Introducing Simple Algebra Equations

I have introduced the process of solving simple equations for a variable with second and third graders, but OFTEN with students in pre-algebra who are intent on solving problems mentally, but lack accuracy and logic skills to show their work.  I have found that the consistency of the process helps students get a good foundation and confidence to progress in algebra.

I use a primary balance scale, green and red bingo spotter counting chips, and a scrap of paper with a X or a letter for the variable.  I color-code the counting chips with green for positive integers and red for negative.  I have written an = sign in marker on the fulcrum of the balance.  For more complex equations, an algrebra balance with 2 pans on each side can demonstrate positive and negative numbers.

To show the steps of X + 4 = 9, have the student copy the equation on paper and write down each step as it is done:

1.  Place the X paper on one side of the scale along with 4 green counters.

2.  Place 9 green counters on the other side of the scale.

3.  Introduce the concept of "isolate the variable," by having the student remove the 4 chips, leaving the paper X.  Underneath the + 4 in the original equation, write down – 4 to show that the chips were taken out.

4.  In order to keep the balance balanced, what we do to one side must be done to the other side.  Have the student take out 4 chips from the 9 on the other side of the balance and write – 4 underneath the original equation to show that 4 chips were removed from that side.

5.  Each side now has a subtraction problem.  I have students cross off the 4-4 problem because there are no chips left.  Have the student bring the X down to the next row to show what is still on the balance. 

6.  On the other side, have the student solve 9-4 and write the answer (5) below to show X=5. Circle the answer.

7.  Have the student count the chips on the balance to check.

As your student progress, using algebra tiles or an algebra balance can continue to help those visual students or struggling students with concepts by maintaining manipulatives that accurately demonstrate the principles of equations.  You can find any of these materials at either of our websites: or


Sequence of teaching multiplication facts

The sequence I use in teaching multiplication facts is a bit different than most textbook curricula.  I like to integrate the concepts of arithmetic facts with other areas of math, so when teaching multiplication facts, I usually step away from the curriculum I am using and spend time helping students learn, master, and become automatic with the facts.

The sequence I use is as follows, along with notes on other concepts that I try to integrate into the instruction and practice times:

* x0–immediate success and introducing that we will be making groups and haven't made any yet.

* x1–I introduce the identity property and talk about people's names.  We can go by our first name, last name, and nickname, but they all mean the same person.  When we multiply by 1, we always end up with the same number.  I integrate this into fractions with the numerator and denominator being alike and meaning 1 whole group.  Count pennies.

* x10–This is an extension of x1 and x0.  Think of the number with a 0 behind it.  Practice counting by 10's.  Practice counting dimes.  Review place value.

* x100–This is another extension of x1 and x0.  Think of the number and place two zeros behind it.  Practice counting by 100.  Practice counting dollars.  Review place value.

* x11–Another extension of x1!  For 1-digit numbers, just write or think the number in 2-digits. For 2-digit numbers, split the digits apart.  Then add the digits and insert the answer in the middle (be careful to carry if necessary).  Example:  24×11=264  Split the 2 and 4.  Add to 6 and insert.

* x5–Practice counting by 5.  Practice counting nickels.  Practice telling time to 5 minutes.  Practice counting nickels in groups of 2 to show the relationship of 5 to 10.

* x2–Practice counting by 2.  Practice counting 2 pennies or counters at a time.  Use the term double.  Find pairs around the house.  Introduce the concept of half of various numbers.  Teach even numbers.

* x4–Practice counting by 4.  I often teach the 4 facts with mnemonic memory tricks, but for some students who become proficient at doubling, we play around with "doubling the double."

* x3–Practice counting by 3.  I usually teach the 3 facts with mnemonic memory tricks.  Introduce thirds.

* x6–Practice counting by 6.  I usually teach the 6 facts with mnemonic cues.  Sometimes I introduce the idea that multiples of 6 will always be multiples of both 2 AND 3.  Making a list of x2 and x3 facts and circling the common multiples helps students to see the relationship to 6.

* x9–Practice counting by 9.  This fact list is easy because of the patterns of the multiples.  I either teach the finger trick or the strategy to "back it up and make it add" to 9.

* x7–Practice counting by 7.  I use mnemonic tricks for 7 facts.  I often introduce prime numbers and fractions that cannot be reduced.

* x8–Practice counting by 8.  I use a mnemonic cue for the remaining fact to learn:  "8×8 fell on the floor; pick it up, it's 64."  Or, "8×8 went to the store; bought Nintendo 64."

* x12–Count by 12.  Introduce the concept of dozen.  Make up games with plastic eggs and used egg cartons or with pretend doughnuts.

If you need more ideas or information for teaching multiplication facts, look in the math section of our catalog.  If you would like a workshop presented to your group that is jam-packed with fun ways to teach multiplication facts, e-mail or call Wisdom Seekers at 1-406-771-0069 to make arrangements.

Setting up percent word problems

When teaching students to solve word problems involving percentages, I start by teaching a generic formula that works for tax, tips, simple interest, discount, and finding the percentage.  I always teach students how to solve simple algebraic equations prior to this step, so they have the foundational skills of balancing an equation and solving for a variable.  The percentage formula is:

T x % = n

Total of percent is small number

Teaching both the formula and the words is important because word problems are in linguistic form and must be translated to a numeric equation.

Once the numbers given in a word problem are inserted in the formula, the student simply solves for the variable.  I have found students make less calculation errors by using this formula, than trying to remember whether to multiply or divide and selecting the wrong process.  I also encourage students to check to see if their answers are logical.  If the percentage is less that 100% or 1.00, then the "n" number should be smaller than the "T" Total.  If the percentage is exactly 100% or 1.00, then the 2 numbers should be the same.  If the percentage is over 100%, then the "n" number will be greater than the "T" Total.  Most word problems for upper elementary student will use percentages less than 100%.

When students are proficient at setting up the original equation and solving for the variable, then teach the second-step variations.  For tax and tips, add to the original Total.  For simple interest, multiply by number of years or a fraction of a year, then add to the original Total.  For discount, subtract from the original Total.

Manipulatives Around the House

Here are just a few items that can be found around the house that work great for counters in primary level math programs–without spending a lot of money for a kit of manipulatives:

* Pennies, Dimes, and Dollars

* A clock made from a plastic picnic plate or paper plate

* Raisins

* Tableware–Knives, Forks, and Spoons

* Dried Beans

* Craft Sticks

* Small plastic toy animals

* Buttons

* Beads

* Barrettes

* Plastic Bread Tabs

* Orange Juice Can Tops

* Milk Jug Caps

* Egg cartons and plastic eggs

* Toy Rings

* Small Twigs

* Fish Crackers

 I must add that while there are unlimited manipulatives to be found easily wherever we go, there are certain manipulatives that accurately demonstrate mathematical concepts and principles that are more challenging to create a home-made duplicate.  My favorites are:  Place Value (Base Ten) Blocks, Attribute Blocks, Two-sided counters or green and red counting chips with a simple balance and/or little ziplock bags.

Teaching addition facts

Beyond using manipulatives for counting, there are 2 main approaches to teaching and memorizing addition facts, by the addends or by the sums.  Both approaches have advantages, but usually one approach works better for each individual student.

Using the addend approach, facts are presented by adding gradually increasing numbers.  First, add 1 to each numeral 1 to 19.  Then add 2 to each numeral.  Then add 3 and so on.  Teaching students to add 10 is often easy and fun and can be helpful in teaching students to add 9, because students often can catch on quickly that if you add 10, then give the number before that answer, you have added 9.  For example:  3+10=13, 3+9= 3+10-1 =12.

Using the sum approach, students learn addition facts in "families" and are taught the additive inverse property (which I often call "twins") so that only half of the facts are needed to be learned.  The sum approach also helps students visually see relationships between addition and subtraction when presented in list form, making a smoother transition to subtraction.  Connections are also able to be made to multiplication.

Materials both in curricula and in supplementary resources use one or the other approach.  Knowing the approach you plan to use can help in selecting materials more wisely or in making your own activities and worksheets.

Students who have a strong sense of numeration, will often become excited to discover both approaches and the number relationships they reveal!