I hope this email finds you and yours doing well. I want to take a moment to say I appreciate you using your valuable time to open this email and read what I have to share. It means a lot to me. My intention is to share, what I see, read, and hear in hopes that it makes the work you do more effective, easier, funner (I know this is not a real word), and more joyful! I feel another wonderful week coming on. Have you considered using number lines as warm-ups to develop strategic thinkers and number sense?

Why I love number lines…

I began using number lines quite a few years back when a teacher and I realized how messy it was to use base ten blocks and draw models of them when solving subtraction with regrouping problems.
Several times the teacher and I observed hard-working students get the answer to a subtraction problem incorrect. When students broke up a ten to make more ones, they accidentally crossed off too many ones or failed to cross off enough. Using the tens blocks was not working. We REFUSED to show second graders the standard algorithm for addition and subtraction (I still do). We wanted students to use place value and number sense to solve subtraction problems versus using the standard algorithm as a shortcut.

So what does one do when several students consistently get incorrect answers? We think of another way. We problem-solve.

I said to my fellow teacher, “What would happen if we used number lines?”

Stand back and watch the magic happen.

Oh, the magic of number lines. The students learned to experiment with number lines when solving addition and subtraction problems. Many students wanted to count by ones on the number lines. We let them. However, the students quickly discovered that counting by ones on the number line is time-consuming, messy, and prone to errors. We asked the students if it would be easier to count by fives or tens on the number line? Oh, the look of wonder and excitement that appeared on their faces. Off the students went, practicing their skip counting skills daily and understanding with greater clarity how many tens were in a given number, e.g., six groups of ten in 67.

Soon we realized that with larger numbers counting by fives or tens could also be inefficient. If one is adding 113 + 297, do you want to count by tens? Could you add groups of 100 (200) and groups of tens (90)? Students began to push themselves to make larger and larger jumps. As the students’ number sense grew, so did their accuracy and efficiency. Students could understand that by starting with the greatest value and counting up, they could solve problems more efficiently. Using number lines to solve problems led students to apply the commutative property as an effective strategy versus a rule one has to follow.

What do you mean it works both ways?

Subtraction is where the magic really began to happen. Watching students use number lines to subtract with regrouping was like watching Olympic gymnasts defy gravity as they flip and twirl on  the balance beam. You found yourself watching students and thinking, “Go for it! “Then applauding and saying, “She nailed it!
He stuck the landing!”

Workbooks provide lessons on using fact families and lessons on the relationship between addition and subtraction, but I rarely if ever see students apply those relationships outside of the book. If you want kids to understand properties and use them, have frequent number line talks with the whole class.

Draw a blank number line on the board and ask the students, “If you needed to subtract 152 – 99 on a number line, where would you place the numbers and why?”

Some students will place 152 on the right side of the number line and count down 99.

152 – 99 = ___

Draw another blank number line.

Some students will place 152 on the right side of the number line and count down until they get to 99.

152 – ___ = 99

Draw another blank number line.

Some students who recognize the relationship between addition and subtraction will place 99 on the left side of the number line and count up to 152. Some students who want to “experiment” will unknowingly try the same strategy and “discover” the relationship between addition and subtraction and find the difference between 99 and 152.
99 + ___ = 152

Warning!!!

You will probably discover that many of your students have less number sense than you would hope. You may also discover that many students have trouble counting by fives. You may also discover that many students do not know how to count backwards, very well. Some students will be very confused by the fact that you can count up from a given number to find the difference between two numbers, e.g., use the relationship between addition and subtraction to solve subtraction problems. These are the students who need to continue working with number lines more than any others. You can give students wth limited number sense and relational thinking smaller numbers, patience, and praise. How do any of us get better at anything? Practice, practice, practice, and supportive feedback.

So, what’s next?

It is one thing for a child to use a procedure that will get them the answer sometimes for some problems. It is a completely different thing when a child develops the power to analyze numbers, strategize, and develop strategies that will work for all numbers (including decimals and fractions).

There are powerful skills that come from using and experimenting with the number line: skip counting, number sense, place value, the commutative property, the relationship between addition and subtracting, and rounding.  Can you say the same for the standard algorithm?

By the way, the number line works for large numbers, too.

Try solving the problem below on a number line. Push yourself to try at least two different ways. What did you notice? What did you learn?

2, 352 – 588 = ___