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G6-21 Coordinate Systems To illustrate the idea of a coordinate system you can start with the following card trick:

GOALS

1. First, deal out nine cards—face up—in this arrangement:

Students will locate a point in an array or on a grid given its pair of coordinates.

Row 3

PRIOR KNOWLEDGE REQUIRED Columns, rows

Row 2 VOCABULARY



coordinate system coordinates row column line of symmetry names of special quadrilaterals

Row 1

Column 1

Column 2

Column 3

2. Next, ask a student to select a card in the array and then tell you what column it’s in (but not the name of the card). 3. Gather up the cards, with the three cards in the column your student selected on the top of the deck. Show clearly how you do that. 4. Deal the cards face up in another 3 × 3 array making sure the top three cards of the deck end up in the top row of the array. 5. Ask your student to tell you what column their card is in now. The top card in that column is their card, which you can now identify! 6. Repeat the trick several times and ask your students to try to figure out how it works. You might give them hints by telling them to watch how you place the cards, or even by repeating the trick with a 2 × 2 array. When your students understand how the trick works, you can ask the following questions:

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• Would there be any point to the trick if the subject told the person performing the trick both the row and the column number of the card they had selected? Clearly there would be no trick if the performer knew both numbers. Two pieces of information are enough to unambiguously identify a position in an array or graph. This is why graphs are such an Copyright © 2007, JUMP Math Sample use only - not for sale

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efficient means of representation: two numbers can identify any location in two-dimensional space (in other words, on a flat sheet of paper). This discovery, made over 300 years ago by the French mathematician René Descartes, was one of the simplest and most revolutionary steps in the history of mathematics and science: his idea of representing position using numbers underlies virtually all modern mathematics, science, and technology.  You might ask your students how many numbers would be required to represent the position of an object relative to an origin in three-dimensional space. (The answer is three. Think of the origin as being situated on a plane or flat piece of paper that has a grid or graph on it. You need two numbers to tell you how to travel from the origin along the grid lines on the plane to situate yourself directly above or below the object, and one more number to tell you how far you have to travel up or down from the plane to reach the object.) • Ask your students if the trick would work with a larger array. Have them try the trick with a 4 × 4 array. They should see that as long as the array is square (with an equal number of rows and columns), the trick works for any number of cards. Ask your students to explain why this is so and why the trick doesn’t work if the array isn’t square (for instance, try it with 2 columns and 6 rows). • Ask your students if the original trick (i.e., with a square array) would work if the subject told the performer which row the card was in rather than which column. Have your students show you how the new trick would be performed. The fact that the trick works equally well in both cases illustrates a very deep principle of invariance in mathematics. In a square array, there is no real difference between the rows and columns. In fact, if you rotate the array by a quarter turn, the rows become columns and vice versa. More generally, once you fix an origin in space, it doesn’t matter how you set up your grid (the lines representing the rows and columns). In all cases you need only two numbers to identify a position. Now draw an array of three columns and rows on the board and number the columns and rows: 3 2 1 1 2 3

C O R W L U M N

Point out a row and a column, and stress that we order rows from bottom to top, and columns from right to left. Ask several volunteers to locate the second column, the third row, the point that is in the third row and the first column, etc. Students that have trouble locating any particular dot could join the points in the given row and column prior to circling the dot itself. (The dot they are looking for will be where the lines intersect.) Draw an array of dots on the board, circle a point, and ask your students to write the coordinates of the point: Column___, Row___. Ask your students: Imagine you have to write the coordinates of 100 points. Would you like to write the words “column” and “row” 100 times? What could you do to shorten the notation? Students might suggest making a T-table or even writing a pair of numbers, because the column is always the first number. Ask: How do you know which is first, column or row? What if you have to ask a partner to find a dot given a pair of numbers, without telling them which number belongs to the column and which number is the row number? Give your students a pair of numbers, such as 2, 3 and do not tell them which one is the column number and which one 2

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WORKBOOK 6:2

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is the row number. How many points can they find that could go with these two numbers? Explain to your students that mathematicians have made an international convention: The place of the dot is given by two numbers, in parenthesis, and the column number is always on the left, and the row number is always on the right: (column, row). Give your students several pairs of numbers and ask them to identify the corresponding points on the grid. Explain that the pair of numbers is called “coordinates of the point.” Draw an array of dots and label the columns with letters and the rows with numbers. Ask your students to identify some points, such as (A, 4), (B, 2), (C, 3). Mark the points (B, 3) and (C, 2) and ask your students to identify them. Repeat the exercise with a grid instead of an array. As a variation, you might label both rows and columns with letters. Students need lots of practice. Review the names of special quadrilaterals and triangles before assigning these questions: 1. Graph the vertices A(1,2), B(2,4), C(4,4), D(5,2). Draw lines to join the vertices. What kind of polygon did you draw? How many lines of symmetry does the shape have? How many pairs of parallel sides does it have? 2. On grid paper, draw a coordinate grid. Graph the vertices of the triangle A(1,2), B(1,5), C(4,2). Draw lines to join the vertices. What kind of triangle did you make? Ask your students if they have seen any of these methods of marking points with letters or numbers in real life. You might show them a map with a grid on it.

Assessment 1. Identify the proper column and row for the circled dot: a) b) c) d)

Column Row

Column Row

Column Row

Column Row

2. Mark the positions: (3, 2), (4, 1). 3. Write the coordinates of the marked positions: _____, _____ 4 3 2 1 1 2 3 4

Bonus Draw the points below on a grid, then join the points in the order you’ve drawn them. Join the first point to the last point. What shape did you make? Find the area of the shape. a) (A, 4), (A, 5), (B, 5), (C, 4), (D, 4), (E, 3), (D, 3), (C, 2), (C, 1), (B, 2), (B, 3). b) (0, 1), (2, 3), (2, 6), (3, 7), (4, 6), (4, 3), (6, 1), (6, 0), (5, 0), (4, 1), (3, 0), (2, 1), (1, 0), (0, 0).

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ACTIVITY 1

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Invite students to sit or stand in an array. Identify the “columns” and the “rows” in the array, and make sure each student knows which row and column they are in. Give one of the students a ball and identify a point on the array with a column number and a row number. The student with the ball has to toss it to the student at the given point. Students can continue tossing the ball around by calling out coordinates rather than names.

Students will need a pair of dice of different colours. The player rolls the dice and records the results as a pair of coordinates: (the number on the red die, the number on the blue die). He plots a point that has this pair of coordinates on grid paper. He rolls the dice a second time and obtains a second point in the same way. The player joins the points with a line, then has to draw a rectangle so that the line he drew is a diagonal of the rectangle. There could be several rectangles drawn this way. If the line is neither vertical nor horizontal, the simplest solution is to make the sides of the rectangle horizontal and vertical. In this case, ask your students if they see a pattern in the coordinates of the vertices.

Iso Tra scele pe s zo id

Rectangle

Sq ua re Ge Tra nera l pe zo id

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us

ACTIVITY 2

Advanced: Students will need a pair of dice of different colours and a spinner below.

Parallelogram

The player rolls the dice twice and plots the points the same way he did in the previous activity. He also spins a spinner. He has to draw a quadrilateral of the type the spinner shows, so that the line he drew is the diagonal of the quadrilateral.

Extension The card trick can be modified for non-square arrays if one allows one extra rearrangement. Deal out an array of 3 columns, 9 rows. Have a student select a card and tell you what column it’s in. Re-deal the cards so that all of the nine cards from the chosen column land in the top three rows of the new array. Ask the student to tell you what column their card is in now, and re-deal the top three cards in that column into the top row of a new array. Once the student tells you what column their card is in, you can identify the top card in that column as the one they selected. This version of the trick illustrates a powerful general principle in science and mathematics: when you are looking for a solution to a problem, it is often possible to eliminate a great many possibilities by asking a well-formulated question. In the card trick one is able to single out one of 27 possibilities by asking only three questions. Repeat the trick, asking your students how many possibilities were eliminated by the first question (18), by the second question (6), and by the third (2). 4

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G6-22 Coordinate Systems (Advanced) Review negative numbers on a number line. Remind your students that number lines can be extended in both directions.

GOALS Students will locate points, given their coordinates, in the four quadrants of a coordinate system. They will also identify the coordinates of points on a grid.

PRIOR KNOWLEDGE REQUIRED Negative numbers on a number line Coordinate systems Rows, columns Special quadrilaterals

Draw a coordinate grid on the board (or use the overhead). Make sure the axes are centered on the grid so that all four quadrants are visible. Tell students that each line in this coordinate system is called an axis. (You might mention that the plural of axis is axes.) The horizontal line is called the x-axis, and the vertical line is called the y-axis. Tell students that the point at which the 2 axes intersect is called the origin. Label the axes and the origin. Point out that the axes separate the grid into four parts. Explain that these are called quadrants, and ask your students which words that they know have the same beginning. (quadrilateral, quadruple) Draw several points in various quadrants and ask your students to tell which quadrant the points are in.



VOCABULARY coordinate system coordinates row column line of symmetry names of special quadrilaterals quadrant

II

I

III

IV

Remind your students that in a coordinate pair such as (3, 6), 3 is the column number and 6 is the row number. Tell students that you are going to plot, or place, this pair on the coordinate system. Add numbers to the positive halves of each axis (as you would on a number line) and ask your students how they might find the point (3, 6). Where are the columns and the rows? Point out that numbers on the x-axis correspond to the numbers of columns, and numbers on the y-axis correspond to the numbers of rows. This means that the first number in the coordinate pair should be counted on the x-axis. Ask your students to plot points such as (1, 2), (4, 0), (0, 3), (0, 0). Now make a mark at point (-1, -1) and ask your students what the coordinates of this point might be. Invite volunteers to add numbers to the negative parts of each axis. Mark a point in the second quadrant, such as (-3, 4), and show students how to find the signs of its coordinates—the point is on the negative side of the x-axis and the positive side of the y-axis, so the signs of the point will be (-, +). Repeat with a point in the fourth quadrant. Ask the students to tell the signs of various points on the coordinate system. Students might mark the signs for each quadrant. Ask your students in which quadrants different points appear. EXAMPLES: (-3, 2), (4, -6), (-2, -2). Show your students how to find the complete coordinates of a point in any quadrant (number and sign). Then have students determine the coordinates of points in various parts of the grid. For example, ask your students to name the coordinate pairs for five points in the second quadrant. What do these points have in common? (They have the same signs.) Then do the reverse: give students various coordinates and ask them to mark the points. Start with easier points, such as (2, 3), (5, 4), (-3, -3), then try slightly harder points, such as (-4, -5), and then harder points, such as (-3, 3) and (3, -3). As a last

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challenge, give your students points such as (-4, 7) or (3, -6). Students need lots of practice plotting points in different quadrants. As an additional exercise, let your students plot the following points, join them, and tell what type of quadrilateral they have drawn. a) (1, 2), (-1, -1), (2, -3), (4, 0) b) (-2, -2), (-1, 1), (-2, 4), (-3, 1)

ACTIVITY 2 6

(–, +)

y-axis

ACTIVITY 1

Using masking tape, create a large coordinate grid with the four quadrants on the floor (a tiled floor is particularly useful). Ask students to locate and stand on various ordered pairs in all four quadrants.

(+, +) x-axis

(–, –)

(+, –)

Repeat Activity 2 from G6-21 using both dice and the spinner shown. Students should spin the spinner with each roll of the dice. The spinner will indicate the quadrant that the point is plotted to.

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TEACHER’S GUIDE