home » instructional materials » geometry » clarifying activities

REFINED Resources. The resources on this page have been updated and revised to align with the refined K-12 mathematics TEKS. These refined TEKS were adopted by the Texas State Board of Education in 2005–06 and implemented in 2006–07.

(G.1) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system.

(G.1.A) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems.

Clarifying Activity

Students discuss non-Euclidean geometries to illustrate the importance of precise definitions and careful application of Euclidean postulates. Taxicab geometry is used as an illustration below. Other possibilities include spherical geometry, hyperbolic geometry (which can be investigated nicely using technology) or even finite geometries that students define themselves.

Suppose we define a geometry that lies on a square grid. We might think of a city with streets that lie on a square grid and the only paths available are the paths a taxicab might take, as shown below.

In order to get from point A to point B in the diagram above, a taxicab might take the path drawn. Note that this is not the only way to get from point A to point B, but we cannot go along the diagonal.

Part 1:

Given the information above about taxicab geometry, decide in your groups if the following objects exist. If the object exists, write a definition for the object. Note: Each group's definitions and responses may be different; you are creating your own geometric system. Be prepared to give the reasoning behind your responses.

Objects to Define: point, line, line segment, ray, triangle, square, rectangle, polygon

Part 2:

The definition of distance between two points in this geometry differs from Euclidean geometry. Describe a formula for finding the taxicab distance between two points. When are the taxicab distance between two points and the Euclidean distance between two points the same?

Part 3:

A circle in Euclidean geometry is defined as the set of all points in the plane equidistant from a center point. Plot the point (0, 0) on a taxicab grid. Then plot the set of points that are a distance of 4 from (0, 0). The resulting figure is a taxicab circle with radius 4.

Additional Clarifying Activity

Students apply logical reasoning through arguments such as the one below:

In ΔABC below, prove the theorem that if angle A has a measure greater than that of angle B, then BC > AC.

Proof: If m angle A > m angle B, then, by the definition of "greater than" for angles, there exists a point D on segment BC between B and C such that m angle BAD = m angle B. By the converse of the Isosceles Triangle Theorem (congruent base angles imply an isosceles triangle), D BAD is isosceles with AD = BD. By the Triangle Inequality Theorem, we know that the sum of the lengths of two sides of any triangle is always greater than the length of the third side, so DC + AD > AC. Since AD = BD, the transitive property of equality allows us to say DC + BD > AC. Since D is on segment BC between B and C, DC + BD = BC, and BC > AC.

(G.1.B) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes.

Clarifying Activity

Students verify how Eratosthenes estimated the earth's circumference:

(Information below is derived from D. Burton, The History of Mathematics, An Introduction, pp. 201–207. See Burton for more in-depth information.)

Eratosthenes (276–194 BC) was born in Cyrena, a Greek colony under Ptolemaic domination. He spent most of his time in Alexandria as the chief librarian of the museum, and his titles included geographer, historian, scientist, mathematician, astronomer, poet, and literary critic.

Eratosthenes devised a simple plan to find the earth's circumference. Travelers had commented on the strange fact that in Syene, at the summer solstice, the sun cast no shadow at noon in a well or from an upright stick. This meant that Syene was directly on the Tropic of Cancer. Alexandria and Syene were then thought to be on the same meridian and the distance between them had been measured as 5000 stadia by a trained walker (surveyor).

Since the sun was so far away, Eratosthenes reasoned that the sun's rays could be considered parallel, so that at noon of the summer solstice, a line through the well in Syene would continue through the center of the earth. At the same time in Alexandria, Eratosthenes used a modified sundial and water to determine the sun's angular position from zenith (7.2ƒ). He imagined a line through the vertical pointer of the sundial that would also go through the center of the earth, forming alternate interior angles; thus the angle ÿ at the center of the earth with rays through the sundial pointer at Alexandria and through the well at Syene was also 7.2ƒ.

Eratosthenes inferred that

circumference = 250,000 stadia; 1 mile = 10 stadia (approximately)

circumference approximately = 25,000 miles [Closer estimate of the circumference: 24,907 miles]

Additional Clarifying Activity

Students investigate a historical problem which led to the development of useful geometry such as the problem of the Tunnel of Samos. (A video and teacher guidebook are available from Project MATHEMATICS!)

(G.1.C) Geometric structure. The student understands the structure of, and relationships within, an axiomatic system. The student is expected to compare and contrast the structures and implication of Euclidean and non-Euclidean geometries.

Clarifying Activity

Students explore a non-Euclidean geometry and explain the differences between it and Euclidean geometry. Spherical geometry is used as an example below.

Spherical geometry consists of all the points on the surface of a sphere. Lines in spherical geometry are defined to be great circles of the sphere (that is, circles whose centers are the center of the sphere and whose points lie on the surface of the sphere). Although we spend a great deal of time learning about geometry on a flat surface (Euclidean geometry), the world we live in is actually spherical. Using a globe, we can illustrate the definition of a spherical line. The equator and other longitudinal lines are examples of great circles on a sphere, or spherical lines. Note that latitudinal lines are not actual lines on a sphere because their centers are not the center of the sphere. You can use string to help find a great circle of a sphere by pulling it tight, like a belt, around the sphere. If the circle is not a great circle the string will slip off the sphere. Cartographers create maps to help us see our three dimensional, spherical Earth on a flat, two dimensional paper. There are several different techniques used to make maps, some of which distort size. To see spherical lines (or great circles) on a flat surface, you might look at an airline flight path. On the flat map, the paths the planes fly do not seem to represent the shortest distance between two points; however if these were shown on a globe, we could use a string to show the path indeed lies on a great circle, the shortest distance between two points on a sphere. (As an example, find the great circle on a globe that passes between Dallas, TX and Paris, France. Make a note of the other cities and geographic landmarks you would fly over following that great circle path. Now follow the path on a flat world map. The path you draw on the flat map will appear very curved.)

Use a spherical model, such as a ball or balloon to help you answer the questions below.

Discussion questions:

• One of our postulates in Euclidean geometry states that given a point not on a line, there exists exactly one line through that point parallel to the given line. Does this postulate hold true in spherical geometry? (Can you draw two parallel lines on a sphere? Remember the definition of a line on a sphere.)
• Draw three intersecting lines, or great circles, to make a spherical triangle. Use a protractor to measure and label the angles of the triangle drawn. What is the sum of the angles of your triangle?
• Compare your triangle sum from Question 2 with other groups in the class. What conjecture can you make about the sum of the angles of a spherical triangle?
• Can you draw a triangle with 3 right angles in spherical geometry? In Euclidean geometry?

- Top -

(G.2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures.

(G.2.A) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships.

Clarifying Activity

Students make a construction based on circles and use it to classify triangles as follows:

Segment AB is the diameter of the semicircle and the radius of the quarter circle. Segment DC is the perpendicular bisector of segment AB.

Segment AB will be the longest side of a triangle with the third vertex lying in the region bounded by segments AC, CD, and arc AD, seen as the area in bold below.

Some examples of possible triangles are shown below.

Students make conjectures about what types of triangles are formed based upon the region where the third vertex is located. (For example, the different regions can be listed and conjectures formed for each region as shown below.)

* The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

Extension

Ask students to verify their conjectures by geometric reasoning.

(G.2.B) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures. The student is expected to make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

Clarifying Activity

Using construction technology, students explore the properties of various quadrilaterals. For example, students are given a parallelogram and use the measure function of the technology to find congruent angles and segments. Then the students drag a vertex of the parallelogram, making sure the new quadrilaterals formed are always parallelograms, and see if the congruences hold true for several different parallelograms. Students then make conjectures about the properties of a parallelogram and verify them using logical reasoning to show that the properties hold for every parallelogram.

Additional Clarifying Activity

Students explore the angles formed by a transversal passing through a pair of parallel lines with an activity like the following.

Using construction technology, or using the lines on lined paper as a guide, draw a pair of parallel lines with a transversal passing through the lines (and not perpendicular to the lines). Then draw a pair of non-parallel lines with a transversal passing through the lines. Notice that 8 angles are formed by the three lines, in each case, as shown in the diagram below.

Measure each of the angles formed in each of your diagrams, and answer the following questions:

• Do certain pairs of angles within each diagram have the same measure? Compare your results with the rest of the class. If you have access to geometry exploration technology, drag the transversal to create several different scenarios and see if you get the same results.
• We could think about line b as a translation of line a. Then, angle 6 would be the image of angle 2 under the translation and therefore be congruent. What other angles correspond by a translation?
• Are there other transformations that can be used to verify pairs of congruent angles in the diagram?
• What happens if the transversal is perpendicular to one of the parallel lines?

- Top -

(G.3) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements.

(G.3.A) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to determine the validity of a conditional statement, its converse, inverse, and contrapositive.

Clarifying Activity

Students investigate the validity of the converses of conjectures they make throughout the course. For example, after exploring the conjecture that the median of an isosceles triangle is perpendicular to its base (in the Clarifying Activity in d.2.B), students write the converse of the conjecture (the segment perpendicular to the base of an isosceles triangle through the opposite vertex is a median of the triangle) and investigate its validity.

Additional Clarifying Activity

Students find conditional statements in magazine or newspaper articles and discuss whether they are true or false. Each student then presents the converse of his or her conditional statement to the class and explains why the converse is true or false.

(G.3.B) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to construct and justify statements about geometric figures and their properties.

Clarifying Activity

Students draw three or four different convex polygons and extend the sides of each polygon, creating one exterior angle at each vertex. Using a sheet of patty paper, students consecutively trace the exterior angles so that the first is adjacent to the second, the second is adjacent to the third, and so forth. From the results, students write a conjecture about the sum of the exterior angles of a convex polygon. (For example, in a polygon, the set of exterior angles that "face clockwise' has a sum of 360 degrees.) Students realize that this conjecture can be proven if one knows the sum of the interior angles of the polygon.

Students can repeat the procedure described above, this time tracing the interior angles of a polygon. Students will notice that the sum of the interior angles is always a multiple of 180ƒ, but they must keep track of the number of rotations made in their tracing. Ask students to begin with a triangle, then a quadrilateral, a pentagon, a hexagon, and so forth. By collecting and organizing this information, students should see a pattern emerge and can write a generalization for the sum of the interior angles of an n-sided polygon. (Note: Construction technology can be used in the explorations described.)

Since this generalization is based on a limited number of examples, students then use definitions and properties to verify the generalization for all convex polygons (see the Clarifying Activity in G.5.A). Students then identify the sum of the linear pairs of interior and "clockwise" exterior angles, n(180), where n is the number of vertices in the polygon, and use the sum of the interior angles of the polygon, (n - 2) (180), to prove that the sum of the "clockwise" exterior angles is n(180) - (n - 2) (180) = 360.

(G.3.C) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use logical reasoning to prove statements are true and find counter examples to disprove statements that are false.

Clarifying Activity

Students understand the importance of proof by investigating the dangers of looking at a pattern formed by only a subset of examples, as illustrated in the problem below.

How many non-overlapping regions can be formed by the segments identified by n points on a circle? (For example, on the circle below, three points are marked and connected. Four nonoverlapping regions are formed.)

To begin the induction process, first draw a circle and mark one point on the circle, and count the regions formed. Then draw a circle for each case below, mark points on the circle, and connect each pair of points. Count the regions formed and record them in the table. Continue the table for up to 5 points, counting the maximum number of regions formed.

What do you predict for 6 points? Check your prediction.

(From the pattern, the prediction is 32. Actually, only 31 regions can be formed.)

(G.3.D) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use inductive reasoning to formulate a conjecture.

Clarifying Activity

Using a compass and straight-edge or construction technology, each student draws a triangle and constructs the segment joining the midpoints of any two sides of the triangle. Each student measures the distance or length of the segment he or she constructed, the angles it forms with the sides of the triangle, and the sides of the original triangle. Students compare their results with the rest of the class and, from the entire set of examples, construct statements about the segment and its relationship to the third side of the triangle (e.g. the newly constructed segment is one-half the length of the third side of the original triangle).

Next, each student draws a trapezoid and constructs the segment joining the midpoints of the two legs, a segment known as the median of the trapezoid. Students measure the distance or length of the median and the angles it forms with the legs of the trapezoid. Students compare their results with the rest of the class and, from the entire set of examples, construct statements about the median and its relationship to the bases of the trapezoid (e.g. the length of the median is the average of the lengths of the two bases).

Note: A triangle can be thought of as a trapezoid with one base of length zero. Then the two conclusions made are the same. In other words, the first conclusion regarding the triangle is a special case of the second conclusion regarding the trapezoid.

(G.3.E) Geometric structure. The student applies logical reasoning to justify and prove mathematical statements. The student is expected to use deductive reasoning to prove a statement.

Clarifying Activity

Students prove a statement like the one below using a variety of methods.

Prove that the base angles of an isosceles triangle are congruent.

Flowchart proof:

Transformational proof:

Construct an isosceles triangle by reflecting a line segment (segment AB) across a line of reflection (line BD) using a MIRA, or other reflective device. Then segment BC is the reflection of segment AB. So, ΔABC is isosceles, since reflection is a congruence mapping. Since A is the pre-image of C and points B and D are on the line of reflection, we can say that ΔCBD is a reflection of ΔABD. Thus, the two triangles are congruent, making angle A approximately equal to angle C by the definition of congruent triangles.

Two-column proof:

Given: ΔABC is isosceles

Prove: angle A approximately equal to angle C

- Top -

(G.4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems.

(G.4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

Clarifying Activity

Students roll up two sheets of notebook paper, one length-wise and one width-wise to form two different-shaped cylinders and fill each cylinder with beans to determine which cylinder has the greatest volume (concrete representation). Students then give a written explanation of the difference in the volumes, including a sketch of the cylinders and the calculations used.

- Top -

(G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems.

(G.5.A) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to develop algebraic expressions representing geometric properties.

Clarifying Activity

Given the theorem that the sum of the angle measures of a triangle is 180°, students draw a conclusion about the angle measures of a four-sided polygon. Students repeat the process for other polygons. They induce a general formula for the sum of the angles of an n-sided convex polygon.

Sample procedure:

Students draw a convex four-sided polygon and draw one diagonal from one vertex, noticing that two triangles are formed. Since the angles of these two triangles combine to form the angles of the original quadrilateral, the sum of the angles of the quadrilateral is 180° + 180° = 360°.

Students can extend this activity to other polygons, making sure to draw the diagonals from only one vertex. Students should notice that the number of triangles formed is always two less than the number of sides (n - 2 for an n-sided polygon) and that the angle-measure sum for each triangle is 180°. Therefore, students can induce a general formula: the sum of the angles of an n-sided convex polygon = (n - 2)(180°).

(G.5.B) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.

Clarifying Activity

Students use equilateral triangle pattern blocks and cubes to make generalizations about the ratios of sides, areas and volumes in similar figures using an activity like the one below.

• Given an equilateral triangle, create a similar triangle so that the ratio of side lengths is 2:1. What is the ratio of areas of the two similar triangles? Now create a triangle similar to the original triangle so that the ratio of side lengths is 3:1. What is the ratio of the areas of these two similar triangles?

  Sample sketches:

  

• Use other pattern block shapes to investigate other similar polygons in the same manner as described in Question 1 and record your findings in the table below.

  

• Based on your investigations in Questions 1 and 2, make a generalization. If the ratio of sides of two similar polygons is n:1, what would the ratio of areas be?
• Given a cube, create a similar cube with ratio of edges 2:1. What is the ratio of volumes? Create a similar cube with ratio of edges 3:1. What is the ratio of volumes? If the edges of two cubes were in a ratio of n:1, what would the ratio of volumes be?

Possible extensions:

• Use construction technology to investigate the ratio of areas of similar non-regular polygons.
• Explore relationships between the investigations in Question 1 and the Sierpinski triangle.

Additional Clarifying Activity

Students use pattern blocks to investigate tessellations of the plane. Students determine which regular polygons tile the plane (fit together around a point with no gaps or overlaps). [Note: Many pattern block sets only have regular polygons which do in fact tile the plane. Be sure to give your students cardboard cut-outs of regular pentagons, heptagons, nonagons and decagons if your pattern blocks don't include these shapes.] Once the students feel confident they have found all the possible tilings, they should justify why these shapes are, in fact, the only regular polygons which tile the plane.

(Students should see that only regular 3-, 4-, and 6-gons work. Once they analyze the interior angle measure of these polygons, they will see that 60ƒ, 90ƒ and 120ƒ are the only factors of 360 which can exist as angle measures in a regular polygon.)

Extension: Students investigate non-regular polygons that will tile the plane.

(G.5.C) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations.

Clarifying Activities

Based on the exploration of regular polygons that tile the plane in the Additional Clarifying Activity in G.5.B, students look for examples of regular tessellations in the world around them and discuss the strengths or benefits of various tessellations.

Some examples:

Under many bridges we find tessellations of triangles. Triangles are structurally rigid and are often used as supports.

Bee's honeycombs are tessellations of hexagons. Since these are used to store honey, the hexagon provides a large area while still having the efficiency of a regular tessellation.

Additional Clarifying Activity

Students investigate a miniature golf problem like the one below.

Below is a diagram of a miniature golf hole as seen from above. Find a path that will get the ball in one shot from the tee to the hole. Justify your answer. Where in the tee box should the ball be placed to create the shortest "hole-in-one" path?

Sample student solution:

First we make a reflection of the hole across the bottom side of the box. The shortest path will come from the very bottom of the tee box. We draw a line from the tee to the reflection of the hole. Then we reflect the part of the line outside the box back over the bottom side of the box. Since a reflection is a congruence mapping, we know part of the line outside the box is equal in length to its reflection. Therefore, the path from the tee to the hole is the shortest distance.

(G.5.D) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems. The student is expected to identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

Clarifying Activity

The Davis Bottling Company currently ships rectangular boxes each containing 48 identical bottles, tightly packed, as shown below.

The company is looking for ways to save money and asks their resident mathematician, Gina, if she can find a way to put more bottles into the box. She tells them that she has figured out a way to fit 50 bottles in the box by nesting the rows of bottles.

Explain how it is possible for Gina to rearrange the bottles so that the rectangular box will hold 50 bottles.

Use drawings and justify algebraically. Hint: Use patterns in 30-60-90 triangles to find the width of the nested arrangement with 9 columns of bottles.

Sample Solution

First, Gina removed one bottle from each of the columns 2, 4, 6, and 8 (these are darkly shaded in the picture below).

Then she nested the remaining five bottles from columns 2, 4, 6, and 8 between the adjacent columns (these are lightly shaded in the picture below).

Next, Gina took the four leftover (removed) bottles, added 2 more bottles, and created a nineth column of six bottles at the right. This yielded an arrangement of 50 bottles.

Consider that the original arrangement with eight columns of bottles has a width of 8(2r) = 16r, since each circle has a diameter of 2r.

If the radius of the circles is r, the triangle connecting the centers of three nested bottles is an equilateral triangle because it has three sides with length 2r. Using 30-60-90 triangles, the height of the equilateral triangle is r√3. Therefore, the distance between the centers of the circles in adjacent columns is also r√3. In the nested arrangement of 50 bottles, there are eight such distances between columns. Moreover, there is an additional distance of r + r = 2r at the two ends. Therefore the total width of the nine columns is 8(r√3) + 2r = r(8√3 + 2) ≈ 15.86r. This distance is less than the width of 16r required for the original 8-column arrangement of bottles. Therefore, the nested 9-column arrangement of 50 bottles will fit.

Also, 8 is the smallest number of columns for which this adjustment will work. For 7 or fewer columns, nesting won't create enough room for an extra column. This conclusion follows since r(2 + 7) ≈ 14.12r, which is more than the width of 14r for 7 non-nested columns.

- Top -

(G.6) Dimensionality and geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

(G.6.A) Dimensionality and geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to describe and draw the intersection of a given plane with various three-dimensional geometric figures.

Clarifying Activity

Groups of students work with various solids (hollow plastic models) and water to investigate cross-sections or slices of the solids. Each group fills a solid at least half full of water and tilts the solid to see what polygonal cross-sections can be formed.

What polygons can be formed by taking a cross-section of the following solids:

• cube? (triangle, square, pentagon, hexagon)
• right circular cone? (triangle, circle, ellipse, parabola, hyperbola)
• right circular cylinder? (rectangle, circle, ellipse)
• right square pyramid? (square, triangle, trapezoid)
• sphere? (circle)

Once each group has investigated each of the solids, students can discuss and analyze the results.

(G.6.B) Dimensionality and geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to use nets to represent and construct three-dimensional geometric figures.

Clarifying Activity

Groups of students bring cereal boxes to class. Each group draws at least two different nets for their box. (Students may choose to cut their box and re-tape it to find different nets.) Once each group has drawn their nets, a representative from each group demonstrates their nets on the board using a box with each face cut apart. As a class, all possible nets are found. Next, each group designs a box for an irregular solid (for example, a piece of fruit, a rock, a pine cone, or a boot).

Additional Clarifying Activity

Students discuss problems whose solutions involve the use of nets, such as the "Spider and Fly" problem below.

A hungry spider climbing on a wall of a cubical room sees a fly land on the opposite wall. What is the shortest path from the spider to the fly? (Use a model to help you solve the problem.)

(G.6.C) Dimensionality and geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems. The student is expected to use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

Clarifying Activity

Students investigate the different views of several three-dimensional models. A flashlight can be used to help students see the views. Pairs of students work together to describe solids by their views.

Using interlocking cubes, one student creates a solid, keeping it hidden from his or her partner. The student then describes the top, front, side and corner views to his or her partner. The partner then tries to recreate the solid and compares it to the original one.

Next, pairs of students design a solid that will pass snugly through a series of three holes: a square hole, a circular hole, and a hole shaped like an equilateral triangle.

Additional Clarifying Activity

Student use interlocking cubes to make a solid (e.g. a three-dimensional "L"). Students make the following representations for the solid:

• a net
• top, side and front views
• four different corner views

Students discuss what information each of the different representations gives that the others do not.

- Top -

(G.7) Dimensionality and geometry of location. The student understands that coordinate systems provide a convenient and efficient way of representing geometric figures and uses them accordingly.

(G.7.A) Dimensionality and geometry of location. The student understands that coordinate systems provide a convenient and efficient way of representing geometric figures and uses them accordingly. The student is expected to use one-and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures.

Clarifying Activity

Students investigate the general coordinates of various polygons through an activity like the one below:

• On a coordinate grid, draw a square, with side length 1, such that one vertex lies at the origin and one side lies on the x-axis. Label the coordinates of each vertex. On the same graph, draw a square with side length 2 in the same manner and label the coordinates of each vertex. Continue this process for squares with side lengths 3, 4, 5 and finally n.

  Sample response:

  

• On a new coordinate grid, repeat the above process for an isosceles triangle such that the base lies on the x-axis, one vertex lies on the origin, and the height remains constant at 4.
• On a new coordinate grid, repeat the above process for an equilateral triangle such that one side lies on the x-axis and one vertex lies on the origin.
• On a new coordinate grid, repeat the above process for a non-square rhombus such that one side lies on the x-axis and one vertex lies on the origin.

Additional Clarifying Activity

Students are given directions to a buried treasure. (For example, the treasure is located at the centroid of a triangle whose vertices have coordinates (0, 0), (8, 3), and (4, 6).) Using a coordinate grid, the students create a scaled map of the treasure site. In discussing this problem, the convenience of organizing a problem using a coordinate system should be pointed out.

(G.7.B) Dimensionality and geometry of location. The student understands that coordinate systems provide a convenient and efficient way of representing geometric figures and uses them accordingly. The student is expected to use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons.

Clarifying Activity

Students investigate the question, "Is there a relationship between a median of a triangle and the corresponding base?" On a coordinate plane, pairs of students draw an isosceles, equilateral and scalene triangle and label each vertex with coordinates. Using the distance formula, students can verify the type of triangle drawn. Using the midpoint formula, students label the midpoint of the base of each triangle. Connecting this midpoint to the opposite vertex of each triangle creates a median of the triangle. Students find the slope of the median and the slope of the base of each triangle and make a conjecture about the two lines. Once each pair of students has finished work on their triangles, the class compares their notes and explores the relationship between the median and the base of special types of triangles. The class writes a conjecture and tests the converse, inverse and contrapositive of the conjecture. As a class, the conjecture can be proven using variable coordinates for the original isosceles triangle.

Note: Construction technology can be used to see the relationship in a variety of triangles.

Additional Clarifying Activity

Students discuss the following problem:

The diagram shows a circle of radius 1 with center at the origin of an x-y coordinate system. Use an argument based on slope to show that a triangle inscribed in a semi-circle is a right triangle.

Students see that since the equation of the circle is x² + y² = 1, the point C on the circle has coordinates (x, ). From this information, students find the following lengths: AP = 1 + x; PD = 1 - x; CP = .

Therefore the slopes of lines AC and CD are:

The product of these slopes is

Since the product of the slopes is - 1, the lines are perpendicular, and angle ACD is a right angle.

(G.7.C) Dimensionality and geometry of location. The student understands that coordinate systems provide a convenient and efficient way of representing geometric figures and uses them accordingly. The student is expected to devise and use formulas involving length, slope, and midpoint.

Clarifying Activity

Students are given several different triangles with the coordinates of the vertices. Using the distance formula and the Pythagorean Theorem, the students classify the triangles by sides (scalene, isosceles or equilateral) and angles (acute, right or obtuse).

Additional Clarifying Activity

Students select two points on the coordinate plane. Students locate a third point in order to form a right triangle in which the first two points selected are the endpoints of the hypotenuse of the triangle. For example,

Students use the coordinate system to determine the lengths of the two legs of the right triangle and then apply the Pythagorean Theorem to find the length of the hypotenuse (the distance between the first two points). Students repeat the process for another pair of points. Students look for patterns in their process to determine a method for using the coordinates of two points to determine the distance between the points and thus develop the distance formula.

A similar activity can be used to develop the midpoint formula.

- Top -

(G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations.

(G.8.A) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find areas of regular polygons, circles, and composite figures.

Clarifying Activity

Students verify the formula for the area of a regular hexagon by finding the area of the six triangles that have two of their vertices as adjacent vertices of the hexagon and the third vertex at the center of the hexagon.

Students also explain in writing why the formula for the area of every regular polygon is A = (1/2)ap where a is the length of the apothem and p is the perimeter of the polygon.

Additional Clarifying Activity

Students find the area of a running track around a football field for the purpose of resurfacing the track.

(G.8.B) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find areas of sectors and arc lengths of circles using proportional reasoning.

Clarifying Activity

Each group of students cuts a circle out of paper and finds the area and circumference of the circle. (Note: Each group should have a circle of different radius so that the class can have a variety of examples to compare.) Then each group folds their circle into various fractional pieces to create sectors of the circle. As the circle is folded, students write the central angle measure as well as the proportion of the full circle for each fold. (For example, folding the circle into sixths creates 60 degree central angles, the sector is 1/6 the original circle, and the arc length is 1/6 the circumference.) Using the proportions, students find the area and the arc length of the sector and use drawings and/or tables to record their findings. As a class, students make generalizations for finding the area and arc length of any circle.

Sample student response:

Click here for a larger version of this table.

Additional Clarifying Activities

• Given a cone with a height of 8 cm and a radius of 6 cm, students create a net for the cone.

  Sample student solution:

  

  Using the Pythagorean Theorem, l = 10. The circumference of the base of the cone, 12 π, will be the arc length of the sector net. To find the central angle, A, of the sector:

  

  So, A = 216ƒ. With this information, a net can be constructed.

• See the Additional Clarifying Activity at G.1.B.

(G.8.C) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to derive, extend, and use the Pythagorean Theorem.

Clarifying Activity

Students use a dissection proof to verify the Pythagorean Theorem and are able to explain why the dissection proves the theorem. Below is a classical dissection proof, Leonardo Da Vinci's proof which involves transformations.

• Construct a right triangle with legs a and b and hypotenuse c.
• Construct squares on each of the three sides.
• Construct two additional right triangles congruent to the original and place them as shown in the diagram. (This will create a nine-sided figure.)
• Separate the upper hexagon (containing two right triangles and the two squares built on a and b).
• Draw a segment joining the two farthest corners of the squares to create two congruent quadrilaterals.
• Reflect the upper quadrilateral across the perpendicular bisector of the segment in step 5 to create a new hexagonal shape.
• Rotate this shape 90ƒ and divide into two right triangles (with sides a, b, and c) and a square. Notice that the area of this square is c² since its sides have length c. This new hexagonal shape is the lower part of the nine-sided figure created in step 3.
• Since the upper hexagon has been transformed into the lower hexagon using reflections and rotations, the areas are equal.
• Therefore,
  2…(1/2) ab + a² + b² = 2…(1/2) ab + c²
  a² + b² = c²

Click here for a larger version of this graphic.

Source: Koplas, Sidney J., The Pythagorean Theorem: Eight Classic Proofs, Dale Seymour Publications, 1992, pp. 19–21.

Additional Clarifying Activity

In exploring the Pythagorean Theorem, students construct squares on the sides of a right triangle as shown below.

A² + B² = c²

Students measure the dimensions of the square and show that the sum of the areas of the squares on the legs of the right triangle is the area of the square on the hypotenuse. Notice that the squares on the sides of the triangle are similar. Students explore other similar figures constructed on the sides of the right triangle to see if the sum of the areas results in the same relationship. Students use construction technology to easily construct various similar figures (such as semi-circles, triangles, rectangles etc.) on a variety of right triangles.

Sample student response:

Click here for a larger version of this graphic.

(G.8.D) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations. The student is expected to find surface areas and volumes of prisms, pyramids, spheres, cones, cylinders, and composites of these figures in problem situations.

Clarifying Activity

Given paper circles, all of the same radius, students follow the guidelines below.

• Cut a sector from the circle. Note: two sectors will result from the cut. Using tape, create cones from the two sectors.
• What do all of the cones created by the class have in common? (same slant height)
• Measure the dimensions of the cones and find the volumes of your two cones. Record on the class chart the volumes of your cones and their corresponding heights.
• Which cone in the class has the largest volume?
• Since the classroom sample does not include every possible cone in this family (cones with the same slant height), use a curve of best fit to find the cone that would produce the maximum volume. Make a scatterplot of the height of the cone vs. the volume of the cone. Then, use a regression equation (and graphing calculator capabilities, if available) to find the dimensions of the cone with the maximum volume.
• Which pair of cones produces the largest combined volume?

Additional Clarifying Activities

• Students investigate the following problem:

  The volume of many solids can be expressed in terms of their base area B, and their height h.

  • Describe as many types of solids as you can whose volume can be expressed as V = Bh. (Students will see that the volume of any prism (such as a cube or a rectangular box) or any cylinder can be expressed as V = Bh.)
  • Describe as many types of solids as you can whose volume can be expressed as V = 1/3 Bh. (Similarly, students will see that the volume of any pyramid or any cone can be expressed as 1/3 Bh.)
  • Find a solid whose volume can be expressed as V = 2/3 Bh. (Students will see, therefore, that a hemisphere's volume lies exactly halfway between the prism/cylinder class of solids and the pyramid/cone class of solids.)

    

    Click here for a larger version of this graphic.

• Students discuss the following problem:

  The diagram shows front views of a circular cylinder, a sphere, and a right circular cone. Show that the volumes of these three objects bear whole number ratios to each other that are independent of the value of d. What are these whole number ratios?

  

  Click here for a larger version of this graphic.

  Applying the standard formulas for volume:

  

  Students need to find the relationships between the parameters s, r, and h in these formulas and the dimension d in the diagram. (These relationships are s = d, r = d/2, and h = d.)

  Students use these relationships, to express the volumes in terms of d:

  

  Taking the ratios of these volumes, students show that cylinder : sphere : cone = 3 : 2 :1.

  (This relationship is one that Archimedes and many since his time have found fascinating.)

- Top -

(G.9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures.

(G.9.A) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models.

Clarifying Activity

See the Additional Clarifying Activity in G.2.B.

(G.9.B) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models.

Clarifying Activity

See the Additional Clarifying Activity in G.3.B.

(G.9.C) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models.

Clarifying Activity

Using construction technology or a circular geoboard, students construct a circle. Students select points A and B on the circle and construct a secant line. Students find the measure of the angle formed by the secant and the radius to point A. Students use the technology, or the rubber bands on the geoboard, to move point B until it coincides with point A, creating a tangent line. Students find the measure of the angle formed by this "secant" (tangent) and the radius to point A. Students use the data they have observed to make a conjecture. (Note: When point B actually does coincide with point A, another point on the line must be used in place of B to define the angle.)

Sample student response:

m∠DAC = 90°

Conjecture: For any point on a circle, the tangent and radius that contains the point of tangency are perpendicular.

(G.9.D) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures. The student is expected to analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.

Clarifying Activity

Students investigate the questions below to analyze the design of juice boxes and their straws.

What is the length of the shortest straw that will not fall into the juice box?

Where should the hole for the straw be to minimize the length of the straw?

If the hole on your juice box is not where you find it should be for Question 2, explain the manufacturer's possible reasoning for the placement of the hole.

Additional Clarifying Activity

Students make models of the five Platonic solids. Students prove that these are the only possible solids using regular polygons with the same number of polygons around each vertex. Students determine the number of faces, vertices, and edges each solid has and organize this information in a table. Students then create mathematical expressions to describe the number of vertices and edges each solid has and look for a pattern relating the number of faces, vertices, and edges. Students study several other polyhedra--a pyramid, a prism, a cuboctahedron, and so forth—to see if the patterns still hold true. Students make conjectures about inequalities that are always true for polyhedra. (For example, two times the number of edges is greater than or equal to three times the number of faces.)

As an extension, students truncate each of the Platonic solids, explain what the truncated piece will look like, what shape the new face created is, and what the resulting solid will look like (e.g. how the number of faces, edges, and vertices will change).

- Top -

(G.10) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems.

(G.10.A) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to use congruence transformations to make conjectures and justify properties of geometric figures including figures represented on a coordinate plane.

Clarifying Activity

Students use rotations to prove the sum of the angles of a triangle is 180ƒ.

Sample student response:

Let M1 and M2 be the midpoints of segments AB and BC, respectively. Rotate angle CAB about M1 180ƒ to angle EBA and rotate angle BCA about M2 180ƒ to angle DBC. Since rotations preserve congruence angle CAB is approximately equal to angle EBA and angle BCA is approximately equal to angle DBC. These are each alternate interior angles formed by two lines and a transversal, so lines ED and AC are parallel. This assures us that m angle EBD=180ƒ and m angle EBA + m angle ABC + m angle DBC = 180ƒ. Substituting the congruent angles, we get m angle CAB + m angle BCA + m angle ABC = 180ƒ.

Additional Clarifying Activity

See the Clarifying Activity in G.8.C.

(G.10.B) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems. The student is expected to justify and apply triangle congruence relationships.

Clarifying Activity

Students use paper strips, brads and cut-out angle measures to investigate the sufficient (SSS, SAS, ASA, AAS) and insufficient (SSA and AAA) conditions for proving triangles congruent as follows:

Each group will need: two each of three different length paper strips, 6 brads, and two each of three different angle cut-outs.

(Using transparencies to make the angle cut-outs allows for easy re-use of the materials. Computer paper edges can be cut into three different lengths for the paper strips.)

Two triangles are defined to be congruent if the three pairs of corresponding angles and the three pairs of corresponding sides are congruent. However, there are several ways to construct congruent triangles with fewer than the six congruencies described in the definition. As a group, try to use the least amount of information to create a pair of congruent triangles with the material given. Keep marked drawings of your findings as illustrated in the example below. Also keep track of methods which do not produce congruent triangles for a class discussion.

Sample student solution:

Using 2 pairs of congruent paper strips and 1 pair of congruent angles, we create a pair of congruent triangles. We call this method Side-Angle-Side Triangle Congruence. See the diagram below.

- Top -

(G.11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems.

(G.11.A) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use and extend similarity properties and transformations to explore and justify conjectures about geometric figures.

Clarifying Activity

Students design a rectangle so that when it is folded in half the new rectangle is similar to the original rectangle.

Additional Clarifying Activity

An overhead projector can be used to demonstrate similar polygons (provided the screen is not slanted on the wall). Use a polygon cut-out to trace a polygon on the overhead. The image produced on the screen will be similar to the original polygon cut-out. By moving the polygon cut-out around the image on the screen, the angles can be shown to be congruent. Ask a student to measure the sides of the cut-out and another student to measure the sides of the image on the screen. The class can find that the ratio of corresponding sides is the same. Then ask students to move the overhead so that the two similar polygons are in a ratio of 1:5 (see the Clarifying Activity in G.11.B).

(G.11.B) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to use ratios to solve problems involving similar figures.

Clarifying Activity

Students investigate the following problem:

A spotlight at point P throws out a beam of light.

The light shines on a screen that can be moved closer to or farther from the light. The screen at position A is a distance A from the light, and at position B is a distance B.

The lengths a and b indicate the lengths of the light patch on the screen.

Show that the ratio of the length b to the length a depends only on the distances A and B, and not on the angle y that the beam is off the perpendicular, nor on the angle x the beam itself makes.

(Students can see that, by similar triangles, . This means a/A + c/A = b/B + d/B. But c/A = d/B, also by similar triangles. Hence a/A = b/B, which is equivalent to b/a = B/A. The ratio B/A is independent of the angles x and y. It is the scale factor relating the distances of the two screens, and the sizes of the images on the two screens.

Additional Clarifying Activity

See the Clarifying Activity at G.5.B.

(G.11.C) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods.

Clarifying Activity

Students create the following construction:

In the construction above, segment AB is the diameter of the semicircle constructed, and segment EC is perpendicular to segment AB.

Find the length of segment EC in terms of x.

(EC = √x, since in ΔAEB, EC is the geometric mean of AC and CB.)

Extension

Students investigate the similarity of the three triangles shown above.

(G.11.D) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems. The student is expected to describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems.

Clarifying Activity

Students find the circumference of the widest circle on a basketball (measured in feet). Students predict what would happen if they took a wire or string 1 foot longer than this circumference and formed a ring around the middle of the basketball, like the rings around Saturn. Students discuss questions such as:

• Would I be able to fit a pencil between the ring and the basketball?
• Would I be able to fit my finger between the ring and the basketball?
• Would I be able to fit my fist between the ring and the basketball?

Using a thin metal wire, such as electrical wire, students cut a piece that is 1 foot longer than the circumference of the basketball. This wire is shaped around the basketball to create a "ring" around the middle of the basketball. Students revisit their predictions about what could fit under the wire.

Students then calculate the actual distance from the surface of the basketball to the ring of wire.

Next, students calculate the circumference of the equator of the earth (see Clarifying Acrivity G.1.B), add 1 foot to this distance and consider the result of making a ring around the earth with a wire this long. For example, "Would I be able to walk under the ring?

Students calculate the actual distance from the surface of the earth to the imaginary ring and compare the results with those from the basketball questions.

Additional Clarifying Activities

• Each group of students is given the dimensions of a different rectangular prism and students calculate the total surface area and volume. Students double the height and compare the new total surface area and volume to those of the original solid. They repeat this process for the width, and then again for the length. Students write a conjecture about the effects on the total surface area and volume when one dimension of the prism is doubled. Next, students investigate doubling two dimensions of the prism and the effects on the total surface area and volume. Finally, students double all three dimensions. As a class, students discuss their conjectures and generalize to other solids. Students then discuss how they would double the surface area of a cone or a cylinder and test their conjectures.
• Given a wedge of cheese, like the one shown below, students find three different ways to cut the wedge into halves.

  

  Students are encouraged to use a model in their explorations. Once everyone has arrived at a solution, students must use geometric properties and measurement principles to justify their cuts to the class.

- Top -