What Do You Do in Your Math Intervention Group?
So, I have a math intervention group. I have done intervention lots of ways…and the thing is, there are always core things that kids struggle with. Those things without a double are always addition, subtraction, and multiplication facts. Next, they struggle with the standard regrouping algorithm. And, why do they struggle? BECAUSE, of course, no one sits with them at home to help them learn these things if the concepts don’t sink in during school time.
Enter me. I have been working with some students the past few weeks on subtraction regrouping…with success! Here is what I have done, and what I have discovered. First of all, several of the intervention students were able to regroup UNTIL they had to regroup across zeros. They weren’t sure what to do when they had to borrow two places over. How did I figure this out you ask? Well, with my group of four students, I gave them a worksheet. (gasp! a worksheet??!!) Yes, I gave them a worksheet and had them work a few and checked to see which ones they were getting correct and which ones they were missing. I would have them work one problem and hand me the sheet to check. This way they were getting immediate feedback. During this time, I realized that they weren’t getting the answers right unless they borrowed across zeros or had to borrow two places over. I used and am so thankful for Super Teacher Worksheets subtraction worksheet generator! This conveniently allowed me to print a new worksheet (complete with answer key) when I felt they needed practice.
Now when I realized they needed help with regrouping across zeros, I realized there was a regrouping misunderstanding. So, I used the Singapore math number discs method to show them what was happening when they were regrouping. After showing them and having them do one with me, the next day they performed a lot better on their subtraction regrouping problems. I have a SMART board lesson and worksheets if you would like some for students to practice with. The grid is already made for the students…these however do not have seven digits like the worksheet above.
A few other things I did to help the students think about the regrouping process were.
- Say this little rhyme…”More on the floor, go next door, and get 10 more”. This way they would always know they were bringing ten over…not 9, not 8.
- Sometimes when students want to skip over a place value column, I would describe it as driving in traffic. Your car doesn’t just fly over the other column, it has to change lanes one at a time…it can’t be a helicopter.
- Another idea I mention is place value columns in relation to the drawers in a cash register. If you cash in your $100 bill for others, you trade it in for 10 $10, then you trade in the $10 for 10 $1 bills.
Try these things and soon you will be on your way to having expert subtraction regroupers!
No Regrouping Needed!
Why does this alternative to regrouping work? I have noticed an image similar to this on Pinterest/Facebook.
The comments go something like:
- Wow!
- Cool, I’ve never seen this before!
- How does this work?!
- Why does this work?!
I thought I would take a moment to explain why this works. A simple piece of ribbon gives us a chance to explore this concept. Above I used smaller numbers to demonstrate. 100-88=12, but subtracting one from the minuend and subtrahend gives us the same answer. 99-87=12 also. When moving the ribbon down the number line we can see how the distance on the number line stays the same because we took the same amount from both the numbers. Hence, the ribbon remains the same size because the distance doesn’t change.
The distance = the difference. As long as the distance is constant between the numbers this will work.
Now, you tell me, will this work if, instead of subtracting one, I subtract five from the minuend and subtrahend?
This Helped a SPED Kid Learn Subtraction Regrouping!
I posted earlier about a strategy that helped a struggling SPED student add with regrouping. Now, I am sharing the strategy I taught this same child to learn subtraction with and without regrouping.
Thank you to Lovin Lit and Educasong for the clip art!
Above there are two examples. One of the examples is with regrouping and the last one is without regrouping. This strategy will work both ways. I admit I have been using the rhyme that a couple of teachers have gotten from Pinterest…”More on top, no need to stop. More on the floor, go next door, and get ten more.” I have kids recite this first. The rhyme works very well so I use it. Anyhow, I have the kids always circle the number on the bottom in subtraction. The circle represents their head. Then they make dots to count up until they get to the top number. The dots are like their fingers. To get the difference they count how many dots they drew. Simple, easy, and if kids can make the jump to use their fingers, they can go ahead and don’t have to draw dots. I did this because I found some students don’t know how to effectively use their fingers to count up yet.
You may also like Addition Intervention Strategies:
3 Ways to Check a Subtraction Problem
Here are 3 ways we are teaching the kids to check their subtraction problems. First, they start out checking it with estimation to see if their answer is reasonable. Next, they can check their regrouping if it is a regrouping problem by adding their regrouping numbers back together to make sure it equals up to the minuend (the top number in a subtraction problem). Finally they can check their accuracy by using the inverse operation of addition and subtraction. They can add the difference to the subtrahend (the number on the bottom of the problem). I remember this because of a sub-marine is on the bottom like a subtrahend. Sub means under. See the ways kids can check this below. Hmmmm…could be a great anchor chart!! 🙂
3 Ways to Check a Subtraction Problem
How Can You Teach Subtraction Regrouping with Understanding?
Our third and fourth grades have been making the uphill trek to teach subtraction regrouping. We have taught kids to first use expanded form/decomposing numbers so that understanding may accompany all of the crossing out/regrouping that occurs. I started out calling it expanded form, but then I realized that having the plus signs in the expanded form would cause confusion when kids are actuallysubtracting. More accurate language would be “decomposing the numbers”. Anyhow, we used the method of decomposing numbers after students had lots of experience with base ten block subtracting. Some students are still using the base ten blocks to subtract because their understanding of the number decomposition isn’t there yet.
Subtraction Regrouping with Number Decomposition
Subtraction regrouping takes at least two weeks to perfect in my experience. After the students have had ample experience decomposing numbers with subtraction regrouping, then we move to the traditional algorithm using the same language we did when the numbers were decomposed–that is we make sure that we call the tens we are marking out tens etc. For example, if we cross out a seven in the tens place to write a six above it, we call it six tens NOT six. To help students’ understanding, it is important that the teacher language is precise and that the students are held accountable to use precise language as well.
One last side note: This week when I was working with an intervention group, I noticed that some students looked confused when I said the word regroup, so I changed my language. I called regrouping exchanging. They still looked confused so I changed my vocabulary again to the word trade. Students seemed to understand that better. I think that is something that kids are familiar with. They are always talking about trading a baseball card, something on their lunch plate, a piece of candy etc.
Happy teaching!
How Do you Teach Regrouping with Understanding?
If kids aren’t building with base ten blocks to add and problem solve FOR WEEKS initially, they will get no where with their number sense understanding for regrouping. Under common core standards, we are heading towards the understanding of the traditional algorithm in the 4th grade standard–not just being quick at a procedure of crossing out numbers and writing new ones above them. Mistakenly 2 years ago we tried to rush the traditional algorithm with our second graders. As a result they are still struggling with this in 4th grade. So here is the success story of what we did in second grade last year. Nearing the end of the year, there were several skills that hadn’t been taught to the degree that they needed to be such as geometry etc. I knew without the foundation of addition, counting, and problem solving we would be up against a wall again in 3rd grade, so we focused on these skills. Throughout the year we spent a lot of time filling out number charts and discussing pattens on the hundreds chart above 100 using these:
Counting you ask? Yes, we spent time counting and looking at patterns in numbers. I know it is in the standard so that is part of why we counted, but counting is so much more important than teaching because it is the standard. How can students reason about whether their answer makes sense if they can’t count? Reasoning about math is in the mathematical practices several times. Students who can’t count, can’t estimate and can’t round because they have NO idea about where the number comes in the whole sequence of numbers.
Second graders last year solved a CGI word problem each day while they were learning addition and subtraction. Students spent several weeks using base ten blocks to solve their addition and subtraction regrouping problems. When students weren’t permitted to use the actual blocks, we prompted them to draw illustrations of the blocks to help them solve their problems. Even after students were shown how the traditional algorithm worked with their blocks, most of them tended towards drawing a picture of the blocks to solve the problem. Most were successful doing this. I was satisfied with this progress because I knew in 3rd and 4th grade that they would again have an opportunity to learn the traditional algorithm and other addition/subtraction strategies.
So here is how we are beginning with the kiddos in 3rd and 4th grades this year to teach addition regrouping. The kids are still given the opportunity to use blocks if needed to formulate understanding. Now I know that in showing them how to regroup the kids aren’t really “discovering” or “constructing” the algorithm themselves, but they are gaining an understanding. I just don’t think we have enough time in the year for the kids to discover everything and they must be shown some things. I haven’t arrived at that place yet where I think in CGI utopia…maybe I will get there someday?? (Don’t get me wrong, I find value in CGI) For right now the kids are getting this method of teaching addition regrouping and making sense of it. I’m happy and the kids are learning.
Traditional Algorithm Behind the Scenes
Now what I’m about to show you is the students’ first experience with regrouping like is pictured above. It isn’t cute at all…not worthy of for sale anywhere…but it is real and handwritten. To make it on a handwritten page was just so much faster than doing it on computer so it is what it is. I wanted to create columns so the students wouldn’t get their numbers confused. This worked well. I didn’t have the kids put pluses between the numbers like true expanded form to keep them from confusion later on when we do subtraction regrouping similar to this.
We discovered that students had a difficult time in the hundreds column when they had a number regroup to the thousands place. They weren’t used to putting two numbers together that weren’t zeros so this seemed to confuse them. If we had three digit adding to do over, we would have the kids include a thousands column so that they could regroup their thousands there at first until they made the connection that they could put two digits other than zeros in the left hand column. In other words, we would have them add one column more than the number of digits that there were in the number. For example…
Later on last week, we taught the kids to regroup without the columns drawn and without the numbers being decomposed into hundreds, tens, and ones. We continued to have the kids draw the arrows and to estimate their answer. It was rocky at first and about half of the class got regrouping with numbers written in standard form (just normal). They will be working on regrouping again early next week.
Are You Teaching Branching? Make Sure You Do This First…
Several years ago I worked at a charter school the first year it opened. The school implemented Singapore math, so that was my first year to test the waters of Singapore math. Our trainer instructed the 3rd grade teachers to go ahead and teach branching even though it was a skill the students should have learned in second grade. To teach children the procedure of branching, it took about four weeks total, and then not all of the students perfected the ‘procedure’ of addition and subtraction branching. The students had more success learning addition than subtraction branching. With the mandates of testing, we weren’t able to solely use Singapore math, but I had to supplement with other materials. Then as you are all familiar with, testing approached and likewise the pressure along with it. Then we didn’t have ‘time’ to teach number sense SO deeply since other skills are tested. Unsurprisingly, the teaching of Singapore Math somewhat fell apart midyear. Please don’t take this wrong I LOVE Singapore math because it works, but the conditions of testing hindered us from teaching it wholly.
Fast forward to four years later. After teaching small groups today, I have reflected on the year that I taught branching and its effectiveness. Yesterday I pulled small groups of average math students to teach them regrouping for the second day in a row, I had them build double digit numbers with base ten blocks blocks. I repeated this process today with the same group of students. After that I started notating their thinking with branching representation on a small white board. Students intently watched and helped me notate the thinking they had done with the blocks in (abstract) numbers . They began to understand grouping with tens and how to decompose numbers to build more tens or hundreds. Then I told them that they couldn’t use paper or blocks, but could only look at the addends I was about to write on the board. I asked them to whisper the answer in my ear so that others could still think. I was amazed! Half of them could answer the question correctly doing mental math. The other half were only 1 away from the correct answer. I was so proud.
I shared the above to really say the kids taught me something in just two days because of their adept ability to add mentally. Teaching branching worked so much better four years later–all I had to do was provide an experience with branching directly after building with base ten blocks. Why didn’t I start out with the concrete blocks first before I threw abstract numbers at them…duh me! Branching made so much more sense to them after building a concrete foundation. Reflection is priceless!
Math Cardsorts…Free Addition With No Regrouping Sort
I just finished a product that I posted on Teachers Pay Teachers. If you buy it before tomorrow, you will still get a chance to get it at the sale price. This product contains 8 sorts with addition and subtraction both with regrouping and without regrouping. Some of the sorts contain matching word problems and number disc picture cards. Others contain matching equations and number disc picture cards. In each sort, there is an extra card so that much discussion among student pairs can revolve around the common errors that confuse students with regrouping. These sorts were designed for use after adding and subtracting with number discs. Number discs are one of my favorite ways to teach addition and subtraction with regrouping. They are right up there with base ten blocks. I can’t decide which tool I like better. Number discs are a bit more abstract for students than the base ten blocks. Base ten blocks are the size of actual ones, tens, or hundreds which make them more concrete. Number discs are all the same size–but much easier to draw. For a FREE sort, you can click on the sort below to download the addition sort with no regrouping. The link will take you to the TPT site. Just download the preview for the free sort. I hope you enjoy it.
Are Your Students Struggling with Learning Subtraction Regrouping?
Two years ago I was introduced to math number discs and began using them in my classroom. I have come to rely on teaching regrouping using the math number discs after modeling regrouping with base ten blocks on a mat. These number discs (which are really expensive to purchase) are marked with 1’s, 10’s, and 100’s. An inexpensive alternative to using the ready made number discs is buying colored bingo chips and writing numbers on these yourself. Every place value position is a different color. The ones are white, the tens are red, the hundreds are orange, and the thousands are yellow. Students group the discs to represent a number on their place value mats and then take away the needed discs. Moving the discs around on the mat themselves does not seem to help students make the connection as much as having them draw and mark out the discs as they subtract. When they notice there are no more discs to mark out in the tens place for example, students realize they have to borrow from the hundreds place, mark out a hundred disc, and draw ten tens discs. If you scaffold this understanding to the actual borrowing and show students that when you borrow from the hundreds place to bring over ten tens, students have a light bulb moment and see the connection to all the marking out and rewriting of numbers that occurs in the abstract algorithm we call subtraction with regrouping.
Also, I am including a link below to my July 14th post in which I am showcasing a Smart Board lesson and practice pages that I created using interactive number discs.
How Can You Be Successful When Teaching Subtraction with Regrouping?
I just read a fabulous article from the periodical Teaching Children Mathematics (March 2011 issue) about the effectiveness of teaching subtraction with regrouping. A group of students was given a pretest beforehand and scored about 16% proficient at subtraction with regrouping from the instruction they had received the year before. The teacher showed students examples of the error patterns they were making. Next, to teach students about the errors they were making, the teacher gave students magnifying glasses and investigator hats so that they could become investigators to find a particular error pattern. Students relished the idea of finding the mistakes. As a result, the post test revealed a dramatic–more than 60% increase in proficiency of subtraction with regrouping. This article is not available for free online, however you can purchase it at http://www.nctm.org/eresources/toc.asp?journal_id=4&Issue_id=973 or your library may have a copy.
