Open Sentences, Symbols and Missing Values for JSS 1

Use of Symbols

Performance Objectives

By the end of the lesson, students should be able to:

1. state the meaning of open sentences;
2. solve problems expressed in open sentences;
3. identify relationship between addition and subtraction in open sentences;
4. identify relationship between multiplication and division in open sentences;
5. use letters to represent symbols in open sentences.

Meaning of Open Sentences

In mathematics, statements can be written using numbers, symbols, and operation signs. These statements are called mathematical sentences. Mathematical sentences can be classified as either closed sentences or open sentences.

A closed sentence is a mathematical statement that is known to be completely true or completely false.

Examples of true closed sentences include:

1. $$5 + 3 = 8$$
2. $$12 \div 3 = 4$$
3. $$7 \times 2 = 14$$

Examples of false closed sentences include:

1. $$4 + 9 = 15$$
2. $$20 – 5 = 10$$
3. $$3 \times 6 = 21$$

An open sentence is a mathematical statement that contains an unknown value or symbol and can be either true or false depending on the value put in place of the symbol.

In an open sentence, the unknown value is often represented by a shape, a symbol, or an empty box. Examples of open sentences include:

1. $$\square + 6 = 10$$
2. $$\triangle – 4 = 12$$
3. $$\bigcirc \times 3 = 15$$

The sentence $$\square + 6 = 10$$ is open because its truth depends on what number is placed inside the box. If the number $4$ is placed in the box, the sentence becomes true because $4 + 6 = 10$. If any other number is placed in the box, such as $5$, the sentence becomes false because $5 + 6 = 11$, not $10$.

The process of finding the correct number that makes an open sentence true is called solving the open sentence. The correct number is called the solution to the open sentence.

Observe the diagram below to see the difference between closed and open sentences.

Relationship Between Addition and Subtraction in Open Sentences

Addition and subtraction are inverse operations. This means that one operation undoes the other. Understanding this relationship makes it easy to find missing values in open sentences.

The relationship between addition and subtraction can be stated as follows:

1. If $$\square + a = b$$, then the missing value is found by subtracting $a$ from $b$: $$\square = b – a$$
2. If $$\square – a = b$$, then the missing value is found by adding $a$ and $b$: $$\square = b + a$$
3. If $$a – \square = b$$, then the missing value is found by subtracting $b$ from $a$: $$\square = a – b$$

Look at the diagram below to see how addition and subtraction work in reverse directions.

Worked Examples

Example 1: Solve the open sentence $$\square + 9 = 25$$.

Solution:

The sentence states that a number added to $9$ gives $25$. To find the missing number, use subtraction, which is the inverse of addition:

$$\square = 25 – 9$$

$$\square = 16$$

The solution is $16$.

Example 2: Solve the open sentence $$\square – 14 = 8$$.

Solution:

The sentence states that when $14$ is subtracted from a number, the result is $8$. To find the missing number, use addition, which is the inverse of subtraction:

$$\square = 8 + 14$$

$$\square = 22$$

The solution is $22$.

Example 3: Solve the open sentence $$30 – \square = 12$$.

Solution:

The sentence states that when a number is subtracted from $30$, the result is $12$. To find the missing number, subtract $12$ from $30$:

$$\square = 30 – 12$$

$$\square = 18$$

The solution is $18$.

Relationship Between Multiplication and Division in Open Sentences

Multiplication and division are also inverse operations. One operation undoes the other.

The relationship between multiplication and division can be stated as follows:

1. If $$\square \times a = b$$, then the missing value is found by dividing $b$ by $a$: $$\square = b \div a$$
2. If $$\square \div a = b$$, then the missing value is found by multiplying $b$ by $a$: $$\square = b \times a$$
3. If $$a \div \square = b$$, then the missing value is found by dividing $a$ by $b$: $$\square = a \div b$$

Look at the diagram below to see how multiplication and division work in reverse directions.

Worked Examples

Example 1: Solve the open sentence $$\square \times 6 = 42$$.

Solution:

The sentence states that a number multiplied by $6$ gives $42$. To find the missing number, use division, which is the inverse of multiplication:

$$\square = 42 \div 6$$

$$\square = 7$$

The solution is $7$.

Example 2: Solve the open sentence $$\square \div 5 = 9$$.

Solution:

The sentence states that when a number is divided by $5$, the result is $9$. To find the missing number, use multiplication, which is the inverse of division:

$$\square = 9 \times 5$$

$$\square = 45$$

The solution is $45$.

Example 3: Solve the open sentence $$48 \div \square = 8$$.

Solution:

The sentence states that when $48$ is divided by a number, the result is $8$. To find the missing number, divide $48$ by $8$:

$$\square = 48 \div 8$$

$$\square = 6$$

The solution is $6$.

Using Letters to Represent Symbols in Open Sentences

In primary school, empty boxes, circles, or triangles are often used to represent missing numbers. In junior secondary school, these shapes are replaced with letters of the alphabet. This is the foundation of algebra.

Any letter of the alphabet can be used to represent a missing number, but lowercase letters such as $x$, $y$, $a$, $b$, $m$, and $n$ are the most common.

For example, the open sentence $$\square + 5 = 12$$ can be written as:

$$x + 5 = 12$$

In this sentence, the letter $x$ represents the symbol or the missing number. Solving the sentence means finding the numerical value of $x$ that makes the statement true.

Observe the diagram below to see how shapes are replaced by letters in mathematical sentences.

Worked Examples

Example 1: Replace the symbol with a letter and solve the open sentence: $$\square + 11 = 20$$.

Solution:

First, replace the box symbol $$\square$$ with the letter $y$:

$$y + 11 = 20$$

Next, solve for $y$ using the inverse operation of addition, which is subtraction:

$$y = 20 – 11$$

$$y = 9$$

The solution is $y = 9$.

Example 2: Replace the symbol with a letter and solve the open sentence: $$\triangle – 7 = 13$$.

Solution:

First, replace the triangle symbol $$\triangle$$ with the letter $a$:

$$a – 7 = 13$$

Next, solve for $a$ using the inverse operation of subtraction, which is addition:

$$a = 13 + 7$$

$$a = 20$$

The solution is $a = 20$.

Example 3: Replace the symbol with a letter and solve the open sentence: $$\bigcirc \times 4 = 32$$.

Solution:

First, replace the circle symbol $$\bigcirc$$ with the letter $m$:

$$m \times 4 = 32$$

Note that in algebra, $$m \times 4$$ is usually written as $$4m$$. So, the sentence is:

$$4m = 32$$

Next, solve for $m$ using the inverse operation of multiplication, which is division:

$$m = 32 \div 4$$

$$m = 8$$

The solution is $m = 8$.

Example 4: Replace the symbol with a letter and solve the open sentence: $$\square \div 3 = 7$$.

Solution:

First, replace the box symbol $$\square$$ with the letter $k$:

$$k \div 3 = 7$$

This can also be written as a fraction:

$$\frac{k}{3} = 7$$

Next, solve for $k$ using the inverse operation of division, which is multiplication:

$$k = 7 \times 3$$

$$k = 21$$

The solution is $k = 21$.

Summary

The key points covered in this unit are:

1. An open sentence is a mathematical statement containing an unknown value that can be true or false depending on the value substituted.
2. Addition and subtraction are inverse operations. Addition is used to solve subtraction problems, and subtraction is used to solve addition problems.
3. Multiplication and division are inverse operations. Multiplication is used to solve division problems, and division is used to solve multiplication problems.
4. In algebra, letters of the alphabet are used to replace shapes and symbols to represent unknown values in open sentences.

Class Exercises

Section A: Multiple Choice Questions

Choose the correct option for each of the following questions:

1. Which of the following is an open sentence?
   1. $$4 + 5 = 9$$
   2. $$x – 3 = 10$$
   3. $$12 \div 2 = 5$$
   4. $$8 \times 3 = 24$$
2. What is the value of the box in the open sentence $$\square – 15 = 7$$?
   1. $8$$
   2. $12$$
   3. $22$$
   4. $105$$
3. If $$y + 12 = 30$$, what is the value of $y$?
   1. $18$$
   2. $42$$
   3. $15$$
   4. $360$$
4. Which operation is the inverse of division?
   1. Addition
   2. Subtraction
   3. Multiplication
   4. Square root
5. Rewrite the open sentence $$\square \times 8 = 56$$ using the letter $p$.
   1. $$p + 8 = 56$$
   2. $$p – 8 = 56$$
   3. $$8p = 56$$
   4. $$p \div 8 = 56$$

Section B: Theory Questions

Solve the following problems step-by-step:

1. State whether each of the following is a true closed sentence, a false closed sentence, or an open sentence:
   1. $$15 – 7 = 8$$
   2. $$y + 4 = 11$$
   3. $$3 \times 9 = 30$$
   4. $$\square \div 5 = 6$$
2. Find the missing numbers in the following open sentences:
   1. $$\square + 14 = 35$$
   2. $$\square – 9 = 18$$
   3. $$40 – \square = 25$$
3. Solve the following open sentences using the relationship between multiplication and division:
   1. $$\square \times 7 = 63$$
   2. $$\square \div 6 = 12$$
   3. $$72 \div \square = 9$$
4. For each of the following sentences, replace the shape with the letter $x$ and solve for $x$:
   1. $$\square + 18 = 42$$
   2. $$\triangle – 15 = 27$$
   3. $$\bigcirc \times 5 = 45$$
   4. $$\square \div 8 = 4$$