Do You Know The Difference?

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SWBAT use the four operations with whole numbers to solve problems.

Big Idea

In this lesson students will examine word problems to determine the difference between additive and multiplicative reasoning.

Warm Up!

10 minutes

Resource:  Think Sheet.docx

 I want to see if students know the difference between the four basic algorithms, and to determine if they know how they relate to each other.  I ask students what they know about addition and subtraction. Elicit responses from students until a student says that they are opposite operations. I ask students what they know about multiplication and division. Again, elicit responses from students until a student says that they are opposite operations.

I post the following problems on the board. 

4 -  ___   = 1

24 -  ___  =  5

So, I say 4 take away what number equals one. I do the same for the next problem. I point out that the number subtracted gives you the answer one.

24 divided by 4 is the same as  4   x  _____ = 24  

As I am modeling, I ask how do you know, and can you explain.   

I tell students I want  them to start thinking about addition and subtraction and also multiplication and division as opposite operations because it’s important for what they will be learning how to do in the lesson. I write the word algebra on the board, and tell students that they will be learning how to write and solve some algebra problems. I ask students what they know about algebra, and elicit responses. Some students mention letters instead of numbers. I point out that algebra is math in which unknowns are represented with letters. It helps us find what number is left over or missing.

Building a Stage:

When finding the difference between multiplicative comparison and additive comparison, see that multiplicative comparisons focus on comparing two amounts by showing that one amount is a certain number of times larger or smaller than the other. It is denoted by the question, "How many times as many?"

 So, the comparison shown in the question is multiplicative comparison. I provide students with a think sheet to help them focus on the intended learning outcome.

 We will be focusing on the following Mathematical Practices in this lesson: 

MP.2. Reason abstractly and quantitatively.

MP.4. Model with mathematics.


Practice Time!

20 minutes

Material: There are two snakes at the zoo.docx

The purpose of this part of the lesson is to foster a classroom discussion that will highlight the difference between multiplicative and additive reasoning.  To give students the extra time they need to explore both of these concepts, I want them to explore a little on their own, but with me facilitating.  I invite students to the carpet so that we all can be in close proximity. This will help maintain a comfortable learning environment. I post a question on the board.


There are two snakes at the zoo, Jewel and Clyde. Jewel was six feet and Clyde was eight feet. A year later Jewel was eight feet and Clyde was 10 feet. Which one grew more?

Talk it Out:

I ask students to read the question with me aloud.  I give them about five minutes or so to think about how they will solve the problem.  I ask students what we need to know. What information does the problem gives us?  What information is missing? Which operation do we use to solve the problem?  How do you know? I may ask student volunteers to represent the problem with an illustration. This will provide students with alternative ways to solve problems.  Some of my students argue that they grew the same amount (an example of "additive thinking").  On the other hand, some students viewing a multiplicative comparison problems might argue that Jewel grew more since it grew more with respect to its original length (an example of "multiplicative thinking"). This would set the stage for a comparison of the two perspectives.  Now that students are ready to compare the two operations, I draw two large boxes on the board and explain the difference between the two.

Note: When students are unable to bring up both arguments on their own, I will introduce the missing perspective.

To ensure that students are fully understanding the intended concept I ask questions such as: Can you think of a way to solve this problem? Can you explain why this operation will work? Can you represent this problem using an illustration? Explain? To reinforce multiplicative comparisons I ask, “How many times greater is x than y.”  If students seem to be confused I address it by modeling and discussing how to determine the difference between multiplicative and additive reasoning.

 I ask students to share out to see if they can determine the difference between additive and multiplicative reasoning. I take notes to adjust the lesson.

Group Session!

25 minutes

Material:  Group Discussion.docx

In this portion of the lesson I want students to work together to help support each other learning.  To do this I ask students to move into their assigned groups.  I want them to use arrays and repeated addition to get them to thinking the difference between multiplicative and additive reasoning.  I ask students to think of some key words that can help them determine the difference between the two. Some students say how many times larger one is than the other lets you know that it is a multiplicative problem.  Other students tend to stick with repeated addition, so I stop for a moment to explain an example of multiplicative comparison.  I say a 12-foot rattle snake is six times longer than a 2-foot rattle snake. As I say this I began to illustrate my thinking, so that students can see the connection. I ask student volunteers if they can explain why the 12 foot rattle snake is six times longer a 2 foot rattle snake. Some students notice that the array was a 2 by 6 frame, so 2 x 6 is 12.  OK! Now we are working guys! I am going to set the timer for about twenty minutes or so. I want you all to work together on creating illustrations and composing explanations for the given problem.  array model.docx


  Do not forget to compare the difference between multiplicative and additive reasoning this will help you guys as we go deeper into other concepts in math.  As students are working I circle the room to remind them of the purpose of this activity.

For instance, I might say describe how the numbers in the problem can be arranged in an array. How can the array be used to find the total? What is the total number of items in your array? Show how you can use addition to solve the problem. Show how you can use multiplication to find out. How are they different?

I use students’ responses to determine if students have successfully mastered this concept, or if additional practice is needed to support their learning.

Student Sample

Independent Thinkers!

20 minutes

Material:Independent work.docx

In this portion of the lesson I turn student lose just to see are they capable of being independent thinkers.  I talked with students about additive and multiplicative reasoning, and I reviewed the importance of using arrays to help assist them in solving their problems.  My students seem to do better when a word problem is read aloud, but for this activity, I made the decision to only read it to students who seemed to be struggling with their thinking.  I do not plan to read to students all year, however, I feel as it is appropriate for me to scaffold them through class activities to ensure they fully understand later on in the year. 

Some students gave incorrect answers, but I gave them credit for explaining the concept for additive compare.  Some students seem to struggle with their thinking and used inappropriate reasoning; I carefully used guided questions to redirect their thinking towards the concept of this lesson.  I understand that some of the concepts here are going to be difficult in the beginning because students are just now learning how to analyze their errors.  I use student’s responses and averages to determine if they had successfully mastered this objective, or if they need additional support.  


Note overall students responded well to the question, however, they seem to need more time working on critiquing their thinking.   

 Student Sample