Grade 3
  1. Number Sense and Numeration  1.1. Understanding Numbers  1.1.1. Reading and Writing Numbers up to 10,000  1.1.2. Place Value up to Thousands  1.1.3. Comparing and Ordering Numbers  1.1.4. Rounding Numbers to the Nearest Ten and Hundred  1.1.5. Skip Counting by 2s, 5s, 10s, and 100s up to 1,000  1.2. Operations with Whole Numbers  1.2.1. Addition and Subtraction up to 1,000  1.2.2. Multiplication Facts (up to 10 x 10)  1.2.3. Division Facts (up to 100)  1.2.4. Multiplication as Repeated Addition  1.2.5. Division as Equal Sharing  1.2.6. Word Problems with Addition, Subtraction, Multiplication, and Division  1.3. Estimation  1.3.1. Estimating Sums and Differences  1.3.2. Estimating Products and Quotients  1.4. Understanding Even and Odd Numbers
  2. Fractions and Decimals  2.1. Understanding Fractions  2.1.1. Identifying and Representing Fractions  2.1.2. Fractions of a Whole and Part of a Set  2.1.3. Comparing and Ordering Fractions with Like Denominators  2.1.4. Equivalent Fractions  2.1.5. Simple Fraction Word Problems  2.2. Introduction to Decimals  2.2.1. Tenths and Hundredths as Decimals  2.2.2. Relationship between Fractions and Decimals
  3. Measurement  3.1. Length  3.1.1. Measuring in Centimeters and Meters  3.1.2. Comparing and Ordering Lengths  3.1.3. Estimating Lengths  3.1.4. Converting Between Centimeters and Meters  3.2. Mass  3.2.1. Measuring in Grams and Kilograms  3.2.2. Estimating and Comparing Mass  3.3. Capacity  3.3.1. Measuring in Milliliters and Liters  3.3.2. Estimating and Comparing Capacity  3.4. Time  3.4.1. Telling Time to the Nearest Minute  3.4.2. Calculating Elapsed Time  3.4.3. Using a Calendar  3.4.4. Understanding A.M. and P.M.  3.5. Area and Perimeter  3.5.1. Calculating Perimeter of Simple Shapes  3.5.2. Estimating and Measuring Area of Shapes  3.5.3. Finding Area by Counting Squares
  4. Geometry  4.1. Properties of Shapes  4.1.1. Identifying 2D Shapes (Triangles, Quadrilaterals, Circles, etc.)  4.1.2. Identifying 3D Shapes (Cubes, Spheres, Cylinders, etc.)  4.1.3. Describing Attributes of 2D and 3D Shapes  4.1.4. Recognizing Parallel and Perpendicular Lines  4.2. Symmetry and Transformations  4.2.1. Recognizing Lines of Symmetry  4.2.2. Identifying Flips, Slides, and Turns  4.3. Angles  4.3.1. Understanding Right Angles  4.3.2. Comparing Angles (Acute, Right, Obtuse)
  5. Data Management and Probability  5.1. Data Collection and Organization  5.1.1. Collecting Data using Surveys and Experiments  5.1.2. Organizing Data using Tally Charts and Tables  5.2.1. Creating and Interpreting Pictographs  5.2.2. Creating and Interpreting Bar Graphs  5.2.3. Reading Simple Line Graphs  5.3. Probability  5.3.1. Understanding Basic Probability Terms (Certain, Possible, Impossible)  5.3.2. Predicting Outcomes  5.3.3. Simple Probability Experiments
  6. Patterns and Algebra  6.1. Patterns  6.1.1. Identifying and Extending Numeric Patterns  6.1.2. Identifying and Extending Shape Patterns  6.1.3. Creating Patterns Using Rules  6.2. Introduction to Algebra  6.2.1. Understanding Variables as Unknowns  6.2.2. Solving Simple Addition and Subtraction Equations  6.2.3. Using Patterns to Solve Problems
  7. Problem-Solving Skills  7.1. Strategies for Problem-Solving  7.1.1. Using Pictures or Diagrams  7.1.2. Identifying Patterns in Problems  7.1.3. Making Organized Lists  7.1.4. Guessing and Checking  7.1.5. Using Logical Reasoning  7.1.6. Breaking Down Problems into Steps

Grade 3 → Patterns and Algebra → Introduction to Algebra ↓


Understanding Variables as Unknowns


In the world of mathematics, especially in algebra, we often come across the concept of variables. Variables are symbols we use to represent unknown values. They are incredibly useful because they allow us to create mathematical expressions and equations that describe many situations in the real world. In this exploration, we will understand variables as unknowns, learn how to use them in simple equations, and discover how they help us solve problems.

What are variables?

Variables are like placeholders. They are letters or symbols that stand for numbers we don't know yet or numbers that can change. It's like calling someone "friend" instead of using his or her real name until you get to know him or her. In algebra, the most common letters used as variables are x, y and z, but we can use any letter.

Consider a box that contains an unknown number of candies. If we call the number of candies c, then the letter c is a variable, which allows us to talk about candies even if we don't know exactly how many candies there are.

Role of variables in equations

Equations are statements that show equality between two expressions. When we use variables, equations can help us find unknown values. For example, if we want to know how many candies are in two identical boxes, we can write an equation.

Suppose each box has the same number of candies, c. If the total number of candies in both boxes is 12, we can write the equation:

 c + c = 12

This equation tells us that twice the number of candies in a box is equal to 12. By solving this equation, we can find c.

Solving simple equations with variables

Let's solve the equation we discussed:

We begin with:

 c + c = 12

This can be rewritten as:

 2c = 12

To find the value of c, we must perform the opposite operation of multiplication, which is division. We divide both sides by 2:

 c = 12 ÷ 2

and we find

 c = 6

This means that each box contains 6 candies. Here, the variable c was a placeholder for the number we wanted to find, and we used mathematical operations to find its value.

Using view models to understand variables

Let's use a simple visual model to understand how variables work in equations. Imagine scales that represent equality. On one side of the scales, we place weights that represent known numbers, and on the other side, we place a box labeled x, which represents the unknown variable.

This helps us to see:

X 4 8

In this model, the scales balance to indicate:

 x + 4 = 8

To solve for x, we need to eliminate the weighted value of 4. We do this by subtracting 4 from both sides:

 x = 8 – 4

Thus, x = 4. This visual method shows how balancing in equations helps find unknowns.

More examples with variables

Let's look at more examples to get comfortable with variables:

Example 1: Finding unknown age

Mary is twice as old as her sister Sarah. If the sum of their ages is 12 years, how old is Sarah?

Let's use s to represent Sarah's age. Then Mary's age can be expressed as 2s. The equation becomes:

 s + 2s = 12

Simplifying this, we get:

 3s = 12

To find s, divide both sides by 3:

 s = 12 ÷ 3

Therefore, s = 4. Sarah is 4 years old, and Mary is 8 years old.

Example 2: Solving for unknown quantities

A farmer has a certain number of apples. If he gives away 10 apples, he will be left with 25 apples. How many apples did he have originally?

Let a denote the original number of apples. The equation is:

 a – 10 = 25

Adding 10 to both sides gives a:

 a = 25 + 10

Thus, a = 35. The farmer originally had 35 apples.

Communication with variables

Variables help us communicate mathematical ideas clearly and efficiently. We can use them to describe patterns, functions, and relationships, and to solve a variety of problems. In real-life situations, we regularly use variables without even knowing it. For example, when deciding how much fruit to buy based on weight, the price per kilogram is a variable in determining the total cost.

Consider finding the price of k kilograms of apples at $3 per kilogram:

Value expression: 3k

So, if you buy 5 kg, the cost equation is:

 3k = 15

Conclusion

Understanding variables as unknowns is an essential part of learning algebra. Variables allow us to solve problems where some numbers are missing, and they provide a way to describe common mathematical situations. By practicing how to set up and solve equations with variables, students develop logical thinking and problem-solving skills that are fundamental in mathematics and everyday life.

As you progress in mathematics, you will see that variables are important not only in equations but also in functions, graphs, and many other areas. They are the building blocks that make algebra a valuable tool for exploring the world around us.


Grade 3 → 6.2.1


U
username
0%
completed in Grade 3

← Prev (6.2)
Introduction to Algebra
Next (6.2.2) →
Solving Simple Addition and Subtraction Equations

Comments 



Understanding Variables as Unknowns

https://www.buddymath.com/g3/6.2.1?bmal=en