ok so i am really confused on my assignment. So What we are suppose to do is take pattern blocks (a triangle, square, hexagon, rhombus, trapazoid, and small rhombus) and see how many can go around a central point. I figured out that there are 6 possible solutions for a triangle, 4 solutions for a rhombus, 4 solutions for a square, 3 solutions for a hexagon, 4 solutions for a trapazoid, and 4, 8, and 12 solutions for the small rhombus. What I am having trouble on is the discussion questions. I answered a few but I am still kind of confused by some.

On Question 1 the question says which blocks offer a unique solution and why. I think the answer is the tan blocks (small rhombus) offer a unique solution because it can have 4,8, and 12 solutions while the others can t. Is that right?

On Question 2 it says why are the tan block solutions only multiples of 4? I think that the reason the tan block solutions can be multiples of 4 is because it is four sided but Im not sure if thats right.

On Question 3 it says explain why the blue and red blocks are interchangable for the purpose of this assignment. im confused as to why the red and blue blocks can be interchanged for this activity

on question 4 it says how can you figure out the angle measurement of the pattern blocks knowing that there are 360 degrees around a point.

Lastly question 5 says what are some pedagogical advantages of engaging students in this kind of activity.

I have a question for you.....WHY!!

Let's break down each of the discussion questions:

Question 1: Which blocks offer a unique solution and why?

To answer this question, you need to consider the number of possible arrangements around a central point for each block. Based on your information, you correctly identified that the tan blocks (small rhombus) offer a unique solution because they can have 4, 8, and 12 solutions, while the others cannot. This means that the tan blocks have multiple symmetric arrangements around a central point, giving them a unique characteristic compared to the other blocks.

Question 2: Why are the tan block solutions only multiples of 4?

To understand why the tan block solutions are only multiples of 4, you need to examine the properties of the block. The tan block (small rhombus) is a four-sided figure with equal angles. The number of solutions around a central point corresponds to the number of times the angles in the figure can fit evenly into 360 degrees (a full rotation). Since the angles of the tan block are the same, they can be evenly divided into 360 degrees in multiples of 4, explaining why the solutions with this block are only multiples of 4.

Question 3: Explain why the blue and red blocks are interchangeable for the purpose of this assignment.

In this assignment, the interchangeability of the blue and red blocks refers to the square and rhombus. Both the square and rhombus have four equal sides, and when arranged around a central point, they create the same pattern. Therefore, for the purpose of this activity, you can use either the blue block (square) or the red block (rhombus) interchangeably because they offer the same solution.

Question 4: How can you figure out the angle measurement of the pattern blocks knowing that there are 360 degrees around a point?

To determine the angle measurement of the pattern blocks, you should divide 360 degrees by the number of sides of the block. Each internal angle of a regular polygon can be found using the formula "angle = 180 * (n - 2) / n", where "n" represents the number of sides of the polygon. For example, if you have a square (4 sides), the calculation would be "angle = 180 * (4 - 2) / 4", which equals 90 degrees. This formula can be applied to any regular polygon to find the angle measurement of each block.

Question 5: What are some pedagogical advantages of engaging students in this kind of activity?

Engaging students in activities involving pattern blocks can have several pedagogical advantages. Some of them include:

1. Hands-on Learning: Manipulating physical objects like pattern blocks allows students to have concrete experiences, promoting better understanding and retention of mathematical concepts.

2. Spatial Reasoning: Engaging with pattern blocks enhances spatial reasoning skills, which are crucial for various disciplines like geometry, architecture, and engineering.

3. Problem Solving: Exploring different combinations and arrangements of pattern blocks encourages creative thinking and problem-solving skills.

4. Collaboration: Group work with pattern blocks fosters collaboration and communication among students, as they discuss and share their findings and strategies.

5. Multiple Representations: Pattern blocks provide visual representation of mathematical concepts, allowing students to understand different geometric properties and relationships.

Overall, using pattern blocks in activities can make math more engaging and accessible, promoting deeper understanding and critical thinking skills.