Subtract Across Zero Using Number Lines

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SWBAT use blank number lines and place value to subtract across zeros.

Big Idea

Using a number line to subtract across zeros supports students conceptual understanding of crossing decades when subtracting.


5 minutes

The warm up for this lesson is reviewing how to use an open number line for both addition and subtraction.  The use of number lines was taught earlier in the school year, and it is an important math tool that I will continue to use. 

The number line is an "actual" representation of sets of numbers. There is no other model that will carry across a students' entire mathematical development, from number sense to high school statistics and probability.

In order to build student confidence with using the number line, I start with easy computations using multiple addends or subtrahends. 

I contextualize the addition and/or subtraction with a word problem.  For example, The librarian has 10 books to put on shelves, then a class comes to check out new library books, and there are 25 books returned.  At the same time 3 students come in from another grade to return their books.  How many books does the librarian need to put on the shelves? 

For subtraction, the problem is: The librarian has 38 books to put on the library shelves. Three books were returned by students that morning. Later that day, a class of of students came in to return their books and check out new ones.  The librarian had ten books in the return bin at the beginning of the day.  How many students were in the class that returned their books?


Using A Number Line

10 minutes

Because my students vary so much in their mathematics abilities, I review using number lines for everyone.  This includes first determining if the problem is addition or subtraction.  If the problem is addition, I teach the students to start on the left side of the number line, and if it is subtraction, they start on the left side of the number line. 

Using the problems from the warm up section, we model using the number line and different strategies that may have been used. The librarian has 10 books to put on shelves, then a class comes to check out new library books, and there are 25 books returned.  At the same time 3 students come in from another grade to return their books.  How many books does the librarian need to put on the shelves?  

I draw a number line on a whiteboard and label the left side starting point with ten, then I jump about an inch and a half and label it +25 above the curved jump line and draw a straight tick mark.  I ask the students to explain what is the new number under the tick mark.  Students add 10 + 25 mentally to get to 35.  

Number Line 1

Then I repeat with another curved jump line above to add three.  I make the jump a little shorter, and I always label the amount being added above the curved line.   Finally, I draw a  tick mark on the line for the total amount and write 38 beneath the tick mark.

Number Line 

This process is repeated for students to practice using individual whiteboards or paper and pencil until they show confidence and success with this level of difficulty.


Let's Subtract Together

10 minutes

Since this lesson is introducing subtraction across zeros and using regrouping, I use examples that are number based and not focused on a word problem.  The goal of the lesson is to use the number line, and I have found it more successful with my current students to separate word problems when building a skill. The Common Core standards require students to fluently subtract using place value and using number sentences rather than words will allows me to focus on this skills without adding any confusion with language especially for the language learners in my class.

Because I want to continue building students' confidence, I start with a relatively simple problem such as 30 - 15.  Using an open number line and starting on the far right end of it, I mark a short line for 30. (This is a good time to remind students why the work begins all the way over on the right side.)

I then ask the students to use place value to tell me how many tens are in fifteen.  I model how to use the arch line to subtract 10, and mark a tick mark under the number line with 20. This is repeated for the ones digit, five, with an arched line and a tick mark to write 15 beneath the number line.  The arched lines are marked above with - 10 and -5. 

I repeat this with number sentences including:

50 - 38 = ______, 70 - 26 = _______, 90 - 64=_______

When my students are showing success with this type of problem, I move on to using the same steps with problems using hundreds as the minuend.  These would be similar to: 

300 - 130 = ______

In this example, I mark 300 on the number line, subtract 100 and mark 200.  Next I subtract 30.  There are two possibilities that may be demonstrated by you are offered as options by students. If your students have a strong foundation in number sense, they may just move back thirty to 170.  The other possibility is that students will need to subtract the remaining 30 by tens to 190, 180, and ending at 170.  Students may also need support in counting backwards crossing the benchmark from 200 to 190.

I continue with other examples such as;

400 - 160 = ______,  700 - 380 = _______

If students are successful, I continue with problems similar to 

600 -237 = ________ ,  500 - 383 = _______ 


Try It Yourself

15 minutes

We use a worksheet to practice using numbers to solve the subtraction sentences.  This can be done independently, in partners, or in groups of three to four.  I had students work together in partners because I want them to be talking and explaining their thinking with another person. This supports the development of student thinking (Math Practice 1,  Making Sense of Problems and Persevere in Solving Them).  

I observe some of my students using the standard algorithm, with borrowing, instead of the number line.  By having them work with a partner they were given the opportunity to talk through each step of the number line process and use the algorithm to check their work.  My students were not able to explain or prove their thinking with just the algorithm, so they are redirected to go back to using the number line to demonstrate their understanding.

Wrap Up

10 minutes

To close the lesson, I ask the students to return to the carpet are to review the process of using a number line.  I draw a blank number line to display on the document camera, or whiteboard, and I use the same example from earlier in the lesson, 300 - 130 = ______.  I ask the students to show it on their individual whiteboards, or paper and pencil.  They have to explain their process to their partner, and then they create their own problem to solve with a partner. 

I choose the same problem as a support for those students that struggled, and to build the confidence of those who were successful.  When the students create their own problem, it allows me to quickly assess their progress and fluency in subtracting using the number line.