In the last blog post, I provided a rationale for why we must go ALL the way back to where students are at in order to help them build a solid foundation. If you have not read that post, I urge you to do so before reading this one as this is a continuation of that idea.
Partitioning is known by many names but it simply means the understanding of how numbers can be broken up. For example, the number 5 can be partitioned into 2+3, 1+4, 1+1+1+2 etc. This might seem really obvious but many students have not yet developed this understanding. Because this skill is the basis for most of our strategies for adding, subtracting, multiplying and dividing it is important to develop it before teaching the strategies.
A related number sense basic is the idea of benchmarking to 5 and 10. Benchmarking means that we use these numbers to help us estimate. If a student really understands how many 10 is, then when they see a pile of blocks, for example, they can estimate how many groups of 10 there are as a way to estimate the total amount of blocks. It is basically using referents like we would with any measurement. If I know how long 1 meter is then I can estimate a distance in meters. So we need students to really understand how big 5 and 10 are.
Fingers, rekenreks (if you have them), blocks, buttons, beans, are all great manipulatives to use with this skill, along with a 10 frame. If you are teaching kindergarten then you’d start with a 5 frame but I’m jumping right to a 10 frame for this. First of all, we want our students to be VERY familiar with numbers to make 5 and 10 (friends of 10). Start with 5 and have students show all the ways to make 5 on the frame with manipulatives or with their fingers (one hand shows none while the other shows 5, then 1 and 4, etc.).
I’m often shocked at how few middle school kids automatically know their friends of 10 (these are the pairs of numbers that add to make 10: 1+9, 2+8, 3+7, 4+6, 5+5). This needs to be done before we can build onto this to strategies like ‘add by making 10’ or ‘subtract by making 10’. Visuals are really important for making sense of quantity so use the 10 frame and blocks to help students learn these facts. I have found that pairing these visuals along with games work the best to help cement these number pairs for students.
See the freebie attached to this post for some games that will help students to learn how to partition well AND that will help them really understand 5 and 10.
Quick quizzes at the beginning of class can also be used to allow students to practice these important pairs. These can be done orally, in partners, using individual white boards or students showing you their fingers as the answers (in front of their chest so others can’t see). Engage in these games and activities often until they have automaticity with them.
There are also hundreds of activities for this on the internet. When I was in Terrace recently, Lisa showed her class a video that they all sang along to (grade 4). The kids loved it and would sing their ‘part’:
Once 5 and 10 are mastered, we can move along to partitioning other numbers AND partitioning into more than 2 parts. I always attribute my ability to partition well to my grandpa who taught me how to play Cribbage at the age of 8. This game is all about making 15 (and 31) so it is a great way to practice this skill. This is something you can teach in school or suggest to parents at home.
On the freebie, you will see the activity called ‘Make 7 (or any other number)’. This activity is open and accessible to all learners. It is done by giving students some cards, for example: 3 aces, 3 twos, 2 threes, a four, a five and a six and ask them to make 7 as many ways as they can and write a number sentence for each. Some students will find all the ways while others will find a few but with practice and hearing from their peers during sharing out in time they will improve. You can give different cards and ask for 7 again or you can ask for another number.
For students who come up with all the ways very quickly, you can challenge them to use the cards and ANY operations to make 7 (or whatever number you are working on). Because I just say “make 7” some students already start doing other operations, like subtraction, which is great as this means they are working at their challenge level. Our language is very important and I believe in tasks like these, using vague language is actually a benefit because it allows for way more interpretations and therefore more opportunities for thinking mathematically.
If you want to do this as a number talk (if you have no idea what number talks are – watch my vlog on number talks) you can do the same idea – show students some numbers (or dot cards) and ask them to make any given number as many ways as they can. Or you can give them these types of equations:
2 + 6 = ___+___+___
It is interesting to note that MANY middle school students struggle with this task as they want to write 8 instead of a way of writing 8 using other numbers. Most students see the = as meaning ‘the answer’. You can also change the numbers and the amount of addends you are looking for. Also, do some that look this way:
___+___+___+___ = 6 + 5
So they don’t get trained to always see numbers on the left with the unknown on the right because we want them to see that equal sign as meaning ‘the same as’ and we also want our students to be flexible thinkers.
The number talk dot cards that I suggested in the last post also use partitioning. Because we can’t subitize more than 5 items, when we give something like this as a number talk:
They are seeing it as 3 + 4 or 3 + 2 + 2 or whatever combination – they all involve partitioning. So dot number talks are serving this purpose too.
If you have students who know their ‘friends of 10’ then you can give them more challenging questions along the same big idea. For example: 37 + ____ = 100 (complements to 100, 1000, etc.) or you can start with ‘friends of 20’ before jumping to 100. This would be a way of differentiating in your class to allow for those who are behind to catch up while still providing a challenge (especially if done mentally) for the others.
Just like with subitizing, I would ask parents to work on this with their children at home as it is a relatively easy concept for most parents and you won’t have a lot of class time to devote to these skills. The next blog I will tackle 1 and 2 more/less as well as rounding.
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