Activity 2 Examine the patterns and complete each table below. 1. Oblong numbers:
How can you predict the next oblong number? What patterns can you find? Describe.
2. Triangular numbers:
How can you predict the next triangular number? What patterns can you find? Describe
|
|
3. Square numbers:
What patterns can you find? Describe.
4. Can you find a relationship between the oblong and the triangular numbers? Describe it.
How would you explain the relationship by using only the dot representations?
5. What relationships can you find between two consecutive triangular numbers and a square number? Describe.
How would you explain these relationships by only using the dot representations? 6. Optional: Sometimes you will see fruit or cans in grocery stores piled high in the form of a pyramid as shown. The number of pieces of fruit or cans is sometimes called a pyramidal number. Pyramidal numbers: What patterns can you find? Describe.
|
Activity 2B Try to discover patterns in the following figures; describe the patterns.
How are the patterns alike? How are they different? |
Activity 2C  
    With eight tiles, create a pattern to form a square with a hole in the middle. Record the number of vertices and the number of sticks in a table. Using twelve tiles, create a pattern to form a square with a square hole in the middle. Do the same with sixteen tiles. What patterns do you see?
How is this pattern different from the other patterns? |
Activity 2D a. an acute triangleCheck with your neighbors. Are different correct results possible? Explain. 2. In the diagrams below, points A, B, C, and D are called boundary points and points E, F, and G are called interior points.
Construct the following figures on your geoboard and record each result on dot paper: a. a triangle with no interior points and four boundary pointsCheck with your neighbors. Are different correct results possible? Explain. 3. Construct a triangle on your geoboard. Do not show it to your neighbor but give her or him careful instructions on how to create your triangle on her or his geoboard. When finished, compare your triangle with your neighbor's. Are they congruent? If they are not congruent, explain why. Now reverse roles. Let your neighbor construct a quadrilateral on a geoboard (without showing it to you) and then give you instructions to construct the same quadrilateral on your geoboard. Compare results. Are they congruent? If they are not congruent, explain why. 4. How many different-sized squares can you find on your geoboard? Record them on dot paper. (Hint: Be sure to turn your geoboard so that it looks like a diamond shape in order to find more different-sized squares.) |
|
5. Place one rubber band in a vertical position on your geoboard to divide it in half. On one side of the rubber band, construct a scalene right triangle. Think of the rubber band as a mirror, and ask your neighbor to construct a scalene right triangle on the geoboard on the other side of the rubber band so that the new triangle is a reflection of your triangle. Do you agree with your neighbor's construction? Record your triangle and its reflection on dot paper.
6. Challenge: Can you construct an equilateral triangle on a geoboard that has its pegs arranged in a square grid pattern? Why or why not? (Check the length of the sides with a ruler.) |
Activity 2E 1. On graph paper, create five right triangles with legs of the following lengths: a. 3 and 4Find the length of the hypotenuse of each of these triangles (use a strip of graph paper) and record the data in the first three columns of the chart below:
Now complete the last three columns of the chart. Do you see any patterns? Describe.
What conjecture would you make concerning the lengths of the three sides of a right triangle? 2. a. In the middle of a geoboard, create a small right isosceles triangle and construct a square on each side of the triangle. Record the result on dot paper. Is there a relationship among the areas of the three squares? Describe.
b. Repeat the activity above on a geoboard, using a right triangle with legs of 1 and 2 units. c. From a set of tangram pieces, select the middle-sized right triangle. Now build squares on each side of the triangle using the following tangram pieces: two large triangles, two middle-sized triangles, and four small triangles. Is there a relationship among the areas of the three squares? Describe it. |
|
2d. From the visual representation of the triangles below, what conjectures might you make about the relationship of a2 + b2 to c2 for an acute triangle? For an obtuse triangle?
a. If two joggers want to go from A to B in a square-shaped open field, what possible paths could they take (without retracing any direction)? What is the length of the shortest path? What is the longest path? Explain. b. Can a circular table top with diameter 2.7 meters long fit through a doorway 2.5 meters high and 1 meter wide? Why or why not? c. How far up on a wall of a building will a 10-meter ladder reach if the foot of the ladder is 6 meters from the wall? Explain d. What is the length of the longest pole you could put in a rectangular storage room 12 units long, 9 units wide, and 8 units high? Explain. 4. Challenge:In the middle of a sheet of paper, draw a right triangle ABC (right angle at C) with legs of 1 unit. Using segment BA as a leg, draw a right triangle ABD (right angle at A) and leg AD equal to 1 unit. Using segment BD as a leg, draw a right triangle BDE (right angle at D) and leg DE equal to 1 unit. Using segment BE as a leg, draw a right triangle BEF (right angle at E) and leg EF equal to 1 unit. Continue this process at least eight more times.What is the length of each hypotenuse? What patterns do you notice? Describe.
If you continued this process twenty more times, what figure would you get? Make a sketch of the resulting figures. |
3. Real-world applications:
