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Thursday, January 9, 2014

Working Toward Conceptual Understanding of Double Digit Subtraction

I have a large class of third graders that I am working with this year and many of the students are missing some second grade skills that are essential for them to have in order for them to learn what they need to master in grade 3.

Recently we have been working on this Common Core Standard
  • CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 
We started with what I thought would be a review of double digit subtraction before branching into larger numbers.  I quickly found out that half the class had no concept or no strategy of subtracting two double digit numbers.  The lessons for the next few days had to take a major detour.  Second grade students should be FLUENT with adding and subtracting within 100 by the end of the year and should have some strategies for numbers within 1000.  These kids missed something! Here are the second grade standards I am talking about.

  • CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
  • CCSS.Math.Content.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
I had to do a major backtrack and brought out some of my favorite routines, games and activities that I ordinarily do in second grade, often do in intervention groups and now had to do with almost an entire third grade class.  Let's take a look.

Whole group instruction, discussion and sharing of strategies

I am a big fan of problem strings.  I start with a problem I know all of my students can access and build from there.  We do this quite a bit and my students are VERY familiar with the procedure.  The basic idea is that I show a problem and kids give me a ready signal when they know the answer.  When most, if not all kids are showing the ready signal, I do one of three things

1) Pick several kids to give me an answer.  Then I might ask if anyone else has a different answer.  I tend to do this when the problem is one I think will challenge students or one I think might bring out misconceptions.
2) I might pick a kid to give me the answer and tell me how they knew they were right.  I tend to default to this option when I think the question is easy and most kids can quickly and efficiently get it correct.  
3) Often I have kids turn and talk with a neighbor about what the answer is and how they know they are right.  It is amazing how many conversations I can listen to at the same time.  This gives me an idea of what kind of strategies are out there and which strategies I want shared.  I often will choose a pair to share and have one student tell me what the other kid's answer was and how they figured it out.  When I first started doing this, kids could often not tell me what their partner said.  Think-Pair-Share is a formative assessment strategy that I love but it works best when kids are held responsible for what their partner says.  This really makes them listen and try to understand.  

Here is a look at some problem strings I did with this group over the past week. 
This was a concept a LOT of these kids were struggling with.  This is actually just one problem and the equations below show the strategy that different kids did.  Previous to sharing of ideas such as this, most of these kids were defaulting to counting backwards and which is fairly inefficient.  They often were off by one and they really needed a strategy that did not rely on counting!

Here is a problem string that started out with something I knew everyone could be successful with.  After a few subtracting from 10 problems, I jumped to bigger numbers.  With this string, kids were able to see how subtracting 3 from 10 is similar to subtracting 3 from any count by ten number.  This skill will help them with their new strategies they came up with in the previous picture.    
Another pre-requisite skill for double digit subtraction is the ability to subtract a multiple of ten from a two digit number.  Here I lead them though some problems of this type.  I often go back to subtracting 10 before jumping to other multiples of 10.  This allows even the lowest student an access point.  

Games, Games and More Games

When kids are learning something new, bringing a game to it makes it more fun for everyone.  I will do these games sitting around a table with an intervention group or if using them for whole group instruction, I will show them under a document camera and then have folks play them with a table group.  I can assess so much understanding while watching these games.  The cards I used for all of these games are from the Numbers to 120 Place Value Deck
The first game I played with this group is easy and QUICK.  I want to make sure they are fluent with their combinations of 10, especially as they relate to subtraction.  I call the game 10 take away and the goal is to subtract each card from 10.  I pull out the cards under 10 and go through them a few times.  Kids love seeing if they can get faster with each pass through the deck.   See how this game relates to the problem strings and supports those concepts?
Here we are playing take away 10 (or 20 or 30 or 40 or whatever number I choose).  I start by having them subtract 10 from each card.  I flip cards in the middle of the table and they shout out answers as they know them.  After they sound fluent with subtracting 10 from each card, I might have them subtract 20 from the next set as I continue flipping through the deck.  Sometimes I want to hear from individual kids so I will pass cards around the table and each kid will have to subtract the current number (10, 20, etc) from the card I give them.  This really shows me who needs more work and who has it.
Here we are playing subtract 8.  I might change this to 7 or 9 or 6 as well.  Notice that this was a skill that needed a lot of work in the problem strings above.  These place value cards really support their developing strategies because they can see the tens and ones.  For those that don't yet have a mental image, this can mean the difference between having to count backwards and being able to think of a more efficient strategy.  

Reinforcing What We have Discovered and Assessing Understanding

After 2 days of playing these games and doing problem strings, I wanted to see where kids were individually and give them a chance to apply what they had learned.  I made this quick written assignment for them to do independently.  
Here is the front.  Notice how the problems are similar to the problem strings we have been working on.  If you want to try this with your students, you can grab this FREEBIE at my TPT store or click here to get it from Google drive

Here is the back.  Kids did this independently but were allowed to have help.  As they were working, I was roaming around the room with my trusty marker and correcting problems as kids worked.  This lets me see problems right away and help kids find strategies that work without giving them and opportunity to practice the wrong way.  
Notice the last section on page 2 says Exit Ticket.  This is another formative assessment strategy that I use with kids.  Everyone had to do this all alone with no help.  Taking a quick peek at these as the kids were leaving let me know what my instruction needed to focus on the next day.

Now these students are ready to move on to subtracting double digit numbers.  Head over to part 2 of this post to see what I did next with these students!
I also have developed a fun winter themed QR code scavenger hunt to practice double digit subtraction that I will be using with my students next week!

My new QR code scavenger hunt.  My students are going to love this!

What skills do you find your students are lacking that keep them from learning what they need to in your grade?  Please respond in the comments below:)


  1. Let's see, what skills are my students lacking that keep them from learning what they need in my grade? EVERYTHING! I am at a loss! I have a third grade class and we just did their STAR tests to see what their reading and math grade level equivalents third grade I have students who are on a fifth month of Kindergarten level all the way up to a third month of fourth grade level, according to the test, but even those "fourth grade level" students can't add or subtract with regrouping. We are supposed to go on to multiplication next week, but I don't want to start something new when they haven't got what we have been doing for the last month...especially since report cards go home in a week and a half. Am I right to slow it down, review the addition and subtraction, strengthen their skills and start fresh after the Fall Break (a week long break 2 weeks from now)?

    1. Hi Katelyn,
      I am so glad you are reading blogs and looking for solutions! I think you should absolutely slow down and strengthen skills before moving onto multiplication. I know you have a lot to teach and the year seems to be going by to fast but they will now be able to develop the understanding they need of multiplication without a solid foundation in numeracy and additive reasoning. See if you can squeeze more math out of your day.

  2. Thank you for this clear-cut, step-by-step lesson, and for the free practice/assessment. I think this is very helpful.