MATH 107 University of Maryland Mathematics in The Kitchen Algebra Discussion

Mathematics in the KitchenJonathan Russe Gines
Math 107 College Algebra
07/05/21
Baking, frying, boiling, grilling, poaching, simmering. We are all familiar with these
processes that take place in the kitchen. Hence, we are aware that, in order to be successful with a
dish or pastry, proper measurements of various ingredients must be taken. This is usually taken by
volume through cups, teaspoons, tablespoons or mass through grams, kilograms, etc. Recipes are
accessible for most known dishes such as soup, steak, pizza, pasta, and many more. However,
there are times when the recipe you are following may not yield the desire servings that you want.
For example, you are a chef in a restaurant and you must create 2-5 servings of vanilla cake but
the recipe that you know can only yield 1 cake. This is where mathematics will come in handy.
The topic that I would like to write about features mathematics in the kitchen. In order to
produce the number of desired servings using the original recipe, one can simply follow the
equation below:
𝑥 = Original Recipe Yield ∙ Desired Recipe Yield
Below is the measurement for the ingredients of a vanilla cake from McKenney’s Baking
Addiction website blog (2019). For example, to know how much cake flour will be used to yield
3 cakes, use the equation:
𝑥=3
2
cups ∙ 3 cakes
3
𝑥 = 11 cups
Since the measurements are being multiplied to a certain number, then we can expect that
there will be a positive linear trend in our graph.
Table 1. Measurements of Ingredients to Yield a Certain Number of Cakes
Ingredients for 1
Cake
3 and ⅔ cups
cake flour
1 and ⅛
teaspoon salt
2 teaspoons
baking powder
¾ teaspoon
baking soda
3 cups unsalted
butter
2 cups
granulated sugar
3 large eggs
2 egg whites
Ingredients for 2
Cakes
7 and ⅓ cups
cake flour
2 and ¼
teaspoon salt
4 teaspoons
baking powder
1 and ½
teaspoon baking
soda
6 cups unsalted
butter
4 cups
granulated sugar
6 large eggs
4 egg whites
Ingredients for 3
Cakes
11 cups cake
flour
3 and ⅜
teaspoon salt
6 teaspoons
baking powder
2 and ¼
teaspoon baking
soda
9 cups unsalted
butter
6 cups
granulated sugar
9 large eggs
6 egg whites
Ingredients for 4
Cakes
14 and ⅔ cups
cake flour
4 and ½
teaspoon salt
8 teaspoons
baking powder
3 teaspoon
baking soda
12 cups unsalted
butter
8 cups
granulated sugar
12 large eggs
8 egg whites
Ingredients for 5
Cakes
18 and ⅓ cups
cake flour
5 and ⅝
teaspoon salt
10 teaspoons
baking powder
3 and ¾
teaspoon baking
soda
15 cups unsalted
butter
10 cups
granulated sugar
15 large eggs
10 egg whites
1 tablespoon
pure vanilla
extract
1 and ½ cups
buttermilk
5 and ½ cups
confectioners’
sugar
⅓ cup whole
milk
1 and ½
teaspoons pure
vanilla extract
2 tablespoon
pure vanilla
extract
3 cups
buttermilk
11 cups
confectioners’
sugar
⅔ cup whole
milk
3 teaspoons pure
vanilla extract
3 tablespoon
pure vanilla
extract
4 and ½ cups
buttermilk
16 and ½ cups
confectioners’
sugar
1 cup whole
milk
4 and ½
teaspoons pure
vanilla extract
4 tablespoon
pure vanilla
extract
6 cups
buttermilk
22 cups
confectioners’
sugar
1 and ⅓ cup
whole milk
6 teaspoons pure
vanilla extract
5 tablespoon
pure vanilla
extract
7 and ½ cups
buttermilk
27 and ½ cups
confectioners’
sugar
1 and ⅔ cup
whole milk
7 and ½
teaspoons pure
vanilla extract
Using Microsoft Excel, the scatter plot can be observed (see Figure 1).
Figure 1. Scatter Plot with Linear Trend
Scatter Plot with Linear Trend of the Measurements of
Ingredients for a Vanilla Cake
30
Original Measurement
25
20
15
10
5
0
0
1
2
3
4
5
New Measurement
Ingredients for 2 Cakes
Ingredients for 3 Cakes
Ingredients for 4 Cakes
Ingredients for 5 Cakes
Linear (Ingredients for 2 Cakes)
Linear (Ingredients for 3 Cakes)
Linear (Ingredients for 4 Cakes)
Linear (Ingredients for 5 Cakes)
References:
McKenney, S. (2019). The Best Vanilla Cake I’ve Ever Had. Sally’s Baking Addiction.
Retrieved from https://sallysbakingaddiction.com/vanilla-cake
6
Mathematics in the Kitchen
Jonathan Russe Gines
Math 107 College Algebra
07/05/21
Baking, frying, boiling, grilling, poaching, simmering. We are all familiar with these
processes that take place in the kitchen. Hence, we are aware that, in order to be successful with a
dish or pastry, proper measurements of various ingredients must be taken. This is usually taken by
volume through cups, teaspoons, tablespoons or mass through grams, kilograms, etc. Recipes are
accessible for most known dishes such as soup, steak, pizza, pasta, and many more. However,
there are times when the recipe you are following may not yield the desire servings that you want.
For example, you are a chef in a restaurant and you must create 2-5 servings of vanilla cake but
the recipe that you know can only yield 1 cake. This is where mathematics will come in handy.
The topic that I would like to write about features mathematics in the kitchen. In order to
produce the number of desired servings using the original recipe, one can simply follow the
equation below:
𝑥 = Original Recipe Yield ∙ Desired Recipe Yield
Below is the measurement for the ingredients of a vanilla cake from McKenney’s Baking
Addiction website blog (2019). For example, to know how much cake flour will be used to yield
3 cakes, use the equation:
𝑥=3
2
cups ∙ 3 cakes
3
𝑥 = 11 cups
Since the measurements are being multiplied to a certain number, then we can expect that
there will be a positive linear trend in our graph.
Table 1. Measurements of Ingredients to Yield a Certain Number of Cakes
Ingredients for 1
Cake
3 and ⅔ cups
cake flour
1 and ⅛
teaspoon salt
2 teaspoons
baking powder
¾ teaspoon
baking soda
3 cups unsalted
butter
2 cups
granulated sugar
3 large eggs
2 egg whites
Ingredients for 2
Cakes
7 and ⅓ cups
cake flour
2 and ¼
teaspoon salt
4 teaspoons
baking powder
1 and ½
teaspoon baking
soda
6 cups unsalted
butter
4 cups
granulated sugar
6 large eggs
4 egg whites
Ingredients for 3
Cakes
11 cups cake
flour
3 and ⅜
teaspoon salt
6 teaspoons
baking powder
2 and ¼
teaspoon baking
soda
9 cups unsalted
butter
6 cups
granulated sugar
9 large eggs
6 egg whites
Ingredients for 4
Cakes
14 and ⅔ cups
cake flour
4 and ½
teaspoon salt
8 teaspoons
baking powder
3 teaspoon
baking soda
12 cups unsalted
butter
8 cups
granulated sugar
12 large eggs
8 egg whites
Ingredients for 5
Cakes
18 and ⅓ cups
cake flour
5 and ⅝
teaspoon salt
10 teaspoons
baking powder
3 and ¾
teaspoon baking
soda
15 cups unsalted
butter
10 cups
granulated sugar
15 large eggs
10 egg whites
1 tablespoon
pure vanilla
extract
1 and ½ cups
buttermilk
5 and ½ cups
confectioners’
sugar
⅓ cup whole
milk
1 and ½
teaspoons pure
vanilla extract
2 tablespoon
pure vanilla
extract
3 cups
buttermilk
11 cups
confectioners’
sugar
⅔ cup whole
milk
3 teaspoons pure
vanilla extract
3 tablespoon
pure vanilla
extract
4 and ½ cups
buttermilk
16 and ½ cups
confectioners’
sugar
1 cup whole
milk
4 and ½
teaspoons pure
vanilla extract
4 tablespoon
pure vanilla
extract
6 cups
buttermilk
22 cups
confectioners’
sugar
1 and ⅓ cup
whole milk
6 teaspoons pure
vanilla extract
5 tablespoon
pure vanilla
extract
7 and ½ cups
buttermilk
27 and ½ cups
confectioners’
sugar
1 and ⅔ cup
whole milk
7 and ½
teaspoons pure
vanilla extract
Using Microsoft Excel, the scatter plot can be observed (see Figure 1).
Figure 1. Scatter Plot with Linear Trend
Scatter Plot with Linear Trend of the Measurements of
Ingredients for a Vanilla Cake
30
Original Measurement
25
20
15
10
5
0
0
1
2
3
4
5
New Measurement
Ingredients for 2 Cakes
Ingredients for 3 Cakes
Ingredients for 4 Cakes
Ingredients for 5 Cakes
Linear (Ingredients for 2 Cakes)
Linear (Ingredients for 3 Cakes)
Linear (Ingredients for 4 Cakes)
Linear (Ingredients for 5 Cakes)
References:
McKenney, S. (2019). The Best Vanilla Cake I’ve Ever Had. Sally’s Baking Addiction.
Retrieved from https://sallysbakingaddiction.com/vanilla-cake
6
Source of data
Select a source that is credible and free from bias
Be sure to describe how data were obtained by source; are the data comprehensive or a subset
of all available data? A description of where the data came is usually included in the source, so
locate this information and include it in your report.
Include in-text and reference citations for your source in APA format
Setting units (for table and graphs)
For the x-axis, use “years since” rather than “year”. This changes the regression equation to give
a more meaningful y-intercept.
In my example data, the “years since” regression line estimates the percentage of females
enrolled in college is about 24% in 1980, which is “0 years since 1980”, which is useful to
compare to the actual value of 25%. If I use “year” on the x-axis, I get a y-intercept of -1666,
which is meaningless (technically, “In the year 0, female enrollment was negative 1666%”).
Similarly, if you have huge numbers on the y-axis, convert your units to something reasonable
(such as thousands, millions) without rounding your decimal for your calculations. For example,
assume I collected the following data
1965
1970
1975
Number of male smokers in the U.S.
28,930,000
26,410,000
25,790,000
I would create this table, and I would use columns 2 and 4 for my scatter plot:
Year
Number of years
since 1965
Number of male smokers
in the U.S.
Number of male smokers in
the U.S. (millions)
1965
0
28,930,000
28.93
1970
5
26,410,000
26.41
1975
10
25,790,000
25.79
NOTE: Often data is already presented in thousands or millions in the source. The
“millions” is considered a unit and is shown in parentheses in the column header.
Here is the table for my example; I used columns 2 and 3 for my scatter plot. For the column
that contains data with units, I included the units in the column header in parentheses (%).
Figure 1. Percentage of 18- to 24-year-old females enrolled in college, 1980 through 1995
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
Number of years
since 1980
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Females enrolled in college
(%)
25
25
26
25
26
27
28
29
30
32
32
34
36
34
36
35
Creating graphs
Notice that a scatter plot consists of data points only – NO LINES! Do not connect the points,
and be sure your points are visible to the middle-aged eye.
This is the scatter plot for my example data.
When you create a separate scatter plot with a linear regression line, you can choose to include
the regression equation and R2 on the chart (see my examples under “Setting units”), but this is
not required. If you include these, make sure this information does not obscure your data or
trendline and is readable. You still must state and discuss the equation and R2 in the text of
your report.
Here is how to add the regression equation and R2 in Excel:
Interpretation of regression equation
Let’s take one more look at my example with the regression line:
The regression equation is 𝑦 = 0.8534𝑥 + 23.586, where x is the number of years after 1980
and y is the percentage of 18- to 24-year-old females enrolled in college in a given year. Notice
that I DO NOT ROUND the values in the regression equation.
The slope of the regression line is the rate of change, and this will always have units of
(y-units)/(x-units). In this example, the units are % per year.
The regression equation estimates that the percentage of 18- to 24-year-old females
who are enrolled in college is increasing at a rate of approximately 0.85% per year.
As discussed previously, the y-intercept is the percentage when x = 0, which occurs in 1980.
The regression equation estimates that approximately 24% of 18- to 24-year-old
females were enrolled in college in 1980.
If I wanted to predict how many 18- to 24-year-old females would be enrolled in college in 2025
based on this trend, I would first find x (𝑥 = 2025 − 1980 = 45) and use this x value to
calculate y:
𝑦 = 0.8534(45) + 23.586
𝑦 = 61.989
This means that if this trend continues, in the year 2025, 62% of all 18- to 24-year-old females
would be enrolled in college. (What is approximate enrollment today? Does this estimate seem
reasonable? What else might affect this trend?) Notice that I DO round the final answer to the
same degree of precision as the original data, which is to the nearest percent.
Interpretation of regression statistics
The coefficient of determination, R2, for my example data is 0.9429. This means that between
1980 and 1995, 94.29% of the change in the percentage of females enrolled in college can be
explained by the year.
To find the correlation coefficient, r, we need to take the square root of R2. Based on the slope
of the line, r will be positive, so we calculate r = 0.97. Note that r is reported to TWO decimal
places. This value means that there is a very strong positive correlation between the percentage
of females enrolled in college and the year between 1980 and 1995.
For more information on interpreting the correlation coefficient and coefficient of
determination, see https://mathbits.com/MathBits/TISection/Statistics2/correlation.htm
Curve-fitting Project – Linear Model
NOTE: Completing additional discussion topics in Weeks 2, 3, and 4 regarding progress on your
project is REQUIRED and will be part of your grade for this assignment.
Instructions
For this assignment, you will collect data exhibiting a relatively linear trend, find the line of best fit, plot the data
and the line, interpret the slope, and use the linear equation to make a prediction. You will also find r2 (coefficient
of determination) and r (correlation coefficient) and discuss your findings. Your topic should be related to a social
or economic trend that you find interesting. The trend should be linear for at least a portion of the data set
available; this will be the data used for regression analysis. There will also be topic suggestions provided in class.
Tasks for Linear Regression Model
Data collection
•
Describe your topic, including why it may be of interest and the relationship that you expect to find.
(NOTE: See the project discussion in Week 2 for generating ideas.)
•
Provide your data in table format and cite your source. Be sure to collect between 8 – 20 data
points. Label your table appropriately with title, column headings, and units. (NOTE: Each student must
use different data, and this information must be posted in the Week 3 Project discussion. Students may
choose similar topics but must have different data sets.)
Data visualization
•
Plot the points (x, y) to obtain a scatterplot. Use an appropriate scale on the horizontal and vertical axes
and be sure to label carefully. (NOTE: See the project discussion in Week 4)
•
Visually interpret your graph and determine whether the data points exhibit a relatively linear trend over
the range you have selected.
Data analysis
•
Find the line of best fit using linear regression analysis and graph it on the scatterplot.
•
State the equation of the line and describe the method you used to determine this equation.
Interpretation of regression equation
•
State the slope of the line of best fit. Carefully interpret the meaning of the slope as it pertains to your
topic in a sentence or two. Report the y-intercept of the regression equation. Does this value seem
reasonable?
Interpretation of regression statistics
•
Find and state the value of r2, the coefficient of determination, and r, the correlation coefficient. Discuss
your findings in a few sentences, making sure to address the following: Is r positive or negative? Why? Is
a line a good curve to fit to this data? Why or why not? Is the linear relationship very strong, moderately
strong, weak, or nonexistent?
Prediction
•
Choose a value of interest and use the line of best fit to make an estimate or prediction. Show your work
for your calculations.
•
Explain why you chose this value, and if data exists for your prediction, discuss whether your model gave a
good estimate.
Summary
•
Write a brief narrative of a paragraph or two. Summarize your findings in words, and be sure to mention
any aspect of the linear model project (topic, data, scatterplot, line, r, or estimate, etc.) that you found
particularly important or interesting. Include a discussion of any limitations of your findings.
Grading
Submit all of your project in one Word document; a template for your work will be provided. (A single pdf file of
your project should also be submitted to preserve your formatting.)
Projects will be graded on the basis of completeness, correctness, ease in locating all of the checklist items, and
strength of the narrative portions. A grading rubric will be provided.
Rubric for Linear Regression Project
The assessment rubric for the MATH 107 Linear Regression Project is shown below. Six criteria are listed
on the left of the rubric, and total point value is listed on the right. The Project is worth 100 points.
A description of what is expected to meet each criterion is provided below. Note that the underlined
headings refer to a specific section in your project write-up.
Discussion
Completes Week 3 Project Discussion on time, indicating topic, source of data, table of data to be used, and
demonstration that relationship is linear. Follows up on comments until topic and data set are approved.
Introduction/ Table/ Data visualization
Introduction: Describes topic, including why it is of interest and/or relevant to current social or economic
issues. Discusses the type of relationship expected (positive or negative slope, strength of relationship, etc.)
and what you hope to learn by completing this project.
Table: Provides appropriate type/quantity of data in table format with appropriate title, column headings and
units. Uses and cites reliable data source.
Data visualization: Describes graphing tools used. Plots the points to obtain a scatter plot; uses appropriate
scales on the horizontal and vertical axes. Includes chart title and axis labels, including units. Interprets graph
and data trends.
Regression analysis/ Interpretation
Data analysis: Describes tools used for linear regression analysis. Finds and states the line of best fit. Creates
a scatter plot that includes the regression line and discusses visual fit to data.
Interpretation of regression equation: States the slope of the line of best fit. Interprets the meaning of the
slope as it pertains to the data being analyzed. States and interprets the y-intercept of the line of best fit.
Interpretation of regression statistics: Finds and states the value of r2, the coefficient of determination, and r,
the correlation coefficient. Interprets and discusses what each of these values means in terms of this specific
analysis.
Prediction
Prediction: Selects an appropriate value to make a future prediction using the line of best fit. Shows work for
calculations. Explains why this value was chosen, and if data exists for the prediction, discusses whether the
model gave a good estimate. Considers and discusses other factors that may influence the estimated value.
Summary
Summary: Writes a summary of procedures and findings. Discusses results and limitations of work.
Writing/ Organization/ Citations
References: Cites sources (in-text and reference citations) in APA format.
Completes all sections of the project. Work is well-organized and easy to follow. Writing is clear and free of
errors.
1:30 7
A learn.umgc.edu
Bibliography
Yahoo is now a part of Verizon Media.
(2019, December 30). Yahoo Finance.
https://finance.yahoo.com/quote/%5EGS
PC/history?
period1=1483228800&period2=1577750
400&interval=1d&filter=history&frequenc
y=1d&includeAdjusted Close=true
Cindy McCullagh
yesterday at 9:27 PM
Hi Pascal,
Your graph looks pretty good, but you
need to change your data table.
Your data table should have the
columns “Quarter”, “Quarters since Q1-
2017” and “Price (USD)”. The last two
columns are what you will use for your
scatter plot.
Your source will be the source of data
that you used to obtain your data for
analysis, which I think was FRED.
In APA format, your source will be
listed under References, not
Bibliography.
Thanks!
1:30 1
learn.umgc.edu
Ретът
suivi
month increments. For example: Quarter
1 is January – March, Quarter 2 is April –
June, Quarter 3 is July – September, and
Quarter 4 is November – December.
S&P 500
Q1
Q2
2017
6.10%
3.30%
8.90%
Large Cap Value
Large Cap Growth
Mid Cap
Small Cap
5.20%
2.50%
2018
Large Cap Value
Large Cap Growth
-0.80%
2.80%
1.40%
0.50%
Mid Cap
Small Cap
0.10%
13.70%
2019
7.80%
Large Cap Value
Large Cap Growth
5.60%
3.30%
Mid Cap
Small Cap
4.30%
S&P 500 Price vs Quarter
PRICE (USD$)
Tri
$3,250.00
$3,000.00
$2,750.00
PRICE (USD$)
$2,500.00
$2,250.00
$2,000.00
Q1 – Q2 – Q3 –
Q4 – Q1
2017 2017 2017 2017 201
1:30 7
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Week 4
MATH 107: Project Discussion
The topic of my project will focus on the
linear analyses of the stock market boom
in the year of 2019, to be more specific,
the S&P 500. Before diving into the S&P
500’s performance, I would like to give
you a glance at what the S&P 500 really
is. The S&P 500 which is also called the
Standard & Poor’s 500 index is a market
capitalisation weighted index of the 500
largest companies that are publicly traded
in the United States. Some of the notable
companies that fall under the S&P 500 are
Apple, Facebook, Amazon, Tesla, JP
Morgan Chase, Johnson & Johnson, and
hundreds more. In regards to the
performance of the S&P 500, on average,
the annual return for the top 500
companies in the United States range
between 10%-11%. Assessing previous
performance history between the years of
2017 to 2019, the return for the year of
2017 was approximately 21%, the return
for the year of 2018 was -4%, and the
return for the year of 2019 was 31%. The
breakdown for this 3 year performance is
broken down below.
Please be advised that quarterly
performances are broken down into 3
month increments. For example: Quarter
1 is January – March Quarter 2 is April –
2019 Stock Market Performance
Math 107
Pascal
40
1
30
20
Percentages
Increase
10
0
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
NASDAQ
S&P 500
Dow Jones
Months
1:291
learn.umgc.edu
Week 4
MATH 107: Project Discussion
The topic of my project will focus on the linear
analyses of the stock market boom in the year of
2019. For anyone who either follows the market or is
invested in it, it is safe to say that the year of 2019
was a win for investors as well as billion dollar
companies. It was during this monumental year that
the S&P 500 recorded 26 all-time high closing price
levels and over a 27% year to date total return. Put
simply, the S&P 500 in the year of 2019 showed
lucrative promise in the aftermath of its dominance
due to the fact that average bull runs are just under 4
years in duration.
In the data chart below, the major market players are
being broken down in percentage profits focusing on
quarterly performance. Each quarterly breakdown
goes as follows: Quarter 1 is January to March,
Quarter 2 is April to June, Quarter 3 is July to
September, and Quarter 4 is October to December.
A
B
С
1
United States Indices
Q1
Q2
Q3
2.
16.60%
4%
3
16.50%
3.60%
4
5
Nasdaq 100
Nasdaq Composite
S&P 500
Russell 3000
S&P Midcap 400
Russell 2000
Dow Jones
Russell Microcap
6
13.10%
13.50%
14.00%
14.20%
11.20%
12.80%
3.80%
3.60%
2.60%
1.70%
2.60%
0.60%
7
8
9