Ron Paul Curriculum 6th Grade Mathematics Problem Set #136

Ron Paul Curriculum 6th Grade Mathematics

Problem Set #136

1. What basic object in geometry represents a location and has no specific size or

dimension?

2. What is a figure?

3. Fill in the blank: A ______ is a subset of a line made up of two endpoints and all

the points between them.

4. Draw two triangles that intersect in exactly three points.

5. Draw two rays whose intersection is:

a) a ray

b) a line

c) a point


Problem Set #137

1. For each of the angles below:

a) Name the sides of the angle, using correct notation.

b) Name the vertex of the angle.

c) Express the angle as the union of two rays.

d) Name the angle in three ways.

e) Assume the figures below represent angles of 30°, 170°, 92° and 120°, and

match each angle with its measure.

A

B

CA

B

C

L

M

N

L

M

N

F

D

E

F

D

E

U

T

V

U

T

V


Problem Set #138

1. Answer the following questions about the triangle:

a) Name the sides of the triangle below, using

correct notation.

b) Name the vertices of the triangle.

c) Express the triangle as the union of three line

segments.

d) Name the triangle in six different ways.

2. The circle is divided into 16 equal parts. What is

the measure of each angle?

3. If the angles in the quadrilateral PQRS below are

30°, 80°, 90° and 160°, write down the name of

each of the angles and their measures. Identify if

the angles are acute, obtuse, straight or right angles.

4. Answer always, sometimes but not always, or never

for the following statements:

a) The union of two rays is an angle.

b) The intersection of two rays is an angle.

c) An angle is a union of two rays.

d) A triangle has two equal sides.

e) A triangle has two right angles.

5. What distinguishes quadrilaterals from other figures created by connecting four

points?

P

S

R Q

P

S

R Q


Problem Set #139

1. Use what we learned in today’s lesson about quadrilaterals to answer whether our

original statements are true or false:

a) All squares are rectangles.

b) All rectangles are squares.

c) Some rectangles are squares.

2. Draw a Venn diagram showing the relationship between equilateral triangles,

scalene triangles, isosceles triangles and right triangles.

3. Use the Venn diagram from problem 2 to draw the hierarchy of triangles.

4. A trapezoid can be defined as a quadrilateral with at least one pair of parallel sides.

Use this definition to draw a new Venn diagram for the quadrilaterals, including the

trapezoid, and draw the hierarchy of quadrilaterals, including the trapezoid.


Problem Set #141

1. Use the hierarchy we drew in today’s lesson to draw a Venn diagram of the

following sets of numbers: real, rational, irrational, integer, positive, and prime.

2. Use the hierarchy we drew in today’s lesson to draw a Venn diagram of the

following sets of numbers: real, integer, positive, zero, negative and prime.

3. Use the Venn diagrams above to tell whether the following statements are always

true, never true, or sometimes but not always true:

a) If a number is an integer, then it is also a whole number.

b) Whole numbers are integers.

c) Odd numbers are irrational numbers.

d) Prime numbers are not negative numbers.

4. If possible, give an example of the following:

a) A real number that is an integer, but not a whole number

b) A number that is a rational number, but not a whole number

c) A number that is a real number, but not a rational number

d) A number that is a rational number, but not a real number


Problem Set #142

Solve the following equations for the variables using the correct transformation:

1. x ‒ 10 = 15

2. 4d = 32

3. m/5 = 2

4. y + 100 = 42

5. n × 7 = 49

6. a + 2353 = 5896

7. b ‒ 569 = 254

8. 1427

=c

9. 9g = 207

10. h × (‒4) = 96


Problem Set #143

Solve the following equations for the variables using the correct transformations. Then substitute to

check your answers.

1. 3a ‒ 7 = 26

2. 856

1=+− x

3. 13 ‒ 4y = 25

4. 1794

1=−− x

5. 197

18 −=− v

6. 53 = 5x + 11

7. 22 = 4 + 5x

8. 36124

3=+x

9. ‒8 + 3f = 6

10. 2r ‒ 13 = 95


Problem Set #144

Solve the following equations for the variables. Substitute your solutions to check your answers.

1. 7r ‒ 8 ‒ 4r = 25

2. 12 ‒ 7(x ‒ 4) + x = ‒2

3. ‒5c + 17 ‒ 8c = 56

4. 2(x + 3) ‒ 5(x ‒ 1) = 32

5. 58 = 6 ‒ 14x + 12x

6. 3(2x ‒ 5) + 2x = ‒7

7. 8t + 5 ‒ 7t = 22

8. 6x + 3(x + 7) = ‒15

9. 20x + 84 ‒ 8x = 0

10. 0 = 5(7 ‒ x) + 12x


Problem Set #146

Solve the following equations for the variables. If the equation is an identity or has no

solution (that is, it is inconsistent) write that conclusion.

1. 5(x + 4) = 5x ‒ 13

2. 10c ‒ 51 = 7c

3. 29y + 56 = 27y

4. 7(2 ‒ r) + 3 = 17 ‒ 7r

5. 9x = 34 + 8x

6. 7z = ‒16 ‒ 9z

7. 5x + 7 = 4x + 7

8. 3(3x + 1) ‒ (x ‒ 1) = 6(x + 10)

9. ‒6a = a ‒ 70

10. 2(5 ‒ t) + 6t = t + 22


Problem Set #147

1. As you coast downhill on your bike, you find that the speed increases by 2/3 mile per hour each second. When you are going 12 miles/hour, you start a stopwatch. a) Define a variable for the number of seconds that have passed since you started the stopwatch. Then write an expression for the speed you are going. b) How fast will you be going after 15 seconds? c) What will the stopwatch read when you are going 18 miles/hour? d) How long before you started the stopwatch did you start coasting down the hill?

Use algebraic expressions and equations to solve the following word problems. These problems can all be solved using a single variable.

2. There were 85 erasers in Box A and 15 erasers in Box B. When an equal number of erasers was added to each box, the number of erasers in Box A was 3 times as many as that in Box B. How many erasers were added to each? 3. Tom bought a table and six chairs for $440. If the table cost $200, how much did each of the chairs cost? 4. There is a number such that three times the number minus 6 is equal to 45. Find the number. 5. Separate 90 into two parts so that one part is four times the other part. 6. A collection of coins has a value of 64 cents. There are two more nickels than dimes and three times as many pennies as dimes. How many of each kind of coin are there?


Problem Set #148

Solve the following pairs of equations with two variables by adding or subtracting the

equations. Substitute your solutions into the original equations to check your answers.

1. x + 4y = 17

‒x + 7y = 38

2. 7x + y = 21

3x + y = 13

3. x ‒ 4y = 23

x + 5y = ‒4

4. 2x + 5y = 14

7x ‒ 5y = ‒41

5. 6x + 11y = ‒48

x + 11y = ‒8

For problems 6-10, first transform the equation so that either the x coefficients or the y

coefficients are opposites. Then solve by adding or subtracting the equations, and

check your answers by substitution.

6. 3x + 5y = 17

2x + 3y = 11

7. 4x ‒ 5y = ‒19

3x + 7y = 18

8. 6a ‒ 7b = 12

5a ‒ 4b = 10

9. 7c + 10d = ‒13

3c ‒ 2d = 7

10. 7m ‒ 2n = ‒26

5m ‒ 12n = ‒45


Problem Set #149

Find the solutions to the following pairs of equations by graphing the two lines. You

should print multiple copies of the coordinate graph.

1. 53

2+−= xy

y = x ‒ 5

2. 143

5−= xy

y = ‒2x + 8

3. 85

8+= xy

y = ‒2x ‒ 10

4. 62

3−−= xy

y = x ‒ 1


Problem Set #152

Solve the following inequalities for the variables.

1. d + 150 < ‒125 2. 24.3 + z ≥ 59.1

3. 13

2−>+ s

4. x + 102 ≤ 150 5. Having consumed 470 calories at breakfast, 615 calories at lunch, and 730 calories at dinner, Mike wants to know how many calories he needs in a late evening snack to consume more than 2000 calories for the day. 6. Caleb collects toy cars. This week he bought a package of 12 race cars, and traded two emergency vehicles for 6 new trucks. He now has more than 47 cars in his collection. Write an inequality to show how many cards he had at the beginning of the week. 7. James has over $1,000 in his bank account. He began the account with a deposit of D dollars, added A dollars to the account, and withdrew W dollars. a) Relate the variables D, A and W in an inequality that expresses the amount James has in his account today. b) Suppose James withdrew $257 and added $792. Find an inequality that expresses the possible values for D. Then solve the inequality.


Problem Set #153

Solve the following inequalities for the variables.

1. 4d + 150 < ‒125 2. 18 ‒ 6z ≥ 54

3. s3

21 >−

4. 2x + 102 ≤ 150

5. 145

3>+x

6. 625 ≥ 5x + 100 7. Haley bought identical gifts for her six sisters and paid no more than $180 in all. How much did she pay for each gift? 8. A couple budgeted $6,000 for the meal after their wedding. They chose a meal priced at $45 per person. What is the greatest number of people they could host at their wedding dinner?


Problem Set #154

Graph the following inequalities:

1. y > 2x ‒ 1

2. y ≥ x ‒ 3

3. xy4

1≤

2. 33

1+< xy


Problem Set #156

Graph the images of ∆HOG under the translations described, and label them correctly.

1. The image of (x, y) is (x + 3, y).

2. The image of (x, y) is (x ‒ 3, y ‒ 2)

3. Tell what happens to a the graph of a figure when:

a) k is added to the second coordinate and k is negative.

b) h is added to the first coordinate and h is positive.

4. If (x, y) is a preimage point, explain what transformation yields (x + 100, y ‒ 500).

5. ∆A´BĆ´ is a translation image of ∆ABC. A = (0, 0), B = (½, 0), C = (0, ⅓), and C´

= (¼, ). What are the coordinates of A´ and B´.

6. If ∆Q´R´S´ is a translation image of ∆QRS, as a result of the translation (x + 4.7, y ‒

5.3), how would you translate point (a, b) on ∆Q´R´S´ back to ∆QRS?

7. Translate polygon ABCDEFGH in the following graph so that it is moved into the

bottom left of Quadrant I (that is, point A is translated to the origin). Find the

coordinates of A´BĆ´DÉ´F´G´H´. (The figure is on the following page).

51


Problem Set #157

Use the diagram at right to fill in the blanks:

1. l is called the ________________.

2. H is called the ________.

3. BEHm∠ = ______.

4. 'HH is ________________ to BD .

5. H and H´ are the same _________ from BD .

6. The reflection image of B over BD is __________.

7. Draw the reflection images of the following figures:

8. Draw all the lines of symmetry in the following figures:

Use the coordinate graph below to answer the following questions:

9. What are the coordinates of the image of point (1, 3) reflected over the x-axis?

10. What are the coordinates of the image of point (1, 3) reflected over the y-axis?

11. If the point (a, b) is reflected over the x-axis, what are the coordinates of the

image point?

12. If the point (a, b) is reflected over the y-axis, what are the coordinates of the

image point?


Problem Set #158

Use the technique we saw in the video along with a protractor, compass and ruler to

perform the following rotations (note: be careful of the surface you’re working on

with the compass!):

1. Rotate the figure ‒60° around point P:

2. Rotate the figure 180° around point S:

Each of the following designs has n-fold rotational symmetry. Find n.

3. 4. 5.


Problem Set #159

Make a tessellation with at least 4 copies of the figure below. Use the following steps:

a) Label the four interior angles of the figure.

b) Carefully trace the figure onto a new piece of paper. (This may be easier to do on a

window if you don’t have tracing paper).

c) Use rotations about the centers of the sides to join three more copies of the figure

that have all four of the angles sharing the same vertex.

d) Now trace this new figure at least 4 times onto another sheet of paper to create the

tessellation.


Problem Set #161

Graph the images of ∆HOG under the translations described, and label them correctly.

1. The legs of a right triangle measure 5.5 and 30. Find the length of the hypotenuse.

2. One leg of a right triangle measures 6 and the hypotenuse measures 10. Find the

length of the other leg.

The next questions refer to the soccer field below. Assume the soccer field is 100

yards by 60 yards.

3. Find the following distances:

a) From one corner of the field to the center.

b) From one corner of the penalty area by the goal line to the opposite front corner.

c) From one front corner of the goal area to one front corner of the penalty area.

d) From one front corner of the goal area to the center.

e) From one corner of the field to the closest front corner of the penalty area.

f) From one corner of the field to the furthest front corner of the penalty area, on

the other end of the field.


Problem Set #162

1. Find the area of ∆KHJ, and check that it equals the sum of the areas of ∆KHI and

∆KIJ.

2. Find the area of the red, white, and blue regions in the flag of the Czech Republic:

Find the area of each of the following figures. Assume the sides of the squares in the

grid are one unit long. Check your answers by counting the squares inside each figure.

3. 4. 5.


Problem Set #163

For problems 1-3, use the triangle inequality to tell whether a triangle can be formed

from sides of the given lengths.

1. 3 cm, 4 cm, 5 cm

2. 1⅜ in., 2½ in., 3⅞ in.

3. 12 ft., 12 ft., and 1 ft.

4. For a triangle with sides of lengths p, q, r, write down the three inequalities that

must be satisfied.

For problems 5-7, find the range of values that the unknown side may have and write

this as a double inequality.

5. A triangle with sides of lengths 13, 14 and j.

6. A triangle with sides of 3, 4, and x.

7. A triangle with sides 7.05 cm, 9.95 cm, and s.

8. A door hinge is made of two metal plates that are 1¾” wide. Find the values for the

distance between the two outer edges of the hinge, written as a double inequality.

9. The distance from Toledo, Ohio, to Columbus, Ohio is about 138 miles, and the

distance from Columbus to Cincinnati, Ohio, is about 101 miles. Use the triangle

inequality to find the possible values for the distance from Toledo to Cincinnati, based

on this information.

10. Suppose a triangle has two equal sides of length x. What are the possible values of

the third side?


Problem Set #164

1. Obtain 3 or 4 sheets of paper, a pencil, a ruler, and a pair of scissors. On each sheet

of paper, use the ruler and pencil to draw a large triangle. Try to make the triangles as

different as possible from each other. Cut out each triangle with scissors. Tear off each

corner the triangles, and don’t mix the corners from different triangles. Now assemble

the three corners of each triangle to see that the angles add to 180°. Does the order

that you arrange them in matter? Why or why not?

2. Use the information in the drawing below to find the measure of .

For problems 3-7, if two angles of a triangle have the given measures, what is the

measure of the third angle?

3. 45°, 45°

4. 30°, 90°

5. 70°, 80°

6. 2°, 3°

7. x°, 120° ‒ x°

8. Explain why a triangle cannot have two right angles.

9. In an equilateral triangle, all of the angles have the same measure. What is the

measure?

5∠

10. In the figure below, is perpendicular to . Angle BAC is bisected (split into

two equal parts) by , and the measure of B∠ = 70°. Find the measures of angles 1,

2, and 3.

BA AC

AD


Problem Set #166

1. Angles M and N are supplementary. If and , find s.

Find the measures of all the unknown angles in the figures below:

2. 3.

4. 5.

6. Answer the following questions about the figure below:

a) What are the measures of angles 1-5?

b) Which angle is vertical with angle 4?

c) What is the measure of the angle which is

supplementary to angle 2?

d) What is the measure of the angle which is

supplementary to the sum of angles 3 and 4?

°−=∠ 24sMm °=∠ 72Nm

M∠M∠ N∠N∠

34°34°

104°104°

79°79°

111°111°

P

19°

38°

1

2

3

4 5

P

19°

38°

1

2

3

4 5


Problem Set #167

Questions 1-8 refer to the figure to the right.

1. Name the transversal.

2. Name all pairs of alternate interior angles.

3. Name all pairs of same-side interior angles.

4. Name all pairs of alternate exterior angles.

5. Name all pairs of same-side exterior angles.

6. Name all pairs of corresponding angles.

7. Name all pairs of vertical angles.

8. Name all linear pairs.

For problems 9 and 10, find the measures of all the angles given the measure of one

angle. Assume lines l and m are parallel.

9. °=∠ 563m

10. °=∠ 1028m

x

y

z

12

34

56

78

x

y

z

12

34

56

78

n

12

34

56

78

l

m

n

12

34

56

78

l

m


Problem Set #168

1. In parallelogram DIMA below, find x, and state the property that helps you find it.

In the figure below, ABED and CGFB are parallelograms, is perpendicular to ,

and . H, E, B, and F are on the same line. Find the measure of the

indicated angles in problems 2-7.

2. EBA∠

3. CBE∠

4. DEH∠

5. CGF∠

6. GFB∠

7. DAB∠

AB BC

°=∠ 42EDAm

H

D A

E B F

GC

H

D A

E B F

GC

x° + 10°

108°

A

D

I

M

x° + 10°

108°

A

D

I

M


Problem Set #169

Use the techniques learned in the video today to make a copy of the following figures

onto a separate sheet of paper, and then construct its bisector.

1. 2.

3.

4. 5.

6.


Problem Set #171

For each of the following boxes, draw and label a net and find the total surface area

and volume.

1. 2.

3.

4. Dr. Richards’ raised garden bed is 4’ wide, 12’ long, and 1' deep and it only has

sides, no top and bottom. Find the surface area of the material used to make the raised

bed.

5. Dr. Richards decides to add a floor to his raised bed to control weeds. Find the new

surface area of the material used.

2 ft

1½ ft

½ ft

2 ft

1½ ft

½ ft

¾ in

½ in

3 in

¾ in

½ in

3 in

10 cm

90 cm

120 cm

10 cm

90 cm

120 cm


Problem Set #172

1. Mark and label the center, radius, diameter and circumference for the following

figure:

2. Find the diameter, circumference and area of a circle whose radius is 20 inches.

3. Find the radius, circumference, and area of a circle whose diameter is 5.7 miles.

4. Find the radius, diameter and area of a circle whose circumference is 15π.

5. Find the radius, diameter and circumference of a circle whose area is 79.21π.


Problem Set #173

Find the surface areas and volumes of the following prisms or cylinders:

1. 2.

3.

4. A cylinder with diameter 12 inches and height 12 inches.

5. A cylinder with radius 0.5 cm and length 1 m.

5”

6”

22”5”

6”

22” A = 24.2 ft2

6.2 ft

4 ft

A = 24.2 ft2

6.2 ft

4 ft

A = 24.2 ft2

6.2 ft

4 ft

A = 6 cm2

2 cm

1 cm

9 cm

A = 6 cm2

2 cm

1 cm

9 cm


Problem Set #174

Find the new area or volume of the following figures when the dimensions are scaled

by the indicated scale factors. Then calculate the area or volume to check your answer.

1. k = 5 2. k = 100

3. k = 6 4. k = ¼

5. What will happen to the surface area of a solid if it’s dimensions are scaled by a

factor of k? Find a rule, and then check your rule with the solids in problems 3 and 4

above.

5”

6”

22”5”

6”

22”

¾ in

½ in

3 in

¾ in

½ in

3 in

21’

8’

10’17’

21’

8’

10’17’

r = 15 feet

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Ron Paul Curriculum 6th Grade Mathematics Problem Set #136

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