Pauline's Guide to Homeschooling

Is Your Child Ready for Algebra?

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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 6th, 7th, and 8th 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 well-prepared.  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 9th, Geometry in 10th, Algebra II/Trig in 11th, and Calculus or Pre-Calc or Statistics in 12th.  Bright students bound for careers in math, science, engineering, etc. often take Algebra in 8th grade, and take four more years of math in 9th-12th  A few particularly bright, motivated students take Algebra I in 7th 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 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 calc-u-draw 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)2x2= 3x4y2 or simplifying (16-4x)/8, or 49x2/(-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 building-blocks 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 Problem-solving 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 problem-solving strategies and considerable experience in solving a variety of “word problems” is critical to mathematical literacy.  Experience with problem-solving is of increasing importance for success with each year of study in mathematics.  Problem-solving 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 PRE-GEOMETRY Since high school Geometry is usually studied the year after Algebra I, it is wise to cover pre-geometry 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, college-bound students can probably learn many of them on an informal, as-needed 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 6th grade text for giving a good solid year of firming up calculation skills (but the 7th grade book is just a re-hash – 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 Pre-Algebra 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.