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Reflect on each of the following questions about the mathematics related to the Representation Standard that you saw in Ms. Dale's class. After you've formulated your own answers, select "Show Answer" to see our response.
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Question: How does Ms. Dale introduce this problem to the class?
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Sample Answer:
She begins by reviewing the term "array," which students appear to have been studying in class. Then she presents number problems written on index cards for the students to solve with a buddy. Although she does not connect the numbers to any situation or story, this is another approach to introducing the problem.
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Question: How might you help the student who calls a 2-by-7 array a "7 by 7"?
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Sample Answer:
The teacher helps the students name the array by saying, "How many down by how many across?" before naming it a 2-by-7 array. She also asked students to write the dimensions outside their pictures. The boy who calls it a 7-by-7 array is probably noticing that it is made of 7 plus 7 tiles. He needs to focus more on the number of rows and the number of tiles in each row. A game where you practice drawing arrays according to the rolls on a dice –– one roll for the number of rows, and one roll for the number of items in each row –– might help him understand how arrays are named.
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Question: One boy, Jay, is asked to find 1/7 of 14; he says the answer is 1, and he draws a 1-by-7 array. What difficulties might he be having with representations?
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Sample Answer:
Jay may not have a strong understanding of what it means to find 1/7 of 14 tiles. He resorts to representing the fraction 1/7 by using seven tiles. It is also possible that he is confusing the 1-by-7 language of arrays and the fraction 1/7 that is written on his problem card.
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Question: What other language would you suggest using besides describing a 2-by-5 array as "2 going down and 5 going across"? How might you explain the symbols used to label an array?
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Sample Answer:
Arrays are often used to help students make sense of multiplication sentences, such as 2 x 5 = 10. Students need to know that one way of showing 2 x 5 is to make two rows with five objects in each row. It is important for all students to recognize that every row has the same number "in each." The symbols "2 x 5" can be read as "2 rows with 5 objects in each row" or "5 columns with 2 objects in each column," and the words "2-by-5 array" are a shorter way to state the same idea.
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