Mathematical Model-‘Shape of You’ Investigation Assessment Answer

Introduction

The demo is the best tool for the mathematical modeling of physical structures (Means and Roschelle, 2010). It is the graphing calculator and it is one of the best technology that is employed in mathematics classrooms. The demos calculator can easily access through the internet and it is free of cost.

Aim

The study aims to perform research on any physical structure and use calculus to develop a mathematical model in graphical form.

Background information

Figure 1 shows the picture of a house that is modeled in the demos calculator.

Figure 1: House to be modeled (mathematical) in demos tool

Summary of mathematical method

A single line is graphed in the demos tool by writing the following expression

y =mx + c

Where ‘m’ is the slope and ‘c’ is the intercept

The physical structure of the house is modeled in this study. The house model consists of straight lines, inclined lines, and circles. The sliders are added for undefined parameters or enter the values for b=3 and m=2.

When a value of constant is given to parameters ‘m’ and ‘b’, the calculator automatically allows to adjustment of values with the help of sliders. The slope of the line is changed if ‘m’ is adjusted with the slider and the intercept is changed if ‘b’ is adjusted with the slider.

An inclined is graphed in demos by writing the following equation

(y-y1)=m(x-x1)a

A circle is graphed in demos with the following equation

It is the equation of a circle

(x-h)²+(y-k)²=r²

Results

A house is modeled in a demos calculator. Initially, the drawing of the house is built in demos.

Assumption

2-dimensional house

Outer Structure

The outer structure of the house is constructed with the following values

x= -4 {-4≤y≤4}

x=4 {-4≤y≤4}

y= -4 {-4≤x≤4}

Windows

Then, the windows of the house will be constructed with the following parameters

x= -3 {0≤y≤3}

x=-1 {0≤y≤3}

y= 1 {0≤x≤3}

y= 3 {-0≤x≤3}

y=3 {-3≤x-1, 1≤x3}

y=1.5 {-3≤x-1, 1≤x≤3}

y=0 {-3≤x-1, 1≤x≤3}

x=-2 {0≤y≤3}

x=2 {0≤y≤3}

Door

The equations of the door are

x=-1 {-4≤y-1}

x=1 {-4≤y≤-1}

y= -1 {-1≤x1}

The equation of circular knob is

(x-0.7)²+ (y+2.5)²=-0.1²

Roof

Finally, the rooftop equations are

y=1/2(x-(-4))+4 {-4≤x≤0}

y=-1/2(x-4))+4 {0≤x≤4}

y=4

y=4 {-4≤x≤4}

Figure 2: A developed mathematical model of the house in demos tool

Limitations

The critics suggest that these calculators can shadow the skills of students. Demos calculator constructs graphs and handles mathematical calculations and students have the freedom to pay less attention to cognitive resources, understanding of concepts, and solution strategies.

Conclusion

A reasonable model of the house is developed in demos tool. A model clearly depicts the structure of the house. The model consists of straight and inclined lines and a circular knob. It is very easy to develop a mathematical model of a house in the demos tool. However, some critics suggest that these calculators can shadow the skills of students. Demos calculator constructs graphs and handles mathematical calculations and students have the freedom to pay less attention to cognitive resources, understanding of concepts, and solution strategies.

Introduction

The demo is the best tool for the mathematical modeling of physical structures (Means and Roschelle, 2010). It is the graphing calculator and it is one of the best technology that is employed in mathematics classrooms. The demos calculator can easily access through the internet and it is free of cost.

Aim

The study aims to perform research on any physical structure (dog) and use calculus to develop a mathematical model in graphical form.

Background information

Figure 1 shows the picture of a dog that is modeled in the demos calculator.

 

Figure 1: Dog to be modeled (mathematical) in demos calculator

Summary of mathematical method

A single line is graphed in the demos tool by writing the following expression

y =mx + c

Where ‘m’ is the slope and ‘c’ is the intercept

The physical structure of the dog is modeled in this study. The dog consists of straight lines, inclined lines, and circles. The sliders are added for undefined parameters or enter the values for b=3 and m=2.

When a value of constant is given to parameters ‘m’ and ‘b’, the calculator automatically allows to adjustment of values with the help of sliders. The slope of the line is changed if ‘m’ is adjusted with the slider and the intercept is changed if ‘b’ is adjusted with the slider.

An inclined is graphed in demos by writing the following equation

(y-y1)=m(x-x1)a

A circle is graphed in demos with the following equation

It is the equation of a circle

(x-h)²+(y-k)²=r²

An equation of curve is

y=a(b-(x-h)²)^0.5 +k

An equation of exponential is

y=e^b(x-c) +d

y= axe^b(x-c) +d

Results

A dog is modeled in a demos calculator. Initially, the drawing of the house is built in demos.

Assumption

2-dimensional modeling of dog

Eye of dog

The eye of a dog is modeled by the following three equations

y=a₆((b₆+(x-h₆)²)^0.5 +k {4.4≤x≤5.8}

y=a₁₀x²+bx₁₀+c₁₀ {5.7≤x≤6.4}

y=a₂((b₂+(x-h₂)²)^0.5 +k₂ {4.4≤x≤5.9}

Tongue of Dog

The tongue of a dog is modeled as

y=a₅((b₅+(x-h₅)²)² +k₅ {2.3≤x≤3} it is for the curved part of the tongue

The straight part of the tongue is modeled by straight line equations

(y-y6)=m6(x-x6) {2.33≤y≤3.5}

y=a₉xe^b9(x-c9)+d₉ {2.62≤y≤3.5}

Nose of Dog

The nose of a dog is modeled as

y=a(b-(x-h)^0.17)^0.5 +k {4.5≤y≤5.66}

The upper part of the nose is modeled as

y=a₇((b₇+(x-h₇)¹)² +k₇ {5.66≤y≤6.2}

Ears of Dog

The ears of the dog are straight. They are modeled with an inclined line equation

(y-y1)=m1(x-x1) {6≤x≤6.6}

(y-y4)=m4(x-x4) {6.6≤y≤8}

x=8 {5.2≤y≤7.1}

(y-y3)=m3(x-x3) {6.4≤x≤7.1}

(y-y3)=m43(x-x3) {6.4≤y≤7.1}

(y-y8)=m8(x-x8) {3≤x≤3.6}

Head of Dog

The head of the dog is modeled as

y=a₄((b₄+(x-h₄)¹)² +k₄ {3.6≤x≤6.2}

Backbone of Dog

The backbone of a dog is modeled by a straight line equation

(y-y5)=m5(x-x5) {8≤x≤10}

Base of Dog

The base of the dog is modeled as

y=a₃((b₃+(x-h₃)^0.5)² +k₃ {6≤x≤10.5}

Neck of Dog

The neck of the dog is modeled by the surge function

y=a₁₁xe^b11(x-c11)+d₁₁ {3≤x≤6}

Hips of Dog

They are modeled as

y=e^b11(x-c11) +d¹¹ {1.4≤x≤4.5}

y=a₁₃((b₁₃+(x-h₁₃)³)^0.5 +k₁₃

Figure 2: A developed mathematical model of the dog in demos tool

Limitations

The critics suggest that these calculators can shadow the skills of students. Demos calculator constructs graphs and handles mathematical calculations and students have the freedom to pay less attention to cognitive resources, understanding of concepts, and solution strategies.

Conclusion

A reasonable model of the dog is developed in the demos tool. A model clearly depicts the structure of the house. The model consists of straight and inclined lines and curves. It is very easy to develop a mathematical model of a house in the demos tool. However, some critics suggest that these calculators can shadow the skills of students. Demos calculator constructs graphs and handles mathematical calculations and students have the freedom to pay less attention to cognitive resources, understanding of concepts, and solution strategies.

References

1millionmonkeystyping.files.wordpress.com. 2022. [online] Available at: <https://1millionmonkeystyping.files.wordpress.com/2021/08/modelling-guide-preview.pdf> [Accessed 31 August 2022].

Means, B. and Roschelle, j., 2010. An Overview of Technology and Learning. In: International Encyclopedia of Education, 3rd ed.

S3.amazonaws.com. 2022. [online] Available at: <https://s3.amazonaws.com/desmos/Desmos_Calculator_User_Guide.pdf> [Accessed 31 August 2022].

1millionmonkeystyping.files.wordpress.com. 2022. [online] Available at: <https://1millionmonkeystyping.files.wordpress.com/2021/08/modelling-guide-preview.pdf> [Accessed 31 August 2022].

Means, B. and Roschelle, j., 2010. An Overview of Technology and Learning. In: International Encyclopedia of Education, 3rd ed.

S3.amazonaws.com. 2022. [online] Available at: <https://s3.amazonaws.com/desmos/Desmos_Calculator_User_Guide.pdf> [Accessed 31 August 2022].

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