1. science
2. mathematics
3. geometry

MATH 1136 COURSE REVIEW
1. Basic geometric concepts and reasoning
•
•
•
•
lines, planes
• compass/straightedge constructions
angles
• reasoning about properties of polygons: simthe parallel postulate
ple proofs, Venn diagrams
polygons (mainly triangles and quadrilaterSuggestions from the Practice Exercises: 10.2.3,
als)
10.2.4, 10.2.6, 10.2.8, 10.4.3, 10.5.7, 10.5.8, additional
• circles, spheres
homework exercises
• angle sum by walking/turning and parallel
postulate
1. Explain the two different ways of defining the meaning of angle.
2. Use basic arithmetic to explain why opposite angles are congruent.
3. State the Parallel Postulate and give an example of using it.
4.
Explain how to determine the sum of the interior angles in a heptagon. Use elementary reasoning
instead of a formula.
5. Use the Parallel Postulate and angle-sum in triangles to determine the angle marked θ.
100◦
θ
140◦
6. Give two examples of trapezoids that are not parallelograms, and two that are.
7. Give two examples of quadrilaterals that are not trapezoids.
8. Explain why the following construction must be a square: Begin with a line, and draw two intersecting
circles of the same radius on that line. Then draw a line passing through the two points where the circles
intersect, and draw a huge circle centered where the two lines intersect. Connect the four points where the
huge circle intersects the two lines.
(See picture below)
9. Explain why the diagonals of a parallelogram bisect eachother.
10. Explain why the construction illustrated at left must be a right triangle.
Date: 04/23/2015.
1
2. Algebraic expressions
• algebraic expressions from pictures and vice
• strip diagrams as a bridge between written
versa
descriptions and algebraic expressions
• expressions and equations to describe relationships among quantities
Suggestions from the Practice Exercises: 9.1.5, 9.1.6,
• solving equations
9.2.2, 9.2.5, 9.3.1, 9.3.3, 9.3.4, 9.4.4, 9.5.1, 9.5.2, 9.5.3
11.
Write different algebraic expressions for the total number of squares shown in the pictures below.
Your expression should reflect structure you see in the pictures.
12. What day of the week will be 365 days from today? What day of the week will it be 3650 days from
today? Use the concept of division with remainders to explain your answers.
13.
Five friends (Alice, Bob, Charley, David, and Eunice) sit in a circle and sing a song that has 21
syllables (something like “eenie-meenie-miney-moe...”). Starting with Alice, and continuing around the circle,
they point to a new person on each syllable. Which person do they end on? How many syllables should they
sing to end on David? Use the concept of division with remainders to explain your answers.
14. If 100 people give eachother high-fives, how many high fives will take place?
15. If 100 people engage in some complicated three-person handshake, in all possible ways, how may of
these handshakes will occur?
16. Suppose you make cheese-and-meat sandwiches with one type of cheese and two different types of
meat. The cheese choices are provolone, Swiss, and American; the meat choices are ham, turkey, roast beef,
or salami. How many different sandwiches are there? How does the problem change if you are allowed to
choose the same type of meat twice?
17.
There are G gallons of iced tea in a container. If 1 12 gallons are poured out, and then 13 of the
remaining tea is poured out, then how much tea is left in the container?
(a) Show two different correct expressions that students might formulate, and indicate what reasoning
might lead to each expression.
(b) Show an incorrect yet plausible expression that students might formulate, and explain why students
might mistakenly formulate it that way.
(c) Evaluate your expression when G = 6.
18.
Bob had m marbles in his marble collection. After Bob got another 15 marbles from a friend, he
gave away 15 of his marbles to another friend. After this, Bob had 32 marbles. Write an equation involving
m that corresponds to this situation.
19. There are 450 coins divided into 3 piles. The second pile has twice as many coins as the first. The
third pile has 25 more coins than the second. How many coins are in the first pile? Solve this problem in
two ways: with the aid of a diagram and with algebraic equations, explaining each briefly. Discuss how the
two solution methods are related.
20. At a book sale, 40% of the books were sold in the first hour. In the next hour, 23 of the remaining
books were sold. At that point there were 80 books left. How many books were there at the beginning of the
book sale? Solve this problem in two ways: with the aid of a diagram and with algebraic equations. Explain
your reasoning.
c 2015 Niles Johnson, The Ohio State University. This document may not be reproduced, modified, or distributed without permission.
3. Functions
• key (defining) property of functions
• proportional relationships as linear functions
• sequences as functions
• arithmetic sequences as linear functions
• using reasoning with remainders to under• equations for linear functions
stand repeating sequences
• other function types: inversely proportional,
• arithmetic and geometric sequences
quadratic, exponential
• shape of graph and qualitative behavior of
Suggestions from the Practice Exercises: 9.7.1, 9.6.2,
function
9.6.3, 9.6.5, 9.7.4, 9.7.6, 9.8.2, 9.8.3, 9.8.4
21. Draw a sequence of figures so the nth figure has 2 + 3n circles and 1 + 6n triangles. Describe how
subsequent figures in your sequence would be formed.
22. Give an expression for the nth entry in the arithmetic sequence
5, 8, 11, 14, 17, ...
23.
Give an expression for the nth entry in the quadratic sequence
4, 7, 12, 19, ...
24. Determine the equation of a line which passes through the points (7,-7), (1,1), and (-2,5). Give two
other points which lie on this line.
25.
Consider the following tables. For each one, use the notions of proportion and slope to explain
whether the entries in the table could be given by a linear function. For those which could be linear, write a
linear equation which matches the values in the table and sketch a graph illustrating the key features of the
corresponding function.
g
h
f
input output
input output
input output
0
3
1
3
3
7
2
2
9
6
5
10
12
9
13
4
4
12
6
24
8
17
18
16
26.
Starting at the bottom of a hill, Ruth runs up the hill, starting out fast, but slowing down more
and more as she nears the top of the hill. At the top of the hill, Ruth stops briefly, then turns around and
runs back down to the bottom of the hill, maintaining a steady speed all the way down until she stops.
(a) Sketch a graph that could be the graph of the speed function whose input is time elapsed since Ruth
began running up the hill and whose output is Ruth’s speed at that time. Indicate how the graph
corresponds with the story.
(b) Sketch a graph that could be the graph of the distance function whose input is time elapsed since
Ruth began running up the hill and whose output is the total distance Ruth has run at that time.
Indicate how the graph corresponds with the story.
(c) Sketch a graph that could be the graph of the height function whose input is the time elapsed since
Ruth began running up the hill and whose output is Ruth’s height above the base of the hill at that
time. Indicate how the graph corresponds with the story.
27.
A mail-order company sells 2 pounds of beads for $10. Each additional pound of beads costs $3.
Write an expression for the cost of P pounds of beads.
c 2015 Niles Johnson, The Ohio State University. This document may not be reproduced, modified, or distributed without permission.
4. Measurement
• Fundamentals of measurement
– meaning of measurement
– measurable v.s. non-measureable attributes
– units
• Length, area, volume
– similarities and differences between the
three
• Error and precision in measurement
• Unit conversion
28.
Students often say “area is length times width”. Explain why this statement can lead to a misunderstanding of the concept of area.
29. Give examples of different shapes that all have an area of 2 square-centimeters.
30. Give examples of different shapes that all have a volume of 4 cubic-centimeters.
31.
A certain ingredient weighs 13 pound per cup and costs $2.00 per pound. Explain how to use the
meaning of multiplication and division to say how much 5 quarts of the ingredient would cost.
32. Use unit conversion to say how many kilometers are in a mile.
33. Let’s say tobit is 5 inches. Does it therefore follow that there are 5 square-inches in a square-tobit?
Explain why or why not? How many cubic-inches are in a cubic-tobit?
5. Area
• Area of rectangles
– meaning of area
– length x width
• Moving and additivity principles
• Areas of triangles, parallelograms
• Cavalieri’s principle
• Similarity
– internal v.s. external scale factors
– square constant and circle constant
• Circle area, estimates of pi and tau
• Areas of irregular shapes
• The Pythagorean theorem
Suggestions from the Practice Exercises:
11.1.7, 11.1.8, 11.4.3, 11.4.4
12.1.2, 12.2.3, 12.3.2, 12.3.3, 12.5.1, 14.5.1, 14.5.4,
14.5.7
12.6.2, 12.6.5, 12.7.2, 12.8.3, 12.9.2, 12.9.5
Also review homework exercises, as these are somewhat different from the available practice exercises.
34.
Determine the areas of the shapes shown here. Determine the permimeters of the blue and green
shapes.
35.
36.
37.
Use Cavalieri’s Principle to explain how to see the area of a parallelogram as the area of a rectangle.
Explain how to see the area of a triangle as half the area of a parallelogram.
Explain at least one of the three proofs of the Pythagorean theorem that we’ve covered.
c 2015 Niles Johnson, The Ohio State University. This document may not be reproduced, modified, or distributed without permission.
6. 3D Shapes: Patterns, Cross-sections, Surface Area, Volume
• Identifying corresponding measurements on
• Explain surface area and volume formulas
2D patterns and constructed 3D shapes; deducing related measurements
Suggestions from the Practice Exercises: 13.2.1,
• Cavalieri’s Principle for volume
13.2.2, 13.2.3, 13.2.5, 13.3.5, 13.3.6, 13.3.8
38.
Draw a pattern for a right pyramid with a regular pentagon base. Give precise measurements for
the pattern so that the pyramid has a height of 5 cm (there are many possible answers). What is the volume
of your pyramid? What is the lateral surface area of your pyramid?
39. Draw a pattern for a right circular cone so that the base circle has a circumference of 3π in and a
height of 2 in.
40.
Give an example of a squared-based prism and a circular cone which both have volumes of 6
cubic-inches.
7. Transformations: Theory and applications
•
•
•
•
Transformations in the plane
Symmetry, congruence
Similarity
Scaling area and volume
41. Consider a line with slope
passing through the point (0,2).
Rotate this line 90◦ around the origin (counter-clockwise) and determine the slope of the resulting line.
42. Show the result of first reflecting the triangle shown below,
left, through the line y = x and
then rotating it 270◦ around the
origin (counter-clockwise).
2
5
43.
Review the three congruence criteria for triangles.
44.
Explain why a pair of
intersecting transversals crossing
parallel lines always create similar
triangles. (There are two cases to
consider.)
Suggestions from the Practice Exercises: 14.2.1,
14.2.2, 14.2.3, 14.2.4, 14.3.1, 14.3.2, 14.4.3, 14.5.4,
14.5.5, 14.6.2, 14.6.3
45.
Consider a triangle with
side lengths 4cm, 5cm, 6cm. Explain where to cut off a corner so
that the cut off piece has exactly 49
of the starting area.
46.
Consider a square-based
pyramid with a base width of 2 feet
and a slant height of 8 feet. How
much should be cut from the top
so that 41 of the pyramid’s area remains?
47. Determine the volumes of
the following shapes, and put them
in order from smallest to largest
volume.
48. Imagine a region obtained
by adding and taking away infinitely many heart shapes, each a 1/2
scaling of the previous one (see below, right).
Write two expressions for the
area of this region, as a fraction of
the total area of the largest heart
shape. First, use use an infinite
sequence of additions and subtractions. Second, use a scaling relationship between the total region
remaining and the total region subtracted, and the fact that they
combine to form the largest heart
shape. Deduce the value of the infinite sum!
(a) A sphere with radius 2 feet
(b) A cube with side-length 3
feet
(c) A right circular cylinder
with base radius of 2 feet
and a height of 3 feet.
(d ) A right circular cone with
a base radius of 4 feet and
a height of 2 feet.
(e) A right square-based pyramid with a base width of 5
feet and a height of 4 feet.
c 2015 Niles Johnson, The Ohio State University. This document may not be reproduced, modified, or distributed without permission.
8. Probability
• Key principles of probability
• Theoretical v.s. empirical probabilities
• Counting techniques
• Probabilities of multistage/compound events
Suggestions from the Practice Exercises:
16.2.5, 16.3.3, 16.3.4, 16.4.3
16.1.1,
49. Design a spinner with red, green, yellow, and blue regions so that the probability of spinning red or
green is 13 of the others, but red is twice as likely as green, and yellow is twice as likely as blue.
50. How many 3-letter codes (all uppercase) are there that do not contain any duplicate letters? How
many 10-letter codes?
51.
Suppose you have a basket containing an apple, a bananna, a cookie, a doughnut, an egg, and a
fish stick. You make a snack by grabbing three different items and then eating them in alphabetical order.
How many different snacks could you make this way? What is the probability that your snack will consist
of cookie, doughnut, fish stick?
52. What is the probability of rolling a 7 with two dice?
53. What is the probability of flipping 3 heads out of 6 coins?
c 2015 Niles Johnson, The Ohio State University. This document may not be reproduced, modified, or distributed without permission.