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| XY Encounter |
Funded by the
US Department of Education
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Teacher's Guide: Part 2: Introduction - "So close, but yet so far."
Watch Video Clips @ XY Encounter Streaming Video Page
Setting:
Against her better judgment, Keisha enters the cave with Max. Mysteriously, the door closes behind them. Using their trusty flashlights, the kids begin to make their way in the eerie darkness. Soon they come to a straight path of bright orange lights which leads to an open part of the cave where they stumble upon several displays on the cave walls. There are several neatly arranged rectangles on a blue background holding reflective silver squares. Oddly, the pictures look similarly arranged yet different somehow. Keisha and Max find a pile of the squares on the floor and a blank background on the wall. The secret to opening the door is in placing the squares correctly on the blank background. Revisiting the clue boards, Max and Keisha look for a pattern and discover that each board always uses 36 squares. They record notes to create a table, and then test a solution that fails. Thanks to Keisha’s notes, they discover that 36 is the area and each shape is a rectangle. To solve the puzzle, both things have to be true. In the process they realize that a square is a rectangle.
- Identify, describe and extend geometric and numeric patterns into growing and shrinking patterns.
- Represent and record patterns using tools such as tables and graphs.
- Students discover rectangles with the same area can have different dimensions.
- Students discover the pattern must satisfy two attributes:
1) the shape must be a rectangle, and 2) the area must be 36.
Prepare student copies of the activity sheet, “So Close, But Yet So Far,” (.pdf file) and an overhead transparency, if desired. Supply square tiles or have students cut the 36 square grid into squares. Students will need scissors to cut out the squares.
Download PDF Activity Sheet |
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As students study multiplication, they learn: factor x factor = product, and use manipulatives demonstrating this operation by an array. The array in multiplication can always be represented as a rectangle. Squares are also rectangles, and if the product can be expressed as a square, the product is called a square number. (Ex: A 3 3 3 array creates a square of 3 rows, with 3 in each row, to show 9 objects. Nine is a “square number.”) If manipulatives are the same shape, size and number, then no matter how you arrange them (even an irregular shape), they will have exactly the same area.
Sometimes solving the next step in a pattern seems obvious, but you can’t tell if your solution is correct until you test it. Keisha’s table shows:
| 1 |
3 |
36 |
| 2 |
3 |
18 |
| 3 |
3 |
12 |
| 4 |
3 |
9 |
Max reasons that since the pattern listing the number of rows reads: 1,2,3,4… the next must be 5, so they test it. They discover that using the 36 pieces they can form a 5 x 7 rectangle, but there is a leftover piece.
Their test does not fit part of the rule: it does not make a rectangle. So they must continue to “guess and check” until they end up with a 6 x 6 rectangle, a square. This satisfies the rule that it has an area of 36 and is in the form of a rectangle.
Background information:
In order to fully engage students, they’ll need to know the definition of “rectangle,” a quadrilateral with four right angles. While a formal definition is not
necessary, students should understand that a square is one type of rectangle. It is a subset of the set of rectangles, because it has four right angles. It also has equal length sides, but this isn’t a requirement to be a rectangle. In working with Max and Keisha to solve the clues, students will be using the problem-solving strategies: create a table, work backwards, look for relationships, and guess and check.
The XY Encounter, Part 2 lays the groundwork for introducing:
array
two-dimensional
rectangle
square
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factors
length
width
product
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area
square units
square numbers
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