AREA 2013 Judo Math Inc.
6 th grade Problem Solving Discipline: Black Belt Training Order of Mastery: Area 1. Area of triangles by composition 2. Area of quadrilaterals by decomposing 3. Draw polygons on coordinate plane 4. Geometry on the coordinate plane Blue Belt Training Area What comes to mind when you think of the word Geometry? Write ALL of the words that pop into your head in the box below. Geometry!!! In this belt, we are going to build upon some of the thing you listed above that you have discussed in 4 th and 5 th grade. There will be a lot of fun with shapes, but we are bumping it up a notch to make it even more challenging and fun! And now that you have your yellow and orange belt and are becoming excellent with equations, you will be able to do more with shapes than you ever imagined! Good Luck Grasshopper. Standards Included: 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. 2013 Judo Math Inc.
Dog Pen Problem: Mr. Shaddox got a new dog and wants to make the largest possible pen for him. He has 60 feet of fencing. What is the largest area the pen can have? 1
1. Area of right triangles by composition Working with a group, try to figure out what the area of this is? a. What are the challenges in this? b. What shortcuts could you use? c. Discuss this situation with others in the class. Was ANYONE able to figure out an EXACT answer? How did they do it? 2
d. Would finding the area of the triangle on the previous page been easier or harder if there wasn t a coordinate plane behind it? 7 5 e. How is the original triangle related to the rectangle above? Can this help you to find the exact area?! 7 5 3
The exact area of a triangle can be determined by finding the area of a rectangle with the same base and height as the triangle (composing a rectangle) and dividing it by. 1. Compose these triangles into rectangles with the same base and height to show how you could find the exact area simply: 3 4 8 8 4 3 2. Draw and then cut out 5 sets of congruent right triangles on grid paper. Tape each pair of congruent triangles together to form a rectangle. Label both the area of the rectangle and the area of each triangle on each set. (you can staple this sheet to the back of this packet) 4
3. Find an object in the classroom that is in the shape of a right triangle. Measure the sides to the nearest half inch. Then, using grid paper, find the area of the triangle by composing it into a rectangle. (do this on the same grid paper as above) 4. So now, just like many mathematicians who have gone before you, you have discovered the formula for finding the area of a right triangle. Inspect the box below to see if you agree with this summary of our findings: 5
Supposing you agree with the information above, I have a very important question for you. The triangle below is NOT a right triangle, but I have drawn it on a grid for you. How would you find the area for this? 4. What happens when you try to compose it into a rectangle? 5. But what is the area of a parallelogram?! Ideas here: 6
6. But what happens when you take the triangle on the left and move it over to the right? That s right, you get rectangle. Therefore the area of this parallelogram is the same as the area of a rectangle (base x height OR 5x8=40) 7. What does that mean for the area of the triangle from above? So now we have looked at a couple of different types of triangles and determined that for both of them, their area was Because a triangle is half of a rectangle/parallelogram! In this formula, b is the base and h is the height. The thing is, it s important to consider WHAT the b is and what the h is. 7
Let s work through this: 8. Your classmates and you were asked to label a base and a height for this triangle. Three students drew the figures below. Saul Bobby James a. Have any of the students from correctly identified a base and the height that goes with it? Explain what is incorrect. 8 More questions on the next page
b. There are three possible base-height pairs for this triangle. Sketch all three. It turns out that every triangle has 3 base-height combinations! Why do you think this is? Brainstorm with a partner about this here: (hint: say something about the height being PERPENDICULAR to the base!) 9. Now find the three base-height combinations for this triangle: 10. One more triangle, label the three different bases and heights: 9
11. Draw a parallelogram on a piece of grid paper. Find the area of the parallelogram. Then, cut it in half to create two congruent triangles. Label the type of triangles you made and then find the area of each triangle. Glue them here: 12. Think of a type of triangle that can be found in real life. Draw an image of that triangle on grid paper. Then, estimate the size of the triangle. Find the area using the strategies you learned in this video 10
Here are 10 more triangles to practice finding the area of. Complete as many of them as you need to feel like a triangle expert! 11
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3. Area of special quadrilaterals and polygons by decomposing In the last section, we did a lot of work with the area of triangles. And we started by composing the triangle into either squares or parallelograms to help us figure out the formula. In this section, we are going to decompose instead of compose. What does decompose mean? DECOMPOSE the following shapes into squares, rectangles, and/or triangles. Use a different colored pencil to shade each shape. Estimate the area to each shape. 1. 2. 13
3. 4. 5. 14
4 6. 7 7. For these, you might have to think a little differently! (hint: pay attention to the area OUTSIDE the shape!) Find the area of each one.) 15
Now let s take a look at another shape: the Trapezoid. Here is a picture of a trapezoid. In the box to the right, please list anything you know about trapezoids, observe about them now, and any ideas you have about how to find the area of them! Trapezoids! There are a couple of ways to think of the area of a trapezoid 1. Composing 2 trapezoids into a parallelogram How could this help you find the area? 2. Decomposing it into two triangles on the ends and a rectangle in the middle. Try this below. How could this help you find the area? 16
Find the area of these shapes using your new knowledge of trapezoids. strategy! 3. 4. 5. 17
6. Draw a trapezoid on a piece of grid paper. Then, draw another of the same trapezoid (this is called a congruent trapezoid). Connect the two trapezoids to form a parallelogram. Find the area of the parallelogram. Using the area of the parallelogram, find the area of each trapezoid. (you can use the grid below OR use graph paper to make a bigger one) 6. (partner activity) Draw an irregular polygon with 20 sides or more on grid paper. Trade with a neighbor and then decompose their polygon into familiar shapes to find the area. 18
4. Geometry on the coordinate plane Review: Here is a coordinate plane. In the space below, list as many things as you can remember about the coordinate plane. You can even label things on it if you need (including a few points!) 19
It is important to remember that a coordinate plane consists of an x-axis and a y-axis. These axes split the plane up into four quadrants. Even more importantly, it is important to remember that the first number in an ordered pair represents the location along the x-axis and the second number represents the location along the y-axis. (3, 2) means go 3 to the right (along the x-axis) and 2 up (along the y-axis) Label (3, 2) on this coordinate plane to the right. 1. Plot these points then connect them to make a shape. (-4, 8) (-9, 3) (1, 3) (-4, -2) 2. Plot these points then connect them to make a shape. (2, 5) (7, 1) (5, -5) (-1, -5) 20
3. Look at these points. Any ideas on what shape it would make without plotting them? (9, 2) (8, -2) (-5, -2) (4, 2) Now plot them and check to see if you were correct! 4. Look at these points. Any ideas on what shape it would make without plotting them? (-4, 8) (-9, 3) (1, 3) (-4, -2) Now plot them and check to see if you were correct! 5. Find the length and width of the shape that s created by these ordered pairs. Plot them if you need to. (4, 1) (4, -4) (-6, 1) (-6, -4) 21
6. Name the shape then find the area of the polygon created by: (-5, 2) (-5, -5) (3, -5) 7. Find the perimeter of the shape that has vertices at (8, -8) (1, -8) (8, -1) (1, -1) Did you need to plot the points to find the perimeter? How could you find it without plotting the points? 22
8. Here are three vertices for a rectangle. Can you figure out what the last point must be? (-6, 4) (-6, -1) (8, 4) 9. If I give you two points to a rectangle, can you find the other vertices? (2, -3) (-8, -7) What if I tell you the area has to be greater than 20 for this rectangle. Does that help narrow your answer? What would you need to fully narrow your answer down to the rectangle that I m looking for? 23
10. Using a piece of graph paper, draw a rhombus. Then, on a separate coordinate grid, plot 3 of the vertices from your rhombus. Switch with a partner and see if they can label your missing vertices and draw your shape. 11. Without looking at a coordinate plane, come up with ordered pairs that could be vertices for a rectangle. Then, plot the points and draw the rectangle. How did you know how many ordered pairs to create and where to put them? 12. On a piece of graph paper, draw least least 3 shapes that have an area of 10 square units. 24
Mini-Project: Bedroom Design Part 1: Using graph paper, draw a scale model of your bedroom. Find the area of the room with the furniture and without the furniture. Part 2: Now spend some time designing your dream bedroom and drawing a model of that on another piece of paper. Think of how much square footage would be give you enough space but wouldn t require too much cleaning. Then think about furniture and floor space. Be prepared to share your design with the class. 25