The other day I walked into a grade 6 classroom with my big bin of base 10 blocks and there was a buzz of excitement.
Students, who are now accustomed to the style of lessons I generally teach, immediately start clearing their desktops to get ready for their blocks. One girl exclaimed “I love block math, it helps me to learn” and another chimed in “Me too, it makes math easy for me, for once”. Today we are going to explore large number division. When I introduce the goal of the lesson; “to develop conceptual understanding of large number division”, I get a few groans and some kids are visibly stressed as they feel like division is their nemesis. Part of changing the way we teach math is changing the mindsets of how we approach it. So, I told them to take a deep breath and that it’s OK to be frustrated and confused as it is through these struggles that great learning happens. I also let them know that I wasn’t expecting them to be able to do an algorithm by the end of the class, in fact, for this lesson, we weren’t going to write down any procedures at all.
We had conducted a pre-assessment of these students and already knew that the majority of them had no understanding of division with large numbers. A few had the traditional algorithm memorized but couldn’t explain why it worked or what it really meant. A few students had alternate strategies such as drawing bubbles and using tick marks to divvy up the number into the groups, which shows some good conceptual understanding but their approach is not efficient, so the goal for them would be to eventually learn to modify their strategy to make it more efficient. We decided to have all the students follow the same lesson to really make sure they had a solid understanding of what was happening with the numbers as well as to help them begin to develop efficient procedures for dividing.
What we Did
We started with generating an estimate of the quotient (we had spent another whole lesson discovering strategies for how to estimate division questions as they often require a different strategy than estimating the other three operations). Then students used their blocks to model each equation. It was fascinating to see what happened. I started with something like 232 ÷ 4 and the first thing we noticed was that several students didn’t know that this meant dividing 232 into 4 equal groups (or putting 4 in each group and counting the groups- which would have taken a lot of little blocks, so we didn’t use this approach). We had already had many discussions about what the operations meant, but it was clear that some students still hadn’t really understood it. That’s also one of the reasons I love using manipulatives; because you can see immediately how much conceptual understanding a student has (and this is also great formative assessment that you can turn into teachable moments).
It took several examples of physically dividing the blocks for these students to begin to understand that this is what division means. To put this in perspective, for the last 2 years, these students have been struggling to do division procedures without even knowing what it meant and then developed a strong hatred for division as a result.
For those that already understood what division meant, the powerful learning came in the ‘trading in’ of blocks. There were several ‘ah-ha’ moments where students finally understood the concept of trading in 2 hundreds for 20 tens so that they could divide them into the four groups. This reinforces their understanding of place value (which many don’t have, so this lesson was also developing number sense and conceptual understanding of division).
The next common misunderstanding came when students had to deal with the remainder, in this example 3 tens and 2 ones left over after they divided their 20 tens into 4 groups. Many stopped at this point and said the answer was 50 remainder 32. I was a bit surprised at this because they knew to trade in the 2 hundreds blocks for 20 tens blocks but then didn’t see that they could do this again, but trading in 3 tens blocks for 30 ones blocks. Because some of them had known to do this, I asked them to explain their thinking and the students quickly understood their mistake and continued ‘trading in’ and dividing.
We tried more examples, with remainders such as 531÷4. We discussed how you know if the solution is going to be more than one hundred or less than one hundred and how close it will be to one hundred. I used partner talk to ask all these questions as we worked through our exploration. The focus was on developing meaning and understanding. Procedures were not even discussed at this point. Next we explored what remainders meant and how you know when to stop (to avoid the previously mentioned mistake). Because the students are in grade 6, I want them to write their remainders as fractions, which was new to all of the students. Another bonus of using the base 10 blocks is that the students could figure out for themselves that they would take the remaining blocks, in this example, 3 and split them between the 4 groups, so 3/4. This is also reinforcing another way to understand fractions; ¾ can be thought of as 3 wholes divided into 4 parts (rather than ¾ of a whole).
Student Reflections on the Class
The students enjoyed the lesson and they were visibly more confident in their understanding. Some of the reflections were; “I liked using the blocks because it made me understand what I was doing”, “I found this easy with the blocks but I find division really hard with numbers”, “this was review for me but I do feel like now I understand it better than I did before”, “I usually use normal long division and so this was more work than what I am used to”.
The students were engaged and when I asked them to rate themselves as a 1,2 3, or 4 on their level of understanding of what large number division means (1 = I have no idea how to do this, 2 = I understand this but I would still need some help to do it, 3 = I understand this concept fully, 4 = I find this really easy and fully understand it). All rated themselves as 2 or higher, with the majority of the class rating themselves as 3 or 4. If nothing else, these students are now not fearing division as they had and know that they now have a tool (the base 10 blocks) to solve division problems in the future.
Our long-term goal is for students to be able to use a procedure that is efficient and makes sense to them in order to perform large number division symbolically while continuing to develop their conceptual understanding and number sense. In future lessons we can explore ways of dividing using symbols but we can now refer to the blocks. For example, if I was going to use the tradition algorithm (which I only use after I’ve made a few modifications: no arrow pull down and actually using the numbers rather than the digits) I can relate the symbols back to the blocks. For example: 545 ÷6
Ask students: What does this mean?
Create a story problem that would be solved by doing this calculation.
Estimate – will it be less than a hundred or more? How do you know?
Ask: are there enough hundreds to put one in each of the six groups? No
How do you know? Because I would need 6 hundreds in order to put 1 in each group.
What do we now? Trade in the 5 hundreds for 50 tens.
How many tens do we have now? 54, 50 from the 5 hundred and 4.
How many tens can I put in each group? 9
What is that equal to? 540
Now how much do I have left? 5
How many ones can I put in each group? 0
See our lesson on Long Division with Meaning using the Base 10 Blocks for a video tutorial on how to do this in your classroom, along with a lesson plan. Also, for an alternative method, which has been proven to be far easier for many students, try the Partial Quotient Method for Long Divison.
This lesson did not necessarily get these students any closer to doing a procedure for large number division so you may be wondering what the point of it was. My goal is to build number sense and a problem solving mindset in learners. We have technology to do algorithms for us, what we need are thinkers, problem solvers, innovators, creators and collaborators, so this approach develops these skills. They learned to use tools to solve problems (such as base 10 blocks), they are talking and explaining their thinking throughout, which is developing their mathematical reasoning and communication skills. They are working together, striving for deeper understanding and act more confidently and positively towards math.
We also developed a deeper conceptual understanding of what division actually means and students figured out for themselves, some strategies and approaches that help them to divide numbers efficiently. They reinforced or developed their understanding of the place value system and how it relates to division. It will be a whole lot easier to teach procedures to students once they have solid number sense
than it is to continually push procedures onto students who don’t have the number sense to make any sense of the algorithms and so promptly forget them as soon as they are taught them. I believe it is my role to facilitate my students in understanding the relationships that make up mathematics, not to tell them how to follow a procedure to get to the right answer in the fastest way possible.
Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan.