Ants, Ants, Ants! Using discovery to understand the meaning of division.

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SWBAT understand that dividing means breaking apart a given amount into equal portions and use strategies and tools to problem solve.

Big Idea

In this discovery lesson, pairs of students are given a sheet with thousands of ants on it. They will choose the best "tool" to figure out how to count all of the ants on the page.

Opening the Lesson

10 minutes


Prepare Materials Ahead of Time: Ant Sheet, various tools like string, a paper clip, a small counter or cube, a rubber band but including one clear plastic 2x2" square, ( They will more than likely figure out that this is the best tool to count the ants because they can see through it.)

Tip: Be SURE your lights are adjusted so that students can't see the "grid" on the paper. I have my papers laminated, and sometimes the grid is very apparent if the lights are too low. So, I have found that the brighter the light, the grid is less apparent. You don't want them to see the grid unless they can't figure this all out for some reason.

*Just before the lesson, I had students copy the questions that are listed in the SB lesson on page 2, so that they were prepared before we began.

Setting the Stage: I opened today's lesson with this National Geographic video of fire ants creating a raft to survive a flood. I chose this not only because it's pretty amazing, but because it shows evidence of math and engineering in nature. I started the video and asked them to be on the lookout for "math" happening in the video. I wanted to see what they would notice.

After the video was done, I probed my students by asking questions to get them talking about what math they noticed or heard.

 This exercises their awareness of mathematics in the natural world around them and supports MP 4. The connection to engineering principals around how a group of ants can structure themselves so they are able to float  in a mass will not be in the forefront of their discussion, but I think there is value in getting fourth graders to wonder and discuss that wonder. As we applied MP 3, they listened to the reasoning of others and constructed their arguments as to why the structure of ants work and why a fish removing the wrong ant will destroy the structure.


One Hundred Hungry Ants: A story about ants and dividing things to go faster.

10 minutes

One Hundred Hungry Ants

I have chosen this whimsical and fun book to help students transition into thinking about division and apply their prior knowledge of inverse operations.  As the ants divide...they are creating arrays, which students relate back to their experiences with multiplication.

I read the book using my squeeky "ant"voice. I had them echo the "We're going to a picnic, a hey and a heidy ho!" using their squeeky ant voices so they would be drawn into the story and ready for thinking questions. I questioned  them to think about telling me what the next factor pair group pushing them to pay attention to the pattern. I wanted them to connect the process of dividing to multiplication. There was laughter, smiles and the chorus joining in make it obvious that this book is just so fun!






Partnering up and getting ready for fun!

5 minutes

Partnering Using Note Cards and Math Facts

I divided students up into partners using these math facts that they continually struggle with and a little music in the background to get them focused on matching their facts with the correct product. 3x8, 3x9, 4x7, 4x8, 6x7, 6x8, 7x7, 7x8, 7x9, 4 x 12. I also thought about behavior and personalities as I passed out cards face down on their desks. (You can adjust your fact cards to fit your class needs. I used plain note card and keep them in my supplies to pull out for partnering. They always act so surprised every time we do this! I guess it is the challenge and the mystery of who will be their partner.)

After they discovered their partner, I asked students to move to the desk of the partner who is the youngest. This way, they had to discuss months and years and do the math. It went fast. They were good at figuring it out. We use this skill in real life very informally all of the time! These little activities support Math Practices 1, and 5 that exercises their reasoning, problem solving and using tools appropriately. I always keep them thinking with every step. I adjusted my practice using the note cards that fit my students weak fact knowledge. I find that these little drills are often the key in helping them get over a hump in remembering a certain math fact that's hard for them.

Ants Ants! Counting all those ants!

20 minutes

Ant Sheet 


The learning goal for this next part is two fold:

* Students will explore the concept of division and its inverse relationship to multiplication. They already know that it is the inverse, but they do not really understand the relationship yet.

* Students will choose the best tools to count all those ants! (There are 1263 of them...although I have never actually counted all of them!)

After students were settled into their work areas, I purposely told them that one person should be the "getter" and the other should be the "returner". This helps things move smoothly. I allowed a minute for them to decide. I had listed the materials they needed on the whiteboard so that they knew what to get, even though there were just two things. Putting the tools in a plastic cup really helps keep everything together and makes it fast to pick up.

After they were settled, I gave them clear directions and made them wait to turn their ant paper over. I counted so that all started at the same time. I also had them stand and look directly down at the ant paper, so that the light would not give away the lines on the paper.

 I instructed them to use any tool in the cup, but had to decide on the best tool and use only that tool to help them do their counting because I wanted them to make a solid choice. They could experiment with any tool, but had to settle on one. I noticed that curiosity drove some to switch mid stream in counting ants.  I told them that after they decided , one person needed to record and the other needed to count. I directed them to be aware of their strategies and be ready to talk about how they went about counting the ants to help them realize that this was a thought process, a reasoning  and mathematical task. This particular group could get lost and just play if I didn't guide the thinking.

I let them work another 10 to 15 minutes. I roved around the room discussing what strategies were working the best for each group coaching them to keep thinking about division, counting, strategies and problem solving. I asked them if  they had to "pull apart" the whole sheet of ants to count them?  How did the tools help them do that? How is that division?

I got various answers about using the square, and they used strategies using arrays. This is good because this is the step that helps them connect the understanding that arrays are related to division.  This discovery process pushes their thinking from something familiar ( multiplication) to understanding that pulling it apart or sectioning off the mat is division and demonstrates the relationship.

I continued to coach as everyone was engaged, working hard and seemed to enjoy the task of counting the ants. I was hoping this all would come together and their understanding would be deeper through this activity.


How many ants?

10 minutes

I stopped students after I saw that they were finished and seemed ready to share. Students who were finished early researched articles about ants on their iPad to read in partners to reinforce their non fiction reading skills, supporting those CC reading standards. There wasn't a whole lot of time for them to wait, people were finishing just about at the same time.

Ants Ants Ants!.

On the SB file, I had prepared a T chart comparing numbers and tools. Each group shared what tool they used and how many ants they got. I refrained from asking for any other strategies at this point because I just wanted the raw data to compare.

We compared the numbers and discussed what may have created any differences in figures.

I asked students to get out their math notebook, to be ready to talk about and write answers to some questions independently for homework. I turned to the next page on the SB file to reveal the questions to discuss.Having them both on the whiteboard prior to this and on the SB file was helpful for my lower language students. I also copied the SB file for them to physically have a piece of paper, even though I had encouraged them to write the questions down. Two of my students never would have gotten them all down on paper. * When it's essential, I email the PDF version of my SB files to all students after a lesson. Today, the paper copies are important for these particular students

1. Why did you choose the tool you chose?

2. Do you think it worked?

3. How did your count compare to the other 9 numbers?

4. Why was it close or way off?

5. Did you find a strategy and what was it? Did it work? Why or why not?

Using their notebook, they jotted down answers to formulate into a paragraph later.

As we looked at our numbers formulated on the SB, we could see a range of 800-1200. We talked about why there was such a wide range. We decided that it depended on the tool we used and we could see that the square produced the most accurate number knowing that it should be around 1260.  This would also be an opportunity to discuss mode, median and range. But, I didn't attack it today. The majority of groups chose the square.


Closing and Homework

10 minutes

Pick up & Be ready for Closure: To get them settled into closing up the lesson and assigning homework, I stopped everyone and I asked the "returner" to pick up and return the materials.

In closing today I wanted to lead students in thinking about the meaning of division. I explained their assignment for homework. I wanted them to journal and reflect on their lesson at home and be ready to share written work the following day.This entry will be added to their writing portfolio to demonstrate writing across the curriculum.

I asked questions and then interviewed students about what they learned. I think it is important to ask students questions about their learning after a lesson like this to reinforce the idea that this activity is really about learning.

Some questions I hoped to get answers for were:

Did you like the lesson? Was it easy or challenging for you?

What did you learn today?

What did you learn about division and multiplication?

Homework Assignment: The assignment for my students was for them to use the questions from today's class (SB file) and  write a summary on loose leaf paper. They then had to edit work using editing marks, correcting the spelling and punctuation if necessary.