Test Preparation

Before you prepare to take any test, it is important to understand that the test is not an indication of your intelligence or pathway to success. The only thing a test measures is your ability to take a test.

Right now, the only two tests we discuss here are the ACT and SAT. Standardized tests are the most cost-effective method of assessment in learning environments where accountability is measured with specific metrics and results.

The tests are standardized.

Thus, the methods of test item construction and scoring are optimized to minimize cost and maximize reliability. This also means that the diminishing marginal utility of these assessment products can only be restricted to a very specific time period in life. That scarcity of time (and the resulting barrier to opportunity) has led to the increased value of these test scores

Standardized test scores are metrics that help students, parents and educators set goals.

Test preparation requires discipline, integrity and sacrifice. Students must use time management skills to prioritize test preparation over play time. Integrity requires the student to strive for success based on merit, rather than position or privilege. Further, the tradeoffs must be made in order for serious test preparation to yield good results.

Four P’s of Test Prep Success

The four P’s of success in testing environments are preparation, practice, participation, and persistence.

Preparation involves all the “test recon” that includes the details regarding the content and administration of the standardized test. The process of preparation involves self-regulation, which requires goal setting, mental preparation, and and more control over the physical learning environment.

Practice helps build connections in the brain that promote faster retrieval of declarative knowledge and more proficient production of procedural knowledge. Wide receivers in the NFL practice skill position drills that look strange out of context but give an edge against defenders in game time situations.

Participation sounds obvious, but some students never take the standardized tests after preparation and practice.

Persistence requires both participation and perseverance. Some students take the exam, but stop halfway because they are not confident in their ability to earn a passing score. Some students get a low score on the first attempt and refuse to retake a test. In both cases, there is a lack of persistence, which leads to guaranteed bad outcome.

Creating a Frustum for 3D Printing using TINKERCAD

TINKERCAD is web-based 3D modeling software that allows you to create objects for 3D printing. In TINKERCAD, you can build a frustum by starting with a solid pyramid and subtracting a “hole” from the top of that pyramid.

The following video provides a demonstration.

Volumes can be added (like when we finished the pyramid).

In this video demonstration, a cylinder and a hemisphere (half-sphere) are combined to form a new volume.

FUTURE WORK: What equation do we get when we add a hemisphere to cylinder as we did in the video?

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

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

Hershey’s Chocolate Candy Bars and Tape Measures

Candy bars and tape measures are related. Workplace math connects them in many ways, but we will only discuss a specific way in this math practice activity: fractions.

Fractions are overlooked by many math teachers and math tutors, but they are an essential part of the workplace math experience. Due to lack of training and practice, students freeze up when fractions appear on workplace qualification exams or standardized tests.

Fractions are invisible until they are needed. Without a plan, they can be hard to manage, especially in a more stressful testing environment. On the other hand, with better understanding, fractions can be a powerful and useful tool for anyone.

Math is a toolbox full of useful things designed to save time and money. In Hollywood movies, math is fancy and impressive, with a lot of strange symbols and gigantic numbers.

Unlike the Hollywood movies, the most important workplace math happens between the values of 0 and 1 on the number line. More specifically, fractions and decimals are very important because they provide information related to precision and the composition of informative inventory units.

This math practice activity will focus on a food product with sixteen parts: the Hershey’s Milk Chocolate Bar XL.

The tape measures used in manufacturing and construction are marked at least 16 times per inch. Professional tape measures are marked with 32 ticks per inch for the first 12 inches, which allows for even more precise measurement.

Tape measure with 32 ticks per inch.

Some tape measures are marked with fractions.

Specialized tape measure called “fractional read tape” from Stanley Tools

Different rulers have different functions, but we will focus on a basic tape measure this math practice activity.

Workplace Tools
Tape measures and rulers (and similar tools) are used in several industries to measure length. Think of at least 20 industries or professions that rely on similar measurement tools (e.g., the tailor tape used in the fashion industry, the speed square used in carpentry, etc.).

Measurement is how building and machining costs are managed by engineering managers and contractors. The ability to read a tape measure will obviously affect your ability to succeed in the these industries.

Now, let us talk about the chocolate candy bar before return to the topic of workplace math and measurement.

Chocolate Candy Bar Pieces

The Hershey’s Milk Chocolate Bar XL provides us sixteen pieces of chocolate sold together in one massive candy bar. This is an excellent learning opportunity because we can break up the chocolate bar to demonstrate how fractions work on a tape measure.

There are 16 pieces. However, we are going to monitor for the the word “pieces” and change that specific word to “sixteenths” for this exercise. So, when we look at the whole chocolate bar we see “sixteen sixteenths” (16/16), which is equal to 1.

Two Groups

If we break the chocolate bar in half, there will be “eight sixteenths” in each half. In other words, we see that eight sixteenths (8/16) equals one half (1/2) of the big candy bar.

The number 8 on the number line acts like a midpoint for the measurement of all sixteen sixteenths.

If the number of “sixteenths” from the candy bar is less than 8, then there is less than half of the candy bar.

If there are more than “eight sixteenths” available, then we have more than half of the candy bar.

More Groups

If we break the chocolate bar into four equal sixteenths, then we have “four sixteenths” in each new group. In other words, see that four sixteenths (4/16) equals one quarter (1/4) of the big candy bar.

The number line is the same, but there are more marked values now.

The number 4 on the number line equals 25% of the “sixteenths” while the number 12 on the number line represents 75% of the “sixteenths” in the chocolate bar.

Percents represent parts of 100.

As decimals, the number 4 on this number line equals 0.25. This helps us tremendously because now we know that given any decimal between 0 and 1, we can determine its quadrant on the number line based on how we understand fractions.

Notice that “eight sixteenths” still equals half the candy bar.

If we break up the chocolate bar into eight equal sixteenths (not “pieces”), we have “two sixteenths” per group. There will be 8 groups of “two sixteenths” (2/16 = 1/8) that comprise the whole candy bar.

And, of course, if we divide the chocolate bar into individual “sixteenths”, we have only “one sixteenth” (1/16) per group.

The chocolate candy bar has the same subdivisions as many tape measures (and many types of rulers).

Given any number of (sixteenth) “ticks” on a tape measure, you should be able to calculate the number of “sixteenths” within any inch.

In other words, instead of monitoring the word “pieces” for the chocolate bar, we will monitor the word “ticks” for the tape measure.

Each tick will be replaced by the word “sixteenth” for our workplace math measurements.

For example, let us suppose that we measure some sheet metal with the length of three inches plus eleven “ticks” on the tape.

That is half of one inch (“eight sixteenths”) plus 3 more “sixteenths” of an inch. Thus, the sheet metal part is “three and eleven sixteenths” in total length.

CCSS.MATH.CONTENT.5.NF.A.1: Use equivalent fractions as a strategy to add and subtract fractions.

As you can see the chocolate candy bar and the tape measure both share the same properties related to fractions. The candy bar can be divided into 16 pieces and the tape measure can be divided into 16 tick marks. If we understand how pieces a candy bar is divided in to smaller equal-sized groups, we can apply that knowledge to how precision measurements are made in the workplace setting.

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

Shape Algebra

Add two shapes in TINKERCAD and you get a new shape. The sphere and cylinder combine here to form a half-capsule. What is the new equation that best describes this new shape??

CCSS.MATH.CONTENT.8.G.C.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Finishing the Pyramid in TINKERCAD with Math in Wolfram Alpha

The frustum can be seen as a Rorschach test. Some people see a gold bar, and some people see an unfinished pyramid.

Frustum in TINKERCAD

One way to create a frustum in TINKERCAD is to start with a solid pyramid basic shape. The top part of the pyramid is then removed by a “hole” that adds negative volume to the shape. The result is a frustum. The following video demonstrates this method:

Before removing the top part of the pyramid, we need to know the dimensions of the solid pyramid’s base. We also need to know the height of the pyramid.

We can do this by using our prior knowledge and best practices. One of the most basic uses for algebra is providing a simple way to express how two things add up to one thing.

We know that a frustum is the “bottom section” (in blue) of a pyramid. This means that if we add the missing “top section” (in pink) to the frustum, we will have a complete pyramid.

That relationship can be described by the following equation:

We know the formula for the volume of the complete (big) pyramid,

with the base area B, the height of the frustum h, and the unknown height of the top pyramid H.

We also know the volume of the top section of the complete pyramid (which can also be described as a mini pyramid),

with the base area A (the same area as the top area of the frustum), and the unknown height of the top pyramid H.

The volume of the bottom section of the complete pyramid can be described as a frustum.

With a base area B of 6″ x 6″. let us suppose that the height of the frustum h is 10 inches, and the top area of the frustum A is 2″ x 2″. This means that the total height of the complete pyramid H + h provides us clues to the information that we need finish the pyramid.

To help finish the pyramid, we now have a useful equation that can be re-used and re-purposed in different workplace contexts:

In Wolfram Alpha, type this:
h=10;A=2*2;B=6*6;(1/3)(H+h)*B=(1/3)A*H + (h/3)(A+B+SQRT(A*B))
(view query)

Notice that we never actually calculated the volumes in the equations. Instead, we used the additive relationships between the volumes, areas and heights to complete a specific workplace task.

The 3D models related to the “finish the pyramid” math practice activity are available on GitHub. Also, the 3D models can be directly imported into TINKERCAD. NOTE: Model units are in inches.

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

CCSS.MATH.CONTENT.7.EE.B.4: Decorative Cake Box

In this activity, you work in a cloud kitchen, a business that has no dining room — only online orders. Cloud kitchens combine pre-planning with just-in-time preparation to optimize kitchen resources and stay competitive.

As the kitchen shuts down for the business day, you are tasked with the responsibility of storing a slice of special order decorative cake in a 9.6 cup plastic storage container. The cake will not be ready until tomorrow, but you already have the dimensions from the customer order.

The cake is 7″ long and 5″ wide , which is no problem because the plastic storage container has the dimensions of 9.5″ x 6.25″. However, the pastry chef is concerned about the height available when the container is sealed with its lid.

The plastic lid is designed to seal the container and protect the freshness of its contents. However, if the decorative cake is somehow damaged, it will become undeliverable and will be discarded as a loss.

The cross-sectional area of the container is a rounded rectangle with a 1″ radius on each corner. We now have a description of the geometric shapes involved, so building a solution requires us to put the pieces together into a system of equations.

CCSS.MATH.CONTENT.7.EE.B.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

We begin with a review of our prior knowledge and best practices. First, we want to use inches to communicate the usable height. To be consistent, the units we will use for the volume of the container are cubic inches.

In Wolfram Alpha, type this:
9.6 cups in cubic inches (view query)

For voice assistants, try this command:
9.6 cups in cubic inches

We find that volume of the container, which is 9.6 cups, can be converted to 138.6 cubic inches.

Second, we recall the formula for the volume of a prism:

with base area A for the rounded rectangle and height h. The actual area of a rounded rectangle is defined by:

The variables a and b describe the length and width between the rounded regions of the container, with radius r. We can subtract the radii of the corners from the overall width and length of the container.

We now have enough information to solve the problem.

In Wolfram Alpha, type this:
V=138.6;r=1;L=9.5;W=6.25;a=L-2r;b=W-2r;V=(ab+2r(a+b)+πr^2)h
(view query)

We find that as long as the decorative cake is less than 2.37 inches, the lid should close without damage.

Even though we used equalities in this math practice activity, the concern for product quality may have changed the nature of our expressions into inequalities. For example, notice how we compared the dimensions of the cake slice to the dimensions of the container before any other calculation were made.

Putting in extra work:
Try experimenting with the above calculation in Wolfram Alpha with inequalities. What happens?

Additionally, we may want a factor of safety to further protect the decorative cake, so we may decide eventually that all cakes measuring no more than 2.25 inches will be stored in that particular container. Of course, final decision is made by the pastry chef.

Comparison Shopping:
Try shopping for this and similar plastic storage containers on Amazon. What did you find?

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