Mary Ellen Kanthack BROOKWOOD MIDDLE, GENOA CITY, WI
4th Grade Math : Unit #1 - Place Value and Multi-Digit Addition & Subtraction : Lesson #1

# Subtracting Using dots, bars, squares and rectangles as a place value model.

Objective: SWBAT regroup numbers using place value understanding in a mult-digit subtraction algorithm.
Standards: 4.NBT.B.4 MP1 MP4
Subject(s): Math
60 minutes
1 More on the Top...No need to Stop: Warm UP! - 5 minutes

Math Power! Setting them up with the idea of mastery.

This rap song is a great way to refresh their memory about regrouping! The kids really loved it. We played it again and again. I chose this because it uses place value language that helps support the concepts learned in adding. I think the continuum of that language is essential to successful mastery of the standard 4.NBT.B.4 regarding fluency and place value understanding.

When they settled down...

I asked my students to write down what is the most common error they make in subtraction.

They said...

I forget to carry.

I forget and subtract the top from the bottom.

I regroup wrong.

I forget to check my work because I don't know how.

I told them that we would learn together how to avoid those mistakes through understanding the standard and knowing how to master it for Math Power!

When I catch students trying to subtract the wrong number, I continually go back to the rap song to help them remember. This will come in very handy when students subtract standard algorithms. The upcoming lesson graduates them into understanding of subtraction and place value as they learn to use a standard algorithm.

2 Start Change Result and Math Mountain Strategies At Work - 5 minutes

*This tutorial is helpful with understanding Math Mountains and Start, Change, Result strategies that is used in this lesson.

I wrote a 6 digit number on the board and subtracted a 4 digit number without using zeros.

657,235

For example                -    1,879

I asked students to tell me where the start was with a problem written like this. There was complete silence and the deer in the headlights stare. So, I started to write it: 657,235 - 1,879 = n and suddenly a student burst out with: "Math Mountain Time, Mrs. K!" I had realized that I had taught "Start Change and Result" in horizontal equations up to this point, which is logical because students often are presented an unknown in any spot. This is why it dawned on him.

So, I asked him to come up to the board and make the math mountain.  He placed the numbers correctly, making sure the total was on top. Again, I posed the question: Where is the start? Where is the change? What is the result? I wanted to be sure this was in the forefront of their minds.

Students volunteered to put the S , C & R where they would go on the Math Mountain, which connects the two strategies beautifully.  Lower end students see the connection very quickly and I often hear sounds of " I get it!" at this point. Not today.

I asked: Now where does it go in the standard algorithm? I reinforced what "standard algorithm" meant. Another student volunteered and placed the S, C, & R correctly in the standard algorithm.

I asked: What do you notice about the start? I wanted them to connect it all together to really solidify their thinking about placement and the function of each number in the equation.

The majority of the class replied: It is the largest number. ( Example of using place value understanding to compare.)

I kept pressing them onward by asking: How is this different than an addition problem math mountain? I was sneaking a chance to use inverse thinking to help them see that connection. Connection in my teaching helps with the flow of understanding of how numbers work together.

We discussed the difference in the placement and that the "Start" is always the largest number in the subtraction problem, being the total on the math mountain. They clearly saw the pattern. I was hoping now that their accuracy and understanding of place value would all connect.

To facilitate my hopes, I brought it all back to: How is understanding where the Start, Change and Result is going to help you in placing your numbers correctly and avoid the common mistake of subtracting some of the top numbers from the bottom?

I decided to veer off with another idea. The rap song is very helpful in remembering what to subtract, but what if I questioned them about thinking about greater values ( in essence, deciding the total).

How would comparing numbers <> help you do this too?  I got a quick answer, to my surprise! "It helps you remember that the greater number should be on the top! Boom! It is coming along through connections, strategies and good thinking! This is what Common Core is about!

*Good guided questions mean everything to the success of my class. I work hard to formulate the correct line of questions in my class. Sometimes it means thinking on my feet, response and pausing to really think where I need to go.

*I think it is important for me to review Start, Change, Result, because it allows the student to remember that the numbers in the standard algorithm have roles to play in finding the solution. I always want to keep them thinking about "Where is the start? What does it look like? What is the change?" It will transfer into solving future multi-digit word problems and, it shows them exercising MP1 & MP2.

Understanding How Math Mountains and Start Change and Result Strategies help students.
Student Ownership

Math Mountain and Start, Change, Result Strategies:  We so often see students make the mistake of not understanding how their fact families really work. They may have memorized them, but do not understand the role of the total in relationship to the addends. Standards for Practice #1 expect that mathematically proficient students start by "explaining to themselves the meaning of a problem and looking for entry points to its solution." If they don't understand the role each number plays within the equation, they will continue to make mistakes in solving problems. You will see it all in practice as you view the slides on the Smart Board File Resource.I love these two strategies and see my students using them over and over. It has served as a great tool for all levels of students. This makes your higher end students who may avoid thinking or explaining concepts because they just "get it", explain it well. It supports thinking for lower end students who really need guidance in critical thought.

3 Using Dots, Bars, Squares & Rectangles to Understand Regrouping. - 10 minutes

This method of transferring place value blocks to paper to subtract is an awesome way to close any gaps in the understanding of regrouping. It is easily differentiated by using smaller or larger place values. Beyond thousands place is not necessary according to the standard and it would get really messy trying to find shapes to represent place value beyond the thousands place. The idea is that after mastering the thousands place, standard algorithm work should be understood well enough to master.

I used this Smart Board lesson and built upon past strategies to incorporate the place value block ideas. Start, Change and Result as well as math mountains needed to be brought back into the picture to keep their minds going about the placement of numbers because they can easily revert to bad thinking habits and lose sight of place value understanding as the standard requires. Subtracting Using Place Value Models/ Smart Board File

As we went through each slide, the rap song kept coming back in their "math talk." As we subtracted using the place value blocks, we spoke place value language and talked about regrouping. This method holds them accountable to recognizing that you are moving a group over to function differently. They can't say words like "cross off the 5 and make it a 4" because it simply doesn't exist. Instead, the language is, "We cross off a group of hundreds or 10 tens an move it over to the tens place. Now I have 15 tens.

This is great! It holds their thinking where it should be as they master the standard by fluently talking about place value as they subtract a standard algorithm.

We practiced on other problems together before practicing doing independent, but monitored, seat work.

4 Subtraction Practice and Never Losing Sight of Place Value in the Process - 20 minutes

I gave my students dice to roll and create subtraction problems. This is a good way to have them manipulate numbers from the dice to form values in a standard algorithm.

So for example: I gave my high  and middle level students 4 dice for thousands place and had them roll the 4 dice first. They had to create the largest number they could. They rolled three dice next and created the smallest number they could. They set up the standard algorithm and they used their place value drawings to subtract.

The lower level students used only three dice doing the same process. All students showed work on paper, drawing the rectangles for thousands place, squares for hundreds place, lines for tens place, and dots for ones place. I worked with them to make sure there was allignment of the subtrahend correctly. This photo shows how the student corrected that, but he forgot to erase his original placement. He continued to subtract correctly.Place Value Drawing: Student shows a thousands place subtraction problem.

5 Assignment - 10 minutes

Students were assigned 10 problems IXL.com for practice and homework. They were assigned 4th Grade, Subtraction C.1. They were required to show their work using place value blocks and/or standard method. I told them I wanted to see proper placement of regrouping on the tops of their algorithm. If the problem was larger than the thousands place value, they needed to exercise their place value language and regrouping knowledge to solve. I like this website for samples of problems to do! It saves on worksheets and keeps us from dragging a book home.

I have included some samples of student work.  As we continue to add and subtract multi-digit numbers in a standard algorithm, I will continue to expect they use their place value language to have it become a natural part of their thinking.

In sample 1:Placement: Correcting bad habits. Where are the regroupings placed?We see that this student wanted to place the regrouped tens in between numbers. This method is taught to get them to remember to add the other amount. I stay away from this method because it does not support place value understanding. When they do that, it looks like they have created another number with more digits. So then we see him remember in the other problems! That's great!

In the place value sample drawing,Place Value Drawing Sample ,this student used the blocks to symbolize the thousands place instead of the rectangles. It must have taken him awhile! But, nevertheless, we see his understanding of subtraction and regrouping very clearly. Below, we see  the example of our discussion on using the inverse to see if we are correct.

Good Lesson!
Developing a Conceptual Understanding

This lesson is truly a good one! I think the place value block drawings really helped struggling students see the regrouping while supporting the language. I also think it keeps the high level student who really mastered the motions but not the explanation!

I can't think of a better way for this developmental level to transform it all into a solid well calculated standard algorithm. I saw smoothness in their subtraction and clear placement of the regrouped numbers. I am so committed to making sure the problem is clearly demonstrating how the regrouping is taking place by the placement of the regrouped sets of tens. Short cuts are fine, but not when they impede on mastering a standard to the depth of understanding that needs to be accomplished with CCSS! It's math power!