Math Resources

Math Resource

• Adding on a numberline
• Using place value to add
• Using pictures to add
• Subtracting on a numberline
• Using pictures to subtract
• Using place value to subtract

Many of these strategies are different from the methods that you and I were taught as children. I remember the teacher telling us to carry, borrow, or simply write down one number and carry the other. I don't know why we did that or what the mathematical thinking behind those actions were. I did them simply because the teacher told me to and that was what I needed to do to get the answer. I didn't really understand the concept.

Math teaching today has changed for the better. We want students to understand what they are doing and why. We want children to truly understand that when they "carry the one" to the tens column, they are really adding one group of ten but simply writing a one because a one in the tens column is worth ten ones. Students need many opportunities to work hands on with manipulatives, talk with partners, and ask questions to get to that point. Over the course of the year you will see your child become confident in adding large numbers together quickly...and probably using different strategies than you were taught!

I urge you to take a little time and try these strategies out for yourself. Ask your child to teach the strategies to you...they enjoy the chance to be in charge or take the teacher's role every once in a while. Try to use these strategies to help your child with homework. It is a great way to reinforce school learning at home.


Adding on a Numberline



One of the strategies students have been taught to use when adding is using a numberline. This helps when kids are adding large numbers together or have to add numbers together to regroup. This is a fast and easy strategy and the kids usually begin to use the numberline as a "go-to" strategy when working.

Here is how to add on a numberline.

• Look at the number sentence.
• Determine which of the addends (the numbers you are adding together) is greater. This is your starting point on the numberline.
• Draw an open numberline (an open numberline has no number markers on it - you choose the starting and ending point). Make a point on the far left of your numberline for the greater addend.
• Then look at your other addend. That number is how many jumps you will move down the numberline. You can choose to break the number up and jump by ones or jump by tens and ones, or whatever combination is easiest for you. Decide how you will break the number up.
• Start at the left point of your numberline (where the greater addend is) and draw the jumps for your smaller number. Add the total in chunks as you go along.
• Make the ending point of your numberline. This is the point where you land and can make no more jumps. The ending point is your total or sum of the two addends.

Let's take a look at a problem together.

• Let's say our number sentence is 71 + 99. These numbers are too big to add in my head and there is some regrouping (what most adults were taught as "borrowing" or "carrying") involved. I need to show my work in a way that lets whoever looks at my work know that I understood what I was doing and why but is also efficient. I am going to use a numberline.
• Out of the two addends, 99 is greater. I draw an open numberline (just a line) and make 99 as my starting point. I put the starting point at the far left because I know that a numberline goes from least to greatest. Smaller numbers come first. If I am adding a number to 99, the total or sum will be larger so it will fall farther down (to the right) on the numberline.
• I need to make 71 jumps because I am adding 71 to 99. I can break up 71 however I want to. I can choose to do 71 jumps of one, but that wouldn't be very efficient. It would be like counting on my fingers. So I need to think of how I can break up 71 into bigger chunks. I am going to break 71 up into tens and ones.
• I go to my numberline and start at 99. I jump one space (+1) to get to 100. From 100, I jump seven groups of ten. I count as I go along...110, 120, 130, 140, 150, 160, 170.
• When I am out of jumps I have found my answer. I label the numberline with my sum. The sum of these two addends (numbers I am adding together) is 170.

Using a Picture To Add

One of the strategies we have taught children is to draw a picture when adding two numbers together. This is a great strategy to use when the numbers are small (for example, ten bananas), but drawing a picture can become time consuming when dealing with larger numbers. Because of this, we have taught the children to draw a picture of the numbers in place value blocks to help them add. Drawing a picture of tens and ones is a lot faster than drawing 99 bananas.

Here is how to add using the draw a picture strategy.

• Look at the first addend (number you are adding). How can you show that number in place value blocks? What is the most efficient way to show them (using the least number of blocks)? Draw a picture of the first addend in place value blocks.
• Look at the second addend. How could you show that number in place value blocks? What is the most efficient way to show them? Draw a picture of the second addend in place value blocks.
• Now, draw a picture showing all the units from both addends together. Put all the hundreds together, all the tens together, and all the ones together.
• Look at your drawing of both blocks together. Is there more than ten in any unit? Can you regroup?
• Count up the total value of all your blocks. This is your sum (the total of both addends together).

Let's look at a problem together. Let's do 92 + 69.

• The first addend is 92. I see a 9 in the tens place and a 2 in the ones place. I can show that as I I I I I I I I I (9 tens) and * * (2 ones).
• The second addend is 69. I see a 6 in the tens place and a 9 in the ones place. I can show that as I I I I I I (6 tens) and * * * * * * * * * (9 ones).
• When I add both addends together (combine groups), I get a picture that looks like this:      I I I I I I I I I I I I I I I (15 tens) and * * * * * * * * * * * (11 ones).
• When I look at my big picture, I see more than 10 groups of ten. I can trade those ten groups of ten in for one group of one hundred. That leaves me with 5 groups of ten. I also see more than ten ones. I can trade ten ones in for one group of ten. When I am done with all my trading, my picture looks like this: [] (1 hundred) I I I I I I (6 tens) * (1 one).
• I look at my picture after all the trading and count up the total. I see 100 + 60 + 1, for a sum of 161.

 Using Place Value to Add

Another strategy the students have been taught to use when adding large numbers is to use place value. To use this strategy students have to look at the value of each digit in a number. This is a quick and easy strategy that kids can do with smaller and larger numbers.

Here is how to add using place value.

• Look at the first addend (number you are adding). Think about the value of each digit in the number. What is the value of the digit in the hundreds place (in other words, how much is the number worth)? What is the value of the digit in the tens place? What is the value of the digit in the ones place? Write the number in expanded form.
• Look at the second addend. Think about the value of each digit in the number. What is the value of the digit in the hundreds place? What is the value of the digit in the tens place? What is the value of the digit in the ones place? Write the number in expanded form.
• Add the value of each place together. It is OK to start with the hundreds first and work left to right. Add the value of the hundreds in both addends together to find the total. Add the value of the tens in both addends together to find the total. Add the value of the ones in both addends together to find the total.
• Look at your totals for each place (hundreds, tens, and ones). Can you regroup?
• After all regrouping, add the values of each place together to get the sum.

Let's look at a problem together. Let's try the problem 317 + 195.

• If I look at the first addend, I see a 3 in the hundreds place for a value of 300. I see a 1 in the tens place for a value of 10. I see a 7 in the ones place for a value of 7. So if I write this number in expanded form, I get 300 + 10 + 7.
• If I look at the second addend, I see a 1 in the hundreds place for a value of 100. I see a 9 in the tens place for a value of 90. I see a 5 in the ones place for a value of 5. So if I write this number in expanded form, I get 100 + 90 + 5.
• Now I add each place together. In the hundreds place I have 300 + 100 for a sum of 400. In the tens place I have 90 + 10 for a sum of 100. In the ones place I have 7 + 5 for a value of 12.
• My totals are 400 + 100 + 12. I can add those together to get a sum of 512.

Roller Coaster Rounding

Today the kids were introduced to the concept of rounding.Rounding is a concept you can help practice at home. I want to take a minute and give you a little background on the strategy that we use to teach the kids to round. This is an easy strategy that you can try at home.

When we round to the tens place, we think about the two tens that are nearest to (the ones that come before and after) the number we want to round. Write the smaller tens number on the left of the roller coaster. Write the larger tens number on the right of the roller coaster. At the top of the roller coaster write the number that is in the middle of those two tens...the number that ends in 5 (25, 55, 75, etc.).

Then figure out where your number fits on the roller coaster. If it is on the left side of the roller coaster, the number slides back down the hill to the smaller ten. If it is on the right side of the roller coaster, the number rolls up to the next ten. Children should understand that numbers that end in 0, 1, 2, 3 or 4 will round down. Numbers that end in 5, 6, 7, 8 or 9 will round up.

I am posting a few pictures to help demonstrate. This is a helpful strategy to make rounding more concrete to the children as most of them have been on a roller coaster or on a hill before and understand the concept of not having enough momentum to go over the hill and having to slide back down or having enough energy to carry you up and over the top of the hill. Please let me know if you have any questions.