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This Web Page by Pauline
Harding for Art Nurk, hardingpj@yahoo.com.
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Is your child
ready for algebra? I’ve looked
at some different Algebra I texts, along with a several texts for 6^{th},
7^{th}, and 8^{th} grades. Based on this research, I’ve made a list of skills a
student should have before starting Algebra I. Ideally, the skills should be acquired through
understanding of the underlying ideas, combined with familiarity gained
through frequent use. If you
would like to begin Algebra I next year and just a few of these skills are
lacking, targeted instruction and practice the summer before taking a formal
course should ensure that the student is wellprepared. If a lot of these skills are lacking,
another year of basic math skills may be time well spent.
The typical
sequence for high school math is Algebra I in 9^{th}, Geometry in 10^{th},
Algebra II/Trig in 11^{th}, and Calculus or PreCalc or Statistics in
12^{th}. Bright students
bound for careers in math, science, engineering, etc. often take Algebra in 8^{th}
grade, and take four more years of math in 9th12^{th} A few particularly bright, motivated
students take Algebra I in 7^{th} grade.
KEY:
A = You should be
comfortable with this skill before starting an Algebra I class.
G = You should
have a basic understanding of this before starting a high school Geometry
class.
L =This skill is
useful in daily life.
F = This is a fun
topic to explore, but not needed for success in Algebra, Geometry, or daily
life.
Obviously these
are opinions/generalizations! They
are offered as a jumping off point – take what works for you and leave the
rest. Input is welcome – email me
at hardingpj@yahoo.com .
A, L

BASIC OPERATIONS:
Addition, including carrying, of large numbers and/or
several numbers.
Subtraction, with regrouping (aka “borrowing”), of
large numbers.
Multiplication of large numbers.
Division, including expressing remainders as fractions,
and interpreting remainders.

A, G, L

FRACTIONS:
Putting fractions in lowest terms.
Comparing and ordering fractions (1/2 > 1/3)
Changing fractions to mixed numbers and vice versa.
Changing fractions to decimals and percents.
Adding, subtracting, multiplying, and dividing
fractions and mixed numbers.
Canceling (a critical skill).
Vocabulary – “numerator” and “denominator”
These concepts are critical for success in algebra,
where students will have to do similar operations using complex expressions
full of variables, exponents, radicals, etc. Students should not only be able to manipulate fractions
but should have an inherent understanding of why various methods work or
don’t work.
A good resource for learning this is the Keys to
Fractions series  inexpensive workbooks that cover understanding and
working with fractions.

A, L

DECIMALS:
Comparing and ordering decimals (2.5>2.05)
Changing decimals to fractions and percents.
Adding, subtracting, multiplying, and dividing
decimals.
Repeating decimals
A good resource for learning this is the Keys to
Decimals series  inexpensive workbooks that cover understanding and
working with decimals.

A, L

PERCENTS:
Expressing percents as a decimal, a fraction, or a
mixed number. (80% = 4/5 =
0.8)
Finding percents (What is 30% of 78?). Finding the base given the
percent. (23.4 is 30% of what
number?)
Sales tax and commission problems.
A good resource
for learning this is the Keys to Percents series, three inexpensive
workbooks that cover percents thoroughly.

A, G, L

NEGATIVE NUMBERS:
Meaning of the term “integers”.
Addition, subtraction, multiplication and division of
negative (and positive) numbers.
(4  3 = 4 + 3)
Some experience with using negative numbers in word
problems.

A, G

SQUARE ROOTS:
Understanding what square roots are. This is actually a very simple
concept.
Understanding that the square root of 3 times the
square root of 3 equals 3.
To introduce
square roots, take some beans and figure out what numbers of beans can be
made into a square. For
example, 16 beans can be arranged into a 4 x 4 square. (4, 9, 16, 25, etc. can make a
square, but 2, 6, 8, 10, etc can only make rectangles, and some numbers (the
prime ones – 3, 5, 7, 13, etc.) can’t make either.) The numbers that can make a square
are called “square numbers”.
The square root of a square number is simply the length of the side
of the square it makes, e.g. the square root of 9 is 3, because 9 beans can
be arranged as a 3 x 3 square.
Check out the video at http://www.brainpop.com/math/algebra/squareroots

A

FACTORING:
Prime numbers.
Factoring (factor trees).
Divisibility rules (how to know if a given number is
evenly divisible by 2, 3, 4, 5, 6, 9, and/or 10).
Algebra
includes a whole lot of factoring!
It’s important to both know how to factor and to understand what
factoring means. Lots of
previous experience with factoring and using divisibility rules will make algebra
considerably easier. There is
a video about prime factorization at
http://www.brainpop.com/math/numbersoperators/primefactorization

A, G, L

MEASUREMENT
Familiarity with common metric and English measures of
length, area, volume, weight, mass, time, and temperature.
Ability to carry units throughout a problem (rather
than tack them on at the end).
Ability to convert units, given the conversion factor.
In order to do word problems in Algebra, familiarity
with common units is important.
In addition, most students will need to use units in their high
school science classes, especially Physics. Memorization of conversion factors is not needed for
success in Algebra or Geometry, however students should understand how to
convert units when given the conversion factor. “A sensible goal is the automatic recall of the most
commonly used facts plus competence in the use of reference sources to find
the less familiar ones.” (J.
Huston Barleg?, Concepts of Measurement, 1959)
It is also important to understand how to carry units
throughout a calculation, to be sure the final answer is in the desired
units.

A, L

PLOTTING & GRAPHS:
Plotting ordered pairs on a graph in all four
quadrants. This is not hard to
learn, but it’s used extensively in algebra.
Reading, interpreting, and making various kinds of
graphs – pictographs, bar graphs, line graphs, circle graphs.
Reading data from tables.
It is important to note that drawing a graph
involves many more skills than simply reading a graph. Students will have to chose
appropriate scales, determine which variables to put on each axis, choose
an appropriate title, label the axis, etc. In particular, creating a circle graph involves
knowledge of angles, circles and percents, and is a much more complex skill
than simply reading circle graphs.
A fun way to practice plotting is the calcudraw books
from Buki. http://www.bukitoys.com


RATIOS & PROPORTIONS


Probability & Statistics
Finding the number of possible combinations
(permutations) or outcomes, including the use of tree diagrams. Experience with this concept can
come through play before formal study. (Example:
dressing magnetic dolls given three tops and three skirts – how many
outfits can you create?).
Probability expressed as a fraction (number of desired
outcomes divided by number of possible outcomes)
A basic understanding of the nature of probability can
be best obtained informally, by playing many games of chance, such as
flipping a coin, board games such as Trouble, dice games, and card games.


PROPERTIES OF NUMBERS
commutative property, associative property,
distributive property, addition property of zero, multiplication properties
of zero, etc..
Some exposure to the various properties of number is
wise. In Algebra I, these will
be applied to complex expressions, and it will be easier to understand if
the student has seen these properties applied to simpler cases. Memorization of the properties is
probably not necessary, but exposure to the ideas and the terminology is
wise.

A

ALGEBRA BASICS:
Order of operations – “Please Excuse My Dear Aunt
Sally” (parentheses, exponents, multiplication & division, addition
& subtraction). There is a
fun little video about this at http://www.brainpop.com/math/numbersoperators/orderofoperations/
Writing expressions (“3 divided by y” = 3/y). Vocabulary like “quotient”,
“product”, “the quantity” etc.
Combining like terms, like 4x + 2 – x = 3x + 2
Solving simple equations, like 4x +8 = 20x
Experience with “plug and chug” – that is, plugging
different values into the variables in an expression, and
evaluating/simplifying the results.
An example would be plugging several values of x into a function
(like y = x + 5) and making a table of the results, then plotting the
results.
Simplifying algebraic expressions including exponents
and fractions, like (3xy)^{2}x^{2}= 3x^{4}y^{2}
or simplifying (164x)/8, or 49x^{2}/(7x/3).
These topics should be covered before starting an
Algebra I course. While some
can be learned “on the fly” if needed, it’s better to start out with some
familiarity with these concepts and skills, since they are basic
buildingblocks for Algebra.
It is very helpful for the student to have some
experience with a physical model of how equations work. For example, you can imagine the
equation 4x + 8 = x + 20 to be a balance scale with three opaque,
weightless bags with identical contents (3x), plus 8 beans, on one side,
with 20 beans on the other.
You can see that it makes sense that you can, for example, take away
8 beans from each side, or subtract one bag from each side, without disturbing
the balance.

A, G, L

Problemsolving Strategies:
Make a table, guess & check, solve a simpler
problem, draw a picture, build a model, write an equation, use a formula,
etc.
Understanding when to use which mathematical
operation.
Experience with complex problems for which there is
not a single “right answer”.
Knowledge of problemsolving strategies and considerable
experience in solving a variety of “word problems” is critical to
mathematical literacy.
Experience with problemsolving is of increasing importance for
success with each year of study in mathematics. Problemsolving is the essence of mathematics, and
should be fully integrated with the entire mathematical curriculum. “Don’t leave home without it.”

A,L

GEOMETRY FOR ALGEBRA
Complex
knowledge of geometry is not needed for most algebra classes. However, students should be
familiar with some basic ideas.
Enough understanding of points and lines to do basic
graphing.
Area and perimeter of square, triangle, rectangle, and
more complex shapes made up of these.
Volume and surface area of simple solids.
The Algebra I
books I have examined generally do not require much knowledge of circles,
though of course this should probably be covered somewhat before Algebra I
as preparation for Geometry, and in order to create circle graphs.

G

PREGEOMETRY
Since high school Geometry
is usually studied the year after Algebra I, it is wise to cover
pregeometry topics before starting Algebra I, even though these topics may
not be needed for Algebra I itself.
What is covered before Geometry varies widely, and indeed Geometry
texts themselves vary widely.
The basics beyond what is needed for Algebra I include:
Points, lines, rays, line segments – definitions,
naming conventions
Parallel and perpendicular
Angles – naming conventions, measurement,
right/acute/obtuse.
Use of a protractor to measure and draw angles.
Area, perimeter, volume, and surface area of more
complex shapes and solids (parallelogram, etc).
Circles – center, radius, diameter, perimeter, area,
naming conventions.
Similar polygons.
Types of solids – pyramid, prisms, regular polyhedra,
sphere.
Additional
topics may include:
Use of a
compass and simple constructions such as bisecting an angle and a line.


Scientific Skills
Scientific notation (0.0004 = 4 x 10^{4 })
Significant digits (including rounding).
These skills are not typically needed for Algebra or
Geometry, but they will be needed in high school science classes. Both are typically taught in the
science classes where they will be used. Scientific notation is also covered in some math
courses.


Work Habits
Students will need to be able to work carefully, show
their work, and carry units.
They will need to copy the problem onto their paper, and lay it out
properly. These skills should
be emphasized in middle school, because Algebra problems can get long and
messy. While many middle
school problems can be done without showing much work, it becomes critical
in Algebra – there’s just too much to keep track of in one’s head.


Consumer math
Interest and loans, use of checkbooks and credit
cards, interpreting utility bills, installment buying, unit pricing, basic
accounting, etc.
These topics are not required for success in algebra,
though they are ones that almost every adult will need to understand. Older math texts put quite a bit of
emphasis on these topics, especially in the days when many students left
school before algebra.
Nowadays, collegebound students can probably learn many of them on
an informal, asneeded basis.
On the other hand, some exposure to these topics, perhaps through
integrating them into the curriculum via word problems, etc, is probably
wise. For students who are not
going to study algebra, a course focusing on consumer math could be
extremely valuable.

misc.

OTHER TOPICS
Roman numerals, and other number systems from cultures
throughout history.
Number bases other then ten – base two, base 8, base
16, etc. These are used in
computer science. In the same
way that studying French grammar can lead to increased understanding of
English grammar, studying base 2 can lead to a deeper understanding of our
base 10 system.
Sets, Venn Diagrams
Topology
Logical reasoning – this will be covered in geometry
classes if they include proofs.
Use of calculators – I honestly believe that, for most
children, calculators are not needed for elementary or middle school, and
their availability can in fact be detrimental. I also believe that in most cases formal instruction in
their use at this level is not needed. Some Algebra I courses will make use of graphing
calculators, but care should be taken that calculator use does not take the
place of basic skills and the understanding that comes through their use.
Estimation – while this skill is very useful in
everyday life and in mathematics in general, no particular estimation
technique is required for success in algebra. Students should of course be growing in their ability to
judge whether the magnitude of their answer is in the ballpark, etc.
Fibonacci sequence, Pascal’s triangle, etc. These topics are fun but not
required for success in Algebra I or Geometry.


Resources & texts
I like the Abeka 6^{th} grade text for giving a
good solid year of firming up calculation skills (but the 7^{th} grade
book is just a rehash – better to move on to NEM). I like the first Singapore New
Elementary Mathematics book for kids who need an extra year of problem
solving and general algebra prep.
I like the Keys To series for filling in any “gaps”, for challenging
younger kids, and for quickly catching up older kids who haven’t done much
math. I like McDougal Littell
for PreAlgebra and Algebra I. I like Jacob’s
Geometry. For younger kids, I
like a mix. Everyone is
different, and a text that works well for one child may not be the best fit
for another.

