Name Date Class Period Activity B Extension 4.1 Modeling Transformations MATERIALS small white boards or paper markers masking tape yarn QUESTION What happens to ordered pairs when a rule is applied to the coordinates? EXPLORE Observe transformations STEP 1 Rectangle Your teacher will have four students form the vertices of a rectangle by standing on a coordinate grid on the floor. STEP 2 Rule Following a rule given by your teacher, students will move to new coordinates. STEP 3 Observe Pay close attention to how different rules change the location and size of the rectangle. Record your observations for each rule in words and symbols. Description Rule in Symbols Rule 1: Add 2 to the x-coordinate. Rule 2: Subtract 3 from the y-coordinate. Rule 3: Add 4 to the x-coordinate. Rule 4: Add 1 to each coordinate. Rule 5: Multiply the x-coordinate by 2. 1 of 8
Rule 6: Multiply the y-coordinate by 1 2. Rule 7: Multiply the x-coordinate by 1 2. Rule 8: Multiply the y-coordinate by 2. DRAW CONCLUSIONS Use your observations to complete these exercises If a transformation is performed on a point A, the new location of point A is indicated by A (read A prime ). In Exercises 1 and 2, use words to describe the transformation of ABCD to A B C D. Draw A B C D. 1. (x, y) > (x - 4, y) 2. (x, y) > (x, 1 2 y) In Exercises 3 8, triangle ΔABC has vertices A(0, 0), B(0, 2), and C(2, 0). Give the coordinates of the transformed triangle ΔA B C. 3. (x, y) > (x, y - 3) 4. (x, y) > (x + 5, y) 5. (x, y) > (x, 2y) A (, ) A (, ) A (, ) B (, ) B (, ) B (, ) 2 of 8
C (, ) C (, ) C (, ) 6. (x, y) > µ 1 2 x, 7. (x, y) > (x, 3y) 8. (x, y) > (x - 3, y + 2) y 9. How would you write the transformation that would shift a figure 2 units right and 4 units down? 3 of 8
Answer Key B STEP 3 Answers in the Description column may vary. Sample answer: Description Rule in Symbols Rule 1: Add 2 to the x-coordinate. Rule 2: Subtract 3 from the y-coordinate. Rule 3: Add 4 to the x-coordinate. Shift horizontally 2 units right (x, y) > (x + 2, y) Shift vertically 3 units down (x, y) > (x, y - 3) Shift horizontally 4 units left (x, y) > (x + 4, y) Rule 4: Add 1 to each coordinate. Shift vertically 1 unit up and horizontally 1 unit right (x, y) > (x + 1, y + 1) Rule 5: Multiply the x-coordinate by 2. Stretch by a factor of 2 horizontally (x, y) > (2x, y) Rule 6: Multiply the y-coordinate by 1 2. Shrink by a factor of 1 2 vertically (x, y) > (x, 1 2 y) Rule 7: Multiply the x-coordinate by } 1 2. Shrink by a factor of 1 2 horizontally and reflect across x-axis (x, y) > ( 1 2 x, y) Rule 8: Multiply the y-coordinate by 2. Stretch by a factor of 2 vertically and reflect across the y-axis (x, y) > (x, 2y) 4 of 8
DRAW CONCLUSIONS 1. shift horizontally 4 units left 2. shrink by a factor of 1 2 vertically 3. A (0, -3), B (0, 1), C (2, 3) 4. A (5, 0), B (5, 2), C (7, 0) 5. A (0, 0), B (0, 4), C (2, 0) 6. A (0, 0), B (0, 2), C ( 1, 0) 7. A (0, 0), B (0, 6), C (2, 0) 8. A ( 3, 2), B ( 3, 4), C ( 3, 2) 9. (x, y) > (x + 2, y - 4) 5 of 8
Teacher Notes ACTIVITY PREPARATION AND MATERIALS Before class prepare a coordinate grid on the floor of your classroom, in a foyer area, or any other open space. You will use a maximum x-value of 8 and a minimum value of 4, a maximum y-value of 4 and a minimum value of 4. Use two strips of masking tape (1/2 inch to 1 inch width) on the floor, each perpendicular to the other to create the axes. If the floor is tiled with square tiles, you can use them as your grid. Use a marker to scale each axes, leaving about 1.5 to 2 feet between integers. You can mark the tape with tick marks, coordinates, or both. You will need 4 small white boards and marker for the four students representing the vertices of the rectangle. If you want to make the activity more visual, use yarn or rope to represent the sides of the rectangle. One long piece (to form the perimeter) is easier for students to manage than four individual pieces. ACTIVITY MANAGEMENT Select four students to form the vertices of a rectangle. Each student holds a small white board or piece of paper with their coordinates recorded on it. They face towards the class. The orientation of the students standing will be the opposite of seated students so be careful with how the rules are given. To begin, hand each student their location (written on the white board or paper), and instruct them to stand on their ordered pair. To reinforce the order of plotting a point, you might have the students begin at the origin and say aloud for the point (5, 2), start at the origin and move right 5 units and up 2 units. Once all four points are plotted, ask the students to hold the yarn so that perimeter of the rectangle can be seen. Give the students a rule to apply to their ordered pair, and on your signal, they move to a new point. Once there, they write the coordinates of the new ordered pair on their white board. Each time a new rule is given, students move from their current position to a new position. Beginning Coordinates: (1, 2), (5, 2), (5, 4), and (1, 4) Rule 1: Add 2 to your x-coordinate Rule 2: Subtract 3 from your y-coordinate Rule 3: Add 4 to your x-coordinate Rule 4: Add 1 to each coordinate Each of the Rules 1 4 produces a vertical and/or horizontal shift. After each rule is plotted, ask students to describe the move, again remembering that students standing have a different orientation than seated students. Students should record their observations in words and symbols in the table provided. For example, the first rule would be recorded (x, y) > (x + 2, y). You might also ask what happened to the area and perimeter of the rectangle. The next set of rules will produce a vertical and/or horizontal stretch or shrink, both with and without a reflection. Be mindful of the current ordered pairs and make sure there is enough room to plot the new ordered pairs. If you want to change the students representing the points, this is a good time to do so. Beginning Coordinates: (0, 2), (4, 2), (4, 4), and (0, 4) Rule 5: Multiply your x-coordinate by 2 Rule 6: Multiply your y-coordinate by 1 2 Rule 7: Multiply your x-coordinate by 1 2 Rule 8: Multiply your y-coordinate by 2 After each rule has been plotted ask students to describe the move. They will have difficulty with 6 of 8
the yarn at this point because it won t stretch to the new location. You can either hand them a longer piece of yarn or omit the yarn at this point. Students should record their observations in words and symbols. For example, the fifth rule would be recorded (x, y) > (2x, y). Again, ask what has happened to the perimeter and area of each new rectangle. You can continue to provide new rules, or ask students what rule should be given to: - Shift the rectangle 5 units to the left. - Stretch the rectangle by a factor of 3 units in the vertical direction. - Shift the rectangle 3 units right and 2 units down. - Shrink the rectangle by a factor of 1 3 in the horizontal direction. - Reflect the rectangle about the y-axis and/or x-axis. An option to bring the activity to closure, reveal the rule only to the standing students (by whispering or by writing the rule on paper). Ask the remainder of the students what rule was just applied. A-Level Alternative After each transformation, discuss as a class how to describe the transformation in words and symbols before students record answers in their tables. C-Level Alternative Have students predict the effect of the transformation before the rectangle moves. 7 of 8
Activity and Closure Questions Ask these questions as a class. In Exercises 1 and 2, rectangle ABCD has vertices A(0, 0), B(0, 4), C(5, 4), and D(5, 0). Describe how the rule will affect the rectangle and give the coordinates of the transformed rectangle A B C D. 1. (x, y) > (x + 2, y - 1) Answer: The rule translates the rectangle two units to the right and one unit down. A (2, 1), B (2, 3), C (7, 3), and D (7, 1) 2. (x, y) > (3x, 1 2 y) Answer: The rule stretches the rectangle by a factor of 3 horizontally and shrinks the rectangle by a factor of 1 vertically. A (0, 0), B (0, 2), C (15, 2), and D (15, 0) 2 3. What operation produces a vertical shift? What operation produces a horizontal shift? Answer: Adding or subtracting a value from the y-coordinate produces a vertical shift. Adding or subtracting a value from the x-coordinate produces a horizontal shift. 4. What operation produces a vertical stretch? What operation produces a horizontal stretch? Answer: Multiply the y-coordinate by a number greater than 1 to produce a vertical stretch. Multiply the x-coordinate by a number greater than 1 to produce a horizontal stretch. 5. What operation produces a vertical shrink? What operation produces a horizontal shrink? Answer: Multiply the y-coordinate by a number between 0 and 1 to produce a vertical shrink. Multiply the x-coordinate by a number between 0 and 1 to produce a horizontal shrink. 6. In order for a figure to be reflected about the x- or y-axis, what has to occur? Answer: To be reflected about the x-axis multiply the x-coordinate by a negative number. To be reflected about the y-axis multiply the y-coordinate by a negative number. 7. What transformation does not change the shape or size of the figure? Answer: A horizontal or a vertical shift. LESSON TRANSITION When students complete this activity they should have a good understanding of horizontal and vertical translations, horizontal and vertical shrinks and stretches, and reflection across the x- and y-axis. After completing this activity students should be ready to start the Exercises on page 214. 8 of 8