. Presentation of Learning-
1. Congruence and triangle congruence
Two objects that are congruent if they have the same shape and the same size. The symbol used for congruence varies but is usually either ≅ or ≃. When objects have congruent parts (sides, segments,and/or angles), congruence markings are used to make this clear. Congruence means that the objects are the same size and shape; it does not mean that they are equal. Triangles that are the same shape and same size are congruent triangles. When triangles are congruent, then the corresponding angles and corresponding sides are congruent.
Congruence was a main factor in my "scaling the world" project during benchmark #2 and #3 because both of those assignments needed to be the same exact shape, the size had to vary either up scaled or down scaled. This was more important for benchmark #2 because that is the one we had to give measurements of the congruent items we were using.
2. Definition of Similarity
When two triangles are similar, we can use proportions to find the lengths of missing sides. When the triangles are similar, we know that corresponding side lengths must be proportional. Similar is a term used in math when discussing geometric figures or shapes, and it means that both figures' corresponding sides are proportional, but the figures themselves are two different sizes. Two objects have to be similar if they both have the same shape (or one has the same shape as the mirror image of the other shape). In geometry, the shape of an object is its form (typically its external boundary or outline) .By definition, mathematicians have agreed that triangle two triangles are similar if all corresponding angles are congruent (have the same measure).
3. Ratio's and Proportions
A ratio is relationship between two numbers indicating how many times the first number contains the second. A ratio can be a whole number.Sometimes ratios are not whole numbers so we use either the “colon notation” (such as 10 : 8) or we use fractions 10/8. There are 3 ways to write a ratio; “a to b”, “a : b” , “16 : 12”. Ratios can have more than two terms ex.( 10: 8: 6). Ratios can be reduced—simplified without affecting the ratio—by dividing each term by the common factors of all the terms. (12 : 8 : 4) reduce by 2 (6 : 4 : 2).
A proportion is an equation that expresses the equality between two ratios.Proportions do not have to involve just numbers. Proportions that involve two ratios can be expressed as an equality of fractions.There are proportions that involve more than two ratios (which is called a continued proportion), the “colon notation”. There are different ways to solve proportions, the most common are cross multiplying and the butterfly method.
4. Proving Similarity
Triangles can be similar, even if one of the triangles is a reflection of the other. If two triangles have two corresponding pairs of sides with the same length and the included angle in both triangles has the same measure, then the two triangles are similar.When objects have congruent parts (sides, segments, angles), congruence markings are used to make this explicitly clear.
5. Dilation
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.The dilated image and the original are (geometrically) similar. A dilation can create a larger image, which is often referring to as expanding, stretching or scaling up, or a smaller image, which is called reducing, shrinking or scaling down. When a stretch dilation occurs, all the points in the image (except the center of dilation if it is a part of the image) move proportionally away from the center of dilation. When a shrink dilation occurs, all the points move proportionally toward the center of dilation. When an image (or object) is dilated, certain properties of the image (or object) do not change. Properties that do not change (are preserved) are said, mathematically speaking, to be invariant. Properties that are invariant under a dilation transformation are: 1. Angle measures (corresponding angles remain congruent) 2. Parallelism (corresponding parallel lines remain parallel) 3. Co-linearity (corresponding points on a line remain on a line) 4. Midpoint (midpoints between any two points remain midpoints)
6. Dilation: Affect on distance and area
Distances (lengths) in shapes are scaled proportionally when an object is dilated. If the scale factor is k, then any distance in the dilated image is the original distance multiplied by k.There is a linear relationship between the new distance and the original distance. There is a pattern with the scale factor as we look at measurements in one and two dimensions. In one dimension, dilated and original distances are proportional to the scale factor; in two dimensions, dilated and original areas are proportional to the square of the scale factor. The relationship between dimension and scale factor may lead you to conjecture that the relationship between dilated volumes and original volumes in three dimensions is proportional to the cube of the scale factor.
Project Description-
The primary focus for this project was for us to focus on proportion and similarity through modeling, reasoning and exploration. We focused on mathematical concepts like congruence and triangle congruence (two objects are the same shape and the same size), the definition of similarity (two objects are the same shape but different sizes), ratios and proportions and proving similarity (using congruent angles and proportionate sides), dilation (when an object shrinks or grows in size), the center of dilation (a point in space), and the effect on distance and area through dilation. The first thing we did was create posters and brainstorm the meaning of congruence and similarity. Then we started to create equations for situations that involve similar figures, and we started to develop techniques to solve those equations with fractions. We also started some experiments where we collected and analyzed data and saw if we could identify any similarities. Then we took the basics we learned and applied them to the worksheets that we were given.
Exhibition-
Benchmark #1
For our first benchmark we had to turn in a project proposal that consisted of who we were working with, what object we were scaling, the scale factor, and how the scale model will be constructed and exhibited. The purpose of this benchmark was to organize our thoughts and get our project going. Below is my benchmark #1 that I turned in.
“I am working with Karin and Ciy'jarah. We plan to scale down a vinyl record that is the average 12x12 inch size and the reduced size will be 1/2 of that so approximately 6x6 inches. We will make this out of ceramic clay and black paint because the ceramic clay has no color to it and we need it to look exactly the same. In order to get our record that size, our scale factor will reduce by 2. ”
Benchmark #2
For our second benchmark we had to create a blueprint that showed what our product was. We had to choose a scale factor in order to dilate the object we chose. Dilating it with our chosen scale factor showed us how wide and how big to make the object. We had to draw two diagrams of the object we are creating. The first one that we had to draw had to include all of the actual dimensions of the object but it was very difficult for us to find the exact measurements for each section of the heart including veins and arteries. It was difficult for us because there are many different size hearts, it always varies from person to person, so we took the average size for every section and based our finding off of that. Then we chose to scale every single dimension we found by 5 to get all the sizes for our model. In our benchmark #2 we made sure to label everything in our diagram.
Benchmark #3
For our final benchmark we had to create our scaled model. We created it out of ceramic clay that had been baked so it would maintain the permanent shape, after that we painted it black and gave it the center ring look that is made out of more clay so it would have the name of the record too . Unfortunately, we ran into a lot of problems while creating it, a record as you know is very thin and we needed to reduce the thinness by half just like everything else and by trying to do that the clay would rip or just not create the perfect round size. The purpose of this benchmark was to make our scaled designs come to life.
Reflection-
When creating my down scaled version of a vinyl I ran into a couple problems when doing this part. The object I scaled was very complex and there were a lot of different parts to scale. I had to take apart and put back together many parts of the center ring we had drawn in the diagrams in order to really understand how to create the model itself. A problem that my group and I ran into was the collaborating part because there were times where someone would do most of the work and other members wouldn’t understand what was going on. There were times where some of our members were missing on days when we would discuss what had to be done. Communication and collaboration was a big problem for our group during this project. Because of that I learned that in order to work well with others as a group there should be good communication between group members in order to succeed. Something that will help me in the future is planning a day in advice to have a group meeting to refocus on the entire final product, rather than having that everyone for themselves idea.
1. Congruence and triangle congruence
Two objects that are congruent if they have the same shape and the same size. The symbol used for congruence varies but is usually either ≅ or ≃. When objects have congruent parts (sides, segments,and/or angles), congruence markings are used to make this clear. Congruence means that the objects are the same size and shape; it does not mean that they are equal. Triangles that are the same shape and same size are congruent triangles. When triangles are congruent, then the corresponding angles and corresponding sides are congruent.
Congruence was a main factor in my "scaling the world" project during benchmark #2 and #3 because both of those assignments needed to be the same exact shape, the size had to vary either up scaled or down scaled. This was more important for benchmark #2 because that is the one we had to give measurements of the congruent items we were using.
2. Definition of Similarity
When two triangles are similar, we can use proportions to find the lengths of missing sides. When the triangles are similar, we know that corresponding side lengths must be proportional. Similar is a term used in math when discussing geometric figures or shapes, and it means that both figures' corresponding sides are proportional, but the figures themselves are two different sizes. Two objects have to be similar if they both have the same shape (or one has the same shape as the mirror image of the other shape). In geometry, the shape of an object is its form (typically its external boundary or outline) .By definition, mathematicians have agreed that triangle two triangles are similar if all corresponding angles are congruent (have the same measure).
3. Ratio's and Proportions
A ratio is relationship between two numbers indicating how many times the first number contains the second. A ratio can be a whole number.Sometimes ratios are not whole numbers so we use either the “colon notation” (such as 10 : 8) or we use fractions 10/8. There are 3 ways to write a ratio; “a to b”, “a : b” , “16 : 12”. Ratios can have more than two terms ex.( 10: 8: 6). Ratios can be reduced—simplified without affecting the ratio—by dividing each term by the common factors of all the terms. (12 : 8 : 4) reduce by 2 (6 : 4 : 2).
A proportion is an equation that expresses the equality between two ratios.Proportions do not have to involve just numbers. Proportions that involve two ratios can be expressed as an equality of fractions.There are proportions that involve more than two ratios (which is called a continued proportion), the “colon notation”. There are different ways to solve proportions, the most common are cross multiplying and the butterfly method.
4. Proving Similarity
Triangles can be similar, even if one of the triangles is a reflection of the other. If two triangles have two corresponding pairs of sides with the same length and the included angle in both triangles has the same measure, then the two triangles are similar.When objects have congruent parts (sides, segments, angles), congruence markings are used to make this explicitly clear.
5. Dilation
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.The dilated image and the original are (geometrically) similar. A dilation can create a larger image, which is often referring to as expanding, stretching or scaling up, or a smaller image, which is called reducing, shrinking or scaling down. When a stretch dilation occurs, all the points in the image (except the center of dilation if it is a part of the image) move proportionally away from the center of dilation. When a shrink dilation occurs, all the points move proportionally toward the center of dilation. When an image (or object) is dilated, certain properties of the image (or object) do not change. Properties that do not change (are preserved) are said, mathematically speaking, to be invariant. Properties that are invariant under a dilation transformation are: 1. Angle measures (corresponding angles remain congruent) 2. Parallelism (corresponding parallel lines remain parallel) 3. Co-linearity (corresponding points on a line remain on a line) 4. Midpoint (midpoints between any two points remain midpoints)
6. Dilation: Affect on distance and area
Distances (lengths) in shapes are scaled proportionally when an object is dilated. If the scale factor is k, then any distance in the dilated image is the original distance multiplied by k.There is a linear relationship between the new distance and the original distance. There is a pattern with the scale factor as we look at measurements in one and two dimensions. In one dimension, dilated and original distances are proportional to the scale factor; in two dimensions, dilated and original areas are proportional to the square of the scale factor. The relationship between dimension and scale factor may lead you to conjecture that the relationship between dilated volumes and original volumes in three dimensions is proportional to the cube of the scale factor.
Project Description-
The primary focus for this project was for us to focus on proportion and similarity through modeling, reasoning and exploration. We focused on mathematical concepts like congruence and triangle congruence (two objects are the same shape and the same size), the definition of similarity (two objects are the same shape but different sizes), ratios and proportions and proving similarity (using congruent angles and proportionate sides), dilation (when an object shrinks or grows in size), the center of dilation (a point in space), and the effect on distance and area through dilation. The first thing we did was create posters and brainstorm the meaning of congruence and similarity. Then we started to create equations for situations that involve similar figures, and we started to develop techniques to solve those equations with fractions. We also started some experiments where we collected and analyzed data and saw if we could identify any similarities. Then we took the basics we learned and applied them to the worksheets that we were given.
Exhibition-
Benchmark #1
For our first benchmark we had to turn in a project proposal that consisted of who we were working with, what object we were scaling, the scale factor, and how the scale model will be constructed and exhibited. The purpose of this benchmark was to organize our thoughts and get our project going. Below is my benchmark #1 that I turned in.
“I am working with Karin and Ciy'jarah. We plan to scale down a vinyl record that is the average 12x12 inch size and the reduced size will be 1/2 of that so approximately 6x6 inches. We will make this out of ceramic clay and black paint because the ceramic clay has no color to it and we need it to look exactly the same. In order to get our record that size, our scale factor will reduce by 2. ”
Benchmark #2
For our second benchmark we had to create a blueprint that showed what our product was. We had to choose a scale factor in order to dilate the object we chose. Dilating it with our chosen scale factor showed us how wide and how big to make the object. We had to draw two diagrams of the object we are creating. The first one that we had to draw had to include all of the actual dimensions of the object but it was very difficult for us to find the exact measurements for each section of the heart including veins and arteries. It was difficult for us because there are many different size hearts, it always varies from person to person, so we took the average size for every section and based our finding off of that. Then we chose to scale every single dimension we found by 5 to get all the sizes for our model. In our benchmark #2 we made sure to label everything in our diagram.
Benchmark #3
For our final benchmark we had to create our scaled model. We created it out of ceramic clay that had been baked so it would maintain the permanent shape, after that we painted it black and gave it the center ring look that is made out of more clay so it would have the name of the record too . Unfortunately, we ran into a lot of problems while creating it, a record as you know is very thin and we needed to reduce the thinness by half just like everything else and by trying to do that the clay would rip or just not create the perfect round size. The purpose of this benchmark was to make our scaled designs come to life.
Reflection-
When creating my down scaled version of a vinyl I ran into a couple problems when doing this part. The object I scaled was very complex and there were a lot of different parts to scale. I had to take apart and put back together many parts of the center ring we had drawn in the diagrams in order to really understand how to create the model itself. A problem that my group and I ran into was the collaborating part because there were times where someone would do most of the work and other members wouldn’t understand what was going on. There were times where some of our members were missing on days when we would discuss what had to be done. Communication and collaboration was a big problem for our group during this project. Because of that I learned that in order to work well with others as a group there should be good communication between group members in order to succeed. Something that will help me in the future is planning a day in advice to have a group meeting to refocus on the entire final product, rather than having that everyone for themselves idea.