RTI : Using the Lead Number to Estimate Addition and Subtraction

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SWBAT estimate using the highest place value in a standard algorithm.

Big Idea

Students use the concept of a train to understand the lead number and how to estimate using a standard multi-digit algorithm.

Everybody's Doin' the Locomotion!

10 minutes

When the kids came in from recess today, I was playing "Locomotion" on iTunes. I grabbed one of my girls and we started the train chain and began moving around the room. There was laughter of course and I stirred up some real energy. 

I stopped and told them that we were actually modeling a math concept!  I wrote one 5 digit number on the board and asked them what it meant to round the "lead digit" or "lead number." 

One student explained that it was the first number. I wanted them to connect the idea of a train and the lead digit ( the engine of the train to rounding, so I questioned) :

How is multi-digit number like a train? A student answered by saying that the number is long like a train.Another mentioned that it is a chain like a train. We live in an area where they see lots of very long freight trains and so this connection was really clicking in their minds.  In order to direct their thinking further, I asked: What pulls the train? Of course they knew that the engine pulled the train. I underlined the lead number on the board.

I wanted them to think again hoping they would make the connection: How is a lead number like an engine of a train? My one very mathematically challenged student looked up at me and said "Oh! It's the first number like the engine is the first thing on the train!" 

Wow. Connections like these are great! I try to make them as often as I can. I knew now that we were ready for the heart of the lesson.


Revisiting the Concept: Using Learnzillion and Direct Instruction to Review Rounding

10 minutes

I pulled out my students who are struggling with estimation to reteach and find another way to reach them. My other students were busy with a individualized MAPs based computer program called eSpark, that we use. They were busy with this throughout the period and they worked independently. Sometimes I do this so that I have time to reach those who are struggling, even though eSpark is part of a daily routine.

I began again today with Ginny Baldwin's Learnzillion lesson. I think they need a direct instruction approach since the critical thinking process is needing support as we transition into Common Core. I had  them sit down on the floor with graph paper, clipboards and a set of subtraction and addition problems I had written by hand and copied. I used various place values of problems to help them feel successful. I know that the large digits overwhelm them and that they would need to be talked through the process for those. I am using graph paper with these students because they struggle with lining numbers up on regular loose leaf lined paper. The graph paper gives them the neatness and structure, but I still have to insist on using the boxes!  Our practice would be done on the graph paper throughout the lesson to reinforce neatness and accuracy.

I stopped the Learnzillion lesson in places to create some estimation algorithms on the white board to practice, leaving the numberline/benchmark examples on the SB and referring back to the number line  as we rounded and estimated on the whiteboard. I used benchmarks, midpoints and the rhyme to help them round pointing out that it is always the "lead digit" or the "engine" of our train. I was back and forth from the whiteboard to the SB with the Learnzillion lesson as a guide. We did four addition algorithms and I increased the place values with each algorithm.

As we finished up, I heard students say, "I get it now!" I asked for thumbs up if they thought they were ready to practice together on their assignments.


We took a minute to discuss the zeros behind the lead digit. I referred to the lead digit as the engine.

I asked: What about the cars behind that engine? I explained that train cars hold things like coal, iron ore, products being shipped, etc. and that trains unload their cars. I drew a train in front of the number I was rounding. I told them that  I think the empty cars are like the zeros we see in our rounded number. I placed the lead digit in the engine spot after deciding if we round up or keep it the same and then the zeros in the cars.

Again I heard: " I get it!".

I reinforced that they needed to remember that all of the place values behind the lead number must be zero. I asked them to check every rounded number before they figured out their answer.




Guiding and Using Student Constructed Notes to Reinforce Learning

20 minutes

To be sure that they understood the concepts, I decided to put the responsibility of them teaching all the steps back to me through the use of student constructed notes.

I brought a blank new page up on the Smart Board  to create the student constructed review of what we had just learned. If you look at the example of the lesson, of a Student Constructed Notes About Estimation  you can see how I just guided the process along to reinforce their own understanding of the lesson. ( I have typed some explanatory notes within it for your use to explain what was being said in the class). This notebook file is intended as an example of what a student constructed review looks like.   In your class, it may look very different depending upon where your students are at in the learning process and what you may decide they need to remember about using lead numbers and estimation. It is a form of "guided constructivism" ( if that isn't an oxymoron) used to engage the learner in taking responsibility and also examining any misconceptions that arise. It works out the glitches with support from each other, guided by the teacher!

As I questioned them through the review, and they answered, I wrote down what they said and guided the lesson along, so that I could understand their thinking better.The initials stand for which student was talking. They like to see their initials on the page when they share. I was really  pleased that one student brought out the train idea again when I asked what the lead digit was!

We worked through each step to estimation and reviewed  the number line and benchmark concept for rounding again ( as brought out in the Learnzillion lesson).   You can see this on page 3. On page 4, I brought it all back to the concept of rounding the lead digit. On page 5, I started the discussion about thinking about subtraction and estimation. Was it any different? Page 6 resolves that it isn't and that the process is the same, except you subtract.

As we were working through all of this, they were working at taking notes on their iPad 'Notes" ap.

*If you try the student constructed notes in an RTI lesson, just understand that anything they say needs to be written down, right or wrong, because they often see what is wrong with their thinking when it is right in front of them and open to discussion. A classroom has to have established trust in order to do this.


After this review, I let them independently work on their graph paper practice sheet, but seated on the floor in front of the SB so that they could refer to it and to their own notes. It also made it easy for me to monitor their progress.grid paper in class practice assignment This copy is very primitive looking! But, it is very basic and structured. This is what I think RTI kids need. You will notice photos of a copy of a student using just plain paper. These students were able to line up work properly.

Warning: There were two mistakes that surfaced  with most of my students.

This may happen to you too. The printed graph paper assignment corrects this problem.

1. During the Learnzillion part of the lesson: Even with the graph paper, the place values were not lining up correctly when they were copying their examples as I stopped the Learnzillion lesson in the review. I noticed if the value of the number on top was ten to hundred thousands and the bottom was thousands or less. They wanted to  move the number over to line up with the largest digit. Them copying it to the graph paper wasn't the complete answer to my neatness problem. It showed me that their concept of place value was still not mastered!

2. During any work time: Rounding to the number next to the lead number. Even though the lead number was underlined, they went to the circled number to round and used the digit to the right of that.  I needed to bring back the arrow underneath pointing to the number being rounded. You will see that in the student examples in the reflection.




5 minutes

After practicing awhile and I could see that they were fairly stable in their work. It looked like the RTI was working and the direct instruction approach was the right decision for this group. Nevertheless, I stopped them from working and reviewed the steps to estimation in both addition and subtraction because I wanted them to hear it once more.

I told them to remember these things and that to be sure that their iPad "Notes" ap contained all the notes.

1. Round both numbers you are working with by rounding the lead digit. Underline and circle it. Use arrows to show the number being rounded.

2. Then add or subtract.

3. If you forget what a lead digit or number is, think of a train engine and all the cars behind being empty. 

 I closed with one final question to solidify and wrap up their thinking: "Why do we round numbers to the lead digit when we estimate?" I emphasized the why so that they continue to understand that we are always looking at understanding the "why" of everything we do.

A student responded. "Making the only number the engine makes it easier to add or subtract because all of the rest of the numbers are zero."