3.1: Basics of Geometry
Points, Lines, and Planes
Connections to Art
Tangrams are a fun way to introduce Geometry to any student. If you have tangrams, let students play with them, make designs or animals. If you do not have access to tangrams, take them to the computer lab to do online tangrams: http://pbskids.org/cyberchase/games/area/tangram.html
Point of points of intersection, lines, triangles, quadrilaterals, and planes to students as they play. You could even take picture of students creations and put them up in your classroom.
(To purchase, see: http://www.amazon.com/Learning-Advantage-Tangrams-Classroom-Set/dp/B000XURP14/ref=sr_1_20?s=toys-and-games&ie=UTF8&qid=1305226538&sr=1-20)
Connections to Map Reading
A practical application of points, lines, and planes are maps. Show students a map of a state and see if they can determine what represents a city (point), a highway (line), and the state itself (plane).
Taking this a step further, ask students what points (cities) are collinear (on the same highway) and which are coplanar (all the cities). Have a discussion as to what sort of representation would be “space.” Students should see that it would be a globe.
Challenge
1. Three planes intersect in three different ways. Draw all three.
2. One line can divide a plane into two regions. Two lines can divide a plane into four regions. Three lines that intersect at a point can divide a plane into six regions, but you can get more regions if the lines do not all intersect at the same point. If you have six lines, what is the maximum number of regions into which you can divide a plane? BONUS: What if you used \begin{align*}n\end{align*} lines? What is the maximum number of regions into which you can divide the plane?
Answers
1. See the pictures below. Notice that we did not include options where all the planes were parallel (no intersection) or when one plane intersected the other two (two planes parallel).
2. For six lines, the maximum number of regions is 22. You can decide if you would like students to generate a pattern or draw pictures for 1-6 lines (the pattern is: 2, 4, 7, 11, 16, 22, 29, ...). For \begin{align*}n\end{align*} lines, there will be \begin{align*}\frac{1}{2} n^2 + \frac{1}{2} n+1\end{align*} regions.
Segments and Distance
Extension
Using the Know What? as a guide, have students measure their own “head” width and length. Then, have them find their height in heads. Students can also find: the length from the wrist to the elbow, the length from the top of the neck to the hip, and the width from shoulder to shoulder, and hip height. Some of these are given in the picture in the text, but have students verify the measurements for their own bodies. Taking it one step further, you could have students create a scale drawing of him/her based on the head measurements.
Connections to History
Describe the advantages of using the metric system to measure length over the English system. Use the examples of the two rulers (one in inches and one in centimeters) to aid in your description.
Research the origins of ancient measurement units such as the cubit. Research the origins of the units of measure we use today such as: foot, inch, mile, meter. Why are standard units important? Student answers should include that the cubit was the first recorded unit of measure and it was integral to the building of the Egyptian pyramids.
Research the facial proportions that da Vinci used to create his Vitruvian man. Write a summary of your findings. Student answers should comment on the “ideal” proportions found in the human face and how these correspond to our perception of beauty.
Challenge
1. Draw four points, \begin{align*}A, B, C,\end{align*} and \begin{align*}D\end{align*} such that \begin{align*}AB = BC = AC = AD = BD\end{align*}.
(HINT: \begin{align*}A, B, C\end{align*} and \begin{align*}D\end{align*} should NOT be collinear)
2. \begin{align*}\overline{AB} \cong \overline{BC}\end{align*} and \begin{align*}\overline{AC} \cong \overline{CD}\end{align*}. If \begin{align*}AD = 20\end{align*}, find the length of all the segments in the diagram.
Answers
1. Here is one possible answer.
2. \begin{align*}AC = CD = 10, AB = BC = 5, BD = 15\end{align*}
Angles and Measurement
Connections to Astronomy
Give students pictures of the Big Dipper and Orion (below). For homework, ask students to observer the night sky and try to find these two constellations. Have them make note of any other constellations. To see a full list of constellations, visit http://en.wikipedia.org/wiki/List_of_constellations. Ask the students to find 2-3 constellations on line or in a book and to bring the hard copy of a picture of the constellation to class.
The next day, in class, begin a whole discussion about the angles in the constellations and the distance between stars. Ancient astronomers used to measure the degrees between stars using their hands. Here are the approximations:
- Extend your little finger; its width is approximately 1 degree.
- Extend your three middle fingers; this is about 5 degrees.
- A clenched fist (thumb to little finger) is about 10 degrees.
- From the tip of the little finger to the tip of the thumb, an extended hand with fingers and thumb splayed is about 20 degrees.
Now, for homework, have students see if they can find the angle measures between the stars in each constellation they found the night before. Remind students that these hand measurements are approximations. They only work because the stars are so far away.
Challenge
- You are measuring \begin{align*}\angle ABC\end{align*} with a protractor. When you line up \begin{align*}\overrightarrow{BC}\end{align*} with the \begin{align*}10^\circ\end{align*} mark, \begin{align*}\overrightarrow{AB}\end{align*} lines up with the \begin{align*}90^\circ\end{align*} mark. Then you move the protractor so that \begin{align*}\overrightarrow{BC}\end{align*} lines up with the \begin{align*}25^\circ\end{align*} mark. What mark should \begin{align*}\overrightarrow{AB}\end{align*} line up with?
- Why do you think the degree measure of a straight line is 180, the degree measure of a right angle is 90, etc? In other words, answer the question, “Why is the straight line exactly \begin{align*}180^\circ\end{align*} and not some other number of degrees?”
Answers
- It should line up with \begin{align*}115^\circ\end{align*}.
- Answers will vary. Students should comment about the necessity to have a number of degrees in a line that is divisible by 30, 45, 60 and 90 degrees because these degree measures are prevalent in the study of geometrical figures. Basically, setting the measure of a straight line equal to \begin{align*}180^\circ\end{align*} allows us to have more whole number degree measures in common geometrical figures.
Midpoints and Bisectors
Connections to Construction
This is a picture of a wooden frame for a home. Notice that \begin{align*}\overleftrightarrow{BD}\end{align*} bisects the sides of the roof \begin{align*}(\angle ABC)\end{align*} and the other support beams (\begin{align*}\angle FDG\end{align*} and \begin{align*}\angle EDH\end{align*}). It is also perpendicular to \begin{align*}\overline{EH}\end{align*}. Ask students why it is important why \begin{align*}\overleftrightarrow{BD}\end{align*} bisects these angles. What would happen if \begin{align*}\overleftrightarrow{BD}\end{align*} did not perfectly bisect these angles? There are also several congruent segments. Ask students to find these. Then, as a classwork (individually or in pairs) assignment, have students answer the following questions.
1. If \begin{align*}m \angle ABC = 150^\circ\end{align*}, find \begin{align*}m \angle ABD\end{align*}.
2. If \begin{align*}m \angle BDF = 80^\circ\end{align*}, find \begin{align*}m \angle FDG\end{align*}.
3. What is \begin{align*}m \angle EFD\end{align*}?
4. List an example of an acute, obtuse, right and straight angle.
Answers
1. \begin{align*}m \angle ABD = 75^\circ\end{align*}
2. \begin{align*}m \angle FDG = 160^\circ\end{align*}
3. \begin{align*}m \angle EFD = 10^\circ\end{align*}
4. Acute angles: \begin{align*}\angle ABD, \angle DBC, \angle BDF, \angle BDG, \angle FDE, \angle GDH\end{align*}
Obtuse angles: \begin{align*}\angle ABC, \angle FDG, \angle EDG, \angle HDF\end{align*}
Right angles: \begin{align*}\angle BDE, \angle BDH\end{align*}
Straight angle: \begin{align*}\angle EDH\end{align*}
Challenge
1. Construct a \begin{align*}90^\circ\end{align*} angle using the constructions learned in this section.
2. Using #1, construct a \begin{align*}45^\circ\end{align*} angle.
3. Using #1, construct a square (all sides are equal and all angles are \begin{align*}90^\circ\end{align*}).
Answers
1. Construct a perpendicular bisector to create two \begin{align*}90^\circ\end{align*} angles.
2. Bisect one of the angles in #1 to create a \begin{align*}45^\circ\end{align*} angle.
3. The square is a little harder. Students will need to start with #1 and repeat this once more. Then, they can mark the distance between two created \begin{align*}90^\circ\end{align*} angles and repeat these on the created lines (the green and purple arc marks).
Angle Pairs
Connections to Map Reading
Use the image of a street map of Manhattan or Washington DC (www.mapquest.com). Print a copy of this map for students to work with during the activity. There are several different examples of complementary angles, supplementary angles, linear pairs, and vertical angles.Have the students work in pairs with a highlighter, colored pencils or markers and identify examples of each of the types of angles in the map. Ask the students to make a list of the intersections (by street name) on paper and how each angle fits the description. You could also enlarge the map and have students use a protractor to measure the angles. Students can either share the findings as a whole class, as a discussion, or in small groups. An extension could be to have students repeat this activity for the city they live in.
Extension
Ask students if they think three angles can form a linear pair. Can three angles be supplementary? Complementary? By the definition, only two angles can form a linear pair, be supplementary or complementary. However, explain to students that this does not mean that three angles cannot add up to \begin{align*}180^\circ\end{align*} (such as in a triangle) or \begin{align*}90^\circ\end{align*}.
Challenge
1. Find the value of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. You may need to factor or use the square root.
2. What is a congruent linear pair?
Answers
1. \begin{align*}x = 11^\circ, y = 7^\circ\end{align*}
2. A congruent linear pair would be two \begin{align*}90^\circ\end{align*} angles. Congruent linear pairs are created by perpendicular lines.
Classifying Polygons
Connections to Art
To prepare, you will need an assortment of one or more of the following items: gumdrops, marshmallows, toothpicks, tinkertoys, or kynex. Be sure that the students understand the different types of triangles and have an example of each type before beginning. Then have them create an example of each type triangles using the materials provided. The gumdrops or marshmallows would be the vertices and the toothpicks would be the sides, for example. Let students play for a while. You could also extend this activity to polygons (squares and other regular polygons are pretty easy to create). Finally, you could have students create geometric designs using the materials.
Connections to Architecture
Either in class or as a homework assignment, have students search for “polygon architecture” on the internet. Tell students to click “images” on the search engine and have them print out 2-3 images to share with the class. Encourage students to print out pictures of buildings, tile designs, interiors, windows, or playground equipment.
As a class, brainstorm the different types of polygons used in architecture (write these on the board and how they are used) and why certain polygons are used over others. As a homework assignment, students can go home and write down all the polygons they find their home.
Extension
You can introduce students to the concept of an equilateral polygon and an equiangular polygon. Only in the case of a triangle are equilateral and equiangular the same shape. Also, equilateral polygons can be concave (see hexagon below), but equiangular polygons cannot. After drawing examples of equilateral and equiangular quadrilaterals, pentagons, and hexagons, show students regular polygons (when sides and angles are congruent). Encourage to guess the definitions of these new terms and ask for volunteers to come up to draw different polygons. Here are a few examples.
Challenge
- Can an equiangular polygon be concave? Why or why not?
- Find the pattern for the number of diagonals drawn from one vertex. Determine how many diagonals can be drawn from one vertex of an \begin{align*}n\end{align*}-gon.
- Find the pattern for the total number of diagonals in a polygon. Determine how many diagonals are in an \begin{align*}n\end{align*}-gon.
Answers
- An equiangular polygon cannot be concave because all the angles must be the same measure. In a concave polygon, one angle will be larger than \begin{align*}180^\circ\end{align*}.
- The number of diagonals at one vertex increases by one. For an \begin{align*}n\end{align*}-gon, the equation would be \begin{align*}n - 3\end{align*}.
- For the total number of diagonals, the pattern is: 0, 2, 5, 9, 14, 20, 27, ... Here, you add one more than what was previously added. If there are \begin{align*}n\end{align*} vertices and \begin{align*}n - 3\end{align*} diagonals from each vertex, then start with \begin{align*}n(n - 3)\end{align*} as the equation for the total number of diagonals. However, if you plug in \begin{align*}n = 5\end{align*} (or any other number), the answer would be double the correct answer (this is because \begin{align*}n(n-3)\end{align*} counts all of the diagonals twice). Therefore, you need to divide by two. The equation is \begin{align*}\frac{n(n-3)}{2}\end{align*}.