CCSS.MATH.CONTENT.5.MD.A.1: Coffee Jar Flower Vase Activity

The following activity is an opportunity for math practice in situations when measurement requires math-based design solutions. If possible, read the problem statement aloud. After reading the problem statement, we will have a brief discussion.

Problem Statement

An empty coffee jar has been converted into a flower vase. The 3D modeling of this coffee jar flower vase is described as two frusta stacked end-to-end to form an hourglass container. The top areas of both frusta share a common square cross-section, which forms a bottleneck at 3.5″ above the bottom of the jar (sitting on a table). The capacity of the container is 800 mL of water. The total height of the container is 5.5″. The base area for the lower frustum and upper frustum are 3.5″ x 3.5″ and 3″ x 3″, respectively. What are the dimensions of the cross-sectional area at the bottleneck?

Coffee Jar

Discussion

Wow! 😂😂😂 Lots of words. That is the point.

This activity is designed to help you practice reading a lot of words in test items. You will gain experience organizing these words and ideas using computational thinking.

Let us take a moment to visualize the solid model described above with 3D file viewer on GitHub. Or you can simply view the video below:

Before we begin this problem, we need to recall some prior knowledge and best practices. First, we choose and confirm the units of the final answer. In this case, the units are [inches x inches] or [in. x in.].

Note that the question did not ask for the side length or the actual area. The call to action asked for the dimensions. In the workplace, it is very important to pay attention to what the request for proposal (RFP) needs.

Second, we convert all quantities into compatible units. We convert 800 millimeters into cubic inches. The result is about 49 cubic inches.

In Wolfram Alpha, type this:
800 ml in cu in (view)

For voice assistants, try this command:
800 milliliters in cubic 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.

Next, we look at the formula for the volume of a frustum.

V = π/3 h (A + B + sqrt(A B))

There are two frusta in this problem, so we will be using the same formula twice to help solve for a system of equations.

Let us place the container on the table. The table will be our reference point. The total height of the container is the distance from the table to the top of the container (see Figure 1). The total height (in inches) of the container is H = 5.5.

Figure 1: Upper frustum and lower frustum modeled in TINKERCAD

This measurement excludes the the jar mouth section, where the screw top lid has been removed.

For the lower frustum volume Y, we suppose that the base area (in inches squared) is A = 3.5 * 3.5 and the height (in inches) from the table is h = 3.5.

The units of area are square inches. We will let Wolfram Alpha make the calculations for us by leaving the dimensions as multiplicands. This methods lets us make design changes to the dimensions as needed without re-building our solution.

Figure 2: Lower frustum dimensions in TINKERCAD

For the upper frustum volume Z, we conclude that the base area is B = 3 * 3, with a height of H – h.

We know the exact height of the upper frustum, but we will still use variables to express the difference between the total height and the height from the table.

Both frusta share the common bottleneck area X. Their combined volume is 49 cubic inches; that is, Y + Z = 49.

Our analysis is complete. We now know enough information to solve for the unknown bottleneck area X.

In Wolfram Alpha, type this:
Y + Z = 49; H = 5.5; A = 3.5 * 3.5; B = 3 * 3; h = 3.5; Y = (h/3) * (A + X + SQRT(A*X)); Z = ((H-h)/3) * (B + X + SQRT(B*X)); J = SQRT(X);
(view)

The dimensions of the bottleneck cross-section are 2.63″ x 2.63″.

Key vocabulary: request for proposal, units, unit conversion

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.3.OA.B.5: Candy Counting Using Chunking

Counting can be stressful.

When the brain is presented a lot of information at the same time, the filtering process may become overwhelming. This is when cognitive strategies are useful, and these skills are part of any good math test preparation program.

Let us try to count all the candy in the following photo:

It is difficult. And a little stressful. The direction of counting may feel random, and the multiple colors and positions may be very distracting. Also, if students with disabilities have attention problems, counting the candy in the photo is nearly impossible without some cognitive strategy.

Perhaps we can help improve the counting process by grouping the candy by color. We do that, and try counting again.

Better.

However, this grouping may still be putting pressure on working memory. The brain is still making decisions about where to start counting and which direction to follow. That bandwidth is taking “processing power” away from our main counting task.

If we could break up the candy into smaller groups, the chunking process could make counting much easier.

Let’s find out.

Much better.

We can now count the smaller groups of candy using the chunking facility in our working memory. We can see the groups at a glance, and we can count the number of groups with minimal effort.

Grouped by color, we can now take a running tally of the candy without overwhelming the working memory.

ColorCount
Red2 groups of 4, 1 group of 5
(4 x 2) + (5 x 1)
Green2 groups of 5
(5 x 2)
Blue1 group of 3, 1 group of 4, 1 group of 5
(3 x 1) + (4 x 1) + (5 x 1)
Brown1 group of 4
(4 x 1)
Yellow1 group of 4, 1 group of 3
(4 x 1) + (3 x 1)
Orange2 groups of 5
(5 x 2)

In Wolfram Alpha, type this:
(4 * 2) + (5 * 1) + (5 * 2) + (3 * 1) + (4 * 1) + (5 * 1) + (4 * 1) + (4 * 1) + (3 * 1) + (5 * 2) (view)

OR type this:
SUM(4*2,5*1,5*2,3*1,4*1,5*1,4*1,4*1,3*1,5*2) (view)

CCSS.MATH.CONTENT.3.OA.B.5: Apply properties of operations as strategies to multiply and divide.

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.3.NF.A.2.A: Clock Math and Unit Fractions

We use the terms seconds, minutes, and hours to provide a context for the numbers related to time. Each of these quantities sit on a number line, which allows us measure the distance between two times. The most common use of this measurement is the expression of the time elapsed since midnight.

In this activity, we are focusing on the language related to time. We will look at each time segment and express that quantity as a unit fraction.

There are 60 seconds in one minute. Therefore, each second is 1/60 (“one-sixtieth”) of a minute. In most contexts, saying “one second” makes more sense than saying “1/60” of a minute. We know that if we add 59 more seconds, we will have a total of 60 seconds. We would have a full minute.

We can use an expression to describe this addition:

In both fractions, the numerator indicates what part of the minute is being considered. Also, the denominator is providing the total number of seconds in every minute. When we add all the parts in the above expression, we get a whole minute.

There are also 60 minutes in one minute. Each minute is 1/60 of an hour. We have 24 hours in a day, and each of those hours equals 1/24 (“one-twenty-fourth”) of a day. There are 7 days in a week, so each day is 1/7 (“one-seventh”) of a week.

CCSS.MATH.CONTENT.3.NF.A.2.A: Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

BONUS: The 12 Unequal Months

There are 12 months in a year. This means each month is considered 1/12 (“one-twelfth”) of a year. Right? Well, yes and no.

The length of a month can vary between 28 and 31 days. This can be confusing because each month of the year can divided into an unequal amount of days.

We know that unit fractions must represent a single part of a whole quantity that has equal parts.

Interesting. Let’s work through this.

The months are containers, and the contents in those containers are days. We have 12 equal containers, but the contents in each of those 12 containers may not be equal.

The good news is that banks have created a special convention to help each month have equal parts. It is called the 360-day convention (or the banker’s year).

To make things easier (and to reduce the cost of extra math), banks say that every month has 30 days. Thus, each banker’s month is 1/12 of a 360-day banker’s year.

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.G.B.4: Circumference and Paper Towel Rolls

Circles have several properties: diameter, radius, area, and circumference. The diameter, radius and area are often described in expressions and equations that are related to π (“pi”). However, circumference is almost never discussed with as much detail.

There are a wide range of math topics covered in the classroom, but not as much depth and exploration of these new ideas. Adding personal thoughts and experience to a math concept is a great way to help student engagement and promote meaningful learning.

When learning is more meaningful, students retain the information in long-term memory and recall those concepts with less difficulty.

Paper towel rolls are great way to use prior knowledge from students to build new ideas about math topics. Daily interactions with common household objects, such as paper towel rolls, can strengthen the relevance of math in everyday life.

Note on Teaching Supplies: Craft paper rolls can be purchased online from Amazon. Or you can use an empty paper towel roll.

From observation, we see that a circumference is the perimeter around a circle. If you walk around the entire perimeter of a circle, we find that the total distance around the circle is equal to the length of a segment that has been “un-rolled” from a round arc into a straight line.

Let us take a cardboard paper towel roll. The cross section is a circle.

If we make a single cut along its longest side, we can “un-roll” the paper towel roll into a flat sheet of cardboard. (The straightness of the cut does not need to be perfect.)

We know that a circumference is the length around a circle. We also know that the length remains unchanged. Also, the surface area of the paper towel roll along its longest length is also unchanged. The only change was the path of the length, which was transformed from a round arc into a straight line.

This activity demonstrates the powerful the idea of π (“pi”), which allows us to relate the diameter of an object to its circumference without the need for modification. We also see that circumference is simply the arc length that travels completely around a circle.

Key Vocabulary: circumference, length, area

CCSS.MATH.CONTENT.7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

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.8.G.C.9: Work TINKERCAD with Cone, Cylinder, Frustum

In this short video, we explore the relationship between cones, cylinders, and frusta (the plural form of frustum). The three shapes are connected by a general equation, and we explore the impact of changing the top radius for each of these shapes. TINKERCAD is the 3D modeling platform used for this activity.

The general equation for the volume of a frustum (volume V) with any cross-sectional area is defined as follows:

V = π/3 h (A + B + sqrt(A B))

where, height h, and areas A and B are the independent variables.

In Wolfram Alpha, type this:
V = (h / 3) * (A + B + SQRT(A * B) (view)

Let us use a more specific cross-sectional area: the area of circle. Now, we have a conical frustum:

where, height h, base radius R, and top radius r are input variables. Also, the top radius must be a value between 0 and R; that is, 0 < r < R.

In Wolfram Alpha, type this:
V = (pi / 3 ) * h * (r^2 + r*R + R^2) (view)

When r = 0, we have the formula for a cone:

V = 1/3 π h R^2

In Wolfram Alpha, type this:
V = (pi / 3 ) * h * (0 + 0 + R^2) (view)

When r = R, we have the formula for a cylinder:

V = π h R^2

In Wolfram Alpha, type this:
V = (pi / 3 ) * h * (R*R + R*R + R*R) (view)

In this activity, we demonstrated that not every math problem has a numerical answer. There is no answer “at the back of the book” in the real world because design decisions require some careful thought and planning.

Math Work is Work.

One very important concept at MathForWork is the practice of working through ideas.

TINKERCAD provides a great opportunity to work through different ideas related to geometry and trigonometry.

Mindless rote learning for standardized testing is not work. Mindful exploratory learning and computational thinking is meaningful work that leads to meaningful learning.

TINKERCAD has a simple interface that creates a wonderful environment for 3D modeling. This job skill requires a good understanding for how to choose and modify basic shapes for tasks in the workplace. Also, the 3D models can be exported for 3D printing in maker spaces, which allow the model to be inspected more closely for future reference.

TINKERCAD is a great entry-level software applications for learning 3D CAD modeling and design. It may feel like a toy, but TINKERCAD is very useful. The 3D CAD software used in basic design does not need to be too fancy, because the exported 3D models are standardized formats.

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.

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

Create Instant QR Codes on any Device with a Bookmarklet: waferQR

Creating QR codes can improve usability for any mobile experience. Many camera apps and web browsers now include features that read can and interpret QR codes, allowing fast mobile-friendly access to web sites, social media, videos, and podcasts.

waferQR is a bookmarklet that converts any web page URL into a QR code by using Wolfram Alpha. The QR code makes the URL scannable by any compatible camera app.

The input command used for Wolfram Alpha is straightforward. Type “qrcode” in the input box followed by the text that will be converted into a QR code image.

waferQR automatically detects the current web page URL in your web browser and runs the query for you.

The following example converts the Northern Arizona University Library home page into a QR code with the following Wolfram Alpha input command:

qrcode https://nau.edu/library/

Wolfram Alpha is a leading technical computing web platform that provides powerful interpretations of command line inputs for mathematics, science, and much more.

In addition to processing complex computations, Wolfram Alpha also provides an input command that generates a QR code image. The resulting QR code can represent any text information, including web addresses.

A bookmarklet is a special kind of bookmark that contains JavaScript, which adds functionality to specific web pages. Bookmarklets are ideal for cloud-based platforms, where bookmarks sync seamlessly across devices. For example, adding the waferQR bookmarklet to Safari on a desktop will allow the quick QR generator to be available on iPhones, iPads, and other compatible devices.

There are many methods of adding bookmarklets to bookmarks folders. Drag-and-drop is a popular method.

More methods are described on the waferQR project page on GitHub.

No sign-up or sign-in is required to use waferQR. However, this means that all updates must be made by manually re-adding the bookmarklet.

You can sync the waferQR bookmarklet in your Chrome, Firefox, and Safari bookmark folders. More cloud-backed browser platforms are being tested and are expected to be supported by the end of October.

The waferQR bookmarklet was developed by Bitwise Thermodynamics to generate QR codes for instructions support for projects like MathForWork.. The QR code is generated by the Wolfram Alpha platform, which belongs to Wolfram Alpha LLC and Wolfram Research.