2. Numerical Expressions
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Chapter 4
2.
Numerical Expressions
This lesson provides a thorough understanding of numerical expressions, emphasizing the importance of the order of operations and the PEMDAS rule. It explains how these principles are crucial for solving mathematical problems accurately. For example, the order of operations ensures that calculations are performed in a specific sequence to avoid errors. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, serves as a helpful mnemonic for remembering this sequence. These concepts are not just theoretical; they have practical applications in various fields such as engineering, finance, and data analysis. Overall, the material is a valuable lesson for anyone interested in mastering the intricacies of numerical expressions and their real-world applications.
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Lesson Settings & Tools
| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Numerical Expressions
Slide of 10
The basic mathematical operations make up most of the mathematical ideas applied to the real world. Generally, these operations are combined in different ways to express different things. The goal of this lesson is to show how to evaluate a set of mathematical operations.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Which Result is Correct?
If a mathematical operation is done correctly, there is always a single correct result. When adding two numbers, there is only one correct value for the sum.
Both times the operations were done correctly, but the results are different. Who evaluated the expression correctly? Now consider more operations, like the set of operations below.
3+2=7×3+2=5✓
The same happens when two numbers are multiplied. If the multiplication is done correctly, the result is correct.
2⋅5=13×2⋅5=10✓
But what happens if multiple operations are combined? Izabella and Kriz are discussing how to solve an expression that includes both addition and multiplication.
3+2⋅5
Izabella says that the operations should be done left to right. Here is how she evaluated the operations. But Kriz thinks that the multiplication should be done first. This is how Kriz evaluated the operations.
{"type":"choice","form":{"alts":["Izabella","Kriz"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
42−448÷8+18
Can the same method be used to evaluate it? What is the resulting value? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2"]}}
Discussion
Numerical Expressions
A numeric expression, or numerical expression, is a sequence of mathematical operations that only involves numbers. Consider the following examples.
| Example | Is It a Numeric Expression? |
|---|---|
| 5+3−2⋅8 | ✓ |
| (9+12)3−4⋅7 | ✓ |
| 73+[(5−1)⋅(7+4)]5−71 | ✓ |
| 9w2+4s+7 | × |
| 1125 | × |
Pop Quiz
Identifying Numerical Expressions
Select whether each given expression is a numerical expression or not.
Discussion
Order of Operations
The order of operations is the order to follow when evaluating an expression that has more than one operation. The order of operations can be described as a series of steps.
Consider this expression to see the steps in action.
(1+2)⋅32−25+5=?
The example expression contains a set of parentheses, an exponent, multiplication, division, addition, and subtraction. The expression is evaluated following the order of operations. | Expression | Simplified | Operation |
|---|---|---|
| (1+2)⋅32−25+5 | 3⋅32−210 | Evaluating Parentheses and Grouping Symbols |
| 3⋅32−210 | 3⋅9−210 | Exponents |
| 3⋅9−210 | 27−5 | Multiplication and Division |
| 27−5 | 22 | Subtraction |
There are a few things to note about this evaluation.
- The addition in the numerator of 25+5 was evaluated at the same time as the expression between parentheses (1+2). This is because fraction bars are grouping symbols like parentheses.
- The order of operations must be followed when evaluating expressions in grouping symbols.
- Operations that are inverse and at the same level must be evaluated from left to right. This rule applies for multiplication and division and for addition and subtraction.
To remember the order of operations, it is useful to memorize the acronym PEMDAS. Each letter of PEMDAS indicates a set of operations. A fun sentence to remember this acronym is Please Excuse My Dear Aunt Sally.
Example
Baseball Practice
While waiting for baseball practice to start, Zain passed the time by counting how many people arrived to the field to practice and how many people left the field.
When Zain arrived and started counting, there were 9 people on the field practicing. Before Zain's practice started, two groups of three people left and four groups of six people arrived to the field. Then Zain's practice started.
a Write a numerical expression of the number of people in the field.
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b How many people were at Zain's practice that day?
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Hint
a Consider the given information carefully. What is the first number of people in the field?
b Follow the order of operations.
Solution
a To write the numerical expression, it is important to consider the given information carefully. First, it is given that there were 9 people on the field when Zain started counting. Write this number as the first part of the numerical expression.
9
Then, 2 groups of 3 people left. The 2 groups of 3 people can be written as the product of 2 times 3. Since these people are leaving, the product is subtracted from 9.
9−2⋅3
Finally, 4 groups of 6 people arrived to the field. This number of people can be written as the multiplication of 4 times 6. This time the product is added because the people are arriving.
9−2⋅3+4⋅6
This is a complete numerical expression to find the number of people at the field when Zain's practice started.
b The order of operations must be followed to evaluate the numerical expression found in Part A. The acronym PEMDAS is useful to recall the order of operations.
9−2⋅3+4⋅6
In this case, the expression does not have any grouping symbols or exponents. The symbols × and ⋅ are used to indicate multiplication, which means that the expression has two multiplications. These multiplications should be done from left to right.
The resulting expression has a subtraction and an addition. Since both of these operations are on the same tier of the order of operations, they are performed at the same time. Remember to do the operations from left to right.
Therefore, there were 27 people at Zain's practice that day.
Example
Baseball Equipment
Zain's baseball team needs new equipment before the season starts. Since Zain lives close to a good baseball equipment store, they were in charge of checking the prices. They noted these prices in a table.
| Item | Price |
|---|---|
| Bat | $200 |
| Glove | $95 |
| Uniform | $130 |
Zain's team need 2 new bats, 8 new gloves, and 4 new uniforms. Luckily, there is a sale going on where bats and gloves are half their regular prices. Zain also has a $100 discount coupon that they will give the coach for equipment.
a Write a numerical expression for the total cost of the equipment that Zain's team needs.
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b Evaluate the expression to find how much money Zain's team needs to buy the equipment.
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Hint
a Consider the given information carefully. The total cost per group of items can be written as a product.
b Remember the order of operations.
Solution
a Start by carefully considering the given information. It is given that Zain's team needs 2 new bats. Since each bat costs $200, the total cost of the bats is the multiplication of 2 and 200.
2⋅200
Next consider the cost of the gloves. Each glove has a price of $95. Since Zain's team needs 8 new gloves, the gloves have a cost the product of 95 and 8. Add this product to the cost of the bats.
2⋅200+95⋅8
Before adding the cost of the uniforms, it is important to remember that there is a sale going on that affects the cost of the bats and the gloves. To group this total, we can add parentheses to the addition.
(2⋅200+95⋅8)
The sale reduces this total by half, which can be written as multiplying the total by 21.
21⋅(2⋅200+95⋅8)
The total cost of the uniforms is the product of the number of uniforms and the price per uniform. The team needs 4 uniforms and the price of each uniform is $130. This product must be added to the expression above.
21⋅(2⋅200+95⋅8)+4⋅130
Lastly, the discount from Zain's coupon reduces the cost by $100. This is written as a subtraction of 100.
21⋅(2⋅200+95⋅8)+4⋅130−100
b The order of operations must be followed to evaluate the numerical expression from Part A. The steps of the order of operations can be remembered considering the acronym PEMDAS.
21⋅(2⋅200+95⋅8)+4⋅130−100
The P of PEMDAS refers to parentheses and other grouping symbols of the expression. The expression has two grouping symbols: a fraction line and a set of parentheses. The first thing to calculate from left to right is the fraction. It can be calculated directly.
21⋅(2⋅200+95⋅8)+4⋅130−100⇓0.5⋅(2⋅200+95⋅8)+4⋅130−100
Next, simplify the expression inside the set of parentheses.
2⋅200+95⋅8
This expression does not have any grouping symbols or exponents. However, it does have two multiplications. Evaluate these multiplications moving from left to right.
The resulting expression is a single addition. 400+760=1160
This result can be substituted for the expression inside the parentheses. 0.5⋅(2⋅200+95⋅8)+4⋅130−100⇓0.5⋅1160+4⋅130−100
Now the expression has two multiplications.
Finally, perform the addition and the subtraction from left to right.
Zain's team needs $1000 to buy the equipment they need!
Extra
Using the Distributive PropertyThe Distributive Property was discussed in previews lessons. The expression can be solved differently by using this property.
The Distributive Property allows to remove the parentheses before simplifying the expression inside. The rest of the expression can be simplified regularly.
The result is the same as the previous one following the order of operations. But remember that the property only holds true when a number is multiplying the result of an addition. It is not always possible to use the Distributive Property, but it can be useful in some cases.
Example
Zain at the Bat
Zain is having a great time at bat in today's baseball game. He is hitting every single ball!
a The height of the ball 2.75 seconds after Zain hit it is given by a numerical expression.
21(88⋅2.75−32⋅2.752)
What is the height of the ball? Round to the nearest integer. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"feet","answer":{"text":["0"]}}
b The distance that the ball traveled after the same 2.75 seconds can be found by using another numerical expression.
0.87⋅147⋅2.75
How far did the ball travel? Round to the nearest integer. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"feet","answer":{"text":["352"]}}
Hint
a Follow the order of operations.
b Note that the only operation in the numerical expression is multiplication.
Solution
a To evaluate the expression, every operation must be done following the order of operations. The acronym PEMDAS is useful to remember each step.
21(88⋅2.75−32⋅2.752)
The numerical expression has two grouping symbols: a fraction line and a set of parentheses. The decimal value of the fraction can be calculated directly.
21(88⋅2.75−32⋅2.752)⇓0.5(88⋅2.75−32⋅2.752)
The second grouping symbol is the set of parentheses. To remove the parentheses, the expression inside must be evaluated. Again, the first thing to evaluate this expression is to identify the operations.
88⋅2.75−32⋅2.752
The expression does not have any more grouping symbols, but it has an exponent. Evaluate it first.
There are two multiplications in this expression, so calculate them moving from left to right.
Now there is only a single subtraction left. 242−242=0
The expression between parentheses equals 0. Substitute this result into the original expression in place of the expression inside the parentheses. The parentheses can be replaced with a multiplication symbol as well.
0.5(88⋅2.75−32⋅2.752)⇓0.5⋅0
Finally, there is a single multiplication left. It is important to remember that the result of multiplying any number by zero is zero.
0.5⋅0=0
Therefore, the height of the ball is of 0 feet. This means that the ball hit the ground 2.75 seconds after Zain hit the ball.
b Looking at the given numerical expression, it can be noted the only operation present is multiplication.
0.87⋅147⋅2.75
These calculations will be done one at a time, from left to right.
0.87⋅147⋅2.75
MultII
Multiply 0.87 by 147
127.89⋅2.75
MultII
Multiply 127.89 by 2.75
351.6975
RoundInt
Round to nearest integer
352
Pop Quiz
Evaluating Numerical Expressions
Write the value of each given numerical expression. Remember the order of operations!
Closure
Finding the Correct Result
Two different numerical expressions were presented at the beginning of this lesson. The second expression has more operations than the first one. The good thing is that any numerical expression can be solved following the order of operations. The acronym PEMDAS is helpful for remembering the order!
The first numerical expression has both addition and multiplication.
3+2⋅5
When evaluating this expression, Izabella and Kriz found different results.
Izabella:Kriz:2513
But the correct result is the one obtained following the order of operations. This means that the multiplication needs to be done first, then the addition.
The answer is 13. This means that Kriz was correct. The second numerical expression is more complex, but the procedure to evaluate it is the same. The first step is always to determine which operations are in the expression.
42−448÷8+18
A fraction line works as a grouping symbol, like a set of parentheses. This means that the expressions of the denominator and the numerator have to be evaluated before dividing. Note that the denominator has an exponent, so this is the first operation to be performed.
42−448÷8+18⇓16−448÷8+18
Next, evaluate the division in the numerator.
16−448÷8+18⇓16−46+18
Next, calculate the addition and the subtraction, then divide the obtained values.
The value of the numeric expression is 2. Remember that it does not matter how simple or complex a numerical expression is — any numerical expression can be solved following the order of operations. That is the beauty of it!
Numerical Expressions
Exercise 1.1
3+5×4
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We want to find the value of the given numerical expression by using the specified order of operations. 3 + 5 * 4 We are told to add first, then to multiply. To make things easier, let's find the addition by itself first. 3 + 5 = 8 The two numbers add up to 8. We can substitute 8 for the addition in the expression so that we have the multiplication alone. 3 + 5 * 4 ⇓ 8 * 4 Now let's find the final value of the expression. 8 * 4 = 32 Therefore, if we follow the given order, the value of the expression is 32.
This time we want to do the multiplication first, then add. We will do something similar to what we did in Part A by calculating the multiplication first.
5* 4 = 20
Now substitute 20 into the expression for the multiplication expression. This will result in an expression with a single addition.
3 + 5 * 4
⇓
3 + 20
Let's perform the addition to find the result!
3 + 20 = 23
The result of evaluating the expression with this order is 23. It is different from the result from Part A!
To determine the correct result, we need to remember the order of operations. The acronym PEMDAS that helps to remember it.
If we consider the order, we can see that the M for multiplication goes before the A for addition. If we follow the order of operations, we need to multiply first, then add. We did this in Part B, which means that the correct result is 23!
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The following expression needs grouping symbols inserted in order to be calculated correctly.
41×32−12=5
Rewrite the expression on the left-hand side of the equal sign using grouping symbols.
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We are given a numerical expression. We want to insert grouping symbols — parentheses — so that the expression has a value of 5. 1/4 * 32 - 12 There should be at least two numbers and an operation inside the parentheses for the grouping to make any changes to the value of the expression. There are two ways we can do this with the given expression. (1/4 * 32) - 12 [0.8em] 1/4 * (32 - 12) According to the order of operations, expressions inside parentheses should always be evaluated first. This is why the parentheses can affect the value of the expression. Here we can see what we simplify next.
Note that operations that are from the same step are performed from left to right. Now let's use these rules to simplify the two expressions, starting with the first one.
(1/4 * 32) - 12
Let's do it!
| Operation | Before Simplification | After Simplification |
|---|---|---|
| Multiplication | ( 1/4 * 32) - 12 | ( 8) - 12 |
| Subtraction | 8-12 | -4 |
The expression simplifies to - 4. This is not our target value, so let's move on to the second one. 1/4 * (32 - 12) Let's simplify it while keeping the order of operations in mind!
| Operation | Before Simplification | After Simplification |
|---|---|---|
| Subtraction | 1/4 * ( 32 - 12) | 1/4 * ( 20) |
| Multiplication | 1/4 * 20 | 5 |
This time the expression does equal 5. Now we know how we need to rewrite the expression with parentheses! 1/4 * (32 - 12)
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Evaluate the following numerical expression.
11+3⋅72−59
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According to the order of operations, expressions inside parentheses are evaluated first, followed by exponents, then multiplication and division, and finally addition and subtraction. Let's carefully consider the given expression. 11 + 3* 7^2 - 59 For the given expression, we will evaluate the exponent first, the multiplication next, and then the addition and subtraction last. Let's do it!
| Operation | Before Simplification | After Simplification |
|---|---|---|
| Exponent | 11 + 3* 7^2 - 59 | 11 + 3* 49 - 59 |
| Multiplication | 11 + 3* 49 - 59 | 11 + 147 - 59 |
| Addition | 11 + 147 - 59 | 158-59 |
| Subtraction | 158-59 | 99 |
The expression is equal to 99.
We can also evaluate this expression using a calculator. To do so, we type the given expression in the window of the calculator, then press Enter.
The result is the same, so we know our answer is correct.
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Evaluate the following numerical expression.
3×[(81−27)×3]
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According to the order of operations, expressions inside parentheses are evaluated first, followed by exponents, then multiplication and division, and finally addition and subtraction. Let's carefully consider the given expression. 3 * [(81-27) * 3 ] For the given expression, we will evaluate the expression inside the square brackets completely before evaluating the product outside the square brackets. Remember to follow the order of operations inside the square brackets as well! (81-27) * 3 Let's evaluate the difference inside parentheses first, then the product.
| Operation | Before Simplification | After Simplification |
|---|---|---|
| Subtraction | 3 * [( 81- 27) * 3 ] | 3 * [ 54 * 3 ] |
| Multiplication Inside Brackets | 3 * [ 54 * 3 ] | 3 * 162 |
| Multiplication Outside Brackets | 3 * 162 | 486 |
The expression is equal to 486.
We can also evaluate this expression using a calculator. Type it into the calculator, then press Enter.
Whenever using a calculator for a complicated expression such as this, be very careful when placing the square brackets and the parentheses. A calculator will only do what it is told to do, not what you meant for it do!
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Paulina needs to read 3 chapters of a book in a week. Each chapter is 14 pages long. The pages Paulina needs to read each day can be expressed with a numerical expression.
3⋅14÷7
Help Paulina evaluate the expression to know how many pages she needs to read each day.
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Paulina needs to read 3 chapters of a book in a week. We can find the number of pages Paulina needs to read every day by solving the given numerical expression. 3* 14 ÷ 7 To evaluate this expression, we need we need to use the order of operations.
Remember that the symbols * and * are used to indicate multiplication. The symbol ÷ is used to indicate division, and we can also interpret a fraction as a division of the numerator by the denominator. In our case, since there are no grouping symbols or exponents, we will start by multiplying 3 by 14. 3 * 14 = 42 Now let's divide this result by 7. 42 ÷ 7 = 6 This means that Paulina needs to read 6 pages a day to stay on schedule for her assignment!
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Numerical Expressions
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