Meaning of Open Sentences and Finding Missing Numbers for Primary 4

Open Sentence

Performance Objectives

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

1. explain open sentence;
2. find missing number in an open sentence involving addition and subtraction operations;
3. solve quantitative reasoning involving open sentence

What is an Open Sentence?

An open sentence is a mathematical statement that contains a missing number, represented by a symbol like a box ($\square$), a question mark (?), or a letter (like $x$ or $y$). The statement is “open” because its truth cannot be decided until the missing number is found.

Think of it like a puzzle. You have a mathematical rule, but one piece of the puzzle is hidden. Your job is to find that hidden piece to make the statement complete and true.

Here are some examples of open sentences:

1. $5 + \square = 12$
2. $15 – x = 7$
3. $y + 8 = 20$
4. $10 – 4 = ?$

In each example, there is a number missing. Once the missing number is put in place, the sentence becomes a true mathematical statement.

Finding Missing Numbers in Open Sentences

To find a missing number in an open sentence, it is often helpful to use the inverse (opposite) operation. Addition is the inverse of subtraction, and subtraction is the inverse of addition.

Missing Numbers in Addition Open Sentences

When an open sentence involves addition, there are two main situations:

1. Finding a missing addend: This is when one of the numbers being added is missing.

   Example: $5 + \square = 12$

   To find the missing number, subtract the known addend (5) from the sum (12).

   Worked Example 1:

   Find the missing number in $5 + \square = 12$.

   Solution:

   $$
   \begin{aligned}
   5 + \square &= 12 \\
   \square &= 12 – 5 \\
   \square &= 7
   \end{aligned}
   $$

   So, $5 + 7 = 12$.

   Worked Example 2:

   Find the missing number in $18 + x = 30$.

   Solution:

   $$
   \begin{aligned}
   18 + x &= 30 \\
   x &= 30 – 18 \\
   x &= 12
   \end{aligned}
   $$

   So, $18 + 12 = 30$.

2. Finding a missing sum: This is when the total after adding is missing.

   Example: $8 + 6 = \square$

   To find the missing number, simply add the two known numbers.

   Worked Example 3:

   Find the missing number in $8 + 6 = \square$.

   Solution:

   $$
   \begin{aligned}
   8 + 6 &= \square \\
   14 &= \square
   \end{aligned}
   $$

   So, $8 + 6 = 14$.

   Worked Example 4:

   Find the missing number in $25 + 15 = y$.

   Solution:

   $$
   \begin{aligned}
   25 + 15 &= y \\
   40 &= y
   \end{aligned}
   $$

   So, $25 + 15 = 40$.

Missing Numbers in Subtraction Open Sentences

When an open sentence involves subtraction, there are three main situations:

1. Finding a missing subtrahend: This is when the number being subtracted is missing.

   Example: $10 – \square = 3$

   To find the missing number, subtract the difference (3) from the minuend (10).

   Worked Example 5:

   Find the missing number in $10 – \square = 3$.

   Solution:

   $$
   \begin{aligned}
   10 – \square &= 3 \\
   \square &= 10 – 3 \\
   \square &= 7
   \end{aligned}
   $$

   So, $10 – 7 = 3$.

   Worked Example 6:

   Find the missing number in $45 – x = 20$.

   Solution:

   $$
   \begin{aligned}
   45 – x &= 20 \\
   x &= 45 – 20 \\
   x &= 25
   \end{aligned}
   $$

   So, $45 – 25 = 20$.

2. Finding a missing minuend: This is when the number from which another number is subtracted is missing.

   Example: $\square – 4 = 6$

   To find the missing number, add the subtrahend (4) and the difference (6).

   Worked Example 7:

   Find the missing number in $\square – 4 = 6$.

   Solution:

   $$
   \begin{aligned}
   \square – 4 &= 6 \\
   \square &= 6 + 4 \\
   \square &= 10
   \end{aligned}
   $$

   So, $10 – 4 = 6$.

   Worked Example 8:

   Find the missing number in $y – 12 = 15$.

   Solution:

   $$
   \begin{aligned}
   y – 12 &= 15 \\
   y &= 15 + 12 \\
   y &= 27
   \end{aligned}
   $$

   So, $27 – 12 = 15$.

3. Finding a missing difference: This is when the result of the subtraction is missing.

   Example: $15 – 7 = \square$

   To find the missing number, simply subtract the second number from the first.

   Worked Example 9:

   Find the missing number in $15 – 7 = \square$.

   Solution:

   $$
   \begin{aligned}
   15 – 7 &= \square \\
   8 &= \square
   \end{aligned}
   $$

   So, $15 – 7 = 8$.

   Worked Example 10:

   Find the missing number in $30 – 10 = z$.

   Solution:

   $$
   \begin{aligned}
   30 – 10 &= z \\
   20 &= z
   \end{aligned}
   $$

   So, $30 – 10 = 20$.

Quantitative Reasoning Involving Open Sentences

Quantitative reasoning involves using numbers and mathematical operations to solve problems, often by identifying patterns or relationships. When these problems include a missing number, they become open sentences.

In quantitative reasoning, you might see a series of numbers or a diagram where you need to figure out the rule or pattern that connects them. Once you find the rule, you can write an open sentence to find the missing number.

Worked Example 11:

Look at the pattern below and find the missing number.

$(3, 5, 8)$

$(4, 6, 10)$

$(7, 9, \square)$

Solution:

First, look at the relationship between the numbers in the first two sets:

1. In $(3, 5, 8)$, we see that $3 + 5 = 8$.
2. In $(4, 6, 10)$, we see that $4 + 6 = 10$.

The pattern is that the first number plus the second number equals the third number. Now, apply this rule to the third set:

$$
\begin{aligned}
7 + 9 &= \square \\
16 &= \square
\end{aligned}
$$

The missing number is 16.

Worked Example 12:

Find the missing number in the sequence:

$20, 17, 14, \square, 8$

Solution:

Look at the difference between the numbers:

1. $20 – 17 = 3$
2. $17 – 14 = 3$

The pattern is that each number is 3 less than the previous number. So, to find the missing number, subtract 3 from 14:

$$
\begin{aligned}
14 – 3 &= \square \\
11 &= \square
\end{aligned}
$$

To check, subtract 3 from 11: $11 – 3 = 8$. This matches the next number in the sequence. The missing number is 11.

Generating Open Sentence Problems

Generating open sentence problems means creating your own mathematical statements with a missing number. This helps you understand how open sentences work and how to solve them.

The procedure involves the following steps:

1. Start with a complete number sentence: Choose two numbers and an operation (addition or subtraction) to form a true number sentence.

   Example: $10 + 5 = 15$

   Example: $20 – 8 = 12$

2. Hide one of the numbers: Replace one of the numbers with a symbol like a box ($\square$), a question mark (?), or a letter ($x$, $y$).

   From $10 + 5 = 15$, you can create:

   1. $10 + \square = 15$ (missing addend)
   2. $\square + 5 = 15$ (missing addend)
   3. $10 + 5 = \square$ (missing sum)

   From $20 – 8 = 12$, you can create:

   1. $20 – \square = 12$ (missing subtrahend)
   2. $\square – 8 = 12$ (missing minuend)
   3. $20 – 8 = \square$ (missing difference)
3. Solve the open sentence: Use the inverse operation or direct calculation to find the missing number and check your answer.

Let’s generate and solve an open sentence problem:

1. Start with a complete sentence: $12 + 7 = 19$.
2. Hide a number: Let’s hide the 7. The open sentence becomes $12 + \square = 19$.
3. Solve the open sentence:
   $$
   \begin{aligned}
   12 + \square &= 19 \\
   \square &= 19 – 12 \\
   \square &= 7
   \end{aligned}
   $$
   The missing number is 7.