Numerical Expressions: Teaching the Language of Math

Numerical Expressions: Teaching the Language of Math

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Teaching the Language of Math

So much of being successful in math is being able to understand math vocabulary and numerical expressions for problem solving and to express one’s ideas and questions about the math concepts.  

Sometimes kids get hung up on the idea that math is only about numbers, but that’s just not true. I like to think of math as its own language that utilizes numbers, symbols, and words. The words can’t be forgotten!


Relate Numerical Expressions to Something They Already Know

Could you imagine if you were playing football but you didn’t know the many different positions and actions required to play the game? Your head would be spinning as the coach told you to pay attention to the snap, to help sack the quarterback, and to be prepared for rushing. Even if you had a great throwing arm and were super fast, if you didn’t know what those terms meant, you probably wouldn’t be the most helpful teammate or the most successful football player. It’s the same for math vocabulary words– you can’t just know how to work with numbers and symbols, to be a fluent mathematician, you’ve got to know the important terms!


Relationship Between Words, Symbols, and Numbers

Sometimes we think it’s only English Language Learners who need to really focus on the language aspects of math, but it’s important for all students to learn and be able to clearly articulate their understanding of math concepts and they need math vocabulary (the words part of the Math language) to do this. They also need to be explicitly shown the relationship between the words, the symbols, and the numbers. After all, no part of the math language exists in isolation. The numbers, symbols, and words are all key components of becoming fluent in the language of math. Studying numerical expressions is a great way for students to gain experience flexibly moving from numbers and symbols to words and back again. 


Establishing a Common Language: Numerical Expressions

While I believe in the importance of explicitly teaching math vocabulary, remember that this section should focus on exposing students to these concepts and providing them with the tools to apply these concepts in their hands-on learning activity. Students will learn much more by applying these concepts in the Translation Challenge, so don’t try to teach for mastery in this one section. Mistakes in the Challenge will make them think more deeply about the concepts and make the learning stick better than anything we say to them! In fact, I like to do this lesson during our double math block so we can stretch the amount of time students get to spend actively writing and communicating numerical expressions. 


Your Little Math Language Translators

When you launch, start by explaining to students that in this lesson, they are going to work as math language translators! They will use language and numbers to write numerical expressions, turn those expressions into written form, and then verbally test out their written form. However, before they can work as translators, they’ve got to make sure they have the groundwork for understanding how numerical expressions work. 


Numerical Expressions 5th Grade Math

Introduce the term numerical expression as a number sentence that does not have an equal sign. Post the vocabulary cards


Numerical Expressions – Vocabulary Cards (Editable)

Numerical Expressions – Vocabulary Cards (PDF)


Give a few examples of numerical expressions:

Help students to see that numerical expressions are a combination of numbers and symbols. Some of those symbols represent a specific operation. However, these expressions can also be represented with words. 


Give the same examples in a different order, but use words to represent them:

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Order of Operations

Remind students of the importance of the rule of Order of Operations. Long ago, mathematicians agreed upon a rule that determines in which order one should solve the operations found in an expression or an equation to make sure that all mathematicians solving the same problem will get to the same answer. In this way, mathematicians made sure they were all speaking the language of math the same way to ensure everyone arrived at the same understanding! Post or remind students of the Order of Operations vocabulary card.


Math and Music: Order of Operations Song

My students always love when I pair math with music, so play this fun music video if students need more support with this rule:  Order of Operations Song. Make sure to rock out along with the video! 



Next,  you need to make sure students are prepared when they see parentheses!

Show students this expression that was given earlier: (9 – 3) ÷ 2

Ask students to note the math symbols called parentheses that are around the 9-3. Have students turn and talk with a partner to try to determine why those parentheses are used in this expression. Then have students do a brief popcorn share of possible reasons for the presence of the parentheses. Some students may make a connection to the Order of Operations and notice that parentheses were mentioned as the first thing that should be solved. 


What is the Purpose of Parentheses?

Explain to students that the purpose of those parentheses is to more clearly communicate how this expression should be solved. In this case, the mathematician who wrote this expression wanted the subtraction to be done before the division.  Since division comes BEFORE subtraction in the order of operations rule, if the problem had been written as 9 – 3 ÷ 2, we would have to divide 3 ÷ 2 and then subtract that quotient from 9.  The only way to get the subtraction done first is to close it inside the parentheses. Then since these parentheses come first in the order of operations, whatever is inside the parentheses must be solved first! Therefore, when we write equations, if we want to do addition or subtraction before multiplication or division, we need to add parentheses around the numbers that are being added or subtracted. Sometimes brackets are used in the same way parentheses are or they can be used to section off an even larger part of an expression that has parentheses within it.  Post the parentheses and brackets vocabulary card


Play This Music Video:

Writing and Interpreting Numerical Expressions Song



Translator Training Materials

Students are almost ready to dive into their roles as translators, but first, I like to make sure they are well prepared with the vocabulary that will help them fluently move between numbers, symbols, and words. 

Just as students saw in the Writing and Interpreting Numerical Expressions song, language is crucial to working with numerical expressions. Quickly review the important vocabulary on the Order of Operations – Clue Words that will help them choose the correct operation (and symbol) to use when writing their expressions. 

Order of Operations – Clue Words (Editable)

Order of Operations – Clue Words (PDF)


Translation Challenge: Numerical Expressions

Now that students have the foundation for working with numerical expressions, they are ready for the Translation Challenge!


Set Up 

Explain to students that they will be working with a partner for this challenge and have partners quickly get together at their work stations. 


Explain to students that each partnership will receive 3 dice (or use a tech device to roll 3 digital dice.) Once they roll the dice, they will follow the Translation Challenge Round Instructions for each round to translate these words and the numbers on the dice into numerical expressions made of numbers and symbols. 


Translation Challenge Round Instructions (Editable)

Translation Challenge Round Instructions (PDF)


Quickly model with this sample round:


RoundDice #1Dice #2Dice #3
Sample RoundNumber that starts your expressionAdd this number to the first numberMultiply the sum of the first two numbers by this number 


When I model this, if I roll the dice and get 5, 2, 4, I would write: 5 + 2 

Then I’d think aloud,  “Ok, I need to multiply the sum of 5 + 2 so I could write 5 + 2 x 4. But wait, multiplication comes first in the order of operations, so that would mean I would do 2 x 4 and then add 5, which can’t be right because I’m supposed to multiply the sum. Hmmm…I need to add parentheses so I can add first and then multiply the sum. So it should look like this (5 + 2) x 4. Wow, there’s a lot of thinking I have to do before I can translate something correctly in math!”


Remind students of guidelines for proper use of dice (or use of technology if using digital dice). 


Each Partnership Should Receive:


Part 1

Tell students their goal is do 3 rolls of the 3 dice for each round, writing 3 numerical expressions in each round.  Set the timer for 5 minutes for each round. When the timer goes off, give students time to read the instructions for the next round before starting the timer for another 5 minutes. 


Circulate to support students as needed. 


Part 2

When the three rounds are complete, each pair should partner with another pair and exchange recording charts. 


The pairs will then work to translate their peers’ numerical expressions into words, using their math vocabulary! Remind students that the Operations Clue Words can help. 


Give students about 10 minutes to work on this, with the goal that there is at least one written expression for each round. Remind pairs to talk through the possible vocabulary they could use and emphasize that there can be more than one correct way to write these expressions! 


Part 3

As the final part of the translation challenge, students will find a new pair (not the pair they traded recording sheets with). They will read aloud their written expressions to the new pair, without showing what’s on their recording sheets. The new pair will listen and can ask for it to be repeated. They will then translate the written expression into a numerical expression that is represented with numbers and symbols. They will write these numerical expressions onto a post-it. 


If they “Match”

Once it is written on the post it, they can reveal the recording chart to see if the original numerical expression in the first column of the chart matches the one on the post it. If it does, students can check the “communicated clearly” box in the last column on the recording sheet because that means their translation was effective! 

If they “Don’t Match”

If they don’t match, all four students should work together to revise the written expression (if needed) so it more clearly communicates the numerical expression it matches. Students can also look to see if the post-it numerical expression is where the revision is needed. Remember, translators never give up if they don’t get their message across on the first try! Students should add any revisions in pen so they can be distinguished from the pencil used on the rest of the chart. 

Each pair should read at least 3 total written expressions, one from each round. Then the pairs should switch roles with the new set of expressions. 

Once each pair has taken a turn reading and writing, students can stick the post-its onto the chart next to the expression they match. 


Reflecting on the Translation Experience

As I support students in their journey towards fluency in the language of math, I think it’s important to help them reflect on their learning process and acknowledge their areas of strength, areas of need, and lingering questions. Consider the needs and preferences of your class and decide whether you will have students answer these reflection questions by writing in a math journal, by sharing their thoughts with a partner, or as a small group or whole class share. 

  • Which step was the hardest part of this translating activity for you? Why? 
  • Which step was the easiest part of this translating activity for you? Why?
  • Are there math words used today that feel confusing? List them. 
  • Are there symbols or numbers that still feel confusing? List them. 
  • What is the biggest take away/most important thing you learned from this lesson?


Extra Resources for Teaching Numerical Expressions:


FREE Numerical Expressions Worksheets and Resources

Writing and Interpreting Numerical Expressions 5th Grade Math


Translating Expressions with Parentheses


Order of Operations without Parentheses


Order of Operations with Parentheses


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