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Relevant Learning Objectives
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Know the properties of two-dimensional figures.
Determine the perimeter and area of plane figures.
Solve simple equations.
Plot and identify points in the first quadrant of the coordinate plane.
Determine the number of permutations and combinations of items.
Determine how to approach a problem and then find the solution.
Know the different types of quadrilaterals.
Identify and apply proportional relationships in scale drawings.
Convert measurement units within a measurement system.
Evaluate numerical expressions, including those involving exponents.
Determine the range and mean of a data set.
Understand the concept of regular polygons.
Use the four operations with fractions, decimals, and mixed numbers.
Apply the correct order of operations to solve problems.
Compare and order fractions, decimals and common percents.
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Know the properties of two-dimensional figures.
Students should be able to disuss the definitions of parallel lines, perpendicular lines,
acute angles, obtuse angles, right angles, and straight angles.
Tutorial:
This lesson focuses primarily on understanding the concept of parallel and perpendicular
lines, and types of angles. These concepts are important to understand in order for the
student to identify figures and understand the properties of two and three-dimensional
figures as he or she moves through basic geometric concepts.
Begin by a review of the terms parallel and perpendicular. The student may or may not
already be familiar with these terms. As you determine that the student has a basic
understanding of each concept, move on to the next.
 Parallel Lines: Two lines in the same plane that do not intersect
 Perpendicular Lines: Two lines that intersect to form right angles
Look and/or walk around your house and find parallel and perpendicular lines. For
example, the wall is perpendicular to the floor. The tabletop runs parallel with the floor.
Ask the student if to point out other examples of parallelism and perpendicularity.
Next, review angle types.
 Acute Angle: an angle greater than 0 degrees and less than 90 degrees
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Right Angle: an angle equal to 90 degrees
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Obtuse Angle: an angle greater than 90 degrees, but less than 180 degrees
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Straight Angle: equal to 180 degrees
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If the student doesn’t notice that all right angles are also perpendicular lines, then discuss
it with the student. Point out some of the examples you’ve just looked at in the house.
Again, look and walk through your house to find examples of acute, obtuse, and right
angles. Discuss the fact that all of the examples of perpendicular lines will also be
examples of right angles.
Extended learning: Draw angles of different sizes on a piece of paper and ask the
student to name the type of angle. Don’t worry that the lines aren’t perfectly straight;
use this as an activity to strengthen his or her understanding of the concept of naming
angles.
Quick Tips: This activity can be done while in the car or at any location. Use other
examples, such as the streets running parallel and perpendicular.
Review:
What is a right angle?
What is the difference between parallel and perpendicular lines?
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Determine the perimeter and area of plane figures.
Students should be able to determine the perimeter and area of plane figures.
Tutorial:
For this activity, you will need a basic calculator.
Have the student write these formulas on a note card to keep for easy reference. Help
him or her create a collection of note cards containing math facts for review.
 Area

Perimeter
Explain to the student that problems involving the perimeter and area of squares,
triangles, and rectangles involve placing known or given values for the sides of these
objects into the formulas given above. Remind him or her that problems seeking the area
of an object must have the answer expressed in "square" units, such as square feet,
meters, etc.
Example
Find the area of the triangle pictured below:
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Step 1
Identify the parts of the triangle that are used in the area formula: height and base. What
are the values of "h" and "b"?
(h = 10 and b = 4)
Step 2
Use the formula for area of a triangle: input the known values into the formula.
Step 3
Use the calculator to evaluate the equation
A = 2 X 10
A = 20
Step 4
Identify the units involved.
Since the units involved are inches, and this is an area problem, the final answer will be
in terms of square inches.
A = 20 square inches
OR
Activity
The next time you order a pizza for delivery, save the top of the cardboard box. Assist the
student in cutting the top off, so that a square piece of cardboard is obtained. Have him
or her measure the sides of the box, and then use the formula to calculate the area.
When he or she has completed this task, assist the student in cutting the box in half, but
in a special way. Cut the box in half from one corner to the opposite corner with a
diagonal cut.
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The result is two triangles. Ask the student to recall the area of the original square, and
then ask him or her to tell you the area of one of the triangles that exist as a result of the
cut you just made. If the student has difficulty, remind him or her that you just cut the
box in half. Once he or she understands, have the student examine the formula for the
area of a triangle and compare that to the formula for the area of the square. Show him
or her that the formula "predicted" that the triangle’s area would be one-half that of the
square. This might assist the student in understanding the relationship between triangles
and squares (or rectangles) in future area problems.
Review:
What is the area of a triangle with a height of 20 feet and the length of the base of 12
feet?
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Solve simple equations.
Students should be able to solve simple equations.
Tutorial:
For this tutorial, you will need:
 A sheet of scratch paper
 A basic calculator
 A number line
Solving an equation means finding the value of some unknown quantity (a variable,
usually represented by a letter x) that makes the equation true. Explain to the student
that this will involve isolating the variable on one side of the equal sign "=", and "moving"
all other elements of the equation to the other side. Of course, the "moving" has to be
done the right way. Go through the following example with the student.
Exercise 1
Solve the following equation:
x - 7 = 11
Tell the student that one way to think of an equation is as a scale that is perfectly
balanced. If you add or subtract something from one side, you must do the exact same
thing to the other side or it will be "out of balance."
The directions above say to "isolate the variable" on one side of the equal sign. That
simply means that we have to get the x by itself somehow and still keep the equation
balanced. Point out to the student that one way to do this is to think of opposites.
Looking at the equation, the variable isn’t isolated. There is a (-7) attached to it.
"Thinking of opposites" means we are going to put a (+7) right next to it, because (+7) is
the opposite of (-7).
Step 1
Put in the opposite value.
Ask the student if the equation is balanced. Point out that the left side was changed, so
now the right side has to be changed in exactly the same way.
Step 2
Balance the equation.
Now the opposite has been put in, and the equation has been balanced. Tell the student
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the equation can now be simplified. Ask him or her to look at the left side first and think
of a number line. Since x is just some number, have the student use his or her number
line and pick any number in the middle as a starting point. The student should subtract 7
from that number, arriving at a new number on the line. Now have the student add 7 to
that new number and notice that we’re back where we started. Explain to the student
that this illustrates why we use opposites to solve equations - they cancel each other out.
That means that the left side of the equation x - 7 + 7 really is the same as just x.
Step 3
Simplify the equation.
x - 7 + 7 = 11 + 7
x = 18
Step 4
Check your result in the original equation. That means replace x in the original equation
with the solution you’ve just obtained.
Does 18 - 7 = 11? Yes! So, you know that the solution was correct.
Have the student get in the habit of checking his or her results. It is a great habit that will
improve math grades over time.
More Practice!
Exercise 2
Solve the following equation:
3z - 2 = 16
This equation is a little different in that the variable is being multiplied by a number other
than "1". The method of solution will be exactly the same as example 1 - just use
opposites.
Step 1
Put in the opposite value, and make sure the equation is balanced.
Point out to the student that +2 was the opposite value for this problem and that it was
placed on both sides of the equation to make it balanced.
Step 2
Simplify the equation.
3z - 2 + 2 = 16 + 2
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3z = 18
Have the student remember that the -2 and +2 cancel each other out, leaving just the
3z. Ask the student if the variable is isolated.
The variable is not isolated, because it is being multiplied by 3. Again, think of opposites.
Since the variable is being multiplied by 3, tell the student that the student must divide
by 3 in order to continue, and that division must be done to both sides of the equation to
keep it balanced.
Step 3
Divide both sides by 3.
Step 4
Check your result in the original problem.
Does (3 x 6) - 2 = 16 ?
18 - 2 = 16 ?
16 = 16
The answer checks out, and the problem is solved correctly!
Review:
Solve the following equation:
12m - 4 = 20
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Plot and identify points in the first quadrant of the coordinate plane.
Students should be able to plot and identify points in the first quadrant of the coordinate
plane.
Tutorial:
Items needed for this activity:
 Graph paper (you can draw the graphs freehand if graph paper is not easily
accessible)
 Pencil and paper
Here are a few definitions you will need to be familiar with as you work through the
activities with the student. Use the following to define coordinate plane, ordered pair and
point. Ask the student to read through the definitions and point to each on the example
below.
Coordinate Plane: The plane determined by a horizontal number line, called the x-axis,
and a vertical number line, called the y-axis, intersecting at a point called the origin. Each
point in the coordinate plane can be specified by an ordered pair of numbers.
Ordered Pair: Set of two numbers in which the order has an agreed-upon meaning, such
as the Cartesian coordinates (x,y), where the first coordinate represents the horizontal
position, and the second coordinate represents the vertical position.
Point: A location in a plane or in space, having no dimensions.
Discuss with the student the uses of the coordinate plane. The student might tell you that
we use the coordinate plane in mathematics to graph and solve equations. However, we
also use the coordinate plane in the real world. For example, a landscaper might use
coordinates to determine where she will plant trees on a landscape.
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Using the graph above, she can determine both where and how far apart she will plant
the trees. We can count the units between trees by counting 4 units from one tree to the
next tree vertically. Let’s say that each unit was equal to 3 feet, then we would know that
the trees are 12 feet apart. We can also subtract the y coordinates, since 5 - 1 = 4, we
can determine that the trees are 4 units apart.
Ask the student to think of other ways that graphing in the coordinate plane would be
used in the real world. He or she might think of exploring space, building, art, etc.
Exercise 1
For the following exercise, you can either print the graph or re-draw it on graph paper.
What are the coordinates of point A, B, C and D?
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Solution:
A (2, 2)
B (6, 1)
C (5, 5)
D (8, 7)
Exercise 2
The student will need to plot these points by creating his or her own graph.
Plot the following points on the coordinate plane.
A (3, 5)
B (4, 7)
C (1, 8)
D (8, 6)
Solution:
For enhanced learning, ask the student to plot points in all quadrants (or quarters) of the
coordinate plane. In the exercises above, we have been working only with positive
numbered coordinates. Continue working with the student using negative coordinates in
all four quadrants.
Review:
Define coordinate plane, ordered pair, and point.
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Determine the number of permutations and combinations of items.
Students should be able to determine the number of permutations and combinations of
items.
Tutorial:
Begin by discussing with the student the definitions of permutations and combinations. In
order to understand combinations, the student will first need to understand the difference
between permutations and combinations.
Have the student write on 2 note cards each definition below. The student will later add
other information to his or her note cards.
 Permutation: an arrangement of a group of items in a particular order.
For example, if we want to seat Anna, Jackson, and Dominique in a row of seats,
how many different ways could we arrange them? (Jackson, Dominique, Anna;
Anna, Jackson, Dominque; Jackson, Anna, Dominique...)
 Combination: a group of items in which the order of items is NOT important.
For example, the combination of (Andrea, Jose) is the same as the combination
(Jose, Andrea).
Below is an example illustrating the difference between a combination and permutation.
Work through the example with the student.
Suppose Chang, Harlin, and Jackson run a 100 meter race. The first two to cross the
finish line will make the varsity team.
There are 6 possible ways they can finish, however there are just 3 different possible
results for whom will make the varsity team.
The list on the left of the graph shows 6 permutations and the list on the right shows 3
combinations.
Activity 2: Understanding Equations
Begin by showing the student the following equations for permutations, which will help
him or her quickly solve problems containing permutations.
Suppose there are 6 people who are going to give a speech in class today. In how many
different orders can the speeches be given?
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We can use the counting principle to determine permutations:
We can also use factorial notation to express our permutation. Remember that a factorial
is expressed using an exclamation point, 6!:
However, the equation for a permutation is typically written as:
This would be solved as follows:
Similarly, let’s suppose we have 5 teachers and 2 will be chosen for teachers of the year.
The number of permutations of 5 teachers taken 2 at a time is written as follows:
And we solve this equation as follows:
You can use permutations to find combinations. Using the above example, you can find
the number of combinations of 5 things taken 2 at a time and dividing by the number of
ways 2 things can be arranged, or 2!.
The combination of 5 things taken 2 at a time is written as:
Let’s Practice!
Work through the following examples with the student. Determine if each situation is a
permutation or combination, and find the number of possible outcomes. The solutions are
below the questions. Remember, with a combination, the order is not important.
1. Start 5 soccer players from a team of 12.
2. Arrange 4 students in a row of seats.
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3. Select 3 people for student government from a choice of 10.
Determine the number of combinations for the following problems.
4.
5.
Solutions
1. combination; 792
2. permutation; 24
3. combination; 120
4. 35
5. 84
Review:
Find the possible combinations if you select 3 players from a roster of 9.
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Determine how to approach a problem and then find the solution.
Students should be able to make decisions about how to approach problems, determine a
problem-solving strategy, and then use those strategies to determine solutions.
Tutorial:
For this activity, you will need:
 A sheet of scratch paper
 A basic calculator
 A computer with Internet access
The student will need to have a variety of problem-solving strategies in his or her math
courses, in daily life, and in his or her future career.
Review the following problem-solving strategies with the student:
 First, determine exactly what the problem is asking you to solve for (the unknown).
 Come up with a plan to solve the problem by analyzing the information:
 write down all of the "knowns"
 underline important information within the problem
 convert the problem into a math problem, if possible, using an equation
 draw pictures and/or graphs and label all of the "knowns"
 determine if there is information that is unnecessary
 Solve the problem.
 Then, analyze the solution you have found and determine if it is a reasonable solution
to the problem by looking back at the original problem.
Exercise
Read the following word problem with the student. Print the problem out so he or she can
underline important information. Help the student to determine which problem-solving
strategies to use. Help with the solution is listed below, but work through solutions, and
give the student enough time to consider all strategies.
Rosie loved her garden. She was going to take her prize rose bush to the state fair. She
wanted her rose bush to be as tall as possible when the judges looked at the rose bush at
6:00 pm Saturday evening.
She spent hours and hours trying to figure out which mixtures of plant food would make
her rose bush grow as tall as possible. Finally, she found a mixture that made it grow to
15 feet tall. Rosie thought this was great, but she still wanted her rose bush to be taller.
She went to a magic gardener in the city, Mr. Greenthumbs, and asked for the plant food
that would make her rose bush grow as tall as possible. Mr. Greenthumbs told her that he
had a magic plant food that would make her rose bush grow to the greatest heights, but
she must be very careful! She would have to time it just right in order for her rose bush
to be the right height. Here are the instructions:
"When you feed the rose bush the plant food, it will double in height for the first 5 hours,
then it will shrink by one-third each hour until it is back to its original height. It will have
to fit through 30-foot doors to get into the building where the state fair is held."
Rosie had a soccer game on the same Saturday as the state fair. Her soccer team had
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won their last 3 games. Rosie was very excited to play in this game on Saturday because
they were having a team pizza party right after the game. She would have to hurry back
after the soccer game, but she would have plenty of time to load up her rose bush into
the back of her brother’s truck and go to the state fair grounds. The only time Rosie
would be able to feed the magic to her rose bush was at 6:00 a.m. in the morning
because she will be busy that day.
Will Rosie be able to get her rose bush through the doors if she feeds it the magic plant
food at 6:00 am? (Round all answers to the nearest ones place.)
Solution
1. What is the problem asking? (It is asking if the rose bush will be less than 30 feet
so it can fit through the doors of the building at the state fair.)
2. Come up with a plan. In this scenario, it would probably be most helpful to
underline important information since there is a lot of miscellaneous information
(help the student determine the important pieces of information to underline). It
would probably be helpful to draw some type of graph showing what height the rose
bush will be at each hour. See sample below.
3. Solve.
4. Check our solution to determine if it is reasonable.
Below is a sample of what your graph might look like in order to solve the problem:
At 11:00 am, it has grown for 5 hours. Now it is going to shrink by one-third every hour
until it reaches its original height of 15 feet.
Since it will shrink by one-third, we can multiply each height by one-third and subtract
one-third of the height, or we can multiply by two-thirds.
Problem Area Alert! You might want to multiply by one-third, BUT we actually need to
subtract one-third of the height at that time.
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The rose bush will be 28 feet at the time she takes it to the state fair, so it will fit through
the doors!
Review:
Solve the following problem:
Ticket sales for the school dance started out slow.
On the first day, only 8 tickets were sold.
On the second day, 10 times that many tickets were sold.
During the next 3 days, 150 more tickets were sold than the day before.
What was the total number of dance tickets sold at the end of the week?
Check your answer!
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Know the different types of quadrilaterals.
Students should know different types of quadrilaterals (square, rectangle, rhombus,
trapezoid, kite, and parallelogram).
Tutorial:
For this activity, you will need the following:
 Note cards
 Pencil
Exercise 1
Have the student label a set of note cards with the names of the shapes listed below on
one side and the descriptions and drawings of each on the other side.
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Rectangle: A quadrilateral with two pairs of congruent, parallel sides and four right
angles
Rhombus: A parallelogram with all sides equal in length
Trapezoid: A quadrilateral with exactly one pair of parallel sides
Kite: A four-sided figure in which two adjacent sides are equal to each other and the
other two sides are also equal to each other
Parallelogram: A quadrilateral with two pairs of parallel and congruent sides
Once the student has created the note cards, spend some time having him or her "study"
them and use them as a learning tool. You can play a game by putting all of the cards on
a table. Make sure that the definitions/descriptions and drawings are all facing up. Point
to a card and ask the student if he or she knows which shape is described by the drawing
and definition. If the student gets it right, he or she gets a point. Then let him or her
point to a card for you to identify. Take turns going back and forth. The one of you with
the most points at the end wins.
Save these cards. You can review them together whenever you have a few extra minutes,
in the car on the way to school, on your way to soccer practice, etc. You can also add to
them with new shapes and definitions/descriptions and their drawings as the student is
exposed to them.
Exercise 2
Have the student identify objects in your home that are examples of squares, rectangles,
rhombi, trapezoids, kites, and parallelograms. Make it a contest between the two of you.
Each of you can take a piece of paper and a pencil with you, and start off in different
rooms of your home. (You can also do this activity outside.) See who can come up with
the most different shapes in a certain amount of time, or see who can find 10 objects of
different shapes (at least one of each) first. Make sure that you both write down the
object’s name and what shape it is.
Review:
What is the difference between a square and a rectangle?
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Identify and apply proportional relationships in scale drawings.
Students should be able to identify and apply proportional relationships in scale
drawings.
Tutorial:
Items needed for this activity:
 Note cards
 Pencil and paper
 Map of your state
 Ruler
Have the student write the following definitions on note cards:
 Scale Drawing - a drawing that reduces or enlarges the original object
 Scale Model - a representation of an object that is larger or smaller than the actual
size of the object being represented
It is important that the student understand scale, not only to be able to read and
interpret maps, but also should the student choose a field of work that requires it. There
are many careers, such as architecture, the sciences, and construction that require
understanding of this important skill. In addition to careers that directly use scale in their
day-to-day business, the use of scale is often used in other careers. For example, a
business owner might use a smaller scale in order to sketch out the necessary office
space needed to run his or her business. Discuss these careers that use scale, and
brainstorm with the student other careers that might use scale.
Activity 1
Ask the student if the following image below is a scale model or a scale drawing. (It is a
scale drawing.) Work through the following scale activity with the student. We will use
proportion to solve for the unknown values.
NOTE: We can use our "known values" in a proportion to find our "unknown values." One
fraction represents the scale of the drawing (our "known value"), and the other part
represents our "unknown value." Proportions are in the form:
Below is a scale drawing of a soccer field. Use it to answer sample problems 1 through 3.
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Sample problem 1.
Find the width of the field:
Step 1
Let c represent the actual width of the field. Use scale (1:20) to write a proportion.
QUICK REVIEW: Fractions can be written in the form 1 to 20, 1:20, 1 over 20, and in
the standard fraction form:
In this way, we can use our "known" to determine the "unknown."
Step 2
Solve the proportion.
Answer: The width of the field is 40 feet.
Sample problem 2.
Find the length of the field:
Step 1
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Set up the proportion using the scale. We’ll use m to represent the unknown length.
Step 2
Solve the proportion.
Answer: The length is 70 feet.
Sample problem 3.
Find the width of the goal:
Answer: The width of the goal is 10 feet.
Activity 2
Use a map of your state and tell the student to pick 2 cities. Then have the student
estimate the distance between the 2 cities. Have the student write down his or her
estimate, then measure using a ruler. Compare his or her estimate to the actual distance.
For example, if the scale is 1 cm:5 miles and the measure between the 2 cities is 15 cm,
we set up the proportion as follows:
Scale 1 cm:5 miles
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The actual distance is 75 miles.
For enhanced learning, find someone who has a career or hobby in which they use scale
models. For example, a scientist might use scale models of insects to study insects (or
might study insects as a hobby). Someone you know might build toy car models. In
addition, someone in construction or architecture will sometimes create models of
buildings. Then ask that person to explain to the student the reason for the scale model.
(For example, do they use it to see things that would be too small to see in the original
form?)
Review:
The scale between two cities is 1/2 inch : 10 miles.
The distance between the two cities is 25 miles.
What is the measure between the 2 cities in inches?
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Convert measurement units within a measurement system.
Students should be able to convert measurement units within a measurement system.
Tutorial:
Items Needed for This Activity:
 Pencil and paper
Begin by discussing with the student the ways in which we express measurement. Ask
the student the following leading questions:
 How do we measure your height? Do we use miles, yards, or inches? Why do you
think we use feet and inches to describe your height?
 Why do we use pounds to describe your weight? Would it be easier if we used
ounces or fractions of a ton?
We use specific units of measure for particular situations according to the simplest and
easiest way to describe measurement. In order to describe measurement appropriately,
we sometimes need to convert from one unit to another unit.
Have the student measure his or her height using a tape measure or sewing measure (in
this way his or her height will be measured in inches only). Next, ask the student to
convert his or her height to the appropriate units using the chart below.
LENGTH
1 mile = 1,760 yards
1 yard = 3 feet = 36 inches
1 foot = 12 inches
Since we measure height in feet and inches, the student will divide his or her height in
inches by 12.
For example, if his or her height is 55 inches, 55 divided by 12 equals 4 feet with a
remaining 7 inches. His or her height would be described as 4 feet 7 inches.
Use the chart above to make the following conversions.
Conversion 1:
3 feet and 6 inches to inches. (Multiply 3 times 12 since there are 12 inches in a foot, and
then add the 6 inches)
Answer: 42 inches
Conversion 2:
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2 miles to feet.
Since we know that 1 mile = 1,760 yards, we can multiply 1,760 by 3 to come up with
feet per mile.
1,760 x 3 = 5,280 feet/mile
Now we multiply 5,280 feet/mile times 2 miles.
5,280 x 2 = 10,560 feet
Answer: 10,560 feet
We can also practice liquid conversions. Practice a few conversions with the chart below.
LIQUID VOLUME
1 gallon = 8 pints = 4 quarts
1 quart = 32 ounces = 4 cups = 1/4 gallon
Conversion 3:
5 gallons to pints.
Since we know that 1 gallon = 8 pints, we multiply the number of gallons, 5, times 8
pints/gallon.
5 x 8 = 40 pints
Answer: 40 pints
Conversion 4:
7.5 quarts to pints.
Since we know that 8 pints = 4 quarts, we can divide both sides by 4. Then we know that
2 pints = 1 quart.
Since there are 2 pints in 1 quart, we take 2 times 7.5 quarts.
2 x 7.5 = 15 pints
Answer: 15 pints
If you have Internet access, go to www.onlineconversion.com. The student can make up
his or her own problems and then check answers using the online conversion calculator.
Review:
Convert 5,876 yards to miles.
Convert 8 gallons to quarts.
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Evaluate numerical expressions, including those involving exponents.
Students should be able to evaluate numerical expressions, including those that involve
exponential notation.
Tutorial:
For this activity, you will need:
 A note card
 Pencil and paper
For this activity, the student must learn that when evaluating numerical expressions the
order in which the expressions are calculated is important, much like when putting on
your shoes and socks. It is important that you put on your socks first, and then put on
your shoes. Ask the student what would happen if you put your shoes on before your
socks.
In mathematics, to find the correct value of an expression we must use the correct order
of operations, just like putting our socks on before our shoes. Calculations are performed
from left to right, beginning first with parentheses, then exponents, then multiplication
and division (same weight), and finally addition and subtraction (same weight).
There is an easy memory aid when it comes to the order of operations. Simply remember
the expression Please Excuse My Dear Aunt Sally. The first letter of each word
represents the operation: P= parentheses, E= exponent, M = multiplication, D = division,
A = addition, and S = subtraction. Have the student write this simple but important
sentence on a note card to refer to. Also, make sure the student includes that we perform
these calculations from left to right.
Solve the following problems with the student. See if he or she can solve them using the
note card with the order of operations listed. If not, allow the student to refer to the
solutions listed below the exercises for help.
Exercise 1:
Evaluate:
Step 1
Remember Please Excuse My Dear Aunt Sally! In this problem, we do the multiplication
first because there are no exponents or parentheses.
The student should multiply 4 x 5. We get the answer 20.
Leaving the following to evaluate:
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Step 2
The student should now solve the problem.
2 + 20 = 22
Answer: 22
Next, you can use the same problem to demonstrate why the order of operations is
important. What happens if we do NOT use the correct order?
We get the WRONG answer!
Explain to the student that mathematical expressions have only one correct answer.
Even computers follow the order of operations as defined.
Exercise2:
This one might seem tricky because it contains multiplication and division. Ask the
student which part of the equation gets completed first. (Since multiplication and division
are equal in weight, the student must evaluate from left to right.)
Step 1
This leaves the following to evaluate: 9 + 4 x 8
Step 2
Using Please Excuse My Dear Aunt Sally, we know we have to multiply before we add.
4 x 8 = 32
This leaves 9 + 32
Step 3
9 + 32 = 41
Answer: 41
Exercise 3
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Evaluate:
Ask the student what operation should be done first. He or she should indicate that the
equation within the parentheses should be completed first. Remind him or her that the
order of operations will apply to what is inside the parentheses, too.
Step 1
Evaluate the parenthesis (3+5 x 4)
5 x 4 = 20
(3 + 20)
3 + 20 = 23
Step 2
Now we solve the exponents!
This leaves 4 + 25 - 23
Step 3
Ask the student what’s next. (Since addition and subtraction are equal in weight, evaluate
from left to right.)
4 + 25 = 29
29 - 23 = 6
Review:
Evaluate:
9 x 1 x (30 - 10 x 2) =
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Determine the range and mean of a data set.
Students should be able to determine the range and mean of a data set.
Tutorial:
For this activity you will need:
 A basic calculator
 Note card
Have the student write the following definitions on a note card to keep for easy reference.
 The range of a set of data (numbers) is the difference between the highest and
lowest values of the set.
 The mean is the average value of a numerical data set.
Range Example Activity
To calculate the range of a data set, have the student consider the heights of 10 students
in a fourth grade class. For example:
47in., 51in., 55in., 52in., 48in., 57in., 45in., 53in., 55in., and 49in.
Step 1
Place all of the values in order from least to greatest, leaving off the unit of
measurement. For this data set that result is:
45, 47, 48, 49, 51, 52, 53, 55, 55, 57
Step 2
Subtract the smallest value in the set from the largest value. The result is the range of
the data set.
57 - 45 = 12.
So, 12 is the range of this data set. The fact that two or more values are identical has no
effect on the range.
Mean Example Activity
In order to calculate the mean for a data set, it will be necessary to add together all of
the values that appear in the data set. Then, that total will be divided by the number of
numerical entries in the set. As an example, consider the data set that was used for
range above.
Step 1
Add the values together (verify with a calculator):
45 + 47 + 48 + 49 + 51 + 52 + 53 + 55 + 55 + 57 = 512
Step 2
There were 10 numerical entries in the data set, so divide 512 by 10 (Again, use the
calculator to verify your result).
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So, 51.2 is the mean of this data set.
Note: It is not necessary to place the values in order to determine the mean, even
though it was done in this example. Also, the fact that one number was repeated made
no difference either. The value of 55 was just added twice, since it appeared two times in
the set. Lastly, the mean value will always be between the smallest and largest values of
the set.
Discussion
Ask the student what each way of looking at the data tells us. For example, if we wanted
to know approximately how tall your little brother might be in the fourth grade, the
average would be a good measure. If we wanted to know the difference of how much
more one person might grow versus someone else in the fourth grade, the range would
be a good measure.
Ask the student to think of real life examples of how we use the different ways to
measure data. For example, in sports competitions, we use a variety of measures. In
gymnastics, the judges’ scores are averaged to give the gymnast one score. In
basketball, the game is often summarized by saying "there was a 12 point difference".
This is the range of points scored between the teams.
Extended Practice
The ideas of range and mean can be applied to any set of numbers. As an easy activity to
help master these concepts, the student should ask all of the members of his or her
immediate and extended family their ages. Have the student create his or her own data
set based on the values they give the student. From that set, the student can calculate
the range and mean. Have the student use a calculator to verify his or her results.
Review:
What is the range and mean of the following set of numbers?
15, 15, 12, 9, 18, 19, 21
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Understand the concept of regular polygons.
Students should be able to identify what makes a polygon a regular polygon.
Tutorial:
For this activity, you will need the following:
 Compass
 Ruler
 Protractor
A polygon is a closed plane figure with at least three sides. "Closed" simply means that all
side segments of a polygon connect with each other only at vertices (segment endpoints).
A regular polygon is one that is both equiangular (all interior angles have the same
measure) and equilateral (all sides have the same length).
All regular polygons can be drawn within a circle (circumscribed) so that the vertices of
the polygon (the "corners") are the only parts of the polygon that touch the circle. The
center of the polygon would be the center of the circle that circumscribes it. In such an
arrangement, the distance from the center to any vertex would just be the radius of the
circle. Have the student examine the following image:
Point out to the student that the image above is of a regular hexagon (6 sides)
circumscribed within a circle of radius "r." Angle a is an interior angle. After looking at the
picture, ask the student to tell you if all of the interior angles would measure the same as
angle a. (They would.)
Formula:
The sum (in degrees) of all of the interior angles of any polygon with n sides is found by
using the following formula:
sum = (n - 2) X 180.
Since all of the interior angles of a regular polygon are the same, the student can
calculate the measure of each of these angles. The measure of each of the interior angles
of a regular polygon with n sides is found by using the following formula:
Example Exercises:
Exercise 1
Have the student calculate the sum of all of the interior angles of a polygon with 5 sides
(pentagon) using the first formula above.
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Solution:
For a pentagon, the number of sides (n) is 5, so the sum of the angles (in degrees) is:
(5 - 2) X 180
3 X 180
540
Exercise 2
Ask the student to calculate the measure of each interior angle of a regular octagon (8
sides) using the second formula above.
Solution:
For a regular octagon n = 8. Each angle measurement (in degrees) can be determined as
follows:
Therefore, each angle of a regular octagon is 135 degrees.
Activity: Circumscribing a regular polygon with 4 sides (a square)
Have the student draw a circle using a compass, and then use a ruler to draw a square
within the circle so that the corners (vertices) of the square are the only points that touch
the circle. Now, have the student use the protractor to measure each of the interior
angles of the square, and use a ruler to measure each of the sides. Ask him or her to
make a conclusion based on the measurements. (Angles are all equal and sides are all
same length.)
Point out that this circumscribing activity can be performed on any regular polygon - not
just a square. Have the student understand the two components that make a polygon
regular (equiangular and equilateral).
Review:
What are the characteristics of a "regular" polygon?
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Use the four operations with fractions, decimals, and mixed numbers.
Students should be able to solve problems using addition, subtraction, multiplication and
division of positive and negative fractions, decimals, and mixed numbers.
Tutorial:
For this activity, you will need:
 A number line showing positive and negative numbers
 A basic calculator
 Note cards
 Pencil and Paper
Quick Review:
Have the student recall the rules for operations with fractions. Specifically, have the
student demonstrate the ability to find a common denominator when adding or
subtracting two fractions. Remember, the easiest way to find a common denominator is
to simply multiply the two denominators of the fractions to be added or subtracted in
order to obtain the new denominator for both fractions. This will also be the denominator
of the result. It may help the student to review the following example:
Step 1:
Multiply the left fraction by 7 and the right fraction by 3. The problem now becomes:
Step 2:
To obtain the numerator of the result, subtract 3 from 14. The numerator value is 11.
Step 3:
The denominator has already been determined. It’s 21.
Step 4:
Complete the problem.
Activity 1: Operations with Positive and Negative Numbers:
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Regardless of the number form used (mixed numbers, fractions, decimals, etc.) the
following rules should be observed in order to determine a correct result. Have the
student write the following rules on a note card to refer to while working through
problems.







When using a number line, addition of positive values means moving to the right,
while subtraction of positive values means moving to the left.
Conversely, addition of negative values means moving to the left, while subtraction
of negative values means moving to the right. NOTE: Notice that subtraction of a
negative has the same result as adding a positive: moving to the right on the
number line.
When adding two negative numbers the result is always negative.
When adding two positive numbers, the result is always positive.
When adding two numbers where one is negative and the other is positive, the sign
(negative or positive) of the larger magnitude number will be the sign of the result.
When multiplying and dividing, if both numbers in the problem have the same sign
(both negative or both positive) the result is always positive.
When multiplying and dividing, if both numbers have different signs (one negative,
one positive) the result is always negative.
Let’s Practice!
Work the following problems with the student. Have the student use his or her note cards
to refer back to. When finished, have the student check his or her answers using the
number line.
Example 1:
-7 + 4.5 = ?
Step 1:
Determine which is the largest magnitude number. One way to determine this is to locate
both values on a number line and see which is furthest from zero. That is the number
with the greatest magnitude. In this example, that value is -7.
Step 2:
Determine the sign (positive or negative) of the result. Since -7 had the greatest
magnitude, and it’s negative, the result will be negative.
-7 + 4.5 = -?
Step 3:
Determine the result by starting at -7 on the number line and moving 4.5 units to the
right (because of addition of a positive number). The result is -2.5
-7 + 4.5 = -2.5
Use the following number line, or have the student draw a number line similar to the one
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below in order to check his or her answer. Remember, our rule in this situation tells us to
start at the -7 on the number line and move 4.5 places to the right.
Example 2: Subtraction:
Have the student pay special attention to this skill, as it is the one where most errors
commonly occur. Review the rules for number line use with positive and negative
numbers as stated above. Until the rules are memorized, use of a number line is
recommended.
-8 - (-14) = ?
Step 1:
Determine which direction to move on the number line. Using the rules for number lines
as stated above, this is subtracting a negative. The rules state that this is the same as
adding a positive. That means that the problem can be changed as follows:
-8 + 14 = ?
Step 2:
As in the addition problem, determine which is the largest magnitude number. In this
example, that value is 14.
Step 3:
Determine the sign (positive or negative) of the result. Since 14 had the greatest
magnitude, and it’s positive, the result will be positive.
Step 4:
Determine the result by starting at -8 on the number line and moving 14 units to the
right (because of addition of a positive number). The result is 6.
-8 - (-14) = 6
Note: As proficiency increases, some steps (2 and 3) in this process may be skipped.
Example 3: Multiplication and Division:
These are presented together because the rules for both are the same, and they’re easy
to remember. If both numbers in the problem have the same sign (both negative or both
positive) the result is always positive. If both numbers have different signs (one negative,
one positive) the result is always negative. Consider the following example:
-5.2 x -3.5 = ?
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Step 1:
Inspect the signs of both numbers. They are both negative. Since the signs are the same,
the result is positive.
Step 2:
Multiply the two values.
Step 3:
Verify result using the calculator.
Take the time to point out to the student that an understanding of the proper operations
with positive and negative numbers is essential if he or she desires a career as a teacher
(elementary), or in the sciences, such as chemistry or physics. In addition, knowledge of
this skill will assist the student in the successful completion of his or her academic
objectives through college.
Review:
What is -9 + 3.50? What is 8 - (-12)? What is -6 x 12?
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Apply the correct order of operations to solve problems.
Students should be able to use the correct order of operations to solve problems.
Tutorial:
Have the student write these rules for the order of operations on a note card to keep for
easy reference. Let the student know that when performing arithmetic operations, there
can be only one correct answer.
The rules for order of operations include the following:
 Rule 1:
Perform any calculations within parentheses first.
 Rule 2:
Next, perform any calculations involving multiplication and division, working from
left to right.
 Rule 3:
Finally, perform any calculations involving addition and subtraction, working from
left to right.
Activity
Work through the following problems with the student. He or she should refer back to the
note card while working through the problems. Remind the student to write down the
original problem. Directly under the problem, the student would perform any operations
that involve parentheses (Rule 1). Then the student would perform Rule 2 operations
directly underneath Rule 1 calculations, and so forth. Emphasize to the student the
importance of writing the original problem down and writing down each operation as he
or she solves the problem. This is important in minimizing simple computational errors!
Example 1:
Evaluate the following expression:
Step 1:
Since there are no parentheses, skip Rule 1 and have the student do Rule 2 first.
Step 2:
Rule 3 is all that’s left, so:
3 + 10 = 13.
Answer: 13
Point out to the student that if the proper order of operation rules were not observed, an
answer of 25 might have been obtained, which would have been INCORRECT!
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Example 2:
Evaluate the following expression:
Step 1:
Rule 1 states that the calculations within the parentheses get solved first. So, (8 - 3) = 5,
and (7 + 4) = 11.
Replacing these values for the parentheses in the original problem yields:
Step 2:
Rule 2 states that next, all multiplication and division need to be performed from left to
right. Since:
Step 3:
All that’s left is addition, so the problem becomes: 99 + 1 = 100
Answer = 100.
Additional Exercises:
Evaluate the following expression:
Let the student know that the rules for order of operations don’t lend themselves to real
world activities particularly well. Since they are more abstract, the student should take a
little extra time reviewing the definitions from his or her note cards. Remind the student
to take the time to work through each of the rules for the order of operations, being sure
to replace parts of the original problem as he or she applies each rule. Be sure to mention
that it’s a good habit to make the replacements directly below the original problem as in
the examples shown here.
If the student applies the rules correctly, the student will obtain the correct answer: 96.
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Review:
Find the solution:
50 - (10 x 2) x (8 - 4) = ?
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Compare and order fractions, decimals and common percents.
Students should be able to compare and order fractions, decimals, and common
percents.
Tutorial:
Here are a few points to emphasize when comparing fractions, decimals, and percents:
 A percent is a ratio that compares a number to 100.
 To express a fraction as a percent, convert the fraction to an equivalent fraction
that has a denominator of 100. This does not work well for all fractions because not
every denominator (bottom number of a fraction) can be multiplied or divided
evenly into 100.
 To write a decimal as a percent, move the decimal point 2 places to the right and
write a percent sign.
Activity
We can write a fraction as an equivalent fraction with a denominator equal to 100 by
writing proportions and cross multiplying. Work through the following problem with the
student. Tell the student that you are working on changing a fraction into a percent.
You might notice that we simply need to convert the fraction to a fraction with a
denominator equal to 100. In this case, 100 is a multiple of the denominator, 5. (5 x 20
= 100) We can easily convert this fraction by multiplying the numerator and denominator
by 20 as follows:
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However, if the denominator can not go evenly into 100, we use cross multiplication to
determine the unknown value.
We can now use these principles to compare fractions, decimals, and percents using the
<, >, and = signs. Remember, the large "mouth" of the > symbol always faces the larger
number.
Work through the following exercise with the student using the steps we have discussed.
Fill in the blank using one of the following symbols. Round answers to the nearest tenth.
Use <, >, or =.
First, try converting 25% to a fraction to see if we can compare the values.
In this form, we still can’t easily compare the values. At this point, perform the
conversion so that we can compare our values in percent form.
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For visual learning, demonstrate this by measuring
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of a cup of sugar and measure
25%, or
of a cup of sugar and visually compare the amount of sugar. Discuss with the
student why we use different representations of numbers in certain situations. For
example, why do we use percentages for grades? (Because we can best represent our
grade by comparing to 100).
Additional Exercises
Solve each proportion. Round answers to the nearest tenth.
1.
Answer: y = 3.3
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2.
Answer: c = 15
Fill in the blank using <, >, or =. Round to the nearest tenth.
3.
Answer: <
4.
Answer: =
5.
Answer: >
Review:
Which is greater?
2/5 or 30%
How would you write one-fourth as a decimal?
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