Finding Sums

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In the previous section, students learned to model addition with the blocksand Place mats. Through your questions and their recordings, they beganto reflect on this process. Now they start to predict what will happen when

they combine the blocks.

Students need many opportunities to combine collections of blocks. In fact,

they should do so until they can consistently predict the result of any com-

bination before they physically perform the task.This focus on prediction

will lead to successful work without the blocks. When the representations

and expectations are internalized, the actual blocks become unnecessary.

Predicting the Total

As a demonstration, present the example 215 + 362. Have volunteers repre-sent the numbers on the Counter and table and record the example on thewhiteboard. Then say,

Set the dials to tell how many there will be when these two groups are combined.

Once the dials are set, students cover them and then combine the blocks.(Covering the dials keeps the numbers from distracting the students.) Aftercombining the two groups on the Counter, the students uncover the dials tocheck their predictions.

Repeat with an example that requires regrouping, such as 434 + 128. For thisexample, a student might first set the dial for the blocks-of-10 at 5, thenchange that dial to 6 after looking more closely at the single blocks.

Note that when predictions are checked and found to be incorrect, the stu-dents can simply reset the dials to show the actual number of blocks on theCounter. Encourage students to realize that we all find predictions difficult

to make, but that we get better with practice. If you notice consistent errors,however, you might encourage the students to reflect on discrepancies bysaying, for example,

I see that your dial for the blocks-of-10 is 1 less than the number of blocks. Why do you

think that happened?

Students follow a similar procedure when working with the Place mats.Present an example such as 235 + 123. Tell the students to stop once theyhave represented the numbers and recorded the example on paper. Then say,

Set the Digit Flip Cards to tell how many there will be when you combine these two groups.

Focus Predicting the outcome of joining groups of

blocks, and finding sums without the blocks


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After setting the flip cards, studentsturn them face down and then combinethe blocks on the Place mat. Studentsturn the cards back over to check

their predictions.

Repeat with an example that requiresregrouping, such as 334 + 229. Whenregrouping is involved, students mayfind it helpful to record the total numberof blocks in each column at the top ofthe mat. Looking at these numbers canhelp students to predict how the blockswill look when they are packed.

The image of blocks in a holder oftenmakes it easier for students to predicttotals in a column. Thus, you might have students place the blocks in holderswhen working on the Place mats.

Students need repeated opportunities to predict the outcome of joining twogroups with the blocks in view. Lack of attention to this step often leads todifficulty with the development of mental computation strategies and withthe transfer to paper-and-pencil techniques.

Developing Recording Techniques

To support computation with paper and pencil, encourage students torecord their work as they combine the blocks. You can either guide themto record in the conventional manner or allow students to develop theirown recording schemes. Either way, there should be a close associationbetween the written work and the physical actions with the blocks. It canhelp to ask questions such as,

How can you show the new block-of-100 on the written example?

Activity Sheet 3, the Three-Place Recording Sheet, may help students tokeep track of the appropriate place for each digit while they are working. It isnot necessary, however, that all students depict the combining process in thesame manner. In fact, variations promote interesting discussions that can leadto deeper understanding as well as efficient recording techniques. Whatsimportant is that students use recordings that are meaningful to them and thatconnect to the physical models.

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For adding on a Place mat,predicted totals on the Digit FlipCards are turned face down while the students combine

and pack blocks.


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Following are two examples of students different recordings for the sameproblem. The first is from a student who uses lined paper turned on edge andrecords the sum in each column, working right to left, and then regroups in asecond step. The second example is from a student who works left to right,

self-correcting as she discovers the need to regroup.

2 5 6 2 5 6

+ 3 9 + 3 9

2 8 15 2 8 5

2 9 5 2 9 5

When students are comfortable with their recordings for a particular combi-nation, have them explain their work. Then you can present a traditional

recording and ask,

Who can figure out what this person was thinking?

Encourage the students to talk about what the recordings mean and whichtechniques might be easier. For example some students might note thatwith traditional techniques, they dont have to revisit any digits. Over time,students can adopt conventional methods or develop their own reliable andefficient recording schemes.

Working Without the Blocks

When students are able to consistently predict what will happen when com-bining groups of blocks and are able to record their work, they can try addingwithout the blocks. Present a written example in vertical form and ask studentsto find the sum using paper and pencil or mental computation. Over time,students should also use such techniques to solve story problems and to findsums when examples are presented horizontally as well as vertically.

Even when working without blocks, these tools should still be available toeveryone. Students may want to use the blocks to check their thinking or toexplore a more challenging example. As students begin to work on paper, itis important that they continue to use their number sense and their mentalimage of the blocks to judge the correctness of their results. In fact, somestudents, while working on paper and pencil, move their hands as if they werepacking and moving blocks.

Estimation is also helpful here. For example, you might present the combina-tion 368 + 221 and ask,

Do you think the sum will be more or less than 500? more or less than 600? Why?

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You can introduce estimation of sums as the process of reporting only thelargest blocks to tell about how many there are. You might also presentan example such as 372 + 289, which will likely prompt a few students toconsider rounding before estimating. Other students may reason, There are

5 hundreds, but I know there are more than 10 tens, so Ill say 600. Again,encourage students to share their thinking and to maintain their ability towork f lexibly with numbers.

Finally, present students with a challenge. Ask them to imagine bigger andbigger blocks, and Counters or Place mats with more and more columns. Ask,

Suppose we were adding two very large groups of blocks, with some blocks in every col-

umn. What would we do? (Combine the blocks, packing when necessary.)

Provide an addition problem with five- or six-digit numbers and ask,

How would you add these numbers?

While use of calculators is recommended for addition of large numbers,particularly when many addends are involved, it is important that studentsbe able to generalize. We want them to realize that they can add any twonumbers they are given, by operating in each column in exactly the same way.

Practicing Key Ideas

Predict on a Counter

Students work in pairs with one Counter.The first student loads a collection of blocks

on the Counter and uses the whiteboard to record the number

in each place.The second student then writes a number underneath the first and

sets out the corresponding blocks on the table. Students work together to predict

the total number when the two groups are combined. They set the dials to record

their prediction and then cover them.Next the students join the blocks on the

Counter and uncover the dials to check their predicted answer.Students can repeat

this activity many times, reversing roles.

Predict on a Mat

Students work in pairs with one Place mat.They lay a piece of string horizontallyacross the middle of the mat.The first student writes an addition example,and then

represents the first addend by placing blocks above the string.The second student

represents the second addend with blocks placed below the string.

Students then work together to predict the total number of blocks when the two

groups are combined.They set the Digit Flip Cards to represent their prediction.Then

they turn the cards face down,remove the string,and combine the blocks, packing

when needed. Finally they look again at the flip cards to check their predicted answer.

They repeat the activity, taking turns providing the addition example.

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Whats Missing?

Present a vertical addition example with the sum given but with one

of the addends missing. Students first predict the missing number

and then use the blocks to check.

Assessing Learning

1. Ask the student to represent 359 + 124 on the Place mat. Say,

Before you put these blocks together, set the Digit Flip Cards to show how many

blocks you think there will be. Tell me how you decide.After setting the cards, the student should perform the task physically tocheck. Does the student predict the correct total? self-correct, if necessary? clearly explain his or her thinking?

2. Present a written example such as 176 + 253 (in vertical form) and ask thestudent to find the sum without using the blocks. Have the studentexplain his or her technique. Does the student find the correct total? clearly explain his or her thinking?

3. Present 735 + 273 (in vertical form) and ask,

Do you think the sum will be more or less than 900? more or less than 1000?

Why do you think so?Does the student answer correctly? reason correctly? clearly explain his or her thinking?

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Finding Sums