MTHED 305 Final

Perceptual Subitizing

"You Just Know" When you look at something, you know how many there are

Describe how you would solve 29x42 using lattice method

(see lattice picture on desktop)

Suppose you had a rhombus with diagonals of lengths r and s. Write a formula for the area of a rhombus.

(1/2(1/2r)(1/2s)) x 4 = area of a rhombus

Discrete

(fewer) distinct, individual units

Continuous

(less) accuracy within a certain amount "To the nearest..."

Pythagorean Theorem Proof

(see picture on desktop) C^2 = (a + b)^2 - 4ab / 2 C^2 = a^2 + b^2

A point exists in ____ dimension(s)

0

1 mm^2 = _____ cm^2

0.01

19.6 cm^2 = _____ dm^2

0.196

500 ms = _____ s

0.5

0/0

0/0 = undefined because 0/0=c iff 0/0xc, but c can be an infinite amount of whole numbers, so since c is undefined, 0/0 is also undefined

A line exists in ______ dimension(s)

1

A decimeter is _____ times as long as a centimeter, but a square decimeter is ____ times as big as a square centimeter, and a cubic decimeter is _____ times as large as a cubic centimeter.

10, 100, 1000

1 m^2 = _____ cm^2

10,000

A meter is _____ times as long as a centimeter, but a square meter is _____ times as big as a square centimeter, and a cubic meter is _____ times as large as a cubic centimeter.

100, 10000, 1 million

Dodecahedron

12 pentagons

1.2 km = _____ m

1200

How many cubic inches are in a cubic foot?

1728in^3

For a regular 20-gon find the measure of an exterior angle

180 - vertex angle 180 -162 = 18*

For a regular 20-gon find the measure of a vertex angle

180(n - 2) / n 180(20 - 2) / 20 180(18) / 20 = 162*

Icosahedron

20 triangles

2.3 dm^2 = _____ cm^2

230

How many cubic feet are in a cubic yard?

27ft^3

Describe how you would solve 29x42 using partial products

29x42= 18 14 36 8 =1218

3.25 dm = _____ cm

32.5

Describe how you would solve 37/5 using separating tens and ones

37/5 could be turned into 30/5, which would equal 6 and then 7/5, which would equal 1.4 or 1 remainder 2.

Tetrahedron

4 triangles

SQUARE: n, vertex angle, sum of vertex angles, central angle, exterior angle

4, 90, 360, 90, 90

0.05 m^2 = _____ cm^2

500

5.3 m = _____ cm

530

Cube

6 squares

62 mm = _____ cm

6.2

HEPTAGON: n, vertex angle, sum of vertex angles, central angle, exterior angle

7, 128.6, 900, 51.4, 51.4

Octahedron

8 triangles

OCTAGON: n, vertex angle, sum of vertex angles, central angle, exterior angle

8, 135, 1080, 45, 45

NONAGON: n, vertex angle, sum of vertex angles, central angle, exterior angle

9, 140, 1260, 40, 40

How many square feet are in a square yard?

9ft^2

What are the area and circumference of a circle with a diameter of 12mm?

A = (Pi)r^2 = (Pi)6^2 = (Pi)36 = 113.04 C = (Pi)d = (Pi)12 = 37.68

How many square feet of carpet do you need to carpet a room that is 12 ft. 6 in. by 20 ft. 9 in.?

A = 12.5 x 20.75 = 259.375 ft^2 or 259 ft. 4 1/2 in.^2

What is the circumference of a circle with an area of 9(Pi)cm^2?

A = 9(Pi) = (Pi)3^2 radius = 3 C = 2(Pi)r = 2(Pi)3 = 18.84

Total Angle Defect

ALWAYS equals 720*

(This scale shows weight to the nearest 1/2 pound) What weights would give a reading of 128.5 on the scale?

ANything between 128.25 and 128.75. (128.25 <e weight < 128.75)

Compliment

Adding the compliment works because, after adding the compliment, you "cross out" 1 in the thousands place and add 1. This allows the student to convert the subtraction into an addition problem

The Abstraction Principle

Any collection can be counted, whether seen or unseen, concrete or abstract

(This scale shows weight to the nearest 1/2 pound) What weights would give a reading of 162.0 on the scale?

Anything between 161.75 and 162.25. (161.75<e weight < 162.25)

(a+b) + c = a + (b+c)

Associative property of addition

(ab)c = a(bc)

Associative property of multiplication

Using the symbolic representation to help you, say which property (commutative, associative, or distributive) is being used implicitly by the child. Amina solved 15+17+5 by adding 15 and 5 to get 20, then adding 17 to get 37. (15+17+5=15+5+17=37)

Commutative Property

A child solves 8 + 7 + 2, and then adding 7. What property is she implicitly using?

Commutative Property of Addition

a + b = b + a

Commutative property of addition

a x b = b x a

Commutative property of multiplication

(a+b)(c+d) = (ac) + (ad) + (bc) + (bd)

Distributive Property

Using the symbolic representation to help you, say which property (commutative, associative, or distributive) is being used implicitly by the child. Ethan solves 3x14 by laying out 3 tens, and then 3 groups of 4. He counted: "10,20,30,34, 38, 42." (3x14=3x10+3x4=10+10+10+4+4+4)

Distributive Property

True or false. Justify. If the side lengths of a square are doubled, then the area is doubled. If the side lengths of a square are triples, then the area of the square is tripled.

False, because a square with side lengths of 4in. has an area of 16in^2 and a square with side lengths on 8in. has an area of 64in^2, while a square with side lengths of 12in. has an area of 144in^2.

Find the greatest common factor and the least common multiple of 81 and 45.

GCF: 3x3 or 9 LCM: 3x3x3x3x5 or 405

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. _____ raisins weigh 2000 pounds.

Greater than 2000 pounds because every pound of raisins has more than one raisin; one raisin weighs less than 1 pound

Hecto

Hundred (100)

Centi

Hundredths (0.01)

How many unique lines can pass through a given point?

Infinity

What are good characteristics of a good unit of measure for area?

It tessellates and has 90* angles

(IDENTIFY THE PROBLEM TYPE) Lyddie has a scrapbook of Justin Bieber pictures in her desk. Her mother gave her 22 more. Now she has 36 pictures. How many pictures were in her book to start with?

Join Start Unknown

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. 2000 raisins weigh _____ pounds.

Less than 2000 pounds because every pound of raisins has more than one raisin; one raisin weighs less than 1 pound

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. _____ Kilometers = 2000 meters

Less than because a meter is bigger than a kilometer, so the amount of kilometers in 2000 meters will be fewer than 2000

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. 2000 yards _____ miles

Less than because if there are between 1500 and 1800 yards in only one mile, then 2000 yards would not even amount to 2 miles

(IDENTIFY THE PROBLEM TYPE) Milo collects unusually-shaped squash. He has 45 squash in his collection. If he puts 9 squash on each shelf in his bedroom, how many shelves could he fill with squash?

Measurement Division

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. _____ yards = 2000 miles

More than because, there are between 1500 and 1800 yards in 1 ile, so there would be many more than 2000 yards in 2000 miles

(IDENTIFY THE PROBLEM TYPE) Albert's pumpkin patch had 4 rows of pumpkin. In each row, there are 5 pumpkins. How many pumpkins does Albert have in all?

Multiplication

Base Unit (meter, gram, liter, etc.)

Ones (1)

One polyhedron has surface area of 504cm^2 and volume of 648 cm^3. Give the surface area and the volume of the image if the length of each side is decreased to 1/3 its original size.

SA = 1/6 LW, V = 1/12 LWH SA = 56in^2 V = 24 in^3

One polyhedron has surface area of 504cm^2 and volume of 648 cm^3. Give the surface area and the volume of the image if the length of each side is doubled.

SA = 4LW, V = 8LWH SA = 840in^2 V = 1296 in^3

If you were to create a "sample size" cereal box for which the length, width, and height were each 1/4 the original measurements, how would this affect the surface area of the box? How would it affect the volume?

SA: /16 V: /64

If you were to double the length, the width, and the height of a cereal box, how would this affect the surface area of the box? How would it affect the volume?

SA: x4 V: x8

(IDENTIFY THE PROBLEM TYPE) There were 37 dementors hovering around Hogwarts School of Witchcraft and Wizardry . Some flew away. Now there are 13 dementors left. How many dementors flew away?

Separate Compare Unknown

What is the relationship between the sum of the degrees and the number of edges in a polyhedron.

Sum of degrees = 2E

What is the general relationship between V, n, and E for a polyhedra with a common degree at each vertex?

Sum of degrees = 2E = nV

Deka

Ten (10)

Deci

Tenths (0.10)

The Cardinal Principle

The last counting word said in the process of counting tells the cardinality of the set (i.e., how many objects there are)

Explain why the sum of two odd numbers is an even number.

The sum of two odd numbers is always an even number because an odd number is just an even number plus one, and if there are two odd numbers being added, then it will be an even number plus an even number plus two, so those three even numbers wold equal an even number.

Kilo

Thousand (1000)

Milli

Thousandths (0.001)

How many points determine a plane and what is the necessary relationship?

Three noncolinear points

True or False. Justify. A rectangle is never a trapezoid

True because a trapezoid only has one set of parallel line, but a rectangle has to have two sets of parallel lines

Euler's Formula

V - E + F = 2

What does it mean for A to be a multiple of x?

When A can be divided equally by x, A is a multiple of x.

What does it mean for y to be a factor of B?

When B can be divided equally by y.

(This scale shows weight to the nearest 1/2 pound) On consecutive days, one person's readings were 151.0 and 151.0 on the scale. Is it possible that the person had actually gained or lost weight? Explain

Yes, because the scale only weighs to the nearest half-pound, so one day they could weigh 151.27 and then 151.09, but the scale would still show 151.0 for both days

You need to buy a ladder to climb onto your roof. Your rain gutters along the bottom edge of your roof are 13 ft. off the ground. You have bushes around the perimeter of your house, so you will have to place the base of your ladder 5 feet out from your house. Can you buy a 15-ft. ladder and use it to reach the roof of your house? Explain.

Yes, you can buy a 15-ft. ladder and use it because of the pythagorean theorem (a^2 + b^2 = c^2 --> 13^2 + 5^2 = c^2 --> 169 + 25 = c^2 --> 194 = c^2 --> c = 13.93 ft.)

Conceptual Subitizing

You know how many there are because of repeated and practiced patterns

If 260 bas = 1.32 x 10^3 bos, which is larger, a ba or a bo?

a ba is larger because only 260 bas can equal 1.31 x 10^3 bos, thus it takes fewer bas to make the same amount

If 0.19 cons = 0.095 cins, which is larger, a con or a cin?

a cin is larger because only 0.095 cinc can equal 0.19 cons, thus it takes fewer cins to make the same amount

What is a composite number?

a number that is made up of more than two factors

What is a prime number?

a number that is made up of two unique factors, one and itself.

Does the associative property hold true for subtraction? In other words, does a-(b-c)=(a-b)-c? Justify your answer

a-(b-c) does not equal (a-b)-c because some amount (b-c) taken from another group (a) will not equal the same amount if some amount (c) is taken from another group (a-b). Ex: 5-(3-2)=4, but (5-3)-2=0

Does the commutative property hold true for subtraction? In other words, does a-b=b-a? Justify your answer

a-b does not equal b-a because amount(a) taken from another group (b) will not equal the same amount if some amount (b) is taken from another group (a). Ex: 5-3=2, but 3-5=-2

If 150 ags = 100 aps, which is larger, an ag or an ap?

an ap is larger because only 100aps can equal 150ags, thus is takes fewer aps to make the same amount

Without calculation, tell whether the number that belongs in the blank is greater than 2000, equal to 2000, or less than 2000. 2000 kilometers = _____ meters

greater than because a kilometer is smaller than a meter, so the amount of meters in 2000 kilometers will be greater than 2000

Right triangle

one 90* angle

Obtuse triangle

one angle > 90*

Trapezoid

one pair of parallel sides

Parallelogram

opposite sides parallel

Which of the regular polygons tessellate?

square, triangle, hexagon

A number is divisible by 4 if...

the last 2 digits together are divisible by 4

A number is divisible by 8 if...

the last 3 digits together are divisible by 8

A number is divisible by 10 if...

the last digit is 0

A number is divisible by 3 if...

the sum of the digits is divisible by 3

Evidence of base 10 understanding:

1.) Counting a group of 10 as a unit 2.) Showing a unit of ten as equal to 10 individual units 3.) Recognizing groupings in spoken number words 4.) Recognizing the positional values of the digits 0-9 in written numerals 5.) Using invented algorithms/mental strategies that depend on base ten understanding

A plane exists in ______ dimension(s)

2

Explain how the distributive property might be used to more easily find the product of 27x3

27x3 can be separated into (25x3) + (2x3), which would equal 75 + 6 =81

What is the prime factorization of 924?

2x2x3x7x11

Space exists in _______ dimension(s)

3

TRIANGLE: n, vertex angle, sum of vertex angles, central angle, exterior angle

3, 60, 180, 120, 120

For a regular 20-gon find the measure of the central angle

360 / n 360 / 20 = 18*

Describe how you would solve 37/5 using compensating

37/5 could be turned into 35/5, which would equal 7 and then dividing 2/5 to get 0.4 (if necessary) or just recognizing that there will be a remainder of 2.

PENTAGON: n, vertex angle, sum of vertex angles, central angle, exterior angle

5, 108, 540, 72, 72

HEXAGON: n, vertex angle, sum of vertex angles, central angle, exterior angle

6, 120, 720, 60, 60

Equilateral triangle

All congruent sides

Using the symbolic representation to help you, say which property (commutative, associative, or distributive) is being used implicitly by the child. To solve 29+45, Alex said: "I can take 1 from the 45 and give it to the 29, so now I have 30+44, which is 74." (29+45=29+1+45=30+45)

Associative Property

Elle counts out 16 cubes. Then, pulling one cube away from the group at a time, she counts, "15, 14, 13, 12, 11, 10, 9." She counts the cubes she pulled away, "1, 2, 3, 4, 5, 6, 7." My answer is 7." Assuming Elle is using a strategy that corresponds to the structure of the problem, write a story problem that Elle could be trying to solve.

Billy has 16 carrots. He fed some to his pet rabbit, then Billy only had 9 carrots left. How many carrots did Billy feed to his rabbit?

Using the symbolic representation to help you, say which property (commutative, associative, or distributive) is being used implicitly by the child. To solve 29+45, Jessie said: "I did 20 and 40 and got 60. Then I did 9 and 5 and I got 14. That was another ten, so now I have 7 tens and 4 extra ones. That's 74. (29+45=20+9+40+5=20+40+9+5)

Commutative Property

(IDENTIFY THE PROBLEM TYPE) Violet has 29 pet frogs. Crysanthema has 15 pet frogs. How many more pet frogs does Violet have than Crysanthema?

Compare Difference Unknown

(IDENTIFY THE PROBLEM TYPE) Keilani has 27 students in her section of MTHED 305. Brandon has two fewer students than Keilani. How many students does Brandon have?

Compare Referent Unknown

The One-to-One Principle

Exactly one number word (no more, no less) is matched with every object being counted

Formula involving the number of vertices (V), edges (E), and faces (F) of any polyhedron

F + V - 2 = E

Given the base of a prism has n side, how many faces (F), vertices (V), and edges (E) does it have? Reason about the shape itself to explain how you derived each expression.

F = n + 2 because there are two base faces and another base coming from each side to connect the two bases. V =2n because each face will connect at the vertex of each base, and since there are two bases, you multiply the number of sides by 2. E = 3n because you count the edges of each base (2n) and add the edges coming off of each side of the base (n), thus giving you 3n.

True or False. Justify. All right triangles are similar

False because all right triangles have one 90 degree angle, but the other two angles are not necessarily always similar to one another

True or False. Justify. There are six regular polyhedra

False because there are only 5: tetrahedron, cube, octahedron, dodecahedron, and icosahedron

(IDENTIFY THE PROBLEM TYPE) Nineteen children are going on the 2nd grade class field trip to the swamp. If 5 children can ride in each car, how many cars are needed so all the children can go?

Measurement Division

Suppose you want to enlarge a photograph that measures 3 inches by 4 inches in such a way that the shape of the photograph is maintained. The 4-inch side now measures 10 inches. How long does the 3-inch side measure? Explain how you arrived at your answer.

NL - OL = x x / OL = ratio APPLY TO NEW SIDE ratio x OL = b b + OL = NL

Do a hexagon, pentagon, and heptagon together tessellate around a point?

No, because a pentagon's vertex angle measurement is 108*, a hexagon's vertex angle measurement is 120*, and a heptagon's vertex angle measurement is 128.6*. For these three shapes to tessellate, they must add up to 360*, but these three shapes add up to only 356.6*

The Stable Order Principle

Number words being matched with objects must be said in the correct order

(IDENTIFY THE PROBLEM TYPE) There are 13 members in the Chocolate Appreciation Club. Eight club members are girls. How many are boys?

Part-Part-Whole Part Unknown

(IDENTIFY THE PROBLEM TYPE) Virginia decorated 6 T-shirts with neon puff paint. She has 3 best friends. She would like to give each best friend the same number of painted T-shirts. How many T-shirts will each of her best friends get?

Partitive Division

Describe how you might solve the following problem by incrementing: There are 58 geese on the pond. 27 geese fly away. How many geese are left on the pond?

The child would start with 58 and then incrementally decrease by 10 (58, 48, 38) and use his fingers to count down the remaining 7 (38, 37, 36, 35, 34, 33, 32, 31) to get the answer of 31.

The Order Irrelevance Principle

The order in which objects are counted doesn't doesn't affect how many there are

True or False. Justify. Some kites are rectangles

True beaus a rectangle needs four 90 degree angles, and a kite has congruent adjacent ides, and a rectangle is sometimes a square, thus all four sides are congruent, which creates a kite

True or False. Justify. All squares are kites

True because a kite has congruent adjacent sides and since a square has all four congruent side, the adjacent sides will always be congruent, thus making it a kite.

True or False. Justify. All rectangles are parallelograms

True because a parallelogram has two sets of parallel sides and a rectangle, with four 90 degree angles, must have two sets of parallel sides

True or False. Justify. If a parallelogram has at least one right angle, it is a rectangle

True because if there is one right angle, the opposite angle also has to be a right angle in order to keep the opposite sides parallel, thus forcing the shape to have all four 90 degree angles

True or False. Justify. Every rhombus is a parallelogram

True because in order for all sides to be congruent in a rhombus, the opposite side must be parallel

True or False. Justify. Two Regular octagons and squares (together) will tessellate

True because two regular octagons and a square, coming together at a point will add up to 360 degrees, therefore, it will tessellate

True or False. Justify. Every triangle has at least one acute angle

True because, in order for the angles of any triangle to add up to 180 degrees, there must at least be one angle with less than 90 degrees

How many points determine a line?

Two

When using our standard addition algorithm why do we add from the right to left rather than from left to right?

We add from right to left because adding from left to right causes a lot of erasing and crossing out of the digits in each column when the sum of the digits in that column is greater than 9. When we add right to left, it is cleaner and we do not need to cross out any digits of the place value before.

if A and B are both multiples of x, explain why it is true that A + B is also a multiple of x.

When A can be divided equally by x and B can be equally divided by x, when both A and B are added together, they are still able to be equally divided by x, thus showing that A + B is a multiple of x.

Does the associative property hold true for subtraction? In other words, does a/(b/c)=(a/b)/c? Justify your answer

a/(b/c) does not equal (a/b)/c because some amount (a) divided into some amount of groups (b/c) does not equal the same amount if some number (a/b) was divided into some amount of groups (c). Ex: 9/(5/2)=3.6, but (9/5)/2=0.9

a/0

a/0 = undefined because a/0=c iff a=0xc and a cannot equal 0. In this equation, a must be 0, but a cannot equal 0, so a/0 is undefined.

Does the commutative property hold true for division? In other words, does a/b=b/a? Justify your answer

a/b does not equal b/a because some amount (a) divided into some amount of groups (b) does not equal the same amount if some number (b) was divided into some amount of groups (a) Ex: 15/5=3, but 5/15=0.3333...

Rectangle

all 90* angles

Acute triangle

all angles < 90*

Square

all congruent sides

Rhombus

all sides congruent

Isosceles triangle

at least 2 congruent sides

How many faces, vertices, and edges would an n-gonal pyramid have?

n + 1 faces n + 1 vertices 2n edges

N-GON: n, vertex angle, sum of vertex angles, central angle, exterior angle

n, (180 x (n-2))/n, (180 x (n-2)), 360/n, 180-vertex angle

A number is divisible by 5 if...

the last digit is 0 or 5

A number is divisible by 2 if...

the last digit is divisible by 2

A number is divisible by 6 if...

the number is divisible by both 2 and 3; or the number is divisible by 3 and is even

Kite

two pairs of adjacent congruent sides

if y is a factor of B, is y also a factor of 3B? Explain.

y is only a factor of 3B if y is also a factor of 3. If not, then they cannot be a factor of 3B.