MA.5.AR.2.1

Translate written real-world and mathematical descriptions into numerical expressions and numerical expressions into written mathematical descriptions.

Examples

The expression 4.5 + (3×2) in word form is four and five tenths plus the quantity 3 times 2.

Clarifications

Clarification 1: Expressions are limited to any combination of the arithmetic operations, including parentheses, with whole numbers, decimals and fractions. 

Clarification 2: Within this benchmark, the expectation is not to include exponents or nested grouping symbols.

General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 5
Strand: Algebraic Reasoning
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Expression

 

Vertical Alignment

Previous Benchmarks

 

Next Benchmarks

 

Purpose and Instructional Strategies

The purpose of this benchmark is for students to translate between numerical and written mathematical expressions. This builds from previous work where students wrote equations with unknowns in any position of the equation in grade 4 (MA.4.AR.2.2). Algebraic expressions are a major theme in grade 6 starting with MA.6.AR.1.1. Students continue to deepen their understanding of operations with whole numbers (MA.5.NSO.2.1

/MA.5.NSO.2.2).

  • During instruction, teachers should model how to translate numerical expressions into words using correct vocabulary. This includes naming fractions and decimals correctly. Students should use diverse vocabulary to describe expressions. 
    • For example, in the expression 4.5 + (3 × 2) could be read in multiple ways to show its operations. Students should explore them and find connections between their meanings (MTR.3.1, MTR.4.1, MTR.5.1). 
      • 4 and five tenths plus the quantity 3 times 2  
      • 4 and 5 tenths plus the product of 3 and 2 
      • The sum of 4 and 5 tenths and the quantity 3 times 2 
      • The sum of 4 and 5 tenths and the product of 3 and 2 
  • The expectation of this benchmark is to not use exponents or nested grouping symbols. Nested grouping symbols refer to grouping symbols within one another in an expression, like in 3 + [5.2 + (4 × 2)]. Explain to students that parentheses make the expression clearer, but they are not necessary when multiplication comes before addition, as seen in the example.
  • Instruction of this benchmark helps students understand the order of operations, the expectation of MA.5.AR.2.2
  • Teachers can encourage and motivate students to utilize properties to rewrite expressions. Instruction includes a review of the distributive, associative, and commutative properties of addition and multiplication. Reinforce the concept that the distributive property involves the distribution of multiplication over addition, while the commutative property entails a change in the order of the terms.

 

Common Misconceptions or Errors

  • Students can misrepresent decimal and fraction numbers in words. This benchmark helps students practice naming numbers according to place value. 
  • Some students can confuse the difference between what is expected in the expressions 5(9 + 3) and 5 + (9 + 3). Students need practice naming the former as multiplication (e.g., 5 times the sum of 9 and 3) and understanding that in that expression, both 5 and 9 + 3 are factors.
  • Students may have confusion with parentheses. Misplacement or omission of parentheses can lead to incorrect interpretations of expressions. Students might forget to use parentheses where necessary or include unnecessary ones, altering the meaning of the expression. 
  • Additional misinterpretations include misunderstanding the order of operations, where a student mistakenly performs an operation over another (e.g. addition before multiplication with no parentheses present).

 

Strategies to Support Tiered Instruction

  • Instruction includes opportunities to name fractions and decimals correctly according to place value. The teacher provides students a place value chart to support correctly naming decimals. Students use appropriate terminology for naming fractions. 
o For example, write 8.601 in standard form and word form in a place value chart. 

 

 place value chart
    • For example, students write 10.36 in standard form and word form in a place value chart. 
 place value chart
    • For example, students write 2.47 in standard form and word form in a place value chart using place value disks. Place value disks can be used as a visual representation in addition to the place value chart.

    • For example, students write 512 in word form (five twelfths). 
    • For example, students write 2 78 in word form (two and seven eighths). This is repeated with additional fractions and decimals. 
  • Instruction includes opportunities to correctly translate numerical expressions into words using appropriate vocabulary.
    • For example, the teacher has students read aloud the following expression and write in word form. Next, the teacher models one way of reading aloud and has students provide alternate ways while using questioning to facilitate the conversation about the multiple ways the expression can be read aloud to show its operations. 
18.49−(27÷3) 
      • Eighteen and forty-nine hundredths minus the quotient of twenty-seven divided by three.
      • 18 and 49 hundredths minus the quantity 27 divided by 3.
      • The difference between 18 and 49 hundredths and the quotient of 27 divided by 3. 
      • The difference between 18 and 49 hundredths and the quantity 27 divided by 3. 

 

Instructional Tasks

Instructional Task 1 (MTR.4.1

  • Nadia sees the numerical expression 6.5 + 12 (4-2). She translates the expression as, “6 and five tenths plus 1 half times 4, minus 2."
  • Part A: Is her translation correct? Explain. 
  • Part B: Evaluate the expression.

 

Instructional Task 2 (MTR.3.1

Translate the written mathematical description below into a numerical expression: Divide the difference of 20 and 5 by the sum of 4 and 1. 

Instructional Task 3

Melissa spends time each evening designing bracelets for her friends. The number of minutes she spends can be represented by the variable b. Use the variable to complete the following tasks.

1. Write an algebraic expression to represent the number of minutes Melissa’s friend Gabriella spends designing bracelets in an evening if she designs 45 minutes more that evening than Melissa.

2. Write an algebraic expression to represent the number of minutes Melissa will spend designing bracelets at home in 9 weeks.

 

Instructional Task 4
Below is a model that represents 6+4.
model with 10 squares

a. Draw a model that represents 6×(6+4).
b. How much times greater is the value of 6×(6+4) than 6+4?

Instructional Task 5


Alexis is booking a vacation rental for $50 per night.

Part A. Write an expression to determine the total cost of renting the vacation home for h nights.
Part B. What would be the total cost for a 10-night stay?

Instructional Task 6

Jordan is reserving a conference room for $75 per hour.

 
Part A. Write an expression to calculate the total cost of renting the conference room for h hours.
Part B. What would be the total cost to rent the conference room for 9 hours each on three days?

 

Instructional Items

Instructional Item 1 

Translate the numerical expression below into a written mathematical description.
2(53.8 + 4 − 22.9) 

 

Instructional Item 2

Translate the written mathematical description into a numerical expression. “one half the difference of 6 and 8 hundredths and 2” 

 

Instructional Item 3

Which numerical expression is equivalent to the written expression?
4 times the quantity 97 plus 156

a. (4×97)+156
b. 4×97+156
c. 4×(97+156)
d. 2×(4×97+156)

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive. 


Related Courses

This benchmark is part of these courses.
5012070: Grade Five Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
7712060: Access Mathematics Grade 5 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
5012065: Grade 4 Accelerated Mathematics (Specifically in versions: 2019 - 2022, 2022 - 2024, 2024 and beyond (current))
5012015: Foundational Skills in Mathematics 3-5 (Specifically in versions: 2019 - 2022, 2022 - 2024, 2024 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.5.AR.2.AP.1: Translate mathematical descriptions (e.g., five plus two; the product of three and four) into numerical expressions with two terms.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessments

Write the Expression:

Students are presented with a verbal description of a numerical expression and are asked to write the expression and then compare it to a similar expression.

Type: Formative Assessment

Brayden’s Video Game:

Students are asked to write an expression requiring more than one operation and the use of parentheses to model a word problem.

Type: Formative Assessment

How Much Greater Is The Product?:

Students are asked to model an expression that is a multiple of a sum and to compare the expression to the sum.

Type: Formative Assessment

Comparing Products:

Students are asked to analyze and compare two related products.

Type: Formative Assessment

Lesson Plans

Fly Runners Order of Operations MEA:

In this Model Eliciting Activity, MEA, the students will determine the best ad for a tennis shoe company.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

“Express Yourself!” with Math Story Chains:

Students work in small groups to write math story chains (multi-step real world problems) and write expressions or equations for their story chains.

Type: Lesson Plan

Original Student Tutorial

Expressions: Not Just for Faces:

Learn how to write mathematical expressions while making faces in this interactive tutorial!

Type: Original Student Tutorial

Problem-Solving Tasks

Video Game Scores:

This task asks students to exercise both of these complementary skills, writing an expression in part (a) and interpreting a given expression in (b). The numbers given in the problem are deliberately large and "ugly" to discourage students from calculating Eric's and Leila's scores. The focus of this problem is not on numerical answers, but instead on building and interpreting expressions that could be entered in a calculator or communicated to another student unfamiliar with the context.

Type: Problem-Solving Task

Seeing is Believing:

The purpose of this task is to help students see that 4×(9+2) is four times as big as (9+2). Though this task may seem very simple, it provides students and teachers with a very useful visual for interpreting an expression without evaluating it because they can see for themselves that 4×(9+2) is four times as big as (9+2).

Type: Problem-Solving Task

Comparing Products:

The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.

Type: Problem-Solving Task

Words to Expressions 1:

This problem allows student to see words that can describe an expression although the solution requires nested parentheses.  Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

Type: Problem-Solving Task

Tutorials

Translating Expressions with Parentheses:

This Khan Academy tutorial video interprets written statements and writes them as mathematical expressions.

Type: Tutorial

Constructing a Numerical Expression Example:

This Khan Academy tutorial video demonstrates how to write a simple expression from a word problem.

Type: Tutorial

STEM Lessons - Model Eliciting Activity

Fly Runners Order of Operations MEA:

In this Model Eliciting Activity, MEA, the students will determine the best ad for a tennis shoe company.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

MFAS Formative Assessments

Brayden’s Video Game:

Students are asked to write an expression requiring more than one operation and the use of parentheses to model a word problem.

Comparing Products:

Students are asked to analyze and compare two related products.

How Much Greater Is The Product?:

Students are asked to model an expression that is a multiple of a sum and to compare the expression to the sum.

Write the Expression:

Students are presented with a verbal description of a numerical expression and are asked to write the expression and then compare it to a similar expression.

Original Student Tutorials Mathematics - Grades K-5

Expressions: Not Just for Faces:

Learn how to write mathematical expressions while making faces in this interactive tutorial!

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Original Student Tutorial

Expressions: Not Just for Faces:

Learn how to write mathematical expressions while making faces in this interactive tutorial!

Type: Original Student Tutorial

Problem-Solving Tasks

Video Game Scores:

This task asks students to exercise both of these complementary skills, writing an expression in part (a) and interpreting a given expression in (b). The numbers given in the problem are deliberately large and "ugly" to discourage students from calculating Eric's and Leila's scores. The focus of this problem is not on numerical answers, but instead on building and interpreting expressions that could be entered in a calculator or communicated to another student unfamiliar with the context.

Type: Problem-Solving Task

Seeing is Believing:

The purpose of this task is to help students see that 4×(9+2) is four times as big as (9+2). Though this task may seem very simple, it provides students and teachers with a very useful visual for interpreting an expression without evaluating it because they can see for themselves that 4×(9+2) is four times as big as (9+2).

Type: Problem-Solving Task

Comparing Products:

The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.

Type: Problem-Solving Task

Words to Expressions 1:

This problem allows student to see words that can describe an expression although the solution requires nested parentheses.  Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

Type: Problem-Solving Task

Tutorials

Translating Expressions with Parentheses:

This Khan Academy tutorial video interprets written statements and writes them as mathematical expressions.

Type: Tutorial

Constructing a Numerical Expression Example:

This Khan Academy tutorial video demonstrates how to write a simple expression from a word problem.

Type: Tutorial

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Tasks

Video Game Scores:

This task asks students to exercise both of these complementary skills, writing an expression in part (a) and interpreting a given expression in (b). The numbers given in the problem are deliberately large and "ugly" to discourage students from calculating Eric's and Leila's scores. The focus of this problem is not on numerical answers, but instead on building and interpreting expressions that could be entered in a calculator or communicated to another student unfamiliar with the context.

Type: Problem-Solving Task

Seeing is Believing:

The purpose of this task is to help students see that 4×(9+2) is four times as big as (9+2). Though this task may seem very simple, it provides students and teachers with a very useful visual for interpreting an expression without evaluating it because they can see for themselves that 4×(9+2) is four times as big as (9+2).

Type: Problem-Solving Task

Comparing Products:

The purpose of this task is to generate a classroom discussion that helps students synthesize what they have learned about multiplication in previous grades. It builds on applying properties of operations as strategies to multiply and divide and interpreting a multiplication equation as a comparison.

Type: Problem-Solving Task

Words to Expressions 1:

This problem allows student to see words that can describe an expression although the solution requires nested parentheses.  Additionally , the words (add, sum) and (product, multiply) are all strategically used so that the student can see that these words have related meanings.

Type: Problem-Solving Task