Your child's first algebra problems will introduce letters right next to the numbers. For many 4th, 5th, and 6th graders, this can make math feel like a whole new subject.
Although the letters may look new, they build on reasoning your child already knows. Mathematicians apply them in several distinct ways, depending on the problem.
Today, we explain five common uses of letters in algebra, each with a kid-friendly example. Let’s dive in!
In algebra, a letter can stand for a number we don't know yet. The clues in the problem help us figure out what that number is.
In earlier grades, students solve problems with a blank box, such as □ + 3 = 7.
Here, we simply ask which number plus 3 equals 7 and find that the box equals 4. Algebra problems work the same way, with a letter in place of the box.
□ + 3 = 7 becomes x + 3 = 7.
The letter x represents the number students don't know yet.
We find x the same way we found the box, so x = 4.
We can use letters in word problems, too. Let's look at the example:
Maya baked some cookies, and her brother ate 4 of them. If 9 cookies are left, we can write the equation c − 4 = 9, where c stands for the number of cookies Maya baked. Since 13 − 4 = 9, Maya baked 13 cookies.
Some students think letters match numbers in alphabet order. In algebra, any letter can stand for an unknown number.
a doesn't have to equal 1.
b doesn't have to equal 2.
c equals 13 in the cookie problem, but it could equal 50 in the next one.
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In algebra, a variable is a letter that represents a number, and that number can change from one situation to the next. The name connects to the word "vary," which means "change."
Let's look at the example of a video game.
Jonah earns 10 points for every level he finishes. After 3 levels, he has 30 points. After 5 levels, he has 50 points. Jonah can keep playing, so the number of levels keeps changing, and his score changes with it.
We can describe Jonah's score for any number of levels with one expression, 10 × L. The letter L stands for the number of levels Jonah finishes. To find his score, we replace L with a number and multiply.
|
Levels finished (L) |
Expression |
Jonah's score |
|
3 |
10 × 3 |
30 points |
|
5 |
10 × 5 |
50 points |
|
12 |
10 × 12 |
120 points |
We keep the expression the same and change only the value of L. With one short line of math, we can find Jonah's score after any number of levels.
In algebra, we use letters to write math rules that work for every number, so we don't have to check one example at a time.
We already know that 3 + 8 and 8 + 3 both equal 11. We also know that 4 + 6 and 6 + 4 both equal 10. And we could keep writing examples like these, but the list would never end because there is always another pair of numbers to try.
Mathematicians write the rule with letters instead, as a + b = b + a.
The letters a and b can stand for any two numbers. This rule is called the commutative property of addition, and it tells us that we can add two numbers in either order and get the same sum.
Now, let's try the rule with some bigger numbers.
Pick a = 25 and b = 7.
Add them one way, and 25 + 7 = 32.
Flip them around, and 7 + 25 = 32.
The answers match, so the rule works again!
The same idea works with multiplication. For example, a × 1 = a means that multiplying any number by 1 gives you that same number back.
Try it with 9, and 9 × 1 = 9.
Try it with 150, and 150 × 1 = 150.
Try it with any number you like, and you'll get the same number back!
Next, let's see what happens when we try the same pattern with subtraction.
Start with 10 − 4, which equals 6.
Flip the numbers to 4 − 10. The answer isn't 6 anymore!
The answers don't match, so a − b = b − a can't be a rule. A math rule must work for every pair of numbers.
With letters, we can describe a pattern that works for countless numbers at once. That's a lot of math in one small rule!
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Mathematicians use letters in algebra because one letter is much faster to write than a whole sentence about a number.
Remember Jonah's game from earlier? Which of these would you rather write?
In words. Take the number of levels Jonah finishes and multiply it by 10 to find his score.
With a letter. 10 × L
A whole sentence turns into just four symbols!
What happens with a longer problem? Let's try one.
In words. I am thinking of a number. If I double it and then add 5, I get 17.
With a letter. 2 × n + 5 = 17
Who thought of writing math this way? In 1591, a French mathematician named François Viète created the first clear system for writing algebra with letters. More than 400 years later, we build on his idea every time we write an x or an n!
Here's one more shortcut. The times sign × looks a lot like the letter x, and that can get confusing!
Mathematicians often drop the × sign next to a letter or use a different symbol instead. You may see 3 × x written in any of these ways.
3x, with the number right next to the letter
3 · x, with a raised dot
3(x), with parentheses
All three mean "3 times x." And Jonah's score? 10 × L becomes 10L. Even shorter!
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In math, a formula uses letters to show how one amount depends on another. We can use one rule to find the answer for any size or number.
Let's try it with a craft project. Anna wants to glue ribbon around a square picture, and each side is 5 inches long. How much ribbon does she need? We add the four sides.
5 + 5 + 5 + 5 = 20 inches of ribbon
Now Anna picks a bigger picture with 8-inch sides.
8 + 8 + 8 + 8 = 32 inches of ribbon
Do you see a pattern? We add the same side length four times, so we can multiply by 4 instead!
A 5-inch side needs 4 × 5 = 20 inches.
An 8-inch side needs 4 × 8 = 32 inches.
A 12-inch side needs 4 × 12 = 48 inches.
We can write this pattern as a formula, P = 4s.
P stands for the perimeter, or the distance around the square.
s stands for the length of one side.
The formula shows how the two amounts relate. Every time a side grows by 1 inch, the perimeter grows by 4 inches.
What if we forget the formula later? No problem! We can rebuild it by adding the four sides again, because we understand why it works.
Students can use the same thinking with other formulas, such as A = l × w for the area of a rectangle.
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At Mathnasium, students work through algebra step by step, with clear explanations and practice that help them understand why a rule works instead of simply memorizing it.
Mathnasium is a math-only learning center dedicated to helping K–12 students of all skill levels build the understanding they need to work with letters, variables, and equations confidently.
Our proprietary teaching approach, the Mathnasium Method™, is designed to follow each student's learning style and goals to help them truly understand math.
Our approach includes:
Assessment and Personalized Learning Plans: Every student starts with a diagnostic assessment. Here, we discover their relationship with math, their skills, and knowledge gaps. We use this to build a personalized learning plan that matches both their current achievements and future goals.
Teaching for Understanding: Our tutors use natural language and a mix of verbal, visual, mental, tactile, and written techniques to make each concept easy to understand.
Problem-Solving and Critical Thinking: Even though we are always there to help, during sessions we encourage students to try to solve problems on their own. This helps them develop problem-solving skills that are the basis for further development. When we do step in, we show them both the how and the why behind the answer. This helps students develop critical thinking tools they can use in math and beyond.
An Engaging and Fun Learning Environment: Our sessions include fun activities, so students actually look forward to learning. To help boost their confidence, we celebrate and award each milestone.
Our method brings measurable results.
94% of parents report an improvement in their child's math skills and understanding
93% of parents report their child's improved attitude toward math after attending Mathnasium
90% of students saw an improvement in their school grades
We operate over 1,200 learning centers across North America, so there is likely a Mathnasium nearby.
Families in Round Rock East and the neighboring areas can access our proven approach at Mathnasium of Round Rock East.
Whether your student is looking to catch up, keep up, or get ahead in math and algebra, we can support them.
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