## Fractions and Percentages using Little Caesar’s Quattro Pizza

Pizza is a delicious way to talk about fractions and percentages. In the food service industry, catering managers keep track of pizza consumption, which provides important information for future budgeting and planning.

Let us say that a food service manager is catering a pizza party, where the host specifically asks for Little Caesar’s Quattro pizzas.

The Quattro is a specialty pizza that has four sets of toppings: cheese, pepperoni, Italian sausage with bruschetta, and Italian sausage with pepperoni.

## Pizza Slice Counts

During food service, the banquet servers kept the pizzas warm and presentable to the partygoers on multiple serving tables.

The number of empty boxes were counted at the end of the pizza party.

There were 62 boxes delivered to the banquet. When food service concluded, 47 boxes were empty. The remaining boxes had leftover pizza slice that were counted and displayed in the table below.

Box # Cheese Pepperoni Sausage/Bruschetta Sausage/Pepperoni
1 0 0 2 1
2 0 0 1 2
3 1 0 1 2
4 1 1 2 1
5 0 1 2 2
6 0 2 1 1
7 1 2 1 0
8 1 1 1 0
9 0 2 1 1
10 0 1 2 1
11 1 1 2 2
12 1 2 2 0
13 1 2 1 2
14 0 0 1 0
15 1 0 2 1

Pizza is traditionally divided into slices for convenience.

Each slice of the pizza may be considered a sector of the pizza. A sector is a portion of a pizza (or a circle) that has boundaries marked by the two edges of pizza slices and the crust of the pizza (the perimeter).

In the case of the Little Caesar’s Quattro pizza, eight slices are part of one of the four sectors. More precisely, there are two slices of pizza in each sector. Each section contains a specific topping group.

The catering manager can use the leftover slice count data set for predicting customer need, which help control costs. After food service, the manager can also control costs by storing the leftover pizza in the kitchen freezer.

## Pizza Slice Totals

Now, let us look at the slice counts and the totals for each pizza box and totals for each topping group.

# Cheese Pepperoni Sausage/Bruschetta Sausage/Pepperoni Total
1 0 0 2 1 3
2 0 0 1 2 3
3 1 0 1 2 4
4 1 1 2 1 5
5 0 1 2 2 5
6 0 2 1 1 4
7 1 2 1 0 4
8 1 1 1 0 3
9 0 2 1 1 4
10 0 1 2 1 4
11 1 1 2 2 6
12 1 2 2 0 5
13 1 2 1 2 6
14 0 0 1 0 1
15 1 0 2 1 4
T 8 15 22 16 61

There is a total of 61 slices remaining. Eight (8) slices of cheese, fifteen (15) slices of pepperoni, twenty-two (22) slices of sausage/bruschetta and sixteen (16) slices of sausage/pepperoni.

## Fractions and Percentages

We already mentioned the importance of this information for food storage planning. The owner of the catering company wants to know more details about the leftover slices for business analysis.

To deliver that information, we must summarize the leftover slice counts with fractions and percentages.

Toppings Group Counts Fraction Percent
Cheese 8 8 / 61 13%
Pepperoni 15 15 / 61 25%
Sausage/Bruschetta 22 22 / 61 36%
Sausage/Pepperoni 16 16 / 61 26%

Percentages are “fractions of 100” and are best visualized using pie charts.

Let us look at these percentages in the pie chart and see how they relate to the proper fractions that describe each topping group.

First, as soon as we know the total number of leftover slices, we can calculate the unit fraction.

CCSS.MATH.CONTENT.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

For total parts T, we know that the unit fraction is 1/T. There are 61 total (equal) leftover slices. This means the unit fraction is 1/61. Let us calculate his unit fraction in decimal form.

`= 1 / 61 = 0.01639`

## The Percentage Package

Decimals help us calculate percentages. In fact, we can look at percentages as if they were “packages” used to described values.

For example, we can convert any decimal into a percentage by “multiplying the number by 100”:

`= 0.01639 * 100 `

before adding a percent sign (%) to the end:

`= 1.639%`

🤔🤔🤔 Pay attention!!

We are not arbitrarily multiplying a number by 100. Instead, we are multiplying a number by exactly 1. Let us recall the identity property of multiplication.

`A x 1 = A`

It is extremely useful. The property says that we can do anything we want to any number as long as the expression simplifies to that same number. In other words, we can say the same things in a different way.

Another way of saying “one” is by saying “one hundred over one hundred” (100/100).

`= 100 / 100 = (100) * (1/100) = 1`

The “per one hundred” (1/100) portion of the above expression is called percent (“per cent”) for simplicity. In other words, the above expression can be re-written as “100 percent”, which is equal to 1.

The same goes for any number, including fractions and decimals.

`= A x 100 * ( 1 / 100) `

Like many things in mathematics, the percent sign (%) is just another symbol used to save time and resources (and reduce error) for communication of specific values.

`= (A * 100) %`

THINK. THINK. THINK.

How would you describe “percent” in scientific notation?

## Percentages with Unit Fractions

We know the unit fraction that describes the total amount of leftover slices, so we can multiply that equivalent percentage by the total amount from each topping group to find each percentage.

Topping Group Counts Multiply by Unit Fraction Percent
Cheese 8 ( 1 / 61 ) * 8 = 13%
Pepperoni 15 ( 1 / 61 ) * 15 = 25%
Sausage/Bruschetta 22 ( 1 / 61 ) * 22 = 36%
Sausage/Pepperoni 16 ( 1 / 61 ) * 16 = 26%

CCSS.MATH.CONTENT.4.NF.B.4.B: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

## Pizza Storage Memo

The catering manager must figure out how many storage containers are required for placing the leftovers in the kitchen freezers. The leftover pizza slices must be separated by topping group for more effective logistics. The storage containers can only hold 3 slices each.

The more urgent matter is storing the pizza in the kitchen freezers.

We must divide each topping group total `{8, 15, 22, 16}` by 3 to get a quotient. However, we are not concerned about the value of each quotient as we are about the amount of storage containers that must be allocated for this task.

Some of these quantities are not whole numbers.

If there are any quantities with decimals (fractional parts), we want to find the best whole number that makes sense in the real world. To do this, we will use the `=CEILING(number)` function, where the result is the smallest integer greater than an input number.

The `=CEILING` function is used everywhere.

Here are a few example:

Using Wolfram Alpha, simply divide each of the topping group totals by 3 and use the `=CEILING` (or the `=CEIL`) function.

Sample Wolfram Alpha input
`=SUM(CEIL(8/3), CEIL(15/3), CEIL(22/3), CEIL(16/3))`

When you add all the calculations, the result is:

`= 3 + 5 + 8 + 6 = 22`

To store the slices of leftover pizza, we need 22 storage containers.

## Conclusion

The catering manager now has a point of reference for future budgeting and planning of pizza parties. About 88% of the total delivered pizza was consumed. The remaining slices show more detail about the popularity of the topping groups.

The specialty pizza discussed in this math practice activity reorganized 496 slices across four equal topping groups for 62 pizzas.

We were able to regroup these slices for analysis that used fractions and percentages as tools. Analysis of business operations may be helpful for making better decisions. With more data sets, the catering manager may be able to minimize costs and optimize procedures.

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.

## New Worksheet: UPC Barcode Check Digit Algorithm

Use this worksheet for future math activities and bell work.

## UPC Barcodes: World’s Largest Homeschool Math Workbook

Good news! The world’s largest workbook is free and available for anyone who knows where to find it. That workbook is full of real-world math practice problems that use addition, subtraction, multiplication and division.

The workbook is basically all the UPC (Universal Product Code) product identifiers in the known universe.

UPC barcodes are everywhere. We can find UPC barcodes in the house, in the supermarket, and pretty much any place that you may find consumer packaged goods (CPG).

In the CPG industry, UPC barcodes are essential for managing inventory through automated scanning. The beeping sound that you hear at the grocery checkout counter represents millions of dollars that have been invested in supply chain management and customer service.

That same beeping sound confirms the completion of a validation process, where 12 UPC numerical digits (ranging from 0 to 9) are used to make calculations using a specified algorithm.

An algorithm is a series of logical steps that generates a specific and predictable solution. In this case, the algorithm is calculating the UPC check digit, which is an industry standard for validation.

From a broader perspective, algorithms are very important in computer science, business analytics, and supply chain management. We can use algorithms to help manage costs and add opportunities for new revenue streams.

## UPC Check Digit Algorithm

There are many types of UPC barcodes.

In this activity, we will focus on the UPC-A barcode and its corresponding check digit algorithm.

A valid UPC-A barcode has exactly 12 digits. You may see barcodes with only 11 digits recorded for product inventory. If you see 11-digit barcode numbers, simply add a zero to the beginning of the UPC-A character sequence.

We find the UPC-A check digit by applying an algorithm that uses the first 11 digits of the UPC barcode. The check digit should equal the final digit.

• Here is the algorithm for calculating the UPC check digit:
• Step 1: Add the odd-indexed digits. Multiply that first sum by 3.
• Step 2: Add the even-indexed digits for a second sum.
• Step 3: Add both sums from Step 1 and Step 2. Divide that combined sum by 10. Subtract the remainder from 10. The difference is the check digit. If the remainder is zero, the check digit is zero.

Let us use an example to better understand how this algorithm works in the computer whenever the UPC barcode is scanned and validated. Download a blank worksheet, and use it to follow along during the walkthrough below.

## UPC Barcode Check Digit Walkthrough

We will use the following UPC in this example:

`040000527152`

This UPC is probably more familiar when it is printed on CPG products in its iconic barcode visual representation:

Interestingly, you can create UPC barcodes in Wolfram Alpha.

Sample Wolfram Alpha input
`UPC 040000527152`

To begin, enter all the UPC digits in the order they appear. The digits are indexed by their position in the UPC character sequence. There are odd-indexed digits, and there are even-indexed digits.

1 2 3 4 5 6 7 8 9 10 11 12
0 4 0 0 0 0 5 2 7 1 5 2

Remember, we are only using the first 11 digits for this calculation. The final digit (in the 12th position) should always match the check digit for the UPC-A barcode to be considered valid.

Step 1: Add all the odd-indexed digits. Multiply the sum by 3.

1 2 3 4 5 6 7 8 9 10 11 12
0 0 0 5 7 5

`3A = 3 * (0 + 0 + 0 + 5 + 7 + 5)`
`3A = 3 * 17 = 51 `

Step 2: Add all the even-indexed digits (except the last digit).

1 2 3 4 5 6 7 8 9 10 11 12
4 0 0 2 1

`B = (4 + 0 + 0 + 2 + 1) `
`B = 7 `

Step 3: Combine both sums. Find the “modulo 10” of that result. Subtract the “module 10” value (the remainder) from 10 to get the check digit. If the remainder is 0, the check digit is 0. The check digit should equal the final digit of the UPC-A character sequence.

`3A + B = ( 51 + 7 ) = 58`

`check digit `
`= 10 - (3A + B) modulo 10`
`= 10 - (58 modulo 10)`
`= 10 - 8 = 2`

CCSS.MATH.CONTENT.4.OA.A.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted.

NOTE: Markdown tables were generated with Google Sheets using MarkdownTableMaker, another Bitwise Thermodynamics project.

## Discussion

All four operations of arithmetic are used in this math practice activity. Addition, subtraction, multiplication, and division are used to calculate the check digit.

In addition to the four basic arithmetic operations, we had the opportunity to use a specialized operation called the modulo operation. The modulo operation (or function) focuses on the remainder of the division operation instead of the quotient. In other words, modulo is specifically asking the amount leftover after the division operation is completed.

Modulo is very useful for determining properties of numbers. For example, even numbers have a “modulo 2” equal to zero, while odd numbers have a “modulo 2” equal to one.

In computer programming, this modulo operation is often used in algorithms related to the processing and rendering of consecutive data records (e.g., positions of a character sequence, rows in a table, etc.).

In this math activity, we used “modulo 10” to help find the check digit. Computational platforms (such as Wolfram Alpha), search engines and some smart speakers can provide modulo operation results.

Sample Wolfram Alpha input
`88 modulo 2` OR `55 mod 2` OR `55 % 10`

## Conclusion

UPCs are ubiquitous, which provides unlimited opportunity for math practice on shopping trips or activities involving pantry inventory. Every CPG package in the grocery store is now part of the world’s largest math workbook. The math problems are free and the added math practice will pay off with improved recall of multiplication facts.

Every UPC is a math practice problem with the “answer” provided. However, every UPC must be checked because some UPCs fail the validation process, which may cause issues during inventory audits and customer checkout (when the cashier calls for a “price check”).

In addition to the worksheets in this activity, personal math journals are great places to record interesting UPC barcodes found for future reference.

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.

We just discussed a math practice solution for soup ladles in kitchen math.

We focused on the relationship between the diameter of a soup ladle bowl and the standard sizes used in recipes and food service plans.

Now, we will look at how mathematics goes into 3D modeling of soup ladles and similar objects. TINKERCAD is the main tool we will use to visualize some important math concepts.

This activity looks closely at the geometry related to soup ladle bowls.

## Positive and negative volumes

TINKERCAD provides a wonderful platform for applying math knowledge from the classroom. For example, the solid objects represent “positive” volumes (or volume values with positive signs). In contrast, the hole objects represent “negative” volumes (or volume values with negative signs).

During this activity, let us also keep track of the total amount of volume on the workplane. Right now, there are no objects on the workplane. The total volume is zero.

`V_TOTAL = 0`

Let’s dive right into Part 1.

Okay, there is a lot here.

Before we began 3D modeling in Part 1, we chose a system of units. This project used inches.

The hemisphere was placed on the “work plane” with the flat side down. We wanted the flat side up, so we rotated the hemisphere 180 degrees about a single axis.

CCSS.MATH.CONTENT.HSG.GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

The shape that was added from the library was a hemisphere (half-sphere). The length and the width (the diameters) of the hemisphere were equal, but the height was half the diameters.

These rules were applied when the length and width were changed. To maintain the regular hemisphere shape, the height scaled (“proportionally”) at the same rate as the length and width.

Also, when we duplicated the solid hemisphere object, we doubled the total volume of the objects sitting on the “work plane”..

Let us continue with Part 2.

In Part 2, we reduced the length and the width of the second object to 1.8 inches. The height was proportionally reduced to be half that measurement, which was 0.9 inches.

Also, something significant happened.

There was a sign change. The positive volume of the second object was converted into a negative volume.

Now, the total volume on the “work plane” comprises the positive volume of the first object (represented by the solid) and the negative volume of the second object (represented by the hole).

`V_TOTAL = (+)V_SOLID + (-)V_HOLE`

`V_TOTAL = V_SOLID — V_HOLE`

The next step is to combine both volumes. Perhaps adding a negative volume to a positive volume will result in less volume. Let us find out in Part 3.

As we saw in Part 3, when both objects occupy the same space, we can apply additive properties. The (negative) hole object removed material from the (solid) positive object.

The result is a new shape that represents the positive (solid) object minus the negative (hole) object.

This is the same as digging a hole. Whenever we dig a hole, we are adding a negative volume to dirt.

## Bigger holes, thinner walls

Let us suppose we want to decrease the wall thickness of the soup bowl ladle. One solution would be to increase the size of the (negative) hole object. Let us see what happens when the (negative) hole object’s volume is increased while the (positive) solid object’s volume remains unchanged.

The hole is bigger.

The size of the length and width were increased to 1.95 inches — and the diameter (0.975 inches) was changed to proportionally match the new value.

Now, we must add the volumes again to form the new shape.

## Conclusion

This activity required a special type of computational thinking. This is visuospatial computational thinking, which is important in manufacturing, logistics, fashion, and the performing arts.

There are also some important takeaways. Volume and density are related to weight (and mass).

Weight = Density * Volume

If density stays constant, then a decrease in wall thickness (an increase in the hole size) would cause the soup ladle bowl weight to decrease. Less material is less expensive to produce and transport. However, minimal material may affect product quality and user experience.

These are design issues to think about in cooperative and collaborative projects. Math is no longer about the “answer in the back of the book” in the workplace. Instead, math is now about creating new solutions and using critical thinking to continuously improve products and create new opportunities.

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.

## Ultra Instinct Multiplication Tables: Practice and Automaticity for Test Preparation

Ultra Instinct describes the elevated state of mental and physical awareness that Goku is able to achieve after years of rigorous training. At this level, Goku is able to move without thinking, which gives him an overwhelming tactical advantage in battle.

Thinking adds to reaction time. By moving without thinking, Goku’s speed and precision are nearly perfect. Training was the key to unlocking his potential.

Test preparation also requires training.

Practice, retention and recall of multiplication facts (i.e., “times tables”) are essential skills for developing solutions for more complex math problems. The ability to quickly recall multiplication facts saves time and builds one of the most frequently-used skills in all of mathematics.

Mastery of multiplication facts involves the process of improving declarative and procedural math knowledge through practice.

That consistent practice leads to eventual automaticity, where you will be able to recall multiplication facts without even thinking. With time, you may even develop the math equivalent of Ultra Instinct (maybe).

## Testing is just testing. No more, no less.

Standardized testing is nothing more than a cost-effective way of measuring proficiency. Minimizing cost is the main goal of using standardized tests for assessment of skills mastery. As a result, standardized testing environments have limitations. Those limitations includes the complexity and the scoring of test items.

Test takers cannot control the content of the assessments, but they can control the amount of time spent per test item and the probability of selecting the “best answer” from a list of choices.

Many standardized test scores are norm-referenced. Imagine sitting on a giant number line with all the other test takers.

Your test score only reflects your position on that number line. Your position on that number line only indicates how well you were able to complete the test.

The test scores must be validated in order for local education agencies (e.g., school districts) to use them. The validation procedure must minimize the cost related to the revenue generated from test administration.

Your position on the number line can be affected by how many test items you complete and how many “best answers” you select. As you move higher on the number line, your test score improves.

This sounds obvious, but many test takers forget that small changes to test preparation can make a big difference in test performance.

Therefore, if your average response time per test item (RTPT) is less than the person next to you on the number line, then you may have a better chance of moving higher on the giant number line.

More time invested in practice test items, will lower response time on standardized test.

If your multiplication skills are better than the test taker next to you, then you may have an even better chance of moving higher on that giant number line. More time invested in recall of multiplication facts under time pressure will help prepare your mind for better test performance.

Meaningful test preparation improves test scores. Investing more training time to practice multiplication facts will place you ahead of all the other students who have not invested any time at all.

As your training intensifies into focused multiplication practice, you will be able to complete arithmetic operations without thinking – a characteristic of automaticity, and perhaps, Ultra Instinct multiplication.

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.

## Kitchen Math: What is the density of paper towel sheets?

Kitchen math is important math for food service and catering. Gastronomy requires both math and measurement. The related kitchen duties also require some understanding of the physical properties of equipment and supplies.

In the kitchen, paper towels are useful for food preparation and essential for food safety. Like everything in the food service industry, paper towels are resources that must be managed to help control costs.

Food service managers use comparison shopping to find the best deals. For paper towels, density and absorbency are very important product features.

In this math practice activity, we we will focus on the density of paper towel sheets.

TL;DR
The density of the paper towel is 0.00784 pounds per cubic inch in this specific math solution. That is roughly equal to 0.2 grams per cubic centimeter.

## How do you find the density of paper towel sheets?

The answer depends on the quality of the paper towel sheets. Our test sample for this math activity is part of the package called Kirkland Signature Big Roll Paper Towels from Costco Wholesale.

Let us first review the governing equation for density.

Density = Weight / Volume

Density is a property that tells us how closely-packed molecules are within a unit volume. Higher densities tend to be heavier, and lower densities tend to be lighter.

Subways during rush hour are densely-packed with passengers and are heavier. During off-hours, the passenger load is much lighter. The same principle applies for molecules.

We already know that matter is anything that has mass and takes up space. Density helps us better understand how much matter is packed into a specified three-dimensional space.

To achieve that understanding, we need to know the amount of mass, which is affected by gravity (which causes the phenomenon called weight) and the amount of volume.

Units for density are weight per unit volume. For this example, we will specifically use the units pounds per cubic inch [lbs. / cu. in.]

The weight of the paper towel material is unknown, so we will use a digital kitchen scale to take some measurements.

We find that the total weight of an unopened, unused paper towel roll with its packaging is 12.4 ounces.

The cardboard roll that holds the paper towel is 0.2 ounces. The packaging that wraps the paper towel roll is also 0.2 ounces.

In other words, we can calculate the weight of the paper towel material itself by subtracting the weight of everything else (e.g., all that other stuff).

W_material = W_total – W_stuff
W_material = 12.4 – 0.4

The paper towel material weighs 12 ounces. There are 16 ounces in one pound. So, we can say that material weight is 12/16 pounds.

`"how many pounds is 12 ounces?"`

We have calculated the weight of the paper towel material. Now, let us calculate the volume of the paper towel material.

We know that the area of the entire paper towel roll is 85 square feet. We can calculate the area A (in square inches) to be:

A = 85 ft. ft. * 12 * 12 in./ft. in./ft. ==> 12,240 in. in.

In other words, we simply converted the area from square inches into square feet. All the voice assistants can readily convert common units.

`"85 square feet in square inches"`

CCSS.MATH.CONTENT.5.MD.A.1: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

The paper has a thickness, but a ruler won’t be practical for this measurement. We need something that can measure very small fractions of an inch.

Let us use digital calipers.

Using digital calipers, we measure that the thickness t (in inches) of a paper towel sheet is 1/128″.

We now have enough information to calculate the volume of the paper towel material. The volume V (in cubic inches) is:

V = A * t
V = ( 85 * 12 * 12 ) * 1/128

The weight W = 12/16 lbs. Combining all our previous calculations, we find that the density must be:

ρ = W / V = (12/16) / ( 85 * 12 * 12 * (1 / 128))

The density of the paper towel sheets is 0.00784 lbs. / cu. in.

All the calculations above can be organized into a computational solution that can be solved using Wolfram Alpha.

Sample Wolfram Alpha input
`W=(12/16);V=(85*12*12*(1/128));rho=W/V`
View Result

## Conclusion

This information can be recorded and used later for comparison shopping of paper towels that may enter the market at a future date (hint, hint). While price may be an important factor, the density of the paper towel sheets for cooking and cleaning is also a very important metric in food service.

MathForWork delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.