E xpe ri e nc e manipulating triangles on a coordinate grid

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1 1 U n t er r ich t splan Triangles on the Coord inate Grid Altersgruppe: Grade 6 Virginia - Mathematics Standards of Learning (2009): 6.11a, 6.11b Virginia - Mathematics Standards of Learning (2016): 6.8.a, 6.8.b Fairfax County Public Schools Program of Studies: 6.11.a.1, 6.11.a.2, 6.11.a.3, 6.11.a.4, 6.11.a.5, 6.11.b.1 Online-Ressourcen: T he M i ssi ng Vert e x Opening Teacher present s Students pract ice Class discussion Closing 8 min 1 2 min 1 2 min 1 2 min 3 min M at h Obj ect ives E xpe ri e nc e manipulating triangles on a coordinate grid P rac t i c e plotting points Learn to identify different types of triangles De vel o p algebra skills Ope ni ng 8 min

2 2 To each pair of students, hand out the following nine shapes. Ask the students to place the shapes into three categories. Once the students have sorted the shapes, discuss. Ask: What are the titles of your categories? Some students may have categorized by color: blue, purple, and orange. Ask: Does anyone have different categories? Some students may have categorized by shape: circle, rectangle, and triangle. Say: We can categorize these shapes in two separate ways: by color or by shape. Today, we are going to the do the same thing with triangles. We can categorize them by side length or by angle measure. Write on the board: Triangles by Sides Equila t e ra l a triangle with all equal sides I s o s c e le s a triangle with two equal sides S c a le ne a triangle with no equal sides Triangles by Angles A c ut e a triangle with three acute angles Right h t a triangle with one right angle Obt us e a triangle with one obtuse angle

3 3 Say: If we are categorizing by shape, belongs in the circle category. If we are categorizing by color, belongs in the orange category. Likewise, we can classify a single triangle in two ways: by its side length or by its angle measure. T e ac he r prese nt s M at h game : T he M i ssi ng Vert e x - T ri angl e s o n a Gri d 12 min Using Presentation Mode/Smart Board Mode, present Matific s episode T he M i ssi ng Vert e x - T ri angl e s o n a Gri d to the class, using the projector. The goal of the episode is to create different types of triangles on a coordinate grid. Mat hespiel - T riangles on a Grid - Pract ice f or

4 4 Review with the students how to plot a point on the coordinate grid. Remind them that vert i c e s is the plural of vert e x. Say: The episode has presented us with a right triangle. Two of the vertices, (0, 5) and (7, 0) are identified. Look at the triangle. What are the coordinates of its third vertex? Enter the coordinates that the students suggest by clicking on each and then click. If the answer is correct, the episode will proceed to the next problem. If the answer is incorrect, the instructions will wiggle. Say: In this second problem, the two given vertices are not plotted for us. Ask a student to come to the front to plot the given points. Ask: What kind of triangle are we asked to form? Describe that type of triangle. We are asked to form an isosceles triangle. Isosceles triangles have two equal sides. Ask: Where should we plot the third point to create an isosceles triangle? There is more than one answer to this problem. The episode will present a total of four problems. Encourage the students to look for more than one solution to each problem.

5 5 St ude nt s prac t i c e M at h game : T he M i ssi ng Vert e x - T ri angl e s o n a Gri d 12 min Have the students play T he M i ssi ng Vert e x - T ri angl e s o n a Grii d on their personal devices. Circulate, answering questions as necessary. Cl ass di sc ussi o n 12 min Display the following: Ask: What kind of triangle is displayed? How do you know? The triangle is a scalene right triangle. One side is vertical and the other is horizontal, so they form a right angle. A triangle that contains a right angle is a right triangle. All the sides are different lengths, so the triangle is scalene. Ask: What are the lengths of the horizontal and vertical sides of the

6 6 triangle? How do you know? We can find the lengths of the sides of the triangle by counting the grid lines. The base of the triangle is 8 units long, and its he i ght is 3 units. Display the following: Ask a student to come to the board and plot the points (3, 4), (8, 4), (3, 9), and connect them.

7 7 Ask: What kind of triangle is this? How do you know? This is an isosceles right triangle. Both the horizontal and vertical sides are 5 units long, and the horizontal and vertical sides form a right angle. Say: We could also have this isosceles triangle. Display the following: Ask: How can we tell that this triangle is isosceles? The bottom vertices are (2, 2) and (6, 2). The top vertex is (4, 7). To get from either of the bottom vertices to the top vertex, we need to travel 5 units up and 2 units to the side, so the distance is the same. Two of the sides of this triangle are equal in length, so it is isosceles. Ask a student to come to the board to plot the points (6, 1) and (6, 9) and connect them.

8 8 Ask: Where could we place a third vertex to form an isosceles triangle? There are infinite solutions. Solicit different answers by calling on a number of students. If the given line is used as the base of the triangle, then possible responses include: (2, 5), (5, 5), and (11, 5). If the given line is used as a l e g of the triangle, then two more points in Quadrant I are possible: (14, 1) and (14, 9). [Two more solutions, (-2, 1) and (-2, 9), exist in Quadrant II.] Say: We can divide our answers into two groups: one where the given line is the base of the triangle and one where it is one of the legs. When it is the base, then all the points we named have 5 as their y- c o o rdi nat e. Why is this true? The number 5 is exactly in the middle of 1 and 9. So any point that has a y-coordinate of 5 will be equally distant from (6, 1) and (6, 9). Ask a student to come to the board to plot the points (3, 5) and (7, 5) and connect them.

9 9 Ask: Where could we place a third vertex to form a right triangle? There are infinite solutions. Solicit different answers by calling on a number of students. Possible responses include: (3, 1), (3, 10), (7, 2), and (7, 8). [When the given line is the hypo t e nuse of the triangle, then the two points (5, 3) and (5, 7) are also possible.] Ask: If the right angle is at (3, 5), what is true of all possible places to locate the third vertex? The x- c o o rdi nat e is always 3. Ask: If the right angle is at (7, 5), what is true of all possible places to locate the third vertex? The x-coordinate is always 7. Ask a student to come to the board and plot the points (5, 7), (10, 2), and connect them.

10 10 Ask: Where could we place a third vertex to form an obtuse triangle with the obtuse angle at (5, 7)? There are infinite solutions. Solicit different answers by calling on a number of students. Possible responses include: (0, 5), (2, 7), (3, 6), and (4, 11). Ask a student to come to the board and shade the area of the coordinate grid where the third vertex could be placed. Anything to the left of this red line, as long as the point is not c o l l i ne ar with (5, 7) and (10, 2): Display the following:

11 11 Ask: How do we know that this triangle is not equilateral? It looks like all three sides are 4 units long. While the horizontal and vertical lengths are both 4 units, the hypotenuse is not. The lengths of the diagonals of each square in the grid are longer than the edges of each square.

12 12 Cl o si ng 3 min Hand out a small piece of paper. Ask students to write down three points and state the type of triangle formed and how they can identify the triangle. When the students are done working, collect the papers to review later. A possible response: (3, 7) (5, 7) (3, 1) These three points form a scalene right triangle. All three sides have different lengths, and the vertical and horizontal sides form a right angle.

Riverside USD Scope and Sequence: 4.G.2 [4.7 ], 4.G.2 [4.8] Oklahoma Academic Standards Mathematics: 4.A.1.3, 4.GM.1.2,

1 U n t er r ich t splan Classifying Triangles Altersgruppe: Grade 5, Grade 4 Texas - TEKS: G4.6.GM.C, G6.8.E E R.A, G6.8.E E R.B Riverside USD Scope and Sequence: 4.G.2 [4.7 ], 4.G.2 [4.8] Oklahoma Academic