Activity Set 1.3PREP PPTX

Visual Algebra for Teachers

Chapter 1

INTEGERS AND INTEGER OPERATIONS

Visual Algebra for Teachers

Activity Set 1.3

Arrays with Black and Red Tiles

Visual Algebra for Teachers

PURPOSETo learn how to model rectangular arrays with black and red tiles and determine the net value of the arrays.

To explore the result of flipping over columns and rows on the net value of a rectangular array.

To learn how to use edge pieces to keep track of flipped columns and rows in a rectangular array of black and red tiles.

To learn about minimal arrays, minimal collections and the empty array.

Black and red tiles with black and red edge pieces

No calculators for this activity set

MATERIALS

INTRODUCTION

Rectangular ArraysA Rectangular Array is a rectangular arrangement of numbers or objects.

Rectangular Array ExamplesThis is a rectangular array of numbers. This array has 3 rows and 4 columns; it is a 3 4 Array (row then column)

Rectangular ArraysRectangular Array ExamplesThis is a rectangular array of black tiles

This array has 2 rows and 5 columns; it is a 2 5 Array

Rectangular Array TermsEdge Pieces and Edge Sets are defined in context, see activity 2

Minimal Arrays, Minimal Collections and Minimal Edge Sets are defined in context, see activity 8.

The Empty Array and the Non-Empty Array are defined in context, see activity 8.

Equivalent ArraysOrientation on the page does not distinguish arrays

A 1 3 array and a 3 1 array with the same edge sets are equivalent arrays and only one should be given as a solution to a question.

Sketching ArraysWhen sketching an array of black and red tiles, show the rectangular shape of the array and the square shape of the tiles. It may be easiest to sketch blank squares, lightly shade the black tiles and label the interiors of the red tiles R or to skip shading and just label both the black and the red squares with a B or an R, respectively.

PREP Question #1aUse your black and red tiles and form the following2 5 array of black tiles. This is Array 1.

a) What is the net value of Array 1?

Solution: There are 10 black tiles, the net value is +10

PREP Question #1bb) Using your model of Array 1, pick one column and flip over all of the tiles in that column; what is the ending net value of the array? Return Array 1 to all black tiles and flip over all of the tiles in another column. Does the ending net value depend on which column you flip over? Sketch two such arrays and explain.Solution: One possibility—Net Value is +6

When flipping one column of Array 1, the net value will always be +6, the column choice does not matter.

PREP Question #1cc) Return Array 1 to all black tiles; pick two columns and flip over all of the tiles in both columns; what is the ending net value of the array? Does the ending net value depend on which two columns you flip? Explain.Solution: One possibility—Net Value is +2

When flipping two columns of Array 1, the net value will always be +2, the column choices do not matter.

PREP Question #1dd) Return Array 1 to all black tiles and flip over all of the tiles in Column 1 (C1) and then flip over all of the tiles in Row 1 (R1).

• Note the tile that is in both C1 and R1 will be flipped twice. What is the ending net value of the array? Does the ending net value depend on which column and row you flip?

• Experiment with several combinations (C2 and R3, C3 and R1, etc.) and several orders (column then row or row then column) and explain what happens if you flip one column and then flip one row.

• Sketch at least one such array.

PREP Question #1dSolution: Flip Column 1

When flipping column 1, then row 1, the net value is 0

Flip Column 2 then flip row 2

The net value is still 0

Flip Row 1

PREP Question #1dFlip Row 1 and then Column 1

The net value is still 0

Does the ending net value change if the order of flipping column, then row, or row then column changes?

Does the array change?

Now do #1 in Activity Set 1.3 in your coursepack if more practice is needed.

Edge SetsNote that it is difficult to look at an array of black and red tiles and determine which columns and rows have been flipped over.

To keep track of the flipping information, we will use edge pieces and edge sets.

Edge pieces indicate whether or not a row or column of an array has been flipped or turned over.

An edge piece is a thin piece of black or red tile that is used to mark the edge dimensions of a black or red tiles (an edge piece has no height).

Edge SetsA red edge piece at the end of a row or column indicates the row or column has been flipped.

A black edge piece at the end of a row or column indicates the row or column has not been flipped.

Edge pieces are designed to keep track of flipping and the dimensions of a rectangular array; the edge pieces themselves are not counted when determining the net value of an array.

Edge Set ExampleStarting with an all black 3 5 array 1, these edge pieces indicate that Column 1, Column 3 and Row 3 have each been flipped over.

The edge sets have been labeled I and II for reference. Edge sets have net values just like tile collections as illustrated in the table:

#B #R NV

Edge I 2B 1R +1

Edge II 3B 2R +1

Array 8B 7R +1

PREP Question #2Write in your coursepack, Activity Set 1.3 #2Use black and red tiles, with edge pieces, to explore the connection between edge pieces and arrays. Use black and red tiles to model each array as you fill out the table. You don’t need to sketch the arrays.

You are now ready for:

PREP QUIZ 1.3

See Moodle

Visual Algebra for Teachers