Simplify Expressions Involving Addition and Subtraction of Positive Rational Numbers

This is going to be an extensive post on the subject. Kind of attacking it at all levels. This is how I approach this section of the curriculum.

TEKS Covered 5.4E 5.4F 5.4B

In Texas this is about the fifth subject we cover in the school year. It is suggested you spend around 5 days on the subject if you have 90 minutes per day.

  • 5.4E Describe the meaning of parentheses and brackets in a numeric expression.
  • 5.4F Simplify numerical expressions that do not involve exponents, including up to two levels of grouping. [Positive Rational Numbers: Addition and Subtraction]
  • 5.4B Represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity. [Addition and subtraction]

Those are the main TEKS you will be covering. Of course these relate to common core as well. For your lesson planning you will also be looking at these process standard TEKS or common core areas.

  • 5.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
  • 5.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
  • 5.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

How Many Times Do I See This On A Test

Here in Texas dating back to 2015 5.4E and 5.4F has always had two types of questions on the state test. However last year they had three questions dealing with these TEKS. Around here we would phrase it as numerical expressions.

Three questions might not sound like a lot, but I know there are times students need to make up three to four questions to make standards. This one TEK could be the one to push them over the edge.

What Students Will be Able to Do

Students will simplify numerical expressions involving addition and subtraction of whole numbers and decimals students will describe the meaning of parentheses and brackets in numeric expressions and will simplify numerical expressions (involving addition and subtraction of whole numbers, decimals, and fractions) that do not involve exponents, including up to two levels of grouping.

Academic Vocabulary Used in Unit

  • Expressions: Does not have an equal sign. See image below
  • Order of Expression: The “operations” are addition, subtraction, multiplication, division, exponentiation, and grouping; the “order” of these operations states which operations take precedence (are taken care of) before which other operations.
  • Algebraic Expression: An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change.
  • Positive Rational Numbers: Rational numbers larger than zero
  • Grouping Symbols: Brackets, parenthesis, etc. show where a group starts and ends, and help to establish the order used to apply math operations. 
  • Simplify: To reduce a fraction to its lowest terms

Recommended Steps of Instruction

My district recommends these steps when it comes to teaching the subject.

  • Use concrete and/or pictorial models to solve whole number, decimal, and/or fraction expressions (involving addition and subtraction) without parentheses and/or brackets.
  • Solve whole number, decimal, and/or fraction expressions (involving addition and subtraction only) with one level of grouping.
  • Solve whole number, decimal, and/or fraction expressions (involving addition and subtraction only) with up to two levels of grouping with parentheses and brackets.
  • Solve and represent real-world, one-step addition and subtraction (of whole number) problems using equations with a letter in the place of the unknown.
  • Solve and represent real-world, two-step addition and subtraction (of whole number) problems using equations with a letter in the place of the unknown.
  • Solve and represent real-world, multi-step addition and subtraction (of whole number) problems using equations with a letter in the place of the unknown.

Common Misconception

One thing that always presents itself in order of operations is the way students attack the problem. Whatever they see first is what they solve. So, I find myself always using this phrase.

Students need to divide OR multiply or multiply OR divide from left to right (whichever operation appears first). Then students need to subtract OR add or add OR subtract from left to right (whichever operation appears first).

Where to Start

We start with one level grouping. We want our students to understand do what is in parenthesis first. I would spend the entire day teaching this and check for understanding.

one level grouping 5th grade math
This would be considered one level grouping

We can expand this out as far as we need to. We can create a multitude of questions for them to answer.

You can break out the whiteboards and have them compete in the classroom.

Of course you mix and match and bring in multiplication and division. Just keep upping the rigor.

One level grouping 5th grade math
One Level Grouping

You can even refer back to what we learned in the unit before. (3/5 + 2.1) – 3.6 =

Show them that it applies to adding and subtracting fractions and decimals.

Depending on your students you can camp out here for a bit and refresh the next day. When it’s time to move on we move into 5.4F Simplify numerical expressions that do not involve exponents, including up to two levels of grouping. [Positive Rational Numbers: Addition and Subtraction] along with 5.4E Describe the meaning of parentheses and brackets in a numeric expression.

Two Level Grouping

I spend more time on one level grouping to front load a bit so I don’t have to spend as long on this portion. If my students get the order of operation, along with one level grouping, then they are ready for this step. If they are still struggling with the first portion then this will set them over the edge.

This is where we introduce the brackets. Explain to them how to read the problem so they can solve the problem.

24.5 – [2 + (15.5 + 3.45)]

24.5 – [2 + (18.95)]

24.5 – 20.95

3.55

Instruct students to record the process of simplifying the expression one line at a time, by solving only one step at a time.

Extend and check for understanding

This is where I would be using a lot of whiteboard check for understanding. For some reason it becomes a competition for those who want to finish first and for those still struggling I can watch other students jump in and help them. Sometimes it takes a peers teaching to bring clarity in a way I couldn’t.

I usually start with simpler problems and increase the difficulty. I am looking more for the right process than the right answer at this moment in time. I just want to see if they are following the right procedure. I can always work on the right answer once I know they understand the process to follow.

So I might write or flash this problem somewhere in the room.

59.5 – 23 + 5.5

Then wait and watch the answers I get and check for understanding. Then give the answer

59.5 – 23 + 5.5 = 62.8

Then just keep upping the rigor.

58 – 9 + 3 + 2 +34 = 88

(6.25 – 4.05) + 3.2 = 5.4

5 + [3 + (6.25 – 3.45)] = 10.8

When you see the majority of the class missing a particular type of question then keep creating those types of problems until you see the majority of your class understanding. Those that don’t, that is who you know who to pull for small group instruction.

Order of Operation

You can also do this with some order of operation. Just start flashing problems to ensure they understand.

4 + 7 – 3 + 2 = 10

28 + 16 – 5 + 3 = 36

24 + 7 + 3 – 2 = 32

Real World Questions

I want to make sure my students have the process nailed down. Those that are still struggling I will be spending more time with them in small group and tutoring.

When we discuss real world questions…these usually are the type of questions we see on state tests and the STAAR test. It might look something like this.

1. Jimmy bought a pair of shoes for $63 and socks for $8. Jimmy paid with cash and received $4 back in change. Which expression, once simplified, can be used to determine the amount of change Jimmy received?  
$75 – ($63 + $8)  
($63 + $8) – $4  
$75 – ($63 – $8)  
($63 + $8) + $4  

I always find that students skim through the problem never placing themselves in the scenario. I usually make them replace the name in the problem with their own and think through what exactly is taking place?

Start breaking this down. Yes you can underline the numbers you see but take them further. Act out what is actually happening here. Picture yourself in the store. Are you going to hand someone money without knowing how much you are spending? What is the first thing we need to find out?

Those are the type of questions I am asking my students. Walking them step by step through the process. I work next to the galleria, one of the largest malls in Texas. I can take them on a journey because most of them have been in it. And just walk them through it in their minds.

I do this throughout the year and by the time they see a real world problem I can see them taking that journey and breaking down the question and what exactly is going on here.

Hope That Helps

If you have any other suggestions that I can add please don’t hesitate to leave a comment below.

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