Kids and adults who struggle with algebra need the best algebra help there is.
Fortunately, help is readily available.
Below I offer additional suggestions that may enable you to provide the best algebra help to your son or daughter if you are a parent or to your students if you are a classroom teacher.
Most of the ideas presented will pertain to the middle school and high school years, although some will pertain to the elementary grades. Especially with these younger students, a visual and kinesthetic approach is essential if they are to experience success with abstract algebraic concepts.
Best Algebra Help Tip #7: Understanding the Distributive Property
Often students have difficulty with expressions such as 2(2x + 3). One way to understand this notation is to realize that the 2 outside of the parentheses informs us that we need two sets of the quantity that is inside the parentheses. Hence, 2(2x+3) is the same as 2x + 3 and another 2x + 3 which is the same as 4x + 6.
This can also be demonstrated concretely. For example, let’s consider 2(3a+b) to symbolize the doubling of the contents of a basket containing three apples and one banana. After the doubling, the basket will contain six apples and two bananas. Hence, 2(3a+b) = 6a + 2b.
Best Algebra Help Tip #8: Understanding the Power of a Counterexample
Whereas the distributive law of multiplication over addition is true, a student may wonder if an exponent can be distributed over a sum, for example, is (a + b)2 the same as a2 + b2? A good first step is to simply take random numbers, say a=2 and b=3, and evaluate both expressions. When we do so we find that (2+3)2=25 whereas 22 plus 32 is 13. This one counterexample is enough to show us that the statement, “(a + b)2 = a2 + b2 for all values of a and b” is false.
When parents or teachers convey to their students the idea that one counterexample is enough to disprove a mathematical proposition, they are communicating a valuable algebra help tip.
Best Algebra Help Tip #9: Understanding the Limitation of Inductive Reasoning
Often students think that multiplication will result in a larger result than addition. But is the product of two numbers, which we will call a and b, always greater than their sum?
Let’s take a few examples and consider whether ab> a+b in these examples:
a = 2, b=3: Yes, their product 6 is greater than their sum of 5
a=5, b=4: Yes, their product of 20 is greater than their sum of 9
a=50, b=20: Yes, their product of 1000 is greater than their sum of 70
From these examples, a young student might wish to propose that the product of two numbers is always greater than their sum. Reasoning from various instances is known as inductive reasoning. One needs to use caution in using inductive reasoning since the conclusion might not be true.
A little reflection will show that the product of two numbers need not be greater than their sum. For example, if a=1, and b=2, their product is 2 but their sum is 3! Hence, in this case the product is less than their sum. After a little reflection the young student will be able to find many other cases where the product of two numbers is less than the sum.
Parents and teachers should encourage their students to use inductive reasoning to formulate algebraic propositions. They should also caution them that the propositions will need further investigation to determine if they are true.
Best Algebra Help Tip #10: Understanding the Power of Deductive Reasoning
Deductive reasoning refers to the presentation of a proof via logical reasoning that will cover all cases under discussion. If the assumptions are sound, and the argument is sound, the conclusion will be true.
Let’s say we are working with a young student and we consider the sum of two odd numbers: 3 + 5, 7 + 9, 55+5, etc. We see that in each instance, the sum is even. Hence, using inductive reasoning the child might propose that “the sum of any two odd numbers is an even number.”
However, we cannot be certain that this statement is true simply because we observe it to be true several times. To prove the statement true, we need a solid logical argument, that is, we need to use deductive reasoning that will show the result to be true for all odd numbers.
First, we need to realize that an odd number is defined as one more than an even number. An even number is one that can be expressed as twice another number. If the odd numbers are x and y, we can express them as x=2n+1 and y=2m+1, where n and m are two integers. Then x+y= 2n+1+2m+1= 2n+2m+2. The latter is an even number since half of it is the integer n+m+1.
In summary, although inductive reasoning can be used to formulate algebraic conjectures, deductive reasoning is required to prove the assertion.
Best Algebra Help Tip #11: Understand the Difference Between a Coefficient and a Constant
Let’s consider an equation such as 4x + 1 = 3x + 4. The student needs to understand that the two instances of the number 4 found in the above equation do not mean the same thing. The first 4 is a coefficient. It simply tells us how many x’s we have on the left side of the equation. Hence, the left side can be rewritten as x + x + x + x + 1, where the value of x is fixed but unknown.
On the other hand, the 4 on the right side is a constant. We know its value; it is 4. This distinction can be demonstrated visually using Hands-On Equations®, where the pawn represents the x and the boxed numbers represents the constants.
The equation 4x + 1 = 3x + 4 can be represented as shown below:
We can now understand the problem as asking for the weight of the pawn that will make both sides balance given that all the pawns have the same weight and that the cubes have the weight of the number shown on them.
I have found that this visual representation does more than almost anything else to help students demystify the meaning of an algebraic linear equation. I would rate it very high among the best algebra help tips that I could provide.
Best Algebra Help Tip #12: Understand the Process of Balancing an Equation
4x + 1 = 3x + 4
In the above example, we can try to find the value of the unknown by trial and error. Let’s consider the possibility that the pawn, x, is worth 5. On the left side we would have a total weight or value of 21, whereas on the right side we would have 19. By continuing in this manner, trying different numbers, we can eventually arrive at the answer to the above problem.
A mathematician, however, is interested in mathematical elegance, that is, in simple ways to obtain the solution so that the work or computation is minimized. We use a principle in mathematics known as the Subtraction Property of Equality to simplify the equation. I have found that students as young as five years of age can be guided to formulate a statement to the effect that removing the same weight from each side of a balanced system maintains the balance of the system.
In particular, if we remove three pawns from each side, the system remains in balance. (The system is balanced to begin with as indicated by the equal sign). After doing this we have:
Now, we can see by inspection that the pawn is worth 3, because 3 plus 1 is 4.
Best Algebra Help Tip #13: Understanding the Importance of Checking the Answer
Students who solve a problem, obtain an answer, and then leave the problem alone never know if they solved it correctly. An important algebra help tip is to convey to a student the need to conduct the check—always! In order to check our answer of x=3, we go back to the original physical setup.
4x + 1 = 3x + 4
When x=3, we see that each side has a value of 13. We write the check as 13=13.
This visual and kinesthetic process for solving algebraic equations, using actual game pieces, has been shown to be successful with students as young as eight years of age as well as with adult learners.
I trust that at least some of these best algebra help tips have been of value to you.
Copyright© 2011 by Henry Borenson, Ed.D.