All use subject to https://about.jstor.org/terms