Mathematics expressions: 3.1: Mathematical Expressions — Mathematics LibreTexts

3.1: Mathematical Expressions — Mathematics LibreTexts

1. Last updated

2. Save as PDF

Page ID

22473

• David Arnold
• College of the Redwoods

Recall the definition of a variable presented in Section 1.6.

Definition: Variable

A variable is a symbol (usually a letter) that stands for a value that may vary.

Let’s add the definition of a mathematical expression.

Definition: Mathematical Expression

When we combine numbers and variables in a valid way, using operations such as addition, subtraction, multiplication, division, exponentiation, and other operations and functions as yet unlearned, the resulting combination of mathematical symbols is called a mathematical expression.

Thus,

2a, x + 5, and y²,

being formed by a combination of numbers, variables, and mathematical operators, are valid mathematical expressions. A mathematical expression must be well-formed. For example,

2 + ÷5x

is not a valid expression because there is no term following the plus sign (it is not valid to write +÷ with nothing between these operators). Similarly,

2 + 3(2

is not well-formed because parentheses are not balanced.

Translating Words into Mathematical Expressions

In this section we turn our attention to translating word phrases into mathematical expressions. We begin with phrases that translate into sums. There is a wide variety of word phrases that translate into sums. Some common examples are given in Table \(\PageIndex{1a}\), though the list is far from complete. In like manner, a number of phrases that translate into differences are shown in Table \(\PageIndex{1b}\).

Table \(\PageIndex{1}\): Translating words into symbols.

Phrase	Translates to:	Phrase	Translates to:
sum of x and 12	x + 12	difference of x and 12	x − 12
4 greater than b	b + 4	4 less than b	b − 4
6 more than y	y + 6	7 subtracted from y	y − 7
44 plus r	44 + r	44 minus r	44 − r
3 larger than z	z + 3	3 smaller than z	z − 3
a) Phrases that are sums		b) Phrases that are differences

Let’s look at some examples, some of which translate into expressions involving sums, and some of which translate into expressions involving differences.

Example 1

Translate the following phrases into mathematical expressions:

1. «12 larger than x,«
2. «11 less than y,» and
3. «r decreased by 9.»

Solution

Here are the translations.

1. “12 larger than x” becomes x + 12.
2. “11 less than y” becomes y − 11.
3. “r decreased by 9” becomes r − 9.

Exercise

Translate the following phrases into mathematical expressions:

1. «13 more than x» and
2. «12 fewer than y«.

Answer

(a) x + 13 and

(b) y − 12

Example 2

Let W represent the width of the rectangle. The length of a rectangle is 4 feet longer than its width. Express the length of the rectangle in terms of its width W.

Solution

We know that the width of the rectangle is W. Because the length of the rectangle is 4 feet longer that the width, we must add 4 to the width to find the length.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length} & \text{is} & \colorbox{cyan}{4} & \text{more than} & \colorbox{cyan}{the width} \\ \text{Length} & = & 4 & + & W \end{array}\nonumber \]

Thus, the length of the rectangle, in terms of its width W, is 4 + W.

Exercise

The width of a rectangle is 5 inches shorter than its length L. Express the width of the rectangle in terms of its length L.

Answer

L − 5

Example 3

A string measures 15 inches is cut into two pieces. Let x represent the length of one of the resulting pieces. Express the length of the second piece in terms of the length x of the first piece.

Solution

The string has original length 15 inches. It is cut into two pieces and the first piece has length x. To find the length of the second piece, we must subtract the length of the first piece from the total length.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length of the second piece} & \text{is} & \colorbox{cyan}{Total length} & \text{minus} & \colorbox{cyan}{the length of the first piece} \\ \text{Length of the second piece} & = & 15 & — & x \end{array}\nonumber \]

Thus, the length of the second piece, in terms of the length x of the first piece, Answer: 12 + x is 15 − x.

Exercise

A string is cut into two pieces, the first of which measures 12 inches. Express the total length of the string as a function of x, where x represents the length of the second piece of string.

Answer

12 + x

There is also a wide variety of phrases that translate into products. Some examples are shown in Table 3.2(a), though again the list is far from complete. In like manner, a number of phrases translate into quotients, as shown in Table 3.2(b).

Table \(\PageIndex{2}\): Translating words into symbols.

Phrase	Translates to:	Phrase	Translates to:
product of x and 12	12x	quotient of x and 12	x/12
4 times b	4b	4 divided by b	4/b
twice r	2r	the ratio of 44 to r	44/r
a) Phrases that are products.		b) Phrases that are differences.

Let’s look at some examples, some of which translate into expressions involving products, and some of which translate into expressions involving quotients.

Example 4

Translate the following phrases into mathematical expressions: (a) “11 times x,” (b) “quotient of y and 4,” and (c) “twice a.”

Solution

Here are the translations. a) “11 times x” becomes 11x. b) “quotient of y and 4” becomes y/4, or equivalently, \(\frac{y}{4}\). c) “twice a” becomes 2a.

Exercise

Translate into mathematical symbols: (a) “the product of 5 and x” and (b) “12 divided by y.”

Answer

(a) 5x and (b) 12/y.

Example 5

A plumber has a pipe of unknown length x. He cuts it into 4 equal pieces. Find the length of each piece in terms of the unknown length x.

Solution

The total length is unknown and equal to x. The plumber divides it into 4 equal pieces. To find the length of each pieces, we must divide the total length by 4.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Length of each piece} & \text{is} & \colorbox{cyan}{Total length} & \text{divided by} & \colorbox{cyan}{4} \\ \text{Length of each piece} & = & x & \div & 4 \end{array}\nonumber \]

Thus, the length of each piece, in terms of the unknown length x, is x/4, or equivalently, \(\frac{x}{4}\).

Exercise

A carpenter cuts a board of unknown length L into three equal pieces. Express the length of each piece in terms of L.

Answer

L/3

Example 6

Mary invests A dollars in a savings account paying 2% interest per year. She invests five times this amount in a certificate of deposit paying 5% per year. How much does she invest in the certificate of deposit, in terms of the amount A in the savings account?

Solution

The amount in the savings account is A dollars. She invests five times this amount in a certificate of deposit.

\[ \begin{array}{c c c c c} \colorbox{cyan}{Amount in CD} & \text{is} & \colorbox{cyan}{5} & \text{times} & \colorbox{cyan}{Amount in savings} \\ \text{Amount in CD} & = & 5 & \cdot & A \end{array}\nonumber \]

Thus, the amount invested in the certificate of deposit, in terms of the amount A in the savings account, is 5A.

Exercise

David invests K dollars in a savings account paying 3% per year. He invests half this amount in a mutual fund paying 4% per year. Express the amount invested in the mutual fund in terms of K, the amount invested in the savings account.

Answer

\(\frac{1}{2}K\)

Combinations

Some phrases require combinations of the mathematical operations employed in previous examples.

Example 7

Let the first number equal x. The second number is 3 more than twice the first number. Express the second number in terms of the first number x.

Solution

The first number is x. The second number is 3 more than twice the first number.

\[ \begin{aligned} \colorbox{cyan}{Second number} & \text{is} & \colorbox{cyan}{3} & \text{more than} & \colorbox{cyan}{twice the first number} \\ \text{Second number} & = & 3 & + & 2x \end{aligned}\nonumber \]

Therefore, the second number, in terms of the first number x, is 3 + 2x.

Exercise

A second number is 4 less than 3 times a first number. Express the second number in terms of the first number y.

Answer

3y − 4

Example 8

The length of a rectangle is L. The width is 15 feet less than 3 times the length. What is the width of the rectangle in terms of the length L?

Solution

The length of the rectangle is L. The width is 15 feet less than 3 times the length.

\[ \begin{aligned} \colorbox{cyan}{Width} & \text{is} & \colorbox{cyan}{3 times the length} & \text{less} & \colorbox{cyan}{15} \\ \text{Width} & = & 3L & — & 15 \end{aligned}\nonumber \]

Therefore, the width, in terms of the length L, is 3L − 15.

Exercise

The width of a rectangle is W. The length is 7 inches longer than twice the width. Express the length of the rectangle in terms of its length L.

Answer

2W + 7

Exercises

In Exercises 1-20, translate the phrase into a mathematical expression involving the given variable.

1. “8 times the width n ”

2. “2 times the length z ”

3. “6 times the sum of the number n and 3”

4. “10 times the sum of the number n and 8”

5. “the demand b quadrupled”

6. “the supply y quadrupled”

7. “the speed y decreased by 33”

8. “the speed u decreased by 30”

9. “10 times the width n ”

10. “10 times the length z ”

11. “9 times the sum of the number z and 2”

12. “14 times the sum of the number n and 10”

13. “the supply y doubled”

14. “the demand n quadrupled”

15. “13 more than 15 times the number p ”

16. “14 less than 5 times the number y ”

17. “4 less than 11 times the number x ”

18. “13 less than 5 times the number p ”

19. “the speed u decreased by 10”

20. “the speed w increased by 32”

---

21. Representing Numbers. Suppose n represents a whole number.

i) What does n + 1 represent?

ii) What does n + 2 represent?

iii) What does n − 1 represent?

22. Suppose 2n represents an even whole number. How could we represent the next even number after 2n?

23. Suppose 2n + 1 represents an odd whole number. How could we represent the next odd number after 2n + 1?

24. There are b bags of mulch produced each month. How many bags of mulch are produced each year?

25. Steve sells twice as many products as Mike. Choose a variable and write an expression for each man’s sales.

26. Find a mathematical expression to represent the values.

i) How many quarters are in d dollars?

ii) How many minutes are in h hours?

iii) How many hours are in d days?

iv) How many days are in y years?

v) How many months are in y years?

vi) How many inches are in f feet?

vii) How many feet are in y yards?

---

Answers

1. 8n

3. 6(n + 3)

5. 4b

7. y − 33

9. 10n

11. 9(z + 2)

13. 2y

15. 15p + 13 17.

11x − 4

19. u − 10

21.

i) n+1 represents the next whole number after n.

ii) n+2 represents the next whole number after n + 1, or, two whole numbers after n.

iii) n − 1 represents the whole number before n.

23. 2n + 3

25. Let Mike sell p products. Then Steve sells 2p products.

---

This page titled 3.1: Mathematical Expressions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by David Arnold.

1. Back to top

• Was this article helpful?

1. Article type

   Section or Page

   Author

   David Arnold

   License

   CC BY-NC-SA

   License Version

   3.0

   Show Page TOC

   no

2. Tags

   This page has no tags.

Math Expressions

Build deep understanding with this essential inquiry-based mathematics curriculum

Mastering math starts with knowing the numbers. We get students started on the right path in Pre-K, but that’s just the beginning. It’s also about looking beyond numbers to understand how and why they work, and learning to ask the questions that lead to a thoughtful, informed approach to solving problems.

Based on research funded by the National Science Foundation (NSF), Math Expressions is a proven Pre-K–6 curriculum that helps children make sense of math by exploring, discussing, and demonstrating their understanding of key concepts. In busy, active lessons, students learn how to look deeper and choose their own path to the answers—skills that will take them far beyond the math classroom.

Jump to…

OverviewResearch & ResultsAuthor

To purchase, please contact your local sales rep

---

Looking deeper produces exceptional results

Every lesson integrates mathematical
processes and practices.

Students use authentic examples to
make sense of mathematics.

Schools see an average gain of
12–15% on state tests.

A Yearlong Study of Success

Students from eight schools across four states showed «statistically significant» improvement.

When children understand math, they’re not relying on memorization. They’re relying on themselves. Dr. Karen Fuson’s comprehensively researched approach to teaching math is based on the way children actually learn and is used across the country because it raises student achievement.

---

Overview

The deep conceptual understanding that’s the hallmark of the program leads to strong skills fluency and the ability to make generalizations within and among mathematical strands. As students grow and learn to think critically, teachers develop their own richer understanding of math through detailed instruction and embedded support.

Math Expressions is based on NSF-funded research.

Contextual Learning

Students develop an understanding of mathematics through real-world situations and visual supports.

Multiple Strategies

Students learn multiple ways to solve problems, including algorithms based on reasoning.

Manageable Instruction

Teaching materials embody a «learn-while-teaching» style.

---

Student technology builds understanding, practice, and problem solving.

• The Math Worlds augmented reality app gives students an amazingly interactive opportunity to apply what they’ve learned.
• Games in the online Math Activity Center develop fluency in math facts and operations.
• Students build concrete and pictorial relationships with digital manipulatives.
• Learners can download and practice working with content offline via HMH Player^®.

Lessons build understanding through active inquiry.

Coherence 
Students make connections between key topics both within and across grades.

Discourse 

Math Talk gives students the chance to ask questions and use their MathBoards to explain and justify their solutions.

Practice 

The online Student Activity Book gives young learners the tools to explore and deepen their understanding of key math concepts.

Making teachers’ lives easier by changing the learning landscape

Hear what Dr. Fuson thinks about early learning in mathematics and how it can help close the equity gap.

---

Classroom Structures Support Understanding and Math Talk

• Students use colorful manipulatives and shape puzzle mats to develop a deep understanding of number and geometry concepts.
• Fun games and puzzle cards engage students and get them excited about mathematics.
• Daily routines include learning the count word sequence and relating to number symbols and quantities through patterns, fingers, and actions.
• Students take turns as student leaders, building confidence over time.

Research tells us that students who start kindergarten with a high level of math understanding do better in both math and reading after kindergarten. With that in mind, we developed Math Expressions Early Learning Resources to provide early learners with the understanding they need to confidently take the first and subsequent steps of their mathematics journey.

Grounded in Research

Students will benefit from teaching-learning paths and structured, repetitive experiences as recommended in “Mathematics Learning in Early Childhood: Paths toward Excellence and Equity.”

Rated as Strong by ESSA
Early Learning Resources is based on the same ESSA strong foundation and research as the rest of Math Expressions, making it an ideal transitional program for pre-kindergarten early learners.

Focused on Numbers and Geometry
Research guidelines led to a strong focus on numbers and geometry, content areas that are particularly important in early math instruction.

Hands-On
Playful activities help children see, discuss, and use mathematical structures, building understanding and fluency gradually and systematically.

---

Math Expressions facilitates learning while teaching.

• Professional development, classroom demonstration, and daily routine videos are at point of use in the Teacher Edition.
• Lessons include teaching and inquiry notes, visual models, and suggestions for Math Talk.
• Mathematical processes, practices, and learning progressions are featured in every lesson.
• Formative assessment reporting helps group students for instruction and identifies appropriate differentiation resources.

Differentiated support to encourage all students.

Differentiated Instruction 
The Math Activity Center supports every child with Practice, Re-teach, and Challenge resources for each lesson.

Adaptive Practice
Customize instruction with the Personal Math Trainer^® Powered by Knewton™, an online personalized learning and assessment system with real-time reporting and adaptive practice.

Math Expressions classrooms are collaborative

Hear from a teacher about how Math Talk engages his class.

---

Instructional Content

The instruction keeps students constantly moving forward while providing teachers with explicit support. Learn more about Dr. Fuson’s approach through the samplers below.

2018 Program Sampler

CCSS Program Sampler

Early Learning Sampler

View the Sampler

---

Outcomes

Deepen mathematical understanding

Develop critical-thinking skills

Enhance teacher effectiveness

Increase student participation

Improve student performance

---

Research & Results

The research driving Math Expressions gives students a full understanding of concepts behind the math and the tools to effectively express what they’ve learned. Learn more about what’s behind the program and its impact on students across the country.

Math Expressions Grades K-5 has been rated as Meets Expectations at Gateways 1, 2, and 3 by EdReports.

---

Math Expressions received a Strong ESSA rating based on the distinctive amount and quality of professional learning teachers receive.

Read the Overview   ESSA Evidence Criteria for Math Expressions

---

Math Expressions is based on the results of the NSF-funded Children’s Math Worlds research project, a 10-year study led by Dr. Fuson.

• Report Type: Research Evidence Base
• Grade Level: Elementary, Middle

Read & Download the Report

• Report Type: Efficacy Study, Publication from External Organization, Study Conducted by Third Party
• Grade Level: Elementary
• District Urbanicity: Urban, Suburban

Read & Download the Report

• Report Type: Efficacy Study, Study Conducted by Third Party
• Grade Level: Elementary
• Region: Midwest, West
• District Urbanicity: Suburban, Rural

Read & Download the Report

Explore More Research

---

About the Author

Dr. Karen Fuson, is a leading voice in math education today

Dr. Karen Fuson, Professor Emerita at Northwestern University, has studied how children understand math ideas for more than 50 years. She designs teaching materials based on how children learn and understand, along with working in schools to help teachers support every child.

---

To purchase, please contact your local sales rep

Numerical and alphabetic expressions — what is a mathematical expression

Home » Grade 5. Mathematics. » Numeric expressions and letter expressions — rules

When solving examples and equations, it is necessary to clearly distinguish between what is a numerical expression and what is a letter expression. Therefore, today we will go through this topic and watch the video. So, learn the rules.

Contents

Rules of mathematics about numeric and alphabetic expressions

A numeric expression is an expression that is made up of numbers and has the signs “+”, “-”, as well as signs of multiplication or division. Numeric expressions can also contain parentheses.

The number resulting from the mathematical operations included in this numeric expression is called the value of the numeric expression.

Literal expressions are expressions containing Latin letters, as well as signs of mathematical operations of addition, subtraction, multiplication and division, or brackets (if necessary).

Numbers that replace a letter are called the values ​​of that letter.

To remember the rules, let’s look at some examples. Examples are the easiest visual way to remember the statements above.

Watch the video:

Examples of Numeric Expressions

— the left side of this equation is a numeric expression, and the right side is the value of the numeric expression.

— on the left side of the equality is a numeric expression, and on the right side is the value of the numeric expression.

See more examples of numeric expressions:

• ,
• ,
• ,
• .

Letter expression examples

Examples of literal expressions:

• ,
• ,
• ,
• ,
• .

There can be many literal expressions. For literal expressions, each letter is a specific number. Or many different numbers.

When literal expressions are used

Letter expressions are used when we need, for example, to enter a formula to find a particular quantity. For example, you know that the perimeter of a rectangle is the sum of all its sides. For the perimeter in general terms, you can write a literal expression:

, where is the width of the rectangle and is its length.

On the right side, you saw a literal expression, the values ​​​​of the letters and — will be numbers — are the values ​​\u200b\u200bof the width and length of the rectangle.

Mathematical expressions

A mathematical expression is an expression that contains both a numeric and a letter expression and their product, sum, difference, or division.

So, for example,

is a mathematical expression.

The number in front of a letter in a math expression is a factor, meaning that the number is multiplied by the letter, which can have a variable value.

Alphabetical, numeric and mathematical expressions — must be distinguished in order to understand the conditions of the problem, for example, you may be asked to simplify a mathematical expression or asked to find the value of a numerical expression. Therefore, it is necessary to know what it is.

If you understand everything, take the Numeric and Letter Expressions test.

Numerical and alphabetic expressions — what is a mathematical expression

Home » Grade 5. Mathematics. » Numeric expressions and letter expressions — rules

When solving examples and equations, it is necessary to clearly distinguish between what is a numerical expression and what is a letter expression. Therefore, today we will go through this topic and watch the video. So, learn the rules.

Contents

Rules of mathematics about numeric and alphabetic expressions

A numeric expression is an expression that is made up of numbers and has the signs “+”, “-”, as well as signs of multiplication or division. Numeric expressions can also contain parentheses.

The number that results from the mathematical operations included in this numeric expression is called the value of the numeric expression.

Letter expressions are expressions containing Latin letters, as well as signs of mathematical operations of addition, subtraction, multiplication and division, or brackets (if necessary).

Numbers that replace a letter are called the values ​​of that letter.

To remember the rules, let’s look at some examples. Examples are the easiest visual way to remember the statements above.

Watch the video:

Numerical examples

— on the left side of this equality is a numeric expression, and on the right side is the value of the numeric expression.

— on the left side of the equality is a numeric expression, and on the right side is the value of the numeric expression.

See more examples of numeric expressions:

• ,
• ,
• ,
• .

Examples of literal expressions

Examples of literal expressions:

• ,
• ,
• ,
• ,
• .

There can be many literal expressions. For literal expressions, each letter is a specific number. Or many different numbers.

When literal expressions are used

Letter expressions are used when we need, for example, to enter a formula to find a particular quantity.