Base 10 Lesson 3: Teaching 2-digit addition and subtraction by demonstrating and explaining.

Teaching by deomonstrating and explaining is more straightforward than teaching using questioning, but there are still a lot of things to learn about how to do it well.

Goals

In each step of teaching mathematics, there are two goal: understanding and doing. Children need to be able to do the task, and children need to understand why the task makes sense and gives the right answer. If we neglect either one, we haven't succeded in anything worth doing:

If children can't do the task, then they haven't gained the mental tool they need to go on. Math builds on itself: you can't just skip addition and successfully learn to multiply.

If children are just following a set of steps blindly that they don't understand, they are likely to forget key steps as time goes on, and not be able to figure out what they were. Understanding why you're doing what you're doing is an important part of making knowledge useful--you're more likely to be able to remember later what to do, and you're more likely to recognize situations where you should use the procedure.

In a lesson where the teacher is demonstrating and explaining what to do, there's more involved than just showing the procedure. Some key goals are:

• Make the steps of what to do and how to do it as clear and simple as possible
• Involve the children in doing whatever it is. This often goes in stages:
  • The teacher shows what to do and how to do it
  • The teacher asks children questions about what to do as s/he shows another example
  • The children do an example while the teacher repeats the instructions
  • The children do problems independently, while the teacher watches and monitors for correctness.
• Explain why each step makes sense. This should also involve the children by asking the children at each step why it makes sense in the later examples.

It's wise to not move too fast in any one lesson. The more you can break down tasks into separate lessons that invlove only one or a few new ideas, the easier it will be to make those few new ideas clear to the students, and the easier it will be to monitor students' understanding.

How this links to the standards

The Common Core State Standards for Mathematics don't require students to learn the standard algorithms for addition and subtraction until fourth grade. What they do ask for in grades 1-3 is that students have strategies to add and subtract 2- and 3-digit numbers that:

• use place value
• students understand why the strategies work
• begin with concrete models or drawings such as base 10 manipulatives
• involve composing and decomposing 10s and 100s

The standard algorithms do satisfy all of those conditions if they are taught in a way that starts out with concrete manipulatives and is taught in a way that students understand why the algorithms work and make sense.

Composing and Decomposing:

The standards use the terms composing and decomposing. Composing means to make a 10 from 10 ones or to make 100 from 10 tens (etc). Decomposing means to make 10 tens from 1 hundred or to make 10 ones from a ten.

There are lots of synonyms for composing and decomposing:

In your assignments, I will be looking for you to scale your language use down to more concrete terms (bundle, take apart, trade, exchange) that fit your choice of manipulatives, and move away from the terms carry and borrow that you are probably in the habit of using.

Addition and Subtraction with manipulatives

Manipulatives can be useful in teaching addition and subtraction in several ways:

• The manipulatives represent the numbers. That means that in order to think about how to add or how to subtract, you (and the children) don't have to hold as much information in your working memory--you don't have to remember the numbers as you work, so you can focus on the action of adding or subtracting.
• Manipulatives are more concrete than written numbers. The size of the number is conveyed by the number of manipulatives (and the size of the manipulatives). Our written number system is beautifully compact--it takes just one symbol to represent 3 or 5 or 8, but that also means that it's harder for children to think about the amount represented by a number than it does when there re 3, 5 or 8 items physically present.
• Base 10 materials show the groupings into 10s and 1's, which makes it easier to show and understand the base 10 groupings you use in adding and subtracting.
• Once children have learned how to add and subtract with manipulatives, you can help them make connections between that and the pencil and paper methods to help them transfer their understanding to those pencil and paper algorithms.

When teaching addition and subtraction with manipulatives, it's important to pay attention to the transition to pencil and paper first, and it's good to have a sequence of lessons that slowly go from concrete to abstract.

The first lessons should be using the materials to add and/or subtract, using pencil and paper only to record the answers.

First, children should be adding and subtracting without regrouping

After children have learned to add and subtract without regrouping (with materials), children should be learning to add and subtract with regrouping using the materials.

A key lesson is to make the connection between the materials work to the pencil and paper algorithm. For this transition lesson, the teacher, and the children should be doing both the work with the materials, and the pencil and paper algorithm, and work should go back and forth between the materials and the pencil and paper work at each step, rather than doing one first followed by the other. At each step the manipulative work (which is more familiar to students at this point) should be performed before the pencil and paper recording.

Finally, later lessons where you are adding and subtracting with only pencil and paper should use similar language to the language you used when doing the work both with manipulatives and pencil and paper.

Children need time at each of these lesson stages for the process to become familiar and well known before moving on to the next step. Ideally, practice should happen over several days in between these lesson stages.

In the examples that follow, please notice:

• How the first example is teacher directed, but with children helping, and the instructions for later examples give more and more work to the children.
• How each lesson is only a small step beyond the previous lesson
• How in the lesson 3 lessons, you go back and forth between the manipulatives and the number work.
• How, in the lesson 4 lessons, the teacher refers back to the manipulatives steps, even though the manipulatives aren't present.

Words spoken by the teacher are in " " marks, and expected answers given by children are in [ ] brackets.

An example of a progression using a set of lessons for addition using base 10 blocks as the manipulative

Addition Lesson 1, Example 1: 32+25 (using base 10 manipulatives, no regrouping)

• write the problem
• put out 3 tens and 2 ones and 2 tens and 5 ones (organizing the materials by place value).
• "We have 32 and 25. When we add, we're going to put those amounts together and find how many in all. First I'm going to find out how many ones I have" (group together the ones). " Can anyone tell me how many I will have when I put together 2 and 5?" [7].
• "Now I'm going to find how many tens." (group together the tens). "Can anyone tell me how many I will have when I put together 3 tens and 2 tens?" [5 tens or 50]
• "So I have 57 in all, and I write that down as my sum" (record the solution: 57)

In example 2, ask children to tell you the steps to place the materials, count and record

In example 3, remind the children and have them do the steps to place the materials, count and record.

Addition Lesson 2, Example 1: 36+29 (using base 10 materials with regrouping)

• write the problem
• put out 3 tens and 6 ones; put out 2 tens and 9 ones. "We have 36 and 29"
• "I'm going start by grouping together the ones" (group together the ones) "Can anyone tell me how many ones I'm going to have when I put 6 and 9 together?" [15]
• "15 is more than ten, so I have to trade in 10 ones for a ten." Show how to count out 10 ones, and line them up next to a ten to show that they are equal, and then make the trade. "I had 15 ones. Who knows how many 1's I have left after the trade?" [5]
• "Now I'm going to group the tens together. I have 3 tens and 2 tens and 1 more ten. Who can tell me how many tens that is?" [6]
• "So I have 6 tens and 5 ones, and that's 65. I'm going to record 65 as my sum." (record 65)

In example 2, ask children to tell you the steps

• count and place the materials
• group the ones
• count out 10 ones and trade them for a ten (ask how many ones you need to trade to get a ten)
• count the tens and ones and record how many

In example 3, remind the children and have them do the steps to place the materials, trade, count and record.

Addition Lesson 3: Example 1: 26+57 (Materials and pencil and paper)

• write the problem
• put out 2 tens and 6 ones; put out 5 tens and 7 ones.
• "I have 26 and 57, and I am going to put together the ones. Can anyone tell me how many ones I will have?" [13] (group together the ones.)
• "Is 13 more than 10" [yes] "So what do I have to do?" [trade in 10 ones for a ten]. (make the trade) "Who can tell me how many ones I have now? [3] 'I'm going to write 3 in the 1's column for the answer." (write 3).
• "When I traded in the 10 ones, what did I get?" (a ten). "I need to record that too. I'm going to write a 1 in the tens column. The tens column shows how many tens there are, so if I put a 1 there it means there is 1 ten in addition to the tens that we already had" (record 1 in the tens column)
• "Now I'm going to put the tens together. I have 1 ten and 2 tens and 5 tens. Can anyone tell me how many tens that will be altogether?" [8] (group together the tens). "So I write 8 in the tens place for the answer.

In example 2, ask children to tell you the steps

• count and place the materials
• group the ones
• count out 10 ones and trade them for a ten
• record in the ones place, and record the newly exchanged ten in the tens place
• count the tens and ones and record how many in the tens place

In example 3, remind the children and have them do the steps to place the materials, trade, count and record.

Addition Lesson 4,

Example 0: 47+38 (Pencil and paper with manipulatives kind of).

The goal of this example is to get children to think about what they would do with the materials without actually using them. This example works well if you are using the same materials the children are using--the physical base 10 blocks. If you are doing this on a SMART board, it isn't going to work, so if you do all of your work on a smart board, you should skip this and go on to example 1. This example can also be used as an effective intervention later with children who can do the manipulative work, but got lost when the rest of the class made the move to the pencil and paper algorithm.

• write the problem
• "If I were going to do this problem using the base 10 blocks, I would start by getting out 4 tens and 7 ones" (put a handful of tens and ones on the table), "and 3 tens and 8 ones" (put another handful of tens and ones on the table). "I haven't counted to see if I have the right number of tens and ones. If we need to, I'll count out the right number. Does anyone know without me counting out 7 ones" point to 7 in the problem) "and 8 ones" (point to 8 in the problem) "how many ones I would have in all?" [15]. "Nice--how many of you knew it was 15?" [many children should know--if not, you're teaching this too early]. (Optional: how did you figure out that there were 15 ones? [invite one or two children to tell a way to get 15]).
• "So we know that we would have 15 ones in all." (push the ones together) "Is that more than 10?" [yes] "If I traded in 10 ones for a ten, how many ones would I have left?" [5] "Yes--how do we know there's 5 left?" [answers might be 15-10=5 or 15 means ten and 5 more]. "So, we can record here with the numbers that we'll have 5 in the ones place, and we traded in 10 ones to get another ten" (take out another ten from the materials box and show it), "so I'm going to write a 1 here in the tens place, because now we have another ten--we have 4 tens and 3 tens and 1 ten" (point to the digits as you say this).
• "We're all done with the ones" (sweep all of the ones in the pile back into the box that holds the materials, and leave the pile of tens out on the table.)
• "Hmmm... who can figure out how many tens we would have? We have 4 tens from the 47, and 3 tens from the 38 and 1 more ten we got when we traded for a ten. Can you figure out how many that would be, or do I need to count out the tens?" [8] "Great--how did you get 8?" [let the student share]. "So I can put 8 in the tens place and I know that I have 85 in all."
• (Sweep the tens off the table) "That was great, and we didn't really use the manipulatives at all, did we?"

Example 1: 28+35 (Pencil and paper)

• Write the problem vertically.
• "If I were doing this with the base 10 blocks I would get out 8 ones and 5 ones. If I had 8 ones" (point to the digit 8) "and 5 ones" (point to 5) "how many would that be all together?" [13]
• "Is 13 more than 10?" [yes]
• "If we trade in 10 ones from 13 to get a ten, how many ones will we have left after the trade?" [3]
• "So, I'm going to write 3 in the ones place because that's how many ones we have, and I'm going to write a 1 in the tens place because after we trade in, we have 1 more ten."
• "Now I have 1 ten and 2 tens and 3 tens. How many tens is that all together?" (point to the numbers in the tens place) [6]
• "So, I'm going to write 6 in the tens place, and my total is 63."

Example 2: 29+45

• Write the problem
• Ask: who knows what we should do first? [add the ones]
• What do we get when we add the ones? [14]
• What do we do with that 14? [trade 10 ones for a ten] (if children don't include this step, and jump to writing 4 in the ones place, ask--why do we write 4 in the ones place and not 14?)
• How many are left over after we trade for a ten? [4]
• What do I do with the 4? [write it in the ones place]
• What do I need to do before I can add up the tens? [write 1 in the tens place]
• Who can tell me where this 1 came from in the tens place? [it came from trading in 10 ones from 14]
• How many tens do I have in all? [7]
• So what's my final answer? [74]

[more examples/problems]

Subtraction lessons

I'm going to show only the first examples for the subtraction lessons. Each lesson, of course, should have additional examples where increasingly the children do more of the thinking.

Subtraction Lesson 1 Example 1 (subtraction with manipulatives, without regrouping) 58-23

• Write the problem
• Lay out 5 tens and 8 ones to show the minuend
• "Now we're going to take away 23 from the 58. First I'm going to take away 3 ones. Who can tell me how many ones will be left after I take away 3?" [5] "Raise your hand if you knew there would be 5 left" (remove 3 ones, optional follow up--how did you know there would be 5 left)
• "Next I'm going to take away 2 tens" (point to the 2 in 23). "Who can tell me how many tens I will have left after I take away 2?" [3]. (remove 2 tens)
• "I have 3 tens left, so I have 3 tens and 5 ones left, and I'm going to record that as my answer." (record 35)

Subtraction Lesson 2 Example 1 (subtraction with manipulatives, with regrouping) 62-35

• Write the problem
• Lay out 6 tens and 2 ones to show the minuend.
• "I want to take away 5 ones from my 62, but I don't have enough ones to take 5 away. The way that I can do this is that I can take one of the tens in 62 and I can trade it in for ten ones." (put back a ten and take out ten ones) "I still have 62 in all after I make the trade, I'm just building that number differently (Line up the new 10-ones so they are in the same shape as a 10).
• "Now I can take away 5 ones. Who can tell me how many ones I will have left after I take away 5 ones? I have 12 right now, and I'm going to take away 5" [7] "Yes" (take away 5 ones) "I will have 7 left."
• "I have 5 tens now (remember that I traded in one of the tens already for 10 ones), and I need to take away 3 tens. Who can tell me how many tens I will have left after I take away 3?" [2] "Yes" (take away 3 tens)
• "So I have 2 tens and 7 ones left left after I take away 35 from 62, so I'm going to record 27 as my answer" (write 27)

[In Examples 2 and 3 ask the children--"what did I have to do before I could take away the ones?" and "how many ones do I get when I trade in a ten?" You may want to make a poster or reminder note on the board using a drawing of the base 10 blocks showing the ten ones is a fair trade for a ten and a ten is a fair trade for 10 ones.]

Subtraction Lesson 3 Example 1 (subtraction with manipulatives and pencil and paper) 73-29

• Write the problem
• Lay out 7 tens and 3 ones to show the minuend.
• "I want to take away 9 ones. Who can tell me what I need to do before I can take away 9 ones?" [Trade in a ten]
• "After I trade in a ten, how many tens will I have left?" [6] (move one ten over to the side) "I'm going to record that 6 with my numbers, and I want you to record your trades just like this when you are working today. I'm going to cross out the 7 tens in my problem and I'm going to write 6 above it becaus after the trade there are only 6 tens" (cross out 7 and write in 6)
• "How many ones do I get when I trade in a ten?" [10]. (remove the ten that you had moved over and take out 10 ones).
• "How many ones will I have after the trade? I had 3 ones before the trade" (point to the 3), "and then I got 10 more ones from the trade. So that's 3 ones and 10 more ones. Who can tell me how many ones that will be in all?" [13] "I'm going to change the 3 in the ones place to a 13. I could do that by crossing out like I did with the 7, or I can sneak a 1 in next to the 3 to make it a 13" (record the change from 3 to 13).
• "Now I'm ready to take away 29. First I'm going to take away 9. Can anyone tell me how many one's I'll have left after I take away 9? I have 13 now, and I'm going to take away 9..." [4] "Yes" (take away 9 ones) "I'll have 4 left, and I write 4 in the ones place of the answer.
• "Next I'm going to take away 2 tens. I have 6 tens now. Who can tell me how many tens I will have after I take away 2 tens?" [4] (take away 2 tens) "Yes, I will have 4 tens left, so I'm going to write 4 in the tens place of the answer" (write in 4).
• "So 73 take away 29 leaves 44".

[In examples 2 and 3 ask children after the trade: what do I need to record before I can start taking away? Insist that children stop and record the trade before doing any taking away in the problem]

Subtraction Lesson 4 (You could do an Example 0 if you are using an overhead projector, or teaching to a small group with physical materials be like Lesson 4, example 0 for addition: Do just what you would do in Lesson 3, but when you get out the manipulatives, don't count them out, and don't lay them out nicely, and instead put out a handful of tens and ones and ask children to imagine what what you would do with the manipulatives.)

Example 1 (pencil and paper): 72-48

• Write the problem
• "In this problem, I only have 1 one, but I do have 7 tens. I want to take away 8 ones. If I had my manipulatives out, what would I do to get more ones?" [trade a ten]
• "I start with 7 tens" (point to 7), "and I trade in a ten. How many tens do I have left after I make the trade?" [6] "So I'm going to cross out the 7 tens and write 6 in--that means I have 6 tens after the trade." (record 6 tens)
• "When I trade in a ten, how many ones do I get" [10]. "I had 2 ones before (point to the 2) and now I have 10 more ones. How many ones is that in all?" [12] "So I'm going to turn that 2 into a 12" (record 12).
• "Now I can take away my ones. I have 12 ones. How many ones will I have after I take away 8 ones?" [4] "So I record 4 in the ones place, because I'm going to have 4 ones left.
• "Next, we take away 4 tens" (point to the 4 in the 48) "I have 6 tens. If I take away 4 tens, how many tens will I have left?" [2] "So I write 2 in the tens place of the answer.

In later examples, when it comes time to take away, ask the children "how many ones/tens do I have now?" "How many am I taking away?" How many will be left?:"