Card Arrays

This activity is intended to develop the concept of multiplication as an array. A key idea underpinning the questions is that of factors. The activity is suited to a co-operative problem-solving approach, with students working in small groups of 3–4 and then reporting back and comparing ideas and solutions with other groups.

Encourage the students to use vocabulary such as “arrays” and “factors” by defining them together, asking for examples, and including the words in your questions and discussion.
Definitions:
• Factor: a number that is multiplied by another number to create a product. If you divide a number by one of its factors, there will be no remainder. (See the activity on pages 16–17.)
• Array: an ordered arrangement of objects by rows and columns.
Some students may need to learn how a deck of cards is organised before beginning the activity. A completed game of Patience both sorts the cards and emphasises the organisation of the pack.
For question 1b, ask the students: How do the arrays you have drawn relate to the factors of 40? Can you use your drawings to make a list of all the factors of 40? (1, 2, 4, 5, 8, 10, 20, and 40) The answers for question 2c ii describe a process for finding all the factors of 40. The same process can be used to find all the factors of 52 (or any other number). Have the students compare the factors of 40 with those of 52. Extend the question by asking: Can you use what you know about factors to predict which of these numbers of cards would give you the most rectangles?
12 or 17 cards? (12) 15 or 11? (15) 12 or 18? (same)
For question 3, encourage students who have a good knowledge of multiplication basic facts to use number properties before checking with their cards or drawings. Ask: How could listing all the factorsof 24 help you to describe all the possible rectangles that could be drawn?
Encourage the students working on question 4 to use imaging or number properties to connect the division operation to arrays. Ask: If you know there are 11 cards across one row, how many rows do you think there will be? Are there any multiplication facts that could help you with this problem?
(4 x 11 = 44)
Extend the question by getting each student in the group to make up their own “How long is the other side of the array?” problem to share with the group. Remind them that they must be able to solve their own problem and explain the solution. This would show you which students are able to apply their knowledge of basic facts to arrays and division problems.
After they have solved the problem in question 5, extend it by telling the students: The numbers of cards you used in your square arrays can be called “square numbers”. The highest square number you got was 49 ( an array with sides 7 by 7). What would the next three square numbers be if you had more cards? (8 x 8 = 64, 9 x 9 = 81, 10 x 10 = 100)

After the activity
Encourage reflective thinking and the discussion of generalisations by asking:
• How did knowing about arrays help with the problems in this activity?
• How did knowing about factors help with this activity?
• What advice and helpful tips could you give someone about to start this activity?

 Answers to Activity

1. a. 40 cards. (Using the ace as 1, 1 to 10 is 10 cards, 4 suits x 10 cards = 40 cards. Or: 4 suits x 3 face cards = 12 face cards taken out: 52 – 12 = 40.)
b. There are 8 ways: 1 by 40, 40 by 1, 2 by 20, 20 by 2, 4 by 10, 10 by 4, 5 by 8, and 8 by 5. (Note that each way has a matching rectangle, for example 1 by 40 and 40 by 1. The rectangles are rotated.)
Drawings: Note that no matter which way the pairs of rectangles are facing, each rectangle contains 40 units. For example, for 5 by 8 and 8 by 5:
2. a. Predictions will vary. The correct answer is fewer.
b. Practical activity. (There are 6 possible rectangles with 52 cards [1 by 52, 52 by 1, 2 by 26, 26 by 2, 4 by 13, 13 by 4] and 8 possible rectangles with 40 cards [see question 1b].)
c. i. Yes, if you said “fewer”. You may have said “more” because there are more cards, but 40 has more factors than 52, and therefore, more rectangular arrays can be made with 40 cards than with 52 cards.
ii. You know you have found all the rectangles when there are no more pairs of possible factors to use. (For example, with 40, after you have found 1 x 40, 2 x 20, 4 x 10, and 5 x 8, the next factor is 8, which you have already paired with 5. [40 ÷ 3, 6, 7, 9, 11, or 12 has a remainder.] With 52, only 1, 2, 4, 13, 26, and 52 are factors.)
3. 8. (The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24, so the possible rectangles are 1 by 24, 24 by 1, 2 by 12, 12 by 2, 3 by 8, 8 by 3, 4 by 6, and 6 by 4.)
4. 4 cards long. (11 x 4 = 44)
5. There are 7 possible square arrays that you can make using up to 52 square cards: 1 by 1, 2 by 2, 3 by 3, 4 by 4, 5 by 5, 6 by 6, and 7 by 7.