Independent Practice (5/22)

MLS                                                                                                              Yolanda Volcimus

Harvard                                                                                                          May 22, 2013


         Question: Write a blog entry for the following prompt: Your neighbor has a

 backyard with an area 450 square feet. He decides to place a sandbox for his

 children in a section of his backyard which is 32 square feet. Determine the

 perimeter of your neighbors back yard, as well as the perimeter of the 

sandbox. (Hint: adding simplified radicals is similar to adding fractions – like

 radicals can be combined!) In your response we sure to clearly articulate your

 thinking as to your process for solving, as well as a justification for your

 reasoning.


         Answer: The perimeter of my neighbor's back yard is 60 Ö2 ft because each side of the yard is 15 * Ö2 feet long. Therefore, by adding 15 * Ö2 four times and the perimeter is  60 Ö2  ft long. 

                                                                  450

                                                            Ö25 * Ö18

                                                  5   *  Ö18

                                                5  *  Ö9  *  Ö2

                                                 5    *    3   *  Ö2

                                                      15  *  Ö2

                                      Perimeter = l+w+l+w     
    
(15  *  Ö2) +  (15  *  Ö2) +  (15  *  Ö2) + (15  *  Ö2) = 60 Ö2 ft

Answer: The perimeter of the sandbox is 16 * Ö2 ft because each side of the sandbox is 4 * Ö2 ft long. That is to say, by adding 4 * Ö2 four times, the perimeter would be 60 * Ö2 . 
                                                                 32

                                                       Ö4    *    Ö8

                                             2    *     Ö8

                                             2    *   Ö4  *  Ö2

                                             2    *    2    *    Ö2

                                                    4   *    Ö2

                                            Perimeter = l+w+l+w 

(4 * Ö2) + (4 * Ö2) + (4 * Ö2) + (4 * Ö2) = 16 Ö2

Volumes of Different Shapes

MLS                                                                                   Yolanda Volcimus

Harvard                                                                                       3/16/13

               Throughout my exploration of volume, I have learned about different relationships of shapes and I had an understanding of these relationships that helped me to remember the formulas used to find each figures volume. I have learned the different uses for the formula V = BH. I learned the relationship between rectangular prisms and square pyramids. Also, I’ve learned the relationship between triangular prisms and triangular pyramids. Lastly, I was able to tell the relationship between cylinders and cones. Therefore, I have learned numerous different associations of figures and by using these relationships, I was able to remember the formula of each shapes’ volume.

            There are several different uses of the formula V = BH. For example, we used the formula V = BH for a cylinder, a rectangular prism, and a triangular pyramid. However, the formula does change depending on the shape I am determine to find the volume of. For example, to find the volume of a cylinder, we use the formula V = BH. To find the value of B, we use the formula B = π r² because the base of a cylinder is a circle and in order to find the volume of the cylinder, we must find the area of that circle. In addition, when we find the volume of a rectangular prism, we also use the formula V = BH. However, we find the B by using the formula B = LW because the base of a rectangular prism is a rectangle and we must find the area of the rectangle first before finding the volume of the whole figure. Lastly, we also use the formula V = BH to find the volume of a triangular pyramid. In the other hand, to find the B, we use the formula B = ½ bh. We use this formula to find B because the base of a triangular pyramid is a triangle and we must find the area of the triangle first which is ½ bh before finding the volume of the whole triangular prism. That is to say, there are various ways we can use the formula V = BH to find the volume of many shapes and each shapes has a different used of the B. 

               A rectangular prism is a 3-d shape that has six faces which are all rectangles. A square pyramid is a pyramid with a square base. The relationship between a rectangular prism and a square pyramid is that a square pyramid is 1/3 of a rectangular prism. The formula used to determine the volume of a rectangular prism is V = BH where B would be represented by the formula B = bh. However, the formula that we use to find the volume of a square pyramid is 1/3 BH. The formula has 1/3 in it because three square pyramids make one rectangular with the same Base and Height. The relationship between these two shapes helps me to remember and understand the formulas used for determining their volumes because since I know that a square pyramid id=s 1/3 of a rectangular, I know that a square pyramid’s volume formula will have 1/3 in front of it. In addition, I also know that to find the volume any prisms or pyramid, I need to multiply the Base by the Height. That is to say, rectangular prisms and square pyramids have a special relationship which helps me to remember the formula of finding their volumes. 

            A triangular prism is a prism whose base is a triangle. Nevertheless, a triangular pyramid is a pyramid with a base of a rectangle. The relationship between a triangular prism and a triangular pyramid is a triangular pyramid is 1/3 of a triangular prism. The formula that is used to determine the volume of a triangular prism is V = BH where B would be represented by the formula B = ½ bh.  On the other hand, the formula for the volume of a triangular pyramid is V = 1/3BH. In this formula, B would also be represented by the formula B = ½ bh. The relationship between these two shapes helps me to remember and understand the formulas used for determining their volume because I know that three triangular prisms will make one triangular pyramid with the same height and base. This also means that the volume formula of a triangular pyramid will also have 1/3 in front of it. That is to say, the relationship between a triangular prism and a triangular pyramid is that a triangular prism is 1/3 of a triangular prism.

A cylinder is a geometry shape with straight parallel sides and a circle for a base. However, a cone is a solid that has a circular base and only one apex. A cone is also considered a pyramid with infinite amount of faces. The relation between a cylinder and a cone is that a cone is 1/3 (one-third) of a cylinder. The formula that is used to find the volume of a cylinder is V = π r²h. Conversely, the formula that is used to determine the volume of a cone is V = 1/3 π r²h. The relationship between these two shapes help me remember and understand the formula used for determining their volume because since I now know that that three cones will equal to one cylinder with the same height and base, I know the volume formula of a cone will contain 1/3 in front of it. Also, I know that the area of any circle is represented by the formula A = π r²h which means when we find the volume of these two figures, we actually finding the area of their bases and multiplying it be the height. In conclusion, the relationship between a cone and a cylinder is that a cylinder is three times a cone.

           To wrap it all up, over the last two weeks in my exploration of volume, I have learned about different relationship of shapes and how they are related to one another. Moreover, I learned that there are various used of the formula V = BH. Also, I learned that a square pyramid is equal to 1/3 of a rectangular prism with same base and height. In addition, a triangular prism is 1/3 of a triangular prism. Lastly, three cones are equal to one cylinder that has the same height and base with the cone. In conclusion, throughout these two weeks, I have learned several different relationships between several different shapes. 

   

MLS                                                                                       Yolanda Volcimus

Harvard                                                                              1/22/13

A percent is used to describe an amount of change in a number. There are numerous strategies that can be used to determine the percent of a number. In some cases, we can even find some real-world applications where percent is used. In addition, these real-world applications are necessary to have an understanding of the "story" behind the situation in order to apply the specific strategies. Therefore, there a various strategies that are involved in percentage.

       Percent of Change is the amount stated as percent; it is one of the strategies to determine the percent of a number. The two types of percent of change are percent increase and percent decrease. Percent increase is when the amount of a number improved; however, percent decrease is when the amount diminishes. The percent of change of a number is calculated as percent of change= amount of change over original amount. For example, in the statement "50 is decreased to 40", you would solve it by doing 50-40/50 based on the formula. Then, it would be 10/50 which is equal to 20%. Therefore, "50 is decreased to 40" is a 20% decreased. In addition, "25 is increased to 40" is a 37.5% increased because I set the problem as 40-25/40= 15/40= 0.375= 37.5%. That is to say, a strategy to find the percent of a number is percent of change.
       Furthermore, people can use discount which is a real-world application to find the percent of a number. Discount is a deduction from the original cost of an item. To find the discount of a number, we use the formula discount/original. It is necessary to have an understanding of the "story" behind the situation in order to apply the specific strategies because if we apply the wrong formula, we will end up with the incorrect percentage. For instant, if we have the story "At a store, an IPod was $50; however, it was on sale for 10% off. What was the new price?" If we have this problem, we know the strategy we would use is discount because 10% is being deducted from $50 and the new price would be $45.            

    Another strategy we can use to determine the percent of a number is commission. Commission is a free paid to a person who makes a sale. Commission is calculated by commission rate x sale = commission. In a real-world application, "A car is sale for $50,000 and Marvin gets 5% of this money? What is the commission?" Based on our understanding of the story, we know the correct strategy is commission because it is a free paid for Marvin. Marvin would receive $2,500 from the original amount.
     The last strategy we can use to find the percent of a number is revenue. Revenue is the amount of money brought in from a sale. After the costs are subtracted from the revenue, the amount remaining is the profit. For example, "the profit of 10 shirts was $50 which is 80% of the revenue. What is the revenue?" The revenue is $62.50 because I set a proportion of 80/100 = 50/x = 80x = 5,000= 5,000/80= 62.5.
        To wrap it all up, percent is an amount of change. The four strategies that are used to find the percent of a number are percent of change, discount, commission, and revenue. These strategies are all used in real-world application and the story is crucial in order to determine which strategy to use. In conclusion, the percentage of a number can be measures in many ways.

MLS                                                                                                    Yolanda Volcimus

Harvard                                                                                             January 10, 2013

For the last two months, I've been studying about linear equations. A linear equation is an equation between two variables that ends up in a straight line when it is graphed. I learned that there are three forms of linear equations; they are Standard Form, Slope-intercept Form, and Point-Slope Form. Each of these equations gave me specific information. That information provides a specific way for each equation form that changes my strategy for graphing. Also, there are situations where I find one equation form that is more useful than the others.

Standard Form equation is a linear equation that us written in the form of Ax+Bx =C. A, B, and C are all real numbers and A and B area not both equal to zero. To graph a linear equation in Standard Form, you can find the x-intercept by substituting zero for y and solving for x. To find the y-intercept, you must substitute zero for x. The information Standard Form equations provide is the y value when substituting zero for x and the x value when substituting zero for y. For instant, if I have this problem {3x + 4y = 12}, I would solve it like this:

Ax + By = C                                                     Ax + By = C    

3x + 4y = 12                                                   3x + 4y = 12

3 (0) + 4y =12                                                3y + 4 (0) = 12

4y=12                                                             3y = 12

4 = 4                    y=3                                      3  =   3                         y=4

This equation can be more useful than the other ones because this equation tells me the value of both x and y when finished solving it and I don’t even have to find the slope while the other equations give me other information except for the value of y and x. Therefore, Standard Form equations provide me with the value of y and x when I substitute zero for x and zero for y and this equation is useful because it provides the value of y and x.

Slope-Intercept Form equation is a linear equation that is written in the form of y= mx+b. The information this equation form provide is both the slope and the y-intercept. The m represents the slope {steepness of a line} of the line and the b is the y-intercept {on the y-axis, the value of x- is always zero}. In Slope-Intercept Form, the y-intercept tells me at what point to start at. Also, the slope tells me where the next point is located. For example, a graph have a y-intercept of 1 and a slope of 2 (Y= 2x + 1) . I can use this equation to find the y-intercept and the slope. The Y-intercept is 1 because the line crosses the y-axis at point 1 and the value of x- was zero. In addition, the slope is 2 because it was substituted from m to two. From point 1, since the slope is 2, you must move two places up and 1 place to the right {Y/X} = {2/1}. By using the slope and the y-intercept, it was easier for me to graph this equation without having to solve the entire equation. I can use this type of equation when I am given two points of a line, I use the slope formula {Y₂-Y₁ / X₂ –X₁} formula this equation to write the equation of a line. Therefore, Slope-Intercept Form equations give me information like the slope and the y-intercept. 

      Point-Slope Form equation is a type of equation that is according to the slope and a point of a line. The information that this equation provide is the slope and the point of the line. The general equation for this form is Y-Y₁= m (X-X₁). Y₁ must be the opposite, m is the slope, and X₁ must also be opposite. For instant, the point of a line is (5, 15) and the slope is 3. I would write this information into point-slope form equation by first using the formula Y-Y₁= m (X-X₁). Then, I would substitute for the y, the slope, and the x in the equation{y-15=3(x-5)}. Therefore, the final equation in Point-Slope Form is y-15=3(x-5). When graphing, I can use this equation because this equation provide us with the slope and the specific point. When a problem only gives me one point of the line and the slope, the equation that is useful to use is this equation because this equation is the only equation I can use when I am given any point of a line with the slope. That is to say, when only given one point of a line and the slope, I can use the Point-Slope form equation to figure out the next point. 

        To wrap it all up, I learned that a linear equation is an equation that provides a straight line when it is graphed. I learned about the three types of linear equations which are Standard Form, Slope-intercept Form, and Point-Slope Form. Each type of equation gives a specific type of information that is unique in order for me to graph it. In addition, there are situations where one equation form is more useful than the other. In conclusion, I can use any of these equations to graph a straight line because they are all linear equations.

 

 

MLS                                                                          Yolanda Volcimus

Harvard                                                                   December 1, 2012

        Relate your understanding of unit rate and proportionality to that of rate of change and slope. How are they similar and how do they differ? How can one use their understanding of unit rate and proportionality to interpret real world rate of change/slope problems presented in graphs.

        A rate, unit rate, proportionality, Rate of Change, and slope all work together to form the understanding of graphs. These relations also have their similarities and differences. We can also use our understanding of unit rate and proportionality to interpret real world Rate of Change/slope problems presented in graphs.   

A rate is a ratio that compares two quantities measured in different units. A rate is similar to a unit rate because a unit rate is a rate whose denominator is 1 when it is written as a fraction. However, a rate can be in any fraction form, whereas, a unit rate has to have a denominator of one. For example, the first chart contains rates only and the second chart has unit rates only. 

                    

Unit Rate

5/1

3/1

9/1

Rates

4/2

7/14

8/5

    

       

      The slope of a line is a measure of its steepness and is a ratio of Rise over Run. There are two types of slopes; a positive slope and a negative slope. A positive slope is when a line in a graph is increasing from left to right.

This picture shows a positive slope. 

However, A negative slope is when the line in a graph is decreasing from left to right when looking at it. This is a picture     of a negative slope.

A Rate of Change is the ratio of two quantities that changes; such as slope. Slope and Rate of Change are similar because the Rate of Change is the ratio for the slope in the simplest form. Without the Rate of Change, the slope wouldn’t be in its simplest form. For instant,

A slope is 6/3. The Rate of Change would be 2 because 6 divided by 3 is 2.

               6 ÷ 3 = 2

               3 ÷ 3 = 1. A slope is different from a Rate of Change because the slope of a line may not necessary be simplified in some situations, however, in a Rate of Change, the ratio is always express in the simplest form.                                                                                                           

         We can use our understanding of unit rate and proportionality to interpret real world Rate of Change/slope problems presented in graphs. For example, some real life examples of slopes are mountains, stairs, and a slide. These things are considered examples of slopes because they are steep which is what slopes measure. If the slope of a stair is 15/5, by using our understanding of using unit rate, we will be able to determine the Rate of Change. We would have to write the slope in the simplest form where the denominator is one. We can do that by dividing both the numerator and the denominator by 5 which gives us an answer of 3.

  15 ÷ 5 = 3

    5 ÷ 5 = 1