News You Can Use, courtesy of Popular Mechanics: The mathematical ways to most efficiently mow your yard, by shape of yard.

https://www.popularmechanics.com/science/math/a28722621/mow-your-lawn-using-math/

News You Can Use, courtesy of Popular Mechanics: The mathematical ways to most efficiently mow your yard, by shape of yard.

https://www.popularmechanics.com/science/math/a28722621/mow-your-lawn-using-math/

*Posted by John Quintanilla on May 15, 2020*

https://meangreenmath.com/2020/05/15/how-to-mow-your-lawn-using-math/

*Posted by John Quintanilla on April 3, 2020*

https://meangreenmath.com/2020/04/03/pythagorean-theorem-and-social-distancing/

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on an interesting math competition problem. This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

Part 1: Statement of the problem.

Part 2: Initial thoughts on getting a handle on the problem.

Part 3: Initial insight.

Part 4: Geometric insight with a Riemann sphere.

Part 5: Roadblock.

Part 6: Getting past the roadblock.

Part 7: Insight.

Part 8: Proof of insight.

Part 9: Alternate solution (now that we know the answer).

*Posted by John Quintanilla on November 4, 2019*

https://meangreenmath.com/2019/11/04/solving-a-math-competition-problem-index/

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Lissette Molina. Her topic, from Geometry: finding the area of a trapezoid.

How could you as a teacher create an activity or project that involves your topic?

I believe most students in America all discovered finding the area of a trapezoid with one very easy and simple activity. Students are to receive a trapezoid of some different sizes. They are then asked to find area by cutting off the triangular sides. The student then finds that all trapezoids are composed of triangles and a rectangle. This is a very quick activity that requires students to come up with a formula that works across all trapezoids. Learning about finding the area of a shape with hands-on discoveries keeps the formula and how it became embedded into studentsâ€™ memories. This activity may also work with most polygons.

How does this topic extend what your students should have learned in previous courses?

Find the area of a trapezoid does not require much information from previous courses. One major topic the student should be able to have learned before coming into a geometry class should be area. However, very rarely, students do not know what area is already. So, the student should be able to apply what they know about area into finding the area of a trapezoid. This involves finding the area of a rectangle and a triangle. It is important that a student understands exactly where a formula is derived, so it is also important that students know that the trapezoid contains two shapes and that finding the area of those two shapes will help them find the area of the resultant trapezoid.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

One helpful website or program is Desmos (desmos.com). There are usually modules made for students often made by teachers. I have not yet come across one already made, but here is what I have in mind. Desmos is primarily made for graphing, but there are so many functions in this website that it can be manipulated to perform other things such as the unit circle. One very helpful idea would be to make a shape of a trapezoid by combining two triangles of different sizes off each of a rectangleâ€™s sides. Since these shapes are placed on top of a graph, students would be able to calculate the area by counting the square units. WIth triangles, students can count the number of half, quarter, etc. square units. This way, students can find the area of a trapezoid by counting the squares, and realize that it would be easiest to find the area of those two triangles and one rectangle and combine them.

*Posted by John Quintanilla on October 14, 2019*

https://meangreenmath.com/2019/10/14/engaging-students-area-of-a-trapezoid/

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cameron Story. His topic, from Precalculus: using right-triangle trigonometry.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Most right-angle trigonometry word problems involve giving two measurements of a triangle (angle, sides or both) and asking the students to solve for the missing piece. I argue that these problems are fine for practice, but one has to admit these problems encourage â€śplugging and chuggingâ€ť along with their formula sheets.

To make things interesting, I would use something along the lines of this word problem from purplemath.com:

â€śYou use a transit to measure the angle of the sun in the sky; the sun fills 34′ of arc. Assuming the sun is 92,919,800 miles away, find the diameter of the sun. Round your answer to the nearest mile,â€ť (Stapel, 2018).

This is incredible! Using trigonometry, students can find out the diameter of the entire sun just by knowing how far away it is and how much of the sky the sun takes up. If you were to use this word problem in a experimental type of project, I strongly recommend using the moon for measurement instead; you can probably guess why measuring the sun in the sky is a BAD idea.

What are the contributions of various cultures to this topic?

One amazing culture to contribute to the study of triangles and trigonometry were the Ancient Babylonians, who lived in what is now Iraq about 4,000 years ago. Archaeologists have found clay tablets from 1800 BC where the Babylonians carved and recorded various formulas and geometric properties. There were several such tablets found to have been lists of Pythagorean triples, which are integer solutions to the famous equation .

The Greeks, while going through their own philosophical and mathematical renaissance, gave the namesake for trigonometry. Melanie Palen, writer for the blog Owlcation, makes is very clear why trigonometry â€ś… sounds triangle-y.â€ťÂ The word trigonometry is derived from two Greek words – ‘trigonon’ which means ‘triangle’ and ‘metron’ meaning ‘measure.’ â€śPut together, the words mean “triangle measuring”â€ť (Palen, 2018).

How can technology (YouTube) be used to effectively engage students with this topic?

In the YouTube video â€śTattoos on Mathâ€ť by the YouTube channel 3Blue1Brown (link: __https://youtu.be/IxNb1WG_Ido__), Grant Sanderson offers a unique perspective on the six main trigonometric functions. In the video. Grant explains how his friend Cam has the initials CSC, which is how we notationaly represent the cosecant function. Not only is this engaging because most students wouldn’t even think of seeing tattoos in math class, but also because Grant always backs up the mathematical content in his videos with beautiful animations.

Students know how sine and cosine functions are represented geometrically; these are just the â€ślegsâ€ť of a right-angled triangle. Most students, however, only see the other four trigonometric functions as formulas to be solved. However, as Grant cleverly explains and visualizes in this video, all of these functions have geometric representations as well when paired with the unit circle. This video (moreover, this entire YouTube channel) can be helpful to those visual-learning students who need more than a formula to be convinced of something like the cosecant function.

References:

3Blue1Brown YouTube Video: __https://youtu.be/IxNb1WG_Ido__

Palen, Melanie. â€śWhat Is Trigonometry? Description & History of Trig.â€ť Owlcation, Owlcation, 25 July 2018, owlcation.com/stem/What-is-Trigonometry.

Stapel, Elizabeth. â€śRight-Triangle Word Problems.â€ť Purplemath, 2018, http://www.purplemath.com/modules/rghtprob.html

*Posted by John Quintanilla on July 19, 2019*

https://meangreenmath.com/2019/07/19/engaging-students-using-right-triangle-trigonometry-2/

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Victor Acevedo. His topic, from Geometry: Proving that the angles of a convex -gon sum to degrees.

How could you as a teacher create an activity or project that involves your topic?

A great activity to try with students would be to look at regular and irregular polygons and the triangles â€śinsideâ€ť of them. Using some string and a few tacks, students could â€śconstructâ€ť regular polygons on pegboard or foam. They could then measure the angles made by the string using a protractor and find the sum. After doing a the first few regular polygons, the students could do the same with irregular convex polygons and notice that the sum of the angles is the same for regular and irregular polygons with the same number of sides. At this point the students might have established a pattern for the sum of all the interior angles of a polygon as the number of sides (n) increases. The next task would be to go back to the regular polygons and make the triangles inside. This would be done by picking one of the vertices as the starting point and connecting that point to all the other vertices. Since the starting point is already connected to two of the other vertices, we wouldnâ€™t have to make those connections again. The students would then see that inside of the regular polygons there are n-2 triangles. Since the sum of every triangleâ€™s interior angles is 180Â°, the sum of the regular polygonsâ€™ interior angles would be 180(n-2). To further prove our original statement, the students would repeat the process of creating triangles with the irregular convex polygons.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This interactive Desmos program helps students work through proving that the sum of the interior angles of convex n-gons is 180(n-2). The program starts with a review of the sum of the angles in triangles. The students would then look at the diagonals of polygons and count the triangles formed. The students get the opportunity of deriving the formula for the sum of interior angles by continuing patterns as the number of sides increase. This program also encourages students to think about the â€ślimitâ€ť to the interior angles of a polygon and why it approaches 180Âş but will never actually reach it. There is also a link to an extension of this activity to looking at the exterior angles of a polygon as well.

https://teacher.desmos.com/activitybuilder/custom/5b75d8d696a0ad0aefe7f3ff

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

RenĂ© Descartes did not necessarily contribute to the discovery of the sum of the interior angles of convex polygons, but he was able to apply some of the outcomes to philosophy. Descartes uses the regular chiliagon (1,000-sided polygon) to demonstrate the differences between intellection and imagination. While we can clearly picture understand a triangle, a chiliagon is not quite as simple to picture due to the large number of angles and edges. To the naked eye, a chiliagon would look nearly identical to a circle. The only possible way to discern any difference would be to zoom in until you can possibly see different vertices. This application to philosophy is great for students to begin thinking about the limit that the interior angles of regular polygons reach as the number of edges increases.

*Posted by John Quintanilla on July 15, 2019*

https://meangreenmath.com/2019/07/15/engaging-students-proving-that-the-angles-of-a-convex-n-gon-sum-to-180n-2-degrees-2/

*engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

This student submission comes from my former student Phuong Trinh. Her topic, from Geometry: proving that two triangles are congruent using SAS.

How does this topic extend what your students should have learned in previous courses?

Before learning how to prove that two triangles are congruent, the students learned about parts of a triangle, congruent segments, congruent angles, angle bisectors, midpoints, perpendicular bisectors, etc. These are some of the tools, if not all, that will aid them in proving two triangles are congruent. The basis of proving two triangles are congruent using SAS is to be able to identify the congruent sides and the congruent angles. That is where their knowledge of congruent segments and angles provide them the information they need. On other hands, not all problems of proving two triangles are congruent are straightforward with all the sides and angles needed are given to us. For example: Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC. In this example, the problem did not clearly state what the congruent angles are. However, since the students have learned about what the angle bisector does to an angle, they can easily identify the congruent angles in this problem. Therefore, in order to successfully approach an exercise of proving two triangles are congruent using SAS, the students must first learn and understand the basics, which are parts of a triangle, angle bisectors, midpoints, etc.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

There are many resources that provide great aid to students in learning about proving two triangles are congruent using SAS. One of them is from ck12.org. The layout of the website is simple and straightforward. The site provides readings and color coded study guide to help the students understand the material of the lesson, such as definitions and properties of congruent triangles. It also provides videos that work out and explain example problems. The videos could potentially be a great resource and aid for students that are visual and/or auditory learners.Â On other hands, the site also gives other practices and activities that help the student estimate how well they understand the material. It is a great resource for not only the students but also the teacher. Under the activity tab, the teacher can find student submitted questions. These questions can be brought up in class for discussion to help the students further understand the topic. Besides materials on SAS triangle congruence, the site also has materials on other cases of triangle congruence. Hence, ck12.org can be used as an aid for students to prepare for the lesson, and/or review on the materials of the lesson.

https://www.ck12.org/geometry/sas-triangle-congruence/

How could you as a teacher create an activity or project that involves your topic?

A three-part activity:

Part 1: At the beginning of the class, I will give the students some cut-out triangles and ask them to find the congruent pair. During this part, the students can easily find the pair by putting the triangles on top of each other to compare the shape and sizes. This is to introduce the students to congruence triangles.

Part 2: The next part will be after I introduce proving triangle congruence by SAS. I will give the students a guide sheet with congruence triangle pairs placed at random places, with side lengths and angles provided. Just like at the beginning, the students must match up the pairs. However, since this time the students cannot move the triangles around, they must utilize the clues provided to them, which are the side lengths and angles, to get the correct answers. Example: Match the congruent pairs by SAS.

Part 3: The last part will be before the end of the lesson. The students will be given a figure and asked to prove the congruent triangles using SAS. However, one of the components necessary for SAS is missing and the students will need to use other provided information to solve the problem. Example:

Given a quadrilateral ABCD as shown below with AB = AD, and AC is the angle bisector of angle BAD. Prove that triangle ABC is congruent to triangle ADC.

Reference:

CK-12 Foundation*. CK-12 Foundation*, CK-12 Foundation, www.ck12.org/geometry/sas-triangle-congruence/.

*Posted by John Quintanilla on July 12, 2019*

https://meangreenmath.com/2019/07/12/engaging-students-proving-that-two-triangles-are-congruent-using-sas/

*engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

This student submission comes from my former student Michael Garcia. His topic, from Geometry: defining the terms *complementary *angles, *supplementary *angles, and *vertical *angles.

**What interesting (i.e., uncontrived) word problems using this topic can your students do now? **

Using complementary, supplementary, and vertical angles, students can do simple angle problems. For example, give them a picture of a slice of pizza (or actual pizza if youâ€™re truly nice). You can then make up questions regarding the pizza. For example, â€ś*Sally and John are going to split half a pizza. After they cut the pizza in two, John goes to wash his hands. Meanwhile, Sally slices herself a pretty generous slice. In fact, her pizza was cut at an angle of 130Ëš. After John realized he was bamboozled, he sadly settled for his piece. What was the angle of Johnâ€™s one pizza slice?â€ť*

When you are working with a pizza, you can modify the scenario/question to fit complementary and vertical angles as well. For this question, the students could draw on a separate pizza pie the 130Ëš by using a protractor. They will hopefully see that these are supplementary angles and subtract 130Ëš from 180Ëš.

**What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?**

If your name is in the title of a subject, activity, or anything else, you more than likely had a tremendous impact on that thing. Euclid of Alexandria was a mathematician who is sometimes known as the â€śfather of Geometry.â€ť Not much information is known about Euclid, but his book *Elements *stands as the foundation of Euclidean Geometry. It is comprised of 13 books based off the work of his predecessors, but that is not to diminish Euclidâ€™s work. He redefined geometry, introduced new concepts such as the Fundamental Theorem of Arithmetic, the intersection of planes and lines in three-dimensional figure, and more. In Book 1 Proposition 13, we see the concept of supplementary and complementary angles. In Book 1 Proposition 15, vertical angles are introduced in this section. Euclid was definitely one of the shoulders of giants upon who Newton, Kepler, and Descartes stood on.

**How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?**

When I took Geometry in high school, I was a huge WWE fan. I thought Shawn Michaels â€śThe Heartbreak Kidâ€ť was the best wrestler on the planet. For his finisher move, he would kick his opponent in the chin (it was very effective), and it was appropriately named â€śSweet Chin Music.â€ť As I grew older, I began to see how Geometry can fit into wrestling.

Below is an image of The Undertaker vs. Shawn Michaels at WrestleMania XXVI. As you look at the dimensions of the ring, notice that there are 4 right angles. If you were to take the consecutive angles of this ring, you would have a pair of angles that are supplementary.

We also have complementary angles. At the beginning of the match, each actor (I mean wrestler) goes to their corner. When the bell rings, they obviously start wrestling. In this match, The Undertaker sprints out of his corner towards Shawn Michaels (see image below). If we were to take his direction and put a ray on top of it, we know have complementary angles. Thanks to the dimension of the ring, we can model supplementary and complementary angles.

Resources:

https://www.youtube.com/watch?v=QaE58Kp806U&t=427s

http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

https://www.britannica.com/biography/Euclid-Greek-mathematician

http://www.storyofmathematics.com/hellenistic_euclid.html

*Posted by John Quintanilla on July 8, 2019*

https://meangreenmath.com/2019/07/08/engaging-students-defining-the-terms-complementary-angles-supplementary-angles-and-vertical-angles-2/

*engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

This student submission comes from my former student Julie Thompson. Her topic, from Geometry: deriving the Pythagorean theorem.

How could you as a teacher create an activity or project that involves your topic?

I believe the best way to convince students that a certain theorem is true is to model it visually. Luckily, the Pythagorean Theorem has several ways to derive it and show that it works. My favorite is showing it with squares. You ask students to consider the numbers 3, 4, and 5. Given paper, ask them to create three squares with each of those dimensions. Then, see if they can form a right triangle out of the three squares they made. Next, ask them if they can find a way to make two squares fit exactly into another square (cutting the squares up if necessary). Hopefully, they will get the squares with dimensions 3 and 4 to fit into the biggest square. Finally, ask them to write a conjecture about what they find. It turns out that the two smaller squares fit perfectly into the bigger square, or, more mathematically, 3^{2}+4^{2}=5^{2}. Generally, a^{2}+b^{2}=c^{2}

I did the activity myself and it is pictured below:

How does this topic extend what your students should have learned in previous courses?

Students learn how to derive the Pythagorean Theorem in Geometry. However, they should have prior knowledge on square numbers, finding the area of a square, and simple algebraic equations. Students should also be able to solve equations and evaluate expressions when given values for the variables. The students will then be able to use all of this prior knowledge and apply it to one fantastic theorem: The Pythagorean Theorem. They can then use the theorem to find missing side lengths of a triangle. This extends their prior knowledge because they are now using their mathematical skills and applying it to the real world.

An example of this extension would be assigning this problem to my students: Think about your rectangular room at home. We want to estimate the length of the diagonal from corner to corner. Estimate that length to 3 decimal places. Then create a model to show why it is true, using the area of squares proof (from my A2 activity). The students are using their prior knowledge of square numbers, area of squares, and solving equations for this problem.

What interesting things can you say about the people who contributed to the discovery?

Pythagoras contributed greatly to the discovery of the Pythagorean Theorem (clearly it is named after him). â€ťIt is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician.â€ť We think of him as having been a very logical man, but he had some very weird, illogical beliefs as well. According to the article, â€śPythagoras imposed odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewelry, never passing an ass lying in the street, never eating or even touching black fava beans, etc.â€ť

The Pythagoreans (Pythagoras and his followers) discovered something very interesting about the number 10. Today, when we wonder why we use base 10, we attribute it to the simple fact that we have ten fingers and ten toes. Our ten fingers are what we use to count with. Pythagoras deemed 10 to be a very special number, but for a more abstract reason. You can form an equilateral triangle with rows of 4, 3, 2 and 1. Altogether this triangle contains 10 points. He called it the tetractys. And 10 was thought to be a very holy number. Of course, he is most known for this theorem. â€śit has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.â€ť

REFERENCES:

https://www.storyofmathematics.com/greek_pythagoras.html

*Posted by John Quintanilla on July 1, 2019*

https://meangreenmath.com/2019/07/01/engaging-students-deriving-the-pythagorean-theorem-8/

*engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

This student submission comes from my former student Christian Oropeza. His topic, from Geometry: finding the area of a square or rectangle.

How can this topic be used in your studentsâ€™ future courses in mathematics or science?

Finding the area of a square or rectangle comes up again later in Geometry when solving for the surface area of a prism. The reason behind this is because any n-sided prism, where n is the number of sides the base of the prism has, will have n many squares or rectangles. Therefore, in order to calculate the surface area of any prism, the use of finding the area of a square or rectangle is required (Reference 1). Another course that involves this topic is Calculus. When approximating the area under a curve, one strategy is to use left or right endpoint approximation which is just the sum of the areas of the rectangles under or over the curve (Reference 2). This topic is also used in physics when covering measurements. The idea of finding the area of a square or rectangle in the measurements section is to precisely and accurately find the area.

How has this topic appeared in the news?

Steiner Ranch is a hair studio that just recently added 1600 square feet, thus bringing their total to 3468 square feet. With the addition of more space the studio now holds: 19 stylist chairs, 8 shampoo bowls, 3 restrooms, and a color mixing room. All in all, this could not have been done without the use of finding the area of a square or rectangle because then the owner, Brian Charles, would not know how much of each studio equipment would be able to fit in a way that was fitting for him (Reference 3). In other news, state deputies of the Legislative Assembly of Rondonia decided to try creating 11 new protected area in the Brazilian Amazon, which amounted to a total of 2,316 square miles. Therefore, the use of the area of a square was used to determine how much area would go to the new protected areas. However, the bacanda ruralista agribusiness lobby opposed this decision and passed a bill that did not allow the process of making the protected areas (Reference 4).

How have different cultures throughout time used this topic in their society?

During 570-495 BC, the use of finding the area of a square impacted math in Greek culture. More specifically, a man by the name of Pythagoras created what is known now to be the Pythagorean Theorem. He discovered this theorem by noticing that the area of the square created by the hypotenuse of a right triangle is equal to the sum of the area of the squares created by the other two sides of the same right triangle (Reference 5). Also, there were different cultures who had discovered the same formula as the Pythagorean Theorem, but were not the first to publish their findings. These different cultures include: Mesopotamian, Indian, and Chinese (Reference 6). Finding the area of a square or a rectangle comes up immensely in computing the cost for installation of hardwood floors. The cost is computed by charging the customer for the price of each square foot of wood used and the labor for each square foot of wood installed (Reference 7).

References:

- https://www.varsitytutors.com/hotmath/hotmath_help/topics/surface-area-of-a-prism
- https://www3.nd.edu/~apilking/Math10560/Calc1Lectures/24.%20Areas%20and%20Distances.pdf
- https://communityimpact.com/austin/lake-travis-westlake/business/2018/10/09/steiner-ranch-area-hair-studio-tacks-on-an-additional-1600-square-feet/
- https://news.mongabay.com/2018/10/brazil-scraps-11-new-amazon-protected-areas-covering-2316-square-miles/
- https://study.com/academy/lesson/development-of-geometry-in-different-cultures.html#lesson
- https://en.wikipedia.org/wiki/Pythagorean_theorem
- https://www.homeadvisor.com/cost/flooring/install-wood-flooring/

*Posted by John Quintanilla on June 28, 2019*

https://meangreenmath.com/2019/06/28/engaging-students-finding-the-area-of-a-square-or-rectangle-5/