MATH EXAM 1
Which of the following describe the same volume, or mean the same as, or are the correct way toread 2 in3?• a 2-inch by 2-inch by 2-inch cube (this is 8 in3)• 2 inches cubed• 2 cubic inches• 2 in x 2 in x 2 in
2 inches cubed• 2 cubic inches• the other 2 would be 8in3
Using a compass and a straightedge, construct an isosceles triangle that is NOT equilateral. Explain how you know that the triangle is isosceles and not equilateral without measuring it.
All the points on a circle are the same distance from the center. In this method, the center is one vertex of the triangle. Choosing the other two vertices to be points on the circle ensures that both are the same distance from the first vertex, giving us two sides of the same length.
Given that the indicated lines in the figure below are parallel, determine angle 𝑎. Explain your reasoning briefly
Angle b is 55° because is opposite to the 55° angle given.Angle c is 125° because is corresponding with the 125° angle given and bythe Parallel Postulate.Angle d is 65° because it forms a straight line with angle c.Now, b + d + e = 180° because the sum of the angles in a triangle is 180°, so,55° + 65° + e = 180° → e = 180° - 55° - 65° = 60°.Therefore a = 120° since it forms a straight line with 𝑒
Given that the indicated lines in the figure below are parallel, determine angle 𝑎. Explain your reasoning briefly
Angle b is 95° because is corresponding with the 95° angle givenand by the Parallel Postulate.Angle c is 85° because it forms a straight line with angle b.Now, c + d + 35° = 180° because the sum of the angles in a triangleis 180°, so, 85° + d + 35° = 180° → d = 180° - 85° - 35° = 60°.Therefore a = 120° since it forms a straight line with d.
Explain clearly why Keisha's and Aaron's "angle explorers" in the figure below show angles of thesame size. Why isn't Keisha's angle bigger? As part of your explanation, discuss how we think about sizes of angles.
Angles measurement is defined in terms of rotation abouta fixed point, the vertex. Both angles show the sameamount of rotation, the length of the sides of the angledoes not affect the amount of rotation about the vertex.
A new Giant Superstore is being planned somewhere in the vicinity of Kneebend and Anklescratch,towns which are 10 miles apart (as shown on the map below). The developers will only say that allthe locations they are considering are less than 7 miles from Kneebend and more than 5 miles fromAnklescratch. Indicate all the places where the Giant Superstore could be located. Explain youranswer
Because the store needs to be locatedless than 7 miles from Kneebend, itmust be inside the circle with radiusof 7 miles with center at Kneebend.Because the store needs to be morethan 5 miles from Anklescratch, itmust be outside the circle with radiusof 5 miles with center at Anklescratch.Therefore, the store could be locatedanywhere inside the circle with centerat Kneebend and outside the circlewith center at Anklescratch.
If you use the diameter of a circle to measure its circumference, what is the result? Does it depend on whether the circle is small or large? Explain
For any circle when the circumference is measured by its diameter the result is always the same number, which is approximately 3.14 (exactly 𝜋). All circles are just scaled versions of each other, either scaled smaller or scaled larger. When we scale shapes, the ratio between the lengths of different parts of the shape remains the same
Explain in detail how the sets of right triangles, equilateral triangles, and isosceles triangles are related using our definitions of these shapes. Make a clear diagram to show how these sets ofshapes are related
Isosceles triangles have at least 2 sides of the same length. All the sides of equilateral triangles are the same length. So, equilateral triangles are isosceles triangles.Right triangles have a right angle and the opposite side to the right angle is called the hypotenuse, which is longer than the other two sides of the triangle. However, these two sides can be of the same size making it an isosceles triangle. So, some right triangles are isosceles triangles.
When we say that a shape has an area of 15 square centimeters, what does that mean?
It means that the shape can be covered, without gaps or overlaps, with a total of fifteen 1-cm-by-1-cm squares, allowing for squares to be cut apart and pieces to be moved if necessary.
When Joe was asked to draw a shape that has an area of 3 square centimeters, he drew a 3 cm by 3cm square. Is Joe right or not? Explain.
Joe is not right. A 3 cm by 3 cm square would have an area of 9 cm2, since it will contain 3 rows of 3one-by-one square cm. A shape that has an area of 3 square centimeters can be represented by any shape that encloses 3 one-by-one square cm.
Students sometimes say, "area is length times width." Explain why this statement is not fully accurate
Not every shape is a rectangle. For shapes that aren't rectangles, the area is not just "length times width". If something has an area of A square inches, that means that it can be thought of as made from A squares, each 1-inch-by-1-inch
Describe one-dimensional, two-dimensional, and three-dimensional parts or aspects of a bottle.What are practical reasons for wanting to know the sizes of these parts or aspects of the bottle?
One-dimensional - the height of the bottle measured in inches or cm, to see if its fits in a particular space. Two-dimensional - the area around the bottle measured in square inches or cm2, to determine the amount of paint needed to paint the bottle. Three-dimensional - the volume of liquid that the bottle can hold measured in oz or milliliters, to determine how much liquid the bottle can hold.
Explain how to determine the sum of the angles 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 in the pentagon below.
f we plot a point inside the pentagon and draw segments from that point toeach vertex of the pentagon, we will be forming 5 triangles. The angles of eachtriangle add to 180° and since we have 5 of them the angles of all triangles addto 5 × 180° = 900°. However, this includes the angles formed at the center ofthe pentagon which are not part of angles a, b, c, d, and e; so, we need tosubtract those angles from 900°. The angles at the center of the pentagonform a full rotation (360°). Therefore a + b + c + d + e = 900° - 360° = 540°.
Jenny wants to know what it means when we say that a tank is 284 cubic feet. What can you tell Jenny?
if you had a 1-feet-by-1-feet-by-1-feet cube that could be filled with water, then you can fill the tank by pouring the water of such cube 284 times.
Suppose that two lines in a plane meet at a point, as in the figure below. Use the fact that the angle formed by a straight line is 180° to explain why a = c and b = d
We know that a straight line has to equal 180. So a +b =180. and we know that b+c=180. Therefore, a= 180-b and c=280-b so they are equal because 180-b = a or c. we can also find b=d this same way to show that they are equal.
Which of the following provides a correct definition of the term circle? Check all that apply. a. A collection of points in a plane that are all one fixed distance away from a point. b. All the points in a plane that are one fixed distance away from each other. c. All the points that are one fixed distance away from a point. d. All the points in a plane that are one fixed distance away from a point
a. (You need all the points to be a circle, not a collection)b. (This will be only 3points) c. (This is a sphere) d. All the points in a plane that are one fixed distance away from a point (ONLY ONE RIGHT)
parallelograms and trapezoids
all parallelograms are trapezoids. but not all trapezoids are parallelograms
Rectangles and parallelograms
all rectangles are parallelograms not all parallelograms are rectangles
Define circle
all the points in a plane that are one fixed distance away from a point
Using a compass and a straightedge, construct an equilateral triangle. Explain how you know thatthe triangle is equilateral without measuring it
The method shown produces an equilateral triangle because of the way it uses circles.Remember that a circle consists of all the points that are the same fixed distance away from the center point. The circle drawn in step2 consists of all points that are the same distance from A as B is; since C is on this circle, the distance from C to A is the same as the distance from B to A. The circle drawn in step 3 consists of all points that are the same distance from B as A is; since C is also on this circle, the distance from C to B is the same as the distance from A to B. Therefore, the three line segments AB, AC, and BC all have the same length, and the triangle ABC is an equilateral triangle
Suppose you use geometry software to construct two circles with centers A and B, in such a way that the circles will always have the same radius, no matter how you move them. Suppose that the two circles meet at points C and D and suppose that you construct line segments to make a quadrilateral ACBD (by connecting A to C, C to B, B to D, and D to A), as shown in the figure below.What kind of special quadrilateral must ACBD be, no matter how you move the points in your construction (as long as the circles still meet at two points)? Explain your answer clearly and in detail, as if you were explaining to someone who was just learning about the geometric concepts involved
Since the two circles have the same radius, the segments AC, AD, BC, and BD (which are radii of the circles) have the same length. This segments form a quadrilateral, with all sides with the same length (the radius of the circle). Therefore, the quadrilateral formed would be a rhombus, no matter how the points are moved in the construction (as long as the circles still meet at two points).
Squares and Rhombuses
Squares and rhombuses All squares are rhombuses. Not all rhombuses are squares
What is a less primitive way for your students to determine the area of the rug and why does this method work?
The tiles can be viewed as organized into equal rows (or columns). Because there are equalgroups of squares, we can multiply to find the total number of squares. When the 3-foot-by-4-foot rectangular rug is covered with square foot tiles, there are 4 rows of squares with 3 squaresin each row. Therefore, there are 4 ∙ 3 squares covering the rug (without overlaps). Each square has area 1 ft2; therefore, the total area of the rug is 4 ∙ 3 square feet, which is 12 square feet.
You have a 3-foot-by-4-foot rectangular rug in your classroom. You also have a bunch of square foottiles and some tape measures.a. What is the most primitive way for your students to determine the area of the rug?
To cover the cover the rug snugly with the square foot tiles and count that it takes 12 tiles.
