Rows & Straws (Multiplication Arrays) Years K - 8

Summary Poly Plug has a built in array structure which many teachers have capitalised on to develop this suite of activities. Rows & Straws is really an investigation with several rich veins that can be visited and revisited at all ages. A whole school approach to planning multiplication and times tables experiences through an array model would bring the best from this material. The array model of multiplication is very powerful. It: allows commutative multiplications (eg: 3 rows of 7 and 7 rows of 3) to be viewed in the same model by changing position rather than changing the picture; sets up a rectangle model which can be used to represent so much mathematics, including fractions and algebra; is a natural organising technique of young children so the development of multiplication can begin from their own play. Materials One Poly Plug set and two straws per person. One calculator per person Poly Plug Paper or Poly Plug Grid or maths journal Poly Plug Grid has many more plug images per page. It can be easily sliced into useful strips.

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. In this case the fresh context is a focus on learning long multiplication in and activity titled Rows, Rectangles and Multiplication. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.

• Visit the Home Page for more Background.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

• Can you check it another way?
• What happens if?
• How many solutions are there?
• How do you know you have found them all?

Make equal rows of plugs (an array), eg: 3 rows of 7 3 x 7 What can the calculator tell us about 3 x 7?

What happens if we use a straw to separate the rows? 1 row of 7 and 2 rows of 7. 1 x 7 + 2 x 7 is the same as 3 rows of 7 1 x 7 + 2 x 7 = 3 x 7 Do we use brackets??? We shouldn't have to. Why? What can the calculator tell us about this picture? (Warning: Very few simple four function calculators correctly evaluate the left hand side of this expression using Order of Operations. Most calculators in schools will get the wrong answer if the expression is entered in the order written. See our comments on Calculators.)

How many ways can we place one straw to separate the rows? 2 rows of 7 and 1 row of 7. 2 x 7 + 1 x 7 is the same as 3 rows of 7 2 x 7 + 1 x 7 = 3 x 7 What can the calculator tell us about this picture?

3 rows of 1 and 3 rows of 6. 3 x 1 + 3 x 6 is the same as 3 rows of 7 3 x 1 + 3 x 6 = 3 x 7 What can the calculator tell us about this picture? Can you also see: 3 rows of (1 + 6) and 3 rows of 1 + 3 rows of 6? 3 x (1 + 6) = 3 x 1 + 3 x 6 3(1 + 6) = 3 + 18

3 rows of 2 and 3 rows of 5. 3 x 2 + 3 x 5 is the same as 3 rows of 7 3 x 2 + 3 x 5 = 3 x 7 What can the calculator tell us about this picture? 3 rows of (2 + 5) and 3 rows of 2 plus 3 rows of 5 3 x (2 + 5) = 3 x 2 + 3 x 5 3(2 + 5) = 6 + 15

3 rows of 3 and 3 rows of 4. 3 x 3 + 3 x 4 is the same as 3 rows of 7 3 x 3 + 3 x 4 = 3 x 7 What can the calculator tell us about this picture? 3 rows of (3 + 4) and 3 rows of 3 plus 3 rows of 4 3 x (3 + 4) = 3 x 3 + 3 x 4 3(3 + 5) = 9 + 12

3 rows of 4 and 3 rows of 3. 3 x 4 + 3 x 3 is the same as 3 rows of 7 3 x 4 + 3 x 3 = 3 x 7 What can the calculator tell us about this picture? 3 rows of (4 + 3) and 3 rows of 4 plus 3 rows of 3 3 x (4 + 3) = 3 x 4 + 3 x 3 3(4 + 3) = 12 + 9

3 rows of 5 and 3 rows of 2. 3 x 5 + 3 x 2 is the same as 3 rows of 7 3 x 5 + 3 x 2 = 3 x 7 What can the calculator tell us about this picture? 3 rows of (5 + 2) and 3 rows of 5 plus 3 rows of 2 3 x (5 + 2) = 3 x 5 + 3 x 2 3(5 + 2) = 15 + 6

3 rows of 6 and 3 rows of 1. 3 x 6 + 3 x 1 is the same as 3 rows of 7 3 x 6 + 3 x 1 = 3 x 7 What can the calculator tell us about this picture? 3 rows of (6 + 1) and 3 rows of 6 plus 3 rows of 1 3 x (6 + 1) = 3 x 6 + 3 x 1 3(6 + 1) = 18 + 3

What happens if we look at the rows the other way?

What happens if we use two straws at the same time? 1 x 1 + 1 x 6 + 2 x 1 + 2 x 6 is the same as 3 rows of 7 1 x 1 + 1 x 6 + 2 x 1 + 2 x 6 = 3 x 7 What can the calculator tell us about this picture?

How many ways can we place the two straws? What are the parts each time? Discuss, record, try the calculator.

Can you also see: (1 + 2) rows of (3 + 4) which is the same as 1 row of 3 + 1 row of 4 + 2 rows of 3 + 2 rows of 4 (1 + 2) (3 + 4) = 1 x 3 + 1 x 4 + 2 x 3 + 2 x 4

As you plan you activities it may help to keep these questions and thoughts in mind.

• What can you find out about it with one straw ... or two straws?
• Are dot diagrams the most efficient way to record?
• Words or symbols or a mixture when recording?
• What happens if the number of plugs in a row and the number of rows are the same?
• Suppose I give you the parts mixed up, eg:
  2 x 2, 1 x 3, 1 x 2 and 2 x 3
  Can make the plug rows (or draw the dot picture)?
• 'One straw thinking' is the conceptual basis of the Distributive Law. It is the understanding which allows you to decide that 3 x 16 can be evaluated as 3 x 10 + 3 x 6.
  
  • How will you lead your students from this visual form of the distributive law to its symbolic form?
  • Of course any placement of one straw is as valid as any other. Why do we prefer the placement above?

• 'Two straw thinking' is the conceptual basis of the broader application of the Distributive Law which allows you to do long multiplication by recognising that 23 x 16 has four parts. (Breaking a problem into smaller more manageable parts is a mathematician's strategy.)
  

  The product is the total of the parts.

  • How will you lead your students from this visual form of the distributive law to its symbolic form?
  • Of course any placement of two straws is as valid as any other. Why do we prefer the placement above?
  • If we understand the visual image above, do we really need to be forced into a way of setting out long multiplication?

Giovana Davies
St. Patrick's School, Cooma

Combining the contributions from Sue and Anne (below), and later contributions from Year 3/4 at Willunga Primary School, South Australia, has led to the development of the sequence on the right which turns The Big Idea into a sequence of lessons for a range of ages. Further contributions to this sequence, or other applications of the array model are welcome.

Sue Rose Numeracy Project Officer, Northern Territory Open Education Centre Through her desire to bring the benefits of learning with Poly Plug to remote students, Sue has come up with a neat idea. One student can't see the array which another has made, but is told, one by one, equations that can be formed from it. Can they guess the array and the placement of the straw(s)? Sue's version is: While I'm sitting at the keyboard, I must tell you what I have been doing on IDL (Interactive Distance Learning). I work with students at Xavier CEC on Bathurst Island. I sent them out a stack of Poly Plugs so that we can do activities on air. Now I can't see what they are doing, but they can see me! They are all 'convincing' me that they have 3 rows of 5 plugs by framing number sentences in as many ways as possible. One student works on a word document on the IDL screen - I can hand over the recording responsibility to him and read what he types on the screen in my studio. Meanwhile the rest of the class are recording their own number sentences on paper and I ask them to check the screen to see if their ideas match those of the typist. It's all interesting use of technology and a great way for me to start using the Poly Plugs in an interactive distance education mode. Many aspects of the Working Mathematically process are involved in this approach. Anne Lawrence Adviser in Numeracy, Mathematics and NCEA Palmerston, New Zealand I have used the straw idea to build a more explicit understanding of number properties in students. Starting with rows of counters we experimented with placing straws and describing the number properties displayed. I then extended this idea from an array of counters to an area model based on a grid. I have only tried this with one small group of Year 9 students but was so pleased with their response that I put the ideas into a PowerPoint so that I could share it with other teachers. I found that using the straws with counters made the number properties very clear to students - it enabled them to generalise what they were seeing using their own words and from there we generated a 'rule' using letters. The area model for multiplication is something we are using in our Numeracy Project work here in New Zealand, so the array and then onto a grid fits nicely with a model we are already familiar with. It is interesting for me because the work in Numeracy focuses very much on building students' number sense and developing different strategies for multiplying. However it seems that the number properties tend to remain implicit, and teachers need strategies (like the straws) that help students make properties explicit.

1. Playing With Rectangles Play with rectangles by popping plugs from the red board. I want you to explore how many ways you can make equal rows of gaps in the red board. (Note: It is better not to use the word 'hole' because Poly Plug is also used to support learning about fractions where the word 'whole' sounds the same but has a different meaning.) Note that each set of equal rows is shaped like a rectangle. You might ask children to record some of their rectangles on Recording Paper by drawing a ring around their rectangle, or in their journal by drawing dot pictures. For older students there is a challenge here. How many rectangles can be made altogether?       Develop agreement about what is meant by the word rows. You might also connect this with the word column. Language use is important in the following activities. For example, the pictures above show: three rows of 5 and four rows of 1. 2. Playing With Straws Choose one of your rectangles and plug it in with yellow or blue plugs. Hand one straw to each child. How many ways can you put your straw between the plugs this way (horizontal) or that way (vertical)? Hand another straw to each child. The straws must always be used at right angles, ie: one horizontally and one vertically. How many ways can you put your straw between the plugs this way (horizontal) or that way (vertical)? Placing either one straw or two straws is called partitioning the number of plugs. This process has links to division, the Distributive Law and Algebra. Play with the idea, perhaps over a number of days with different rows of plugs before recording. 3. Recording Partitions Ask students to record some, or all, of the ways they can place their straw(s). Rows & Straws Paper might help. For older students there is a challenge here. If I tell you any number of rows and any number in each row, can you tell me how many ways you can place your straw(s)? 4. Diverging Into Division Ask students to take any number of plugs out of the yellow/blue board and arrange them in equal rows. Count the total number of plugs and then talk and write equations showing how the rectangle can be divided. Multiple numbers of straws can be used to show the partitions. For example the picture below shows 24 ÷ 6 = 4 and 24 ÷ 4 = 6. Explore whether the same number of plugs can be arranged in equal rows a different way and talk and write more divisions. What about divisions with remainders? Bob's Buttons, Lesson 10, and Tackling Times Tables, Lesson 97, from Maths300 would integrate well with this work. 5. Find My Straws One student hides a plug board which shows an array of equal rows with one or two straws on it. Remember the straws must be at right angles. It is a little quicker organising the game if you use the yellow/blue board and simply turn the plugs over to make a contrasting array. Also, the activity works well if the 'hider' is at the back of the room facing the board. They, and the other students, are then 'oriented' the same way. Hand out Recording Paper and also sketch one of the blank plug pictures on the whiteboard. The other students are allowed to ask questions about the partitions which are of the form: Is there 3 rows of 2? or Is there 1 row of 1?... The 'hider' simply answers yes or no. By analysing and recording the responses the other students have to work out the array and where the straws are placed. Note: It is not required to work out the position of the array within the board - only the dimensions of the array and where the straws have been placed. When a student thinks they have the result, they can show it on the white board so it can be compared to the hidden board.

Kiki hides the rows and places the straws.

Mattias figures out the array and the straws.

Thread this game into the curriculum. That is, use it for a few minutes two or three times a week for several weeks. The students will be comfortable with the familiarity of the game structure, but will find a new challenge each time in the placement of the straws. Start with one straw examples then change to two straws when the time is right.

6. Making Multiplication

1. Begin by exploring particular arrays greater than 5 by 5. Students might put sets of boards together, but it is better to move to graph paper and encourage students to outline the array they are choosing. A straw can still work to partition the array, but something thinner may be better.
2. Use graph paper to discuss and record particular arrays as above in The Big Idea, to familiarise students with this new tool. For example:

Encourage children to see 3 rows of 7 as 3 x 7 and further encourage them to see 3 x 7 as 3 x 5 + 3 x 2. Also suggest that in placing the straw they always try to make one part into rows of 5.

5. A mathematician also asks Can I check this another way?, so suggest alternatives such as moving the straw to a different position and checking with the MathMaster.
6. Thread this activity into the curriculum (for single digit examples). Today I just want you to draw the picture and work out the multiplication for 2 x 8 and 6 x 4.
7. Work towards children being able to immediately write the answer to any of single digit times table. For example: From now on your challenge is to try to put a picture of the rectangle in your head and just write the answer. You can also check it another way.
8. Combine this work with a multiplication version of the calculator activity 6 Plus.
9. When children are developing confidence with a mental image of the times tables, introduce one two digit number and encourage using graph paper and the straw to make rows of 10 ... because the 10 times table is so easy. Later work towards a mental image.
10. When the time is right, move to multiplication of 2 two-digit numbers using two straws and looking for ways to place the straws, like the example above in The Big Idea, which break the problem of multiplication into parts.
11. Record the parts as a step towards the final answer. 'Long multiplication' is only an application of the mathematician's strategy of breaking a problem into smaller parts.
12. The whole process is not about learning to break the problem into the parts the teacher wants, but rather choosing to break the problem into parts which suit the learner.
13. Remember to frequently use the mathematician's question: Can I check this another way?.
14. Enrich this whole process with other challenges and problem-solving experiences involving multiplication such as activities involving measuring and calculating the area of rectangles. There are many available in the Mathematics Task Centre and Maths300. One excellent link is:
    • Lesson 162, Multiplication in a Table Format
    which takes up the two digit multiplication and graph paper idea and borrows from Rows & Straws in suggesting the use of Poly Plug as a way of concretising the experience.

Approaching Algebra

One Straw Images One straw breaks one big rectangle into two parts. Each part is a rectangle and each part has the same width. It doesn't matter what the numbers are, so one big rectangle has the same area as the sum of the two smaller ones:
Two Straw Images Two straws break one big rectangle into four parts and each part is a rectangle. Again it doesn't matter what the numbers are, so one big rectangle has the same area as the sum of the four smaller ones:
Two Straws & Squares Something special happens when the rectangle is a square and we use two straws to make squares within it. The one big square is broken up into two small squares and two rectangles which have the same area. So the area of the big square is the sum of the areas of the two squares and the two rectangles. The squares and rectangles show quite well with four plug boards as shown below. And now we are ready to factorise linear and quadratic equations ... but that's another story!	For complementary thoughts on linking multiplication and algebra through an array model see Generalising from Number Properties to Algebra, a slide show Anne Lawrence, New Zealand, prepared for her professional development work. Anne sets this in context in her letter recorded in the Contributions section above.

Return to Calculating Changes Activities