### MATH113: Second Midterm Review Page

The second midterm exam covers material from page 794 of the textbook through the end of chapter 14, sections 9.1 through 9.3, and possibly section 10.1.

This midterm will be similar to the first midterm. Due to the nature of the material in chapter 9 there will be more emphasis on definitions and less emphasis on calculation.

This exam will cover the objectives listed below. Sample problems are given for each objective.

• Understand and apply concepts of measurement:
• Estimate, calculate and logically verify angle measurements.
• A right triangle has a 30 degree interior angle. What are the measures of its exterior angles?
• Ferdinand brought the following picture to class:

Explain why this cannot be a tiling of the plane by regular pentagons.
• Draw a sketch of a 60 degree angle. How can you verify that the measure of the angle you drew is approximately 60 degrees?
• Given an angle, estimate its measure.
• Use a protractor to measure an angle.
• Understand and apply concepts of geometry:
• Classify and anlayze polygons using attributes of sides and angles.
• Which quadrilaterals must always have two (or more) sides with the same length?
• Is an equilateral triangle also an isosceles triangle? Use the definition of an isosceles triangle to justify your answer.
• Two equilateral triangles are placed edge to edge and vertex to vertex. Is the quadrilateral they form a parallelogram? equiangular? equilateral? a rectangle? a rhombus? a kite?
• Classify and analyze three-dimensional objects using attributes of faces, edges and vertices.
• Sian wants to build a polyhedron which has three pentagons (and no other polygons) at each vertex. Is this possible? If so, do we know a name for this polyhedron?
• Is it possible to construct a polyhedron which has 3 equilateral triangles and a square at each vertex? Bonus: what type of polyhedron would be formed in this way?
• Describe the shape you would get if you:
1. Start with five equal isoceles triangles, each having a 30 degree angle between their congruent sides.
2. Arrange them edge to edge so that the five 30 degree angles meet at a single vertex.
3. Use a planar region to enclose the region inside the assembly of triangles.
• Match three-dimensional figures and their two-dimensional representations -- e.g. nets, projections and perspective drawings.
• Is it possible to construct a polyhedron which has 2 equilateral triangles and 2 squares at each vertex?
• Recognize and apply connections between algebra and geometry (e.g. coordinate systems, area formulas, the Pythagorean theorem).
• A seven sided polygon is called a heptagon. Calculate the angle between two adjacent sides of a regular heptagon.
• A student walked 10 miles north, then turned 30 degrees to her right. After hiking 5 miles she turned 45 degrees to her left. She walked 4 more miles and then turned 15 degrees to her right. What direction was she walking in during the last leg of her walk?
• Understand and apply basic concepts of probability:
• Calculate probabilities of simple and compound events and of independent and dependent events.
• Recognize and apply the concept of conditional probability.
• Apply knowledge of combinations and permutations to the computation of probabilities.
Note: These sample questions use decks of cards so that the questions on the actual midterm can be about student groups and balls in bags.
• Marsha draws 4 hearts from a deck of cars containing 52 cards, 13 of them hearts. What is the probability that the next card she draws will be a heart? Is it likely that the next card she draws will be a heart?
• How many different collections of 5 cards contain 5 hearts?
• What is the probability of being dealt a hand containing 5 hearts from a standard deck of cards?
• What is the probability of drawing a spade or a face card from a standard deck of 52 cards?
• What is the probability of drawing a spade and a face card from a standard deck of 52 cards?
• In how many different ways can a Blackjack player be dealt a Jack, then an Ace, then a Queen from a deck of 52 cards?
• Recognize the difference between experimentally and theoretically determined probabilities in real-world situations.
• Give an example of a question you would answer using an experimental probability.
• Give an examle of a question you would answer using a theoretical probability.
• Marcie calculates that the probability of a randomly selected student in her math class being female is 18/33. Is this a theoretical or experimental probability? Justify your answer.
• After observing the lunch line at the cafeteria, Marcie calculates that the probability of a student choosing jello over pudding is 35%. Is this a theoretical or experimental probability? Justify your answer.
• Marcie caculates that the probability of drawing a red face card from a deck of 52 cards is about 15 percent. Is this a theoretical or experimental probability? Justify your anwer.