Interpreting Remainders!

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SWBAT interpret remainders in word problems.

Big Idea

Students have a hard time determining what to do with the left over numbers, In this lesson students will exploring how to discuss, explain, and interpret remainders in word problems.

Warm Up

20 minutes

Students are aware of how to explain their answers in multi-step word problems using the four basic algorithms. 

Today we begin to explore how to interpret remainders. An important Common Core shift that I am addressing in this lesson is student ability to use models that represent understanding.  I also, want students to begin assessing the reasonableness of their answers using mental computation.

  I begin by going over some ways remainders can be interpreted. My students have a pretty good idea of how these interpretations would look in a word problem, I do not go over each one. However, I leave this list visible for students to use throughout the lesson.

  • Remain as a left over
  • Partitioned into fractions or decimals
  • Discarded leaving only the whole number answer
  • Increase the whole number answer up one
  • Round to the nearest whole number for an approximate result

I post a problem on the board:  

There are 567 students going to the Zoo. If each car holds six students, how many total cars will be needed for the trip?

Students solve the problem independently in their journals while I circle the room noting who has a strategy to solve the problem, and who does not.  I use this time to scaffold students from where they are in their learning, to the next step in developing fluency in interpreting remainders.  This may very well be different for each child.  We read the problem aloud together.  I ask students, “What do we need to know?”  How many total cars will be needed for the trip?  What information does the problem give us?  The problem tells us that there are 567 students going to the Zoo, and each car hold six students. Great!  Now that most students understand what we need to figure out, and what information we will use to solve it, I ask for a student volunteer to share how they solve their problem.  Some students still just want to give the answer, but not explain how they got their answer.  I tell them that it does not matter to me if you have the correct answer; I want to know how you got your answer.  After a response is given, I ask students to decide if the answer makes sense. 

Differentiating Instructions: 

Additional Word Problems.docx

When students do not give a logical explanation of how they problem solved, we work together and construct a written explanation of what steps we used to solve, and what might have been a bit confusing to them.  I ask a couple of students to share how they solve the problem.  I allow students to write their solutions on the board and try to leave as many of them visible as possible.  I ask students to examine the written explanation on the board to see if everyone solved their problem the same way.  I then ask if there is any one right way to problem solve.  I tell students to remember that there is more than one way to solve a problem, and a little later on in the lesson you all will be creating some problems of your own to solve. 

We used the Following Mathematical Practices:

 MP.1. Make sense of problems and persevere in solving them. 

MP.2. Reason abstractly and quantitatively. 

MP.4. Model with mathematics. 

MP.5. Use appropriate tools strategically. 

MP.6. Attend to precision. 

MP.7. Look for and make use

Working In Pairs!

25 minutes

Material: Students FormHow do you interpret a remainder in a division problem.docx

In this portion of the lesson I want students to spend some time working on problem solving. I ask students to move with their assigned partner.  I tell them they will be working with their partner to explore how to interpret remainders.  I tell them to focus on what information needs to be in a math problem if others are going to be able to solve it?  I show them a problem with no question ask if they can solve it? I give them about 3 minutes to turn and discuss with their partner. Some students may need to see the remainders put into context in order for them to interpret them. For instance, it can remain as a left over, partitioned into a fraction/decimal, or rounded.

If you want to split 396 stickers between 5 friends, how many stickers will each friend receive?

Many students immediately want to respond that each student will get 73 stickers, but 8 will be left over.  I post the problem with the question attached to it.

If you want to split 396 stickers between 5 friends, how many stickers will each friend receive? How many will be left over?

 I tell them that my answer is 8. Why do you think we came up with different answers? We realized that the question makes a difference to the solution itself.  We talk about the importance of a question in a word problem.  I use different questions to help them determine the relevance.

  • How many stickers will each person get?
  • How many stickers will be left over?

 I carefully remind students of the different ways a remainder can be interpreted:


  • Remain as a left over
  • Partitioned into fractions or decimals
  • Discarded leaving only the whole number answer
  • Increase the whole number answer up one
  • Round to the nearest whole number for an approximate result

I explain that both of these questions lead to different answers, therefore it is important to examine the question.  I tell them that today they will write word problems that must as a question at the end.  I give them a word problem form that has space for a picture, for the words of the problem, for the questions, and for the answer.  I give students about 15 minutes or so to finish writing their word problem and putting a picture that correctly represents the problem on their papers.  Students who finish early have the option of writing in their math journal, or composing an additional word problem.

Student Sample

Independent Thinkers

25 minutes

Materials: Independent work.docx

In this portion of the lesson, I want students to focus on meeting the objective of this lesson and logically demonstrating and explaining their thinking so far.  I ask students to move back into their assigned seats.  As students are working on solving their answers, I circulate the room reinforcing needed skills. I encourage students to draw illustrations that can assist them in solving, and remind them to show their work. I also, ask what information the problem gives. Are there any key-words? What do we need to find out? Etc…..  I want students to gain a deeper sense of why it is important to explain their answers, and just not give an answer.  However, I tell them to focus on how to interpret a remainder. I give students an opportunity to share their answers and explanation with the class.  The student reads the problem aloud, and then asks for volunteers to tell how they solved the problem.  We stop to discuss a couple of more students’ explanation, but this time I ask them to write their responses and illustrations on the board.  As they are writing I ask students volunteers to explain if their explanations make sense, or not. If they reply with no, I ask them to offer a solution. I invite a couple of more students to share out to ensure that students are able to think logically on their own.

Student Work Sample