23.5-6 Volume of a Cylinder

Step 1
Measure the height (h) of the cylinder. Height is sometimes referred to as the length.

Step 2
Measure the radius (r) of the cylinder. Radius is the distance from the outer edge to the center of the circle.

Step 3
Square the radius. (Multiply the radius by itself.)

Step 4
Mulitply the product (radius squared) from step 3 by pi.
Step 5
Multiply the product from Step 4 by the height

Step 6
Write you answer in the proper cubic unit of measurement.

23.4 volumes of Pyramids

The area of a pyramid can be found by using the following equation:

V= 1/3Bh, where B = l x w

This is read as "volume equals one-third base times height." Watch the following video to see why

23.2 Estimate and find Volume

Things to remember:

• Volume is given in cubic units - a little three above the units
• multiply the area of the base (length times width for rectangles - 1/2 lenght times width f or triangles) times the height to find volume

The robot is back...

23.1 Algebra: Surface Area p.502-505

Here is a video that demonstrates finding surface area of a variety of types of prisms. Please excuse the accent.

This video shows how to find surface area for a cylinder.

22.3 Models of Solid Figures

Go to this link from senteacher.org for printable nets that can be used to construct solid figures.

Please make sure to label your solid figure with the following 5 items:

1. Your name
2. Name of the shape
3. Number of faces
4. Number of edges
5. Number of vertices

22.1 Types of Solid Figures

Polyhedron (Polyhedra - plural): solid figure with flat faces that are polygons

• Prism (named for the shape of its bases): polyhedron with two congruent and parallel bases. - The lateral faces(not bases) are rectangles.
• Pyramid (related to a prism) - has a polygon base and triangular sides

NON POLYHEDRON: curved sides with a circular base

• Cylinder: two flat parallel and congruent sides bases and a curved surface.
• Cone: (related to a cylinder) - has one flat circular base and a curved surface leading to a vertex.

The serious video about circumference

A video about circles from the teacher you wish you had...

Circle Song... another one...

Circle Rap

Enjoy!

Circumference jingle

Don't you wish we did more jingles in class!

20.4 Circumference

borrowed from wikihow.com

Below is an explanation from aaamath.com

The circumference of a circle is the distance around the outside of the circle. It could be called the perimeter of the circle.

How to find the circumference of a circle:

• The circumference of a circle can be found by multiplying pi ( π = 3.14 ) by the diameter of the circle.
• If a circle has a diameter of 4, its circumference is 3.14*4=12.56
• If you know the radius, the diameter is twice as large.

Check your understand with the quiz at the bottom of this tutorial from mathgoodies.com

20.3 Draw a Diagram

The focus of this lesson is to be able to illustrate situations in order to help you find solutions to problems. DRAW A PICTURE

20.2 Perimeter

Find the perimeter by adding up all the external (outside) sides. Watch the video for examples of increasingly difficult problems

19.4 Appropriate Tools and Units (p.428-431)

Precision

• Being very exact and accurate and able reproduce the same measurement over and over again
• Here would be an example of less to more precise units of measurement: miles, yards, feet, inches, 1/2 inches, 1/4 inches... The smaller the unit of measure used, the more precise the measurement.

Try this ruler game from rsinnovative.com to check your understanding of measuring in customary units (inches)

19.3 Relate Customary and Metric

Here are the estimates you need for relating customary and metric

1 in ≈ 2.5 cm

1 ft ≈ 30 cm

1 mi ≈ 1.6 km

1 m ≈ 39 in

1 kg ≈ 2.2 lb

1 oz ≈ 30 g

1 fl oz ≈ 30 ml

1 L ≈ 1 qt

19.1 Customary Conversion

Here is a link that shows nice examples for customary measurement and units for conversion from Mountain City Elementary

Try this matching game made by Harcourt (the people who make our textbooks)

18.4 Symmetry

We will discuss two kinds of symmetry:

1. line symmetry: reflection across a line of symmetry

- the image below has line symmetry, this is shown with the dotted lines

2. rotational symmetry: turning around a center point (point of rotation) and lining up again

-the flowers above also have rotational symmetry because you could turn them and have them line up again.

Check out these links from mathisfun.com for line symmetry and rotational symmetry

• Check out this illustration from innovations learning that shows lines of symmetry in shapes
• Here is game from innovations learning that will check your understanding of lines of symmetry form

18.2 Tessellations (p.397-399)

A tessellation is a design of repeating shapes that fit together without gaps or overlaps.

Shapes that tessellate easily include:

• triangles (three sided)
• quadrilaterals (four sided)
• parallelograms (opposite sides parallel - and congruent)

This link from DIY Tessellations is very helpful for helping you think about how to make your own tessellation.

M. C. Escher is likely the best known artist who used tessellations. Check out this link with some of his tessellations:

Here is a link of "wallpaper patterns" that Escher used as inspiration

18.1 Transformations

Check out these links for more exploration with transformation:

• This link uses shapes to model what we did during story time.
• Link from UTC.edu used during "story time" with examples of transformations.
• http://www.bbc.co.uk/schools/ks3bitesize/maths/shape_space/transformations1/revise1.shtml

17.4 Similar and Congruent Figures

CONGRUENT SHAPES

• Same angles
• Same side lengths
• Can be rotated or a mirror image
• A cut-out of one shape will always fit exactly over the othe

SIMILAR SHAPES

• Same angles
• Sides in the same proportion
• Can be rotated or reflected
• One is an enlargement of the other

17.2 Bisect Line Segments and Angles (p. 380-382)

Drawing perpendicular bisector for a line:
Place the sharp end of a pair of compasses at one end of the line, and open it to just over half of the line. Draw an arc which must intersect the line in the position described. Then put the sharp end at the other of the line and, keeping the compassing at the same length, draw another arc which intersects the first one twice and also the line. Then draw a straight line through the two places where the arcs intersect. This line is the perpendicular bisector.

Drawing perpendicular bisector of angle:
Places the sharp end of the compass at the point of the angle and, after having opened it arbitraily wide, draw an arc which intersects the two lines meeting to form the angle each once in the said position. Then remove the compass and, always keeping it opened at the SAME length, place the sharp end at each of the two places where the previous arc cuts each of the two lines meeting to form the angle. In this position with the described length, draw a small arc at each of the said places, which should cross each other. Draw a straight line from the point of the angle to this crossing. This should be the bisector of the angle.

Donald Duck in Mathmagic Land

Although Donald Duck in Mathmagic Land was made by Disney in 1959, it does an exceptional job of illustrating how math is all around us.

17.1 Congruent Segments and Angles (p.376-379)

I. Congruent: Exactly the same is size and shape

A. Segments: Make congruent line segments by using a compass to measure the length of the original line segment and then marking the same distance with your compass on a ray.

Check out this video from hstutorials.net

Here is another technique from mathopenref.com

B. Angles: Build a congruent angles using a straight edge and compass

1. Draw a ray

2. Draw an arc through the original angle

3. Draw the SAME arc through the ray - label the point of intersection

4. Use a compass to measure the distance between the rays in the original angle

5. Using the same compass opening, mark the distance on the arc through the ray

6. Draw a ray from the endpoint in the original ray through the x formed by the arcs in steps 2 and 5

Check out this video form hstutorials.net

16.6 Circles p.368-369

I. Circle is all the points that are the same distance from a center point – this means every point in a circle is the same distance from the center.
II. Lines through a circle: labeled the same way as line segments – with a line above the two letters
A. Radius: a line segment with one endpoint at the center and the other endpoint on the circle
B. Chord: Line segment with both endpoints on the circle
a. Diameter: a chord that goes through the center of the circle
i. A diameter is twice the length of the radius
ii. A diameter is the longest chord that can be made in a circle

Here is an explanation on how to use a compass from mathsteacher.com

16.3 Find a Pattern p.360-361

I. Regular polygon: polygon in which all sides are congruent and all angles are congruent
II. Diagonal of a polygon: a line segment that joings two nonadjacent vertices of the polygon
III. The measure of each angle in a regular polygon with n sides is found using this following equation:

16.2 Triangles - ACTIVITY

• Use a straight edge to draw an example of each of the following
a. Triangles classified by their ANGLES
i. Right angle
ii. Acute angle
iii. Obtuse angle
b. Triangles classified by their SIDES
i. Scalene triangle
ii. Isosceles triangle
iii. Equilateral triangle
• Label the length or angle measures and write a written explanation for each triangle to show why it is different from the others and classified like it is. Use the example as a guide.

16.2 Triangles (p.356-357)

I. ALL TRIANGLES HAVE INTERIOR ANGLES THAT ADD UP TO 180 degrees
II. Naming Triangles by their ANGLES
A. A. Acute triangle: a triangle with all angles less than 90 degrees
B. B. Obtuse triangle: a triangle with one angle greater than 90 degrees
C. C. Right triangle: a triangle with one right angle

III. Naming Triangles by their sides
A. Equilateral triangle: a triangle with three congruent sides
B. Isosceles triangle: a triangle with exactly two congruent sides
C. Scalene triangle: a triangle with no congruent sides

Check out this helpful link from math.com for pictures and more explanation about each of these types of triangles

16.1 Polygons p.354-355

I. Polygon: Poly- =many, in this case 3 or more; -gon = sides

A. A closed plane figure (2-D) with straight sides that are connected line segments

B. We can use triangles to figure out the amount of total degrees for interior angles – all triangeles= 180 degrees total

C. Decagon: a polygon with ten sides, angles, and vertices

D. n-gon: a polygon with n sides, angles, and vertices

II. Vertex: a point where two sides of a polygon meet

15.3 Activity

Draw a design that includes each of the following items at least one time each:
1.) Point- color black
Kinds of Lines: all lines must be STRAIGHT, COLORED and have AT LEAST TWO POINTS - establishing the direction of the line
2.) Ray- red (using an endpoint and arrow to show that it is a ray)
3.) Line- brown(using arrows on both ends to show that they extend forever)
4.) Line segment- blue
Angles – showing using the name and an arc (or square for complementary)
5.) Vertical angles – yellow
6.) Congruent angles – green – show with arc and dash
7.) Adjacent angles – orange
8.) Complementary angles – purple - indicate with a square in the corner (check with the corner of a piece of paper to make sure it is 90 degree)
9.) Supplementary angles – show with a 180 degree arc
A Key
10.) A key that explains the colors used and the lines used

Measuring angles with a protractor

Amble Side Primary has some good practice resources

Here is a tool from math playground

This site from ixl.com sets up the protractor for you in this practice activity

Ch. 15 Angles

Please use the following file as a reference for this chapter

14.4 Geometric Patterns (p. 319-321)

I. Fractal: an endlessly repeating pattern that looks like the whole, but different sizes
II. Iteration: a step in the process of making a fractal that follows a pattern

14.3 Number and Pattern Functions (p. 315-318)

I. Function: relationship between two quantities in which one quantity depends uniquely on the other every input has one output
II. Use an equation, function table, or function machine

14.2 Patterns in Sequence (p. 312-314)

I. Sequence: an ordered set of numbers - each number in the sequence is called a term
II. If you can find a rule, it can be used to find any number in a sequence
III. Figure out if the values are increasing or decreasing

14.1 Problem Solving: Find a Strategy (p. 310-311)

I. Ways to find patterns
A. Make a table
B. Figure out the relationship between consecutive numbers
C. Find the difference between consecutive numbers

13.5 Inequalities (p.300-303)

I. An inequality is an algebraic sentence that contains the symbol <, >, ≤, ≥, or ≠ because the sides are not equal
II. An inequality can sometimes be solved the same way as an equation
III. When graphing on a number line
A. <> Use an open circle if the value at that spot is not included in the solution
B. ≤ or ≥ Use a filled in circle if the value at that spot is a possible solution
IV. If you multiply both sides of an inequality by a negative number it will reverse the direction of the inequality sign

13.4 Problem Solving: Work Backwards (p.298-299)

13.4 Problem Solving: Work Backwards (p. 298-299)
I. Use when the problem describes a series of actions and tells the result
II. Can be used to solve a problem represented by an equation by undoing the operation
III. When a desired outcome is known, work backwards to figure out what you need to do to make it work

13.3 Use Formulas (p.294-297)

I. A formula is a rule or principle used to find a value: i.e. area, perimeter, volume, etc…
II. Use properties of equations to solve for the variable in a formula
III. Useful formulas:
Converting between Fahrenheit and Celsius :
F = (9/5 x C)+32
C = 5/9 x (F-32)
Average Speed: d = rt (distance equals rate times time)

13.1-2 Solve Multiplication and Division Equations (p.290-293)

I. Focus on getting the variable by itself on one side of the equation – use the opposite operation to cancel and balance the equation.
II. Division Property of Equality: If both sides of an equation are divided by the same nonzero number, the two sides remain equal
III. Multiplication Property of Equality: both sides of an equation are multiplied by the same number, the two sides remain equal.

12.5 Problem Solving Strategy: Write an Equation (p. 282-283)

I. Identify which quantities are equal to each other.
II. Use key words in the problem, identify operations needed to solve.
III. Use a variable to represent any unknown quantity
IV. Write an equation that models what is given in the problem

12.4 Solve Subtraction Equations (p. 280-281)

I. Focus on getting the variable alone on one side of the equation
II. Addition Property of Equality: Add the same number to both sides of an equation, the two sides will remain equal

12.2-3 Solve Addition Equations (p.277-279)

I. Focus on getting the variable alone on one side of the equation
A. Use the subtraction property of equality to subtract an equal amount from BOTH SIDES of the equation

12.1 Words and Equations (p.274-275)

I. Use the same strategies to translate word expressions to equations – similar to the way you translate word expressions to numerical and algebraic expressions.
II. Use variables to represent an unknown number or value
III. “is” translates as “=”

11.3-4 Expressions with Squares and Square Roots (p.264-267)

I. Use order of operations to evaluate any expression
II. In the order of operations, square roots are evaluated at the same time as exponents.

11.2 Evaluate Expressions (p.260-263)

I. Evaluate means to solve or find a value

1. The value of the expression will change, if the value of the variable changes
2. Follow the order of operations as usual
3. Replace any variables with values given before evaluating
4. Combine like terms to make the expression simpler to solve (terms are seperated by addition or subtraction symbols

11.1 Write Expressions (p.258-259)

11.1 Write Expressions
I. Use variables to represent unknown values
II. Operation key words
Addition: sum, increase, more than, plus
Subtraction: difference, decrease, less than, minus
Multiplication: Product, factors, times, multiplied by
Division: quotient, equally shared, divided by

III. Operations are only performed when evaluating, not when writing the expression

10.4-5 Divide Fractions and Mixed Numbers (p.234-239)

1. Change mixed numbers into improper fractions
2. Write whole numbers as fractions (8 = 8/1)
3. Turn the second fraction (the divisor) upside down (the reciprocal) then rewrite the problem as a multiplication sentence.
4. Multiply as usual.
Check out this tutorial from mathisfun.com on how to divide fractions.

10.3 Multiply Mixed Numbers

1. Convert any mixed number into an improper fraction.
2. Multiply the fractions (numerators, then denominators, simplify)
3. Turn the answer back into a mixed number
Check out this tutorial from mathisfun.com

10.2 Multiply Fractions

Three steps for multiplying fractions
1. Multiply the numerators (top)
2. Multiply the denominators (bottom)
3. Simplify

Check out this tutorial and video from mathisfun.com

10.1 Estimate Products and Quotients (p.226-227)

I. Estimate fractions by rounding to 0, 1/2 or 1 (n x 0=0, n x 1/2 = n/2, n x 1 = n)
II. For mixed numbers, round to the nearest whole number
III. For division use compatible numbers