This class may be different from many of the mathematics classes you have taken in the past. You will be encouraged to explore geometrical ideas, solve puzzles, make conjectures, discuss with classmates, make connections between ideas, explain your solutions, and justify your reasoning. You should not expect to be shown rigid methods for solving problems. In fact, there often is not be one single right way to complete problems but rather many acceptable ways of coming to a solution. Your teacher will encourage you to explore your ideas and share alternative approaches with your class.
Perhaps this course would be easier for you if you were simply told how to solve the problems. However, the objective of this course is not to learn facts and methods, but rather to develop your mathematical thinking and communication skills. In this age of technology, computers can do many of the algorithms and processes that were at one time the core of mathematics education. However, humans still have a role in recognizing when mathematics can be applied to real-world situations and in developing creative strategies to solve problems. Developing these reasoning skills will help you more in your future than simply being able to quote a theorem or follow rules. On the other hand, it is still useful to know important geometrical theorems like the Pythagorean Theorem. You should be able to recognize when it can (and cannot) be used to solve both elementary and complex problems. This theorem along with some other basic formulas you have learned in the past (and can look up online) will form a toolbox for our work, but we will give more emphasis to the creative process that is mathematics.
In A Mathematician’s Lament, Paul Lockhart describes a nightmare in which music was taught in the same way that mathematics is currently taught. He says,
Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive.
If we were to teach music via lecture and mere recall, focusing on the notation and structure, but not allowing students to experience and create music, the subject would be dry and uninteresting, the creative process would be stifled, and very few would have an interest in learning music. Yet, that is what we have done with much of our mathematics instruction. We have removed both the exploration and the fun. [1.5.1]
So I encourage you, my readers, to embrace the experience that is mathematics. Approach each puzzle with an open mind, make guesses as to what is happening, and experiment to determine whether your ideas are correct. Play with the materials and the concepts and ask new questions. In this course, you might even be invited to explore a question you asked instead of the one the teacher originally posed! Mathematics is not just about finding the answer; even when there is an answer, our work should evoke new questions: Why? Under what circumstances? What else?
Exploration1.1.1.Tiling Rectangles with Polyominoes.
This task simply asks you to fill a box, but you are encouraged to think outside the box as you do this activity. Take chances and try out different strategies as the challenge increases. Take time to reflect as you respond to the "why" questions and be prepared to share your ideas with your classmates. Have fun!
To use the interactive tool, click and drag the L-shape to create a new copy of the shape. Use the large dot, to rotate the copy. Arrange copies to cover the blue square. This activity may also be done on paper using a \(4 \times 4\) grid and an L-tile formed from four unit squares.
Pictured below are the four other tetrominoes: an I-shape, a Z-shape, a T-shape, and a small square. Like the L-shape, some of these can on their own cover the larger square without overlaps or gaps. Identify which tetrominoes will tile one of the larger squares and show how this can be done. If there are any tetrominoes which cannot tile the square alone, explain why you believe the tiling is impossible.
Once again in the application, copies of the I-shape, Z-shape, T-shape, and small square can be created by clicking the shape and dragging. Four larger squares are provided; try to tile one square grid with the I-shape, one with the Z-shape, etc.
Similarly, we define a pentomino to be a figure formed by joining five congruent squares along edges. One pentomino is pictured. Create new pentominoes by copying and moving the provided purple unit square. Note that if two pentominoes are reflections or rotations of each other, we consider them to be the same pentomino. Be sure to save a copy of your sketch for future reference.
If you do not have access to a set of pentominoes, you may use the interactive Polypad activity below or access a fullscreen version at Polypad by Amplify using the link Pentomino Set.
Exploration1.1.2.More Explorations with Pentominoes.
(a)
To simplify our discussion, we can call each pentomino by the letter it most resembles, namely F, I, L, P, T, U, V, W, X, y, Z, and N. The F and N pentominos are less obvious and are identified below. Sketch and identify the others or refer to and label your earlier sketches.
Choose another pentomino from among F, I, P, T, U, W, y, and Z, and demonstrate how it can be created using four distinct pentominoes. Note that this cannot be done with X or V.
It is possible to create a copy of the V tile that is three times the height and width of the basic V tile by filling it with nine distinct pentominoes.
What happens to the area of a figure when its height and width are both doubled? tripled? How does the number of basic pentominoes needed to fill the figure change?
The purpose of Exploration 1.1.1 and Exploration 1.1.2 is two-fold. They introduce the exploratory nature of the course and suggest a few important geometric facts and definitions that we will use as our initial geometrical foundation. These facts have been chosen because they are intuitively obvious, but they give us some building blocks we will need to justify more complex ideas. We state them here for clarity and future reference.
We noted that each tetromino was made up of four unit squares and each pentomino consisted of five unit squares; thus the area of a tetromino is four square units and the area of a pentomino is five square units. We also applied the area formula for a rectangle.
Another observation is that if you can completely fill a closed region with smaller non-overlapping shapes, then the area of the region is equal to the sum of the areas of the smaller shapes. While this may seem like a simple idea, we will see that it allows us to determine the area formulas of other geometric shapes and to prove the Pythagorean Theorem. We can dissect a shape into smaller shapes, rearrange the pieces into another shape, and assert that the two regions have equal area because they both consist of the same collection of smaller shapes.
We applied these two principles in Task 1.1.1.e where we attempted to construct rectangles using pentominoes. Since pentominoes have an area of five square units, we could only construct rectangles with areas divisible by 5, namely 5, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 60. 2
To construct a rectangle of area 10, the same pentomino must be used twice.
In the same way, any shape filled entirely with non-overlapping tetrominoes must have an area divisible by 4.
Finally, we observed that when the lengths of all sides of a two-dimensional figure are doubled (or tripled), the area of the figure is multiplied by 4 (or 9). This is easy to observe in a rectangle with length \(\ell\) and width \(w\text{.}\) When we create a new rectangle with length \(L=2\ell\) and width \(W=2w\text{,}\) the resulting area is \(LW=(2\ell)(2w)=(2\cdot 2)(\ell w)=4\ell w\) square units.
Hopefully as you worked through these tasks, you allowed yourself to experiment with different arrangements of squares and polyominoes. This openness to creative exploration is essential to mathematical thinking. Phrases like "what if" or "I wonder" arise often as mathematicians create and make sense of new ideas. If you find yourself wanting to explore a question arising from a task but not explicitly asked, jot down your question and share it with your instructor. Your instructor might encourage you to investigate your question or even invite your classmates to consider it. 3
Some questions may be addressed in a later section, too tangential, beyond the scope of this course, or even unsolved by the mathematical community. Thus your instructor will make the decision on which questions to pursue and when.
You may have noticed that you and your peers discovered different ways of completing the same task. Multiple approaches and multiple solutions are often welcome and even encouraged! If you have a strategy that is different than your group members, try both separately and then compare. You have the ability to do the mathematics, and it will be more meaningful if you attack tasks in a way that makes sense to you. This will require you and your peers to take some risks, to challenge each other to speak up, and to be nonjudgmental of others.
In fact, doing mathematics is a social activity. When we notice patterns and make conjectures, we share our ideas with others and use logic to make convincing arguments. For over 2500 years, mathematicians have been adding to the body of mathematical knowledge; some of which has been shared with you during your studies. The field of mathematics continues to grow.
Unfortunately, most students have had little opportunity to creatively explore in their mathematics classes. You may have memorized formulas and performed calculations in many of your mathematics classes. While formulas and calculations are useful tools in mathematics, they are not the essence of mathematics and can often be performed via a computer. In this course, we will develop the creative thinking, reasoning, and problem-solving practices that set humans apart from computers and are valued by employers. We will think outside the box, make predictions, and wonder why. Different approaches and strategies are encouraged. Relax and have fun exploring mathematics!
Throughout the course, you will learn important principles, definitions, formulas, and geometry facts. Create your toolbox so that these tools will be easy to find and meaningful to you when you need to refer to them or review material. Gathering your tools in one section of your notebook would be ideal.
In your own words and illustration(s), describe why the definition of area leads to the formula for the area of a rectangle at least in the case where the length and width are whole numbers.
A hexomino is constructed from six \(1 \times 1\) squares. Explain why it is impossible to construct a solid rectangle with area twenty using non-overlapping hexomino tiles.
Exercises in this section are designed to check that you have understood the main ideas and that you can perform the skills, techniques, and formulas needed to move on in the course. Exercises in this group may also encourage practice of skills that appear in education licensure exams.
Break each figure into shapes for which the area can be computed. Then use Principle 1.1.5 to find the total area of the region. Or think of the figure as residing in a large measurable frame and subtract as appropriate.
Find the area of rectangle \(ABCD\) with height 14 cm. and width 6 cm given in Figure 1.1.8. Then use your answer to find the area of right triangle \(\Delta ACD\text{.}\)
In Figure 1.1.8, parallelogram \(EFGH\) has base 10 cm and height 6 cm, points \(M\) and \(N\) lie on \(\overleftrightarrow{EF}\) with \(\angle{HMF}\) and \(\angle{GNF}\) being right angles.
In Figure 1.1.8, trapezoid \(ILKJ\) has bases \(\overline{JK}\) and \(\overline{IL}\) of lengths 16 cm and 10 cm, respectively. The trapezoid also has right angles at \(L\) and \(K\text{.}\) An altitude is drawn from \(I\) meeting \(\overline{JK}\) at a point \(O\) such that \(m(\overline{JO})=6\) cm and \(m(\overline{IL})=10\) cm.
A triomino consists of three \(1\times 1\) squares joined at edges. How many distinct triominoes exist? Sketch them. Do not include any triominoes that are reflections or rotations of ones you have already sketched.
We do not know how many heptominoes Jay used, but we do know the area of each heptomino. Many rectangles could be built, but their areas will always have what factor?
Show that if you are allowed to use copies of the same pentomino tile multiple times you can construct a rectangle of area \(5n\) for every whole number \(n\text{.}\)
This problem illustrates the Area Model for Multiplication, an effective way of helping elementary school students visualize the operation of multiplication.
Experiment with a set of tetrominoes to sketch all the different rectangles that can be covered with no gaps or overlaps. You don’t have to use all of the tetrominoes in each rectangle and you may use a new set of tetrominoes for each rectangle; however no rectangle will use a single tile more than once.
In Exercise 1.1.5.9, you showed how to use an area model to represent multiplication of numbers. The role of place value in the multiplication of two-digit numbers can also be represented using the area model: for example, to multiply 47 by 25 we can represent 47 as 4 tens and 7 ones and 25 as 2 tens and 5 ones. We then can sketch the product as shown in Figure 1.1.9.
Determine the area of each of the four subrectangles pictured in Figure 1.1.9. Here, the squares on the grid are not \(1\times 1\) but defined by the notations on the sides.
The standard algorithm to multiply \(47 \times 25\) is shown in Table 1.1.10. Identify what the values 235 and 940 in the algorithm represent in the area model pictured at the beginning of this exercise.
Sketch an area model illustration of the product \(1235\times 57\text{,}\) not necessarily drawn to scale, but with each side length broken down into thousands, hundreds, tens, and ones as appropriate. Use your sketch together with words and computations to demonstrate how this area model can be used to compute the product.
Sketch line segments that divide the height of the square equally into thirds and the width equally into fourths. How many subrectangles does this split the square into? What is the area of each subrectangle? What does this tell you about \(\frac{1}{3} \times \frac{1}{4}\text{?}\)
On the unshaded grid, indicate a vertical side length of \(2 \frac{2}{3}\) and a horizontal side length of \(1 \frac{2}{5}\text{.}\) A curly bracket or highlighter might be used to make this visible.
Starting with a sufficiently large unit square, draw an area model of \(\frac{2}{5} \times \frac{3}{4}\text{.}\) Be sure to divide the unit length into fifths along one side and the unit width into fourths along the other. Shade and compute the product. Explain how you find the numerator and denominator of the unsimplified fraction from your sketch.
An algebraic expression involving the variable \(x\) may include numbers and operations like \(+ , -, \cdot, \div\) in addition to the variable \(x\text{.}\) For part a, the algebraic expression is \(7+x\text{.}\)
A tetrahedral die is a triangular pyramid with four sides numbered 1 through 4. Because the die is shaped like a triangular pyramid, there is no top face so the number rolled will correspond to the top point if the dice are labeled like the ones to the right. If three tetrahedral die are rolled the possible sums range from 3 to 12. For example, consider the sum 5 which can be achieved by each of the following six rolls: (1,1,3), (1,2,2), (1,3,1), (2,1,2), (2,2,1), (3,1,1).
The sketch in Figure 1.1.15 is a bar graph or histogram showing the frequency or number of ways that each of the sums from 3 to 12 can be rolled with three tetrahedral dice. Note that the sums are given on the horizontal axis and the frequencies are on the vertical axis. Note how the third vertical bar, centered above the number 5, has height 6 to illustrate the fact that there are six ways of rolling a sum of 5.
The probability of an event is computed by dividing the number of different rolls meeting the criteria for that event by the total number of different possible rolls. What is the probability of rolling a 7 with three tetrahedral dice?
Write an essay (2 to 4 pages) in which you discuss the experiences in your life that have formed you as a mathematician or geometer. This essay will be unique to you, but here are some questions that you might consider in your reflection: Were there any turning points in your education? What experiences have stimulated and which have stimied your mathematical growth? Are there obstacles that you have overcome or would like to overcome? How? What are your personal goals in this course? What do you see as personal challenges to being successful?
The paper should not be a mere timeline (boring!) or a list of answers to the questions above (disjointed), but instead should be a cohesive essay that focuses on the development of your mathematical ability, confidence, and attitude, centering on a few key turning points. Examples of experiences that help illustrate your statements should be included. How did the class format (lecture, testing, activities, interaction, pacing) change you? Did you have experiences beyond the classroom that affected your mathematical growth?
You may be asked to look back on this essay at the end of the semester to assess personal growth. Your instructor may also use this autobiography to better understand you as a student. In case your instructor knows your past teachers, use pseudonyms or letters to refer to them. The essay should focus on you, not your teachers.
