# Mathematics of Change: Ins and Outs or Ups and Downs

Cornelia C. Tierney
Ricardo Nemirovsky

Algebraic thinking encourages examination or conceptualization of number relationships in general. It is a way of going from specific thinking to general cases. -- Carlyle and Moses 1995

We describe here an activity with variations useful throughout the elementary school grades that involve children using small numbers to do a sequence of additions and subtractions that we call "changes." For example, start with 5 cubes in a bag and do three changes: put in 2, take out 1, take out 2; how many cubes are in the bag at the end? Through combining two or more changes, students think about relationships in addition and subtraction. We use small numbers so that the students can focus on the relationships without struggling with the computation. We see algebraic thinking emerge in children's ideas about the general properties of changes, such as those in figure 1.

 Some General Ideas Students Develop The net change, or total change, of several changes does not depend on the order of the changes or the number in the bag at the start. The changes +2 -4 +3 have the same total change as +2 +3 -4. No matter what the starting number, in both these cases the end amount will be one more than the start. Pairs of opposites, such as +3 and -3, have a net change of zero and so can be eliminated from the calculations. A net change can be made by using any number of changes, meaning that it is possible to go from any number to any other number with any number of changes. Positive net changes can be made with some negative changes, and negative net changes can be made with some positive changes.

## In and Out of the Bag

For this basic activity, you will need an opaque container and some small blocks that fit in the container. We use a small paper bag and connecting cubes. Students need to sit where they can see and count the blocks you or another leader are putting in the bag. They may use pencil and paper to keep track of the blocks. Some students will not record at all but will mentally keep a running total, which is fine for this version of the game. Playing the backward game described later will more likely necessitate a recording system for keeping track.

Start with a problem with only a few changes and with one change canceling another, such as this: Start with 3 blocks in the bag. Put in 1, put in 2 more, and take out 2. Tell students to watch closely so that they can figure out how many blocks are in the bag. To act out this problem, start with 3 blocks in the bag and let students know how many are there. You might shake the bag so that they can hear the three blocks rattling around or ask a student to come up and look to check that 3 are there, or even to prepare the 3 in the bag for you. Tell students that you will make changes in the number of blocks in the bag by putting some blocks in and taking some out. They may keep track with pencil and paper if they wish. While students are watching and without talking, pick up 1 block, show it to the students, and drop it in the bag. Allow time for students to record the change or to compute mentally. Then pick up 2 more blocks, show the 2 and drop those in. Exaggerate your motions to enable the students to see and remember what is happening. Finally, take 2 blocks out of the bag, show them to the students, and put them down on the table.

Ask students how many blocks are in the bag and how they know. Students may report a variety of strategies:

• Writing tallies to show the blocks put in and crossing out those taken out (||| | ||)
• Mentally computing or writing down the total number in the bag after each change (3, 4, 6, 4).
• Writing each change with plus and minus signs or another symbol to show putting in and taking out (3 + 1 + 2 -2) and figuring out the total at the end. We have seen young children write "p" for "put in" and "t" for "take out" as in 3p1 p2 t2. See figure 2 for an example by six-year-old Maya before she entered first grade.

The problem posed to Maya was "Go from 4 in the bag to 7 in the bag using exactly four changes." She had a strip of 4 connecting cubes in her hand to which she added 2 and then 1, making 7 in just two changes. She wrote down these changes. Trying to think of two more changes that would leave the total at 7, she took off 1 cubes, counted the 6 cubes, and then put the cube back on. When she realized that she had solved the problem, she drew her hand taking off and putting on a cube.

• Writing down and computing each step. Many children will write as they would tell the story, for example, "Start with 3, put in 1, that's 4; put in 2 more, that's 6; take out 2; so there are 4 in the bag" and, in doing so, misuse equals signs: 3 + 1 = 4 + 2 = 6 -2 = 4. Do not correct students' notes for their own record keeping, but suggest rewriting as separate equations when the procedure is written for others to see: 3 + 1 = 4, 4 + 2 = 6, and 6 - 2 = 4.

Students may notice that 2 blocks are put in and taken out again in this problem, so those do not need to be counted. After a few students have shared their procedures and agreed to an answer, empty out the bag for all to see how many blocks remained.

Try another similar problem, perhaps with more changes: Start with 4 in the bag, put in 1, put in 2, take out 3, take out 1, put in 3.

We have found that young children remain challenged by these problems and enjoy taking turns being the leader who makes up and acts out the problems with the container and blocks. However, most third and fourth graders and older students soon develop strategies so that these straightforward problems become quite routine. You may want to continue to do them from time to time, asking students to solve them mentally. When students have not had a chance to record a problem on paper, precede the discussion of solution strategies by asking a student volunteer to review the sequence of changes while you write them on the chalkboard. For example, for the second problem discussed previously, the student would sayyour actions, "Start with 4, put in 1, put in 2, take out 3,...," while you write4 +1 +2 -3 . . . . This procedure allows students to participate in the discussion of strategies without having to remember all the numbers.

Activities with larger numbers
Pose these problems for students to do as practice with larger numbers, adding and subtracting tens as well as ones. To include tens, use a larger container -- perhaps an opaque plastic box -- and connecting cubes combined in lengths of ten as well as single cubes. Start with 38, subtract 20, add 3, subtract 10. You can expand to larger numbers with hundreds assembled from snap cubes or with base-ten blocks.

Activities with coins
Instead of putting blocks in the bag, you can use coins. Start with a small amount of money in the bag. Tell students the number of each coin you have in the bag, or, for variation, tell them the amount of money but not the exact coins. Put in and take out coins -- one, two, or three at a time of nickels, dimes, and quarters -- showing students the coins each time. For more complicated sums, use pennies as well. After a number of changes, ask students how much money you have in the bag. A sample problem is to start with 55 cents, put in 2 dimes, take out 1 quarter, put in 3 nickels.

## Working-backward Variation

For the working-backward variation, do not tell what number you put in the bag to start with; that number is what students are to figure out. Put the same number of blocks in each of two bags and put one of the bags aside so that you can show students the starting number later. Tell the students what you are doing. These problems are more difficult; limit them to only two or three changes at first so that students can keep the whole problem in mind.

Here is a description of a problem. Start with 7 blocks in the bag but do not show the students. Take 5 blocks out of the bag and put 2 blocks in, showing students what you are doing. Where students can see, perhaps on an overhead projector, pour out all the blocks that are left in the bag (4). As you did with the working-forward problems, allow students time to write down and work out this problem for themselves. Once they have established their own representations, you may want to record each problem as a question mark or a blank in which a number can be written, followed by a series of positive and negative changes, that is, additions and subtractions. This problem would be ? -5 + 2 = 4

After writing the problem but before taking answers, ask: "At the end do you have more or fewer than when you started? How do you know?" You might introduce the idea of net change, which is the total of all the changes done. It is the one change that could substitute for all the changes. In this problem, it is -3 because you take out three more than you put in.

? - 5 +2 = 4

? - 3 = 4

Many students initially do these problems in a way that seems logical but does not work. They work from right to left using the signs as they are: 4 plus 2 minus 5 is 1 (4 + 2 - 5 = 1) so 1 was in the bag at the beginning. If they check this answer, they will see that starting with only 1 is impossible, as you take away more than are there, and 1 - 5 + 2 results in - 2, not +4. Students will find that other strategies give better results: see figure 3 for an explanation by a fourth-grade student who was asked to figure out this problem. The successful strategies that we have seen students use most frequently are (1) try and adjust, (2) undo the net change, and (3) undo from end to beginning.

Try and adjust. Students guess an initial number, add and subtract through all the changes, and check to see if the end numbers match. If the numbers do not match, some students adjust the initial number up or down to compensate. Others try guessing initial numbers randomly. (See fig. 4.)

Undo net change. Some students find the total in and total out and compute the net change. For this problem they would note that the net change is -3, so they must have started with three more than they finished with; to finish with 4, they must have started with 7. By this method, students can manage problems with many changes. (See fig. 3.)

Undo from end to beginning. Some students imagine the story and find the number by starting with the number in the bag at the end and working backward: ? - 5 + 2 = 4. Four items were in the bag at the end. Two were added to get 4, so you must have had 2 before that. Before you had 2, you took out 5; 2 and 5 is 7, so you started with 7. As a shortcut, this sentence can be written from end to beginning, reversing the signs: 4 -2 + 5 =7. (See fig. 5.)

Whatever way they figure out the starting number, most children check their guesses by repeating the story, as shown in figure 6.

Working backward, or undoing, is an important problem-solving strategy for children to develop. We tend to be comfortable going forward, as in adding and multiplying. To understand mathematics relationships, we need to be able to work in both directions.

## Providing Missing Changes

For individual or small-group work, you might challenge students to find ways to use one change, two changes, three changes, and so on, to go from a starting number of blocks in the bag to another number. For example, going from 4 to 7 in one, two, or three or more changes can be done in the following ways:

• One change: 4 + 3 = 7
• Two changes: 4 +1 + 2 = 7
• Three changes: 4 +1 + 1 + 1 = 7
• Four changes: 4 - 1 + 1 + 1 +2 = 7
• Five changes: 4 - 1 - 1 +1 +1 +3 = 7

Children who become facile with this task create patterns in developing one set of changes from another. For example, in this challenge, to go from three to four to five changes, we are making a change of -1 and compensating by adding + 2 and then +3 at the end instead of +1. However, not enough blocks are in the bag at the beginning to continue this pattern of subtracting one more: 4 - 1 - 1 - 1 + 1 + 1 + 4. One could reorder the changes, putting the blocks in first and taking them out later. For six changes, 4 + 4 + 1 + 1 - 1 - 1 - 1 = 7. For seven changes, 4 + 5 + 1 + 1 - 1 - 1 - 1- 1 = 7, and so on.

A six-year-old thought that the only way to get from 4 to 7 was adding 4 + 1 + 2; 4 + 1 +1 +1. When asked for four changes he said 4 + 1 + 1 + 1 + 0; when asked to try not to use a zero, he suggested 4 +1 "and half and another half." With a hint to think about taking out some blocks, he was eager to do any number of changes with taking out as well as putting in.

## Story Problems

Present a story problem about the changing population of a place -- the number of people or animals, other living beings, or fantasy creatures. Adapt such problems to contexts and level of difficulty appropriate for your students.

I counted 14 starlings on a telephone wire. Some flew away when a car went by. Three came back. Then I counted 10. How many flew away when the car went by?

When my neighbor, Sonia, and I got on the bus this morning, it looked more crowded than usual, but we both got seats. Three people got on at the next stop and 4 at the stop after that. At the drugstore corner, 5 kids got on and 2 adults got off. At city hall, 6 people got off and none got on. When we got to the school stop, all of the 38 passenger seats in the bus were full and 2 people were standing. How many people were already on the bus when Sonia and I got on?

To make story problems manageable for students for whom reading English is difficult, show the written problem on the overhead projector or written large on chart paper and pair these students with others who can reread the problem aloud to them as needed.

Students can invent problems of this kind. They make up a story of comings and goings and then leave out or cover up one of the numbers and pose a question about it. Keep a collection of problems that pairs of students can do in free time or that you can use with the whole class.

## Moving on a Number Line

So far, we have described the activity as one for a whole group with the teacher or a student acting as leader. For games for small groups of students working for a longer time, you can provide number lines or numbered tracks and cards with "changes" numbers. In a unit for third grade called "Up and Down the Number Line," we use a vertical number line that we describe as the elevator shaft in a skyscraper that has infinite floors above and below the ground. The ground floor is labeled zero. The buttons that you "push" in the elevator denote how far you will move up or down rather than at what floor you will arrive. We provide decks of 28 changes cards, with four of each number, from -3 to +3. Activities that are suggested include these:

• Find the net result of a large number of changes. Each child writes a change on a sticky note. When all changes are arranged in a line on the wall, the students working in groups figure the net change and explain how they are sure of their answer.
• Find a variety of ways to go from one floor to another to make a given net change.
• Arrange the order of a randomly drawn group of changes to stop on as many different floors as possible. When specifying -2 +2 +1 -3 produces stops at only two floors other than the starting floor; with the changes arranged in a different order, four floors different from the start can be reached.
• Make a graph or diagram to show an elevator trip of several changes for someone else to guess your trip; use no words or abbreviations.

Having changes on separate cards allows students to move them around to plan combinations of changes for problem solutions. Moving on the number line provides a visual way to keep track of steps of a problem.

In solving arithmetic problems, it is useful to distinguish two areas of knowledge: (1) counting and computing, and (2) relations between numbers and operations. In doing the activities described in this article, students practice their skills in the first area while investigating the second. They make generalizations about such ideas as the commutativity of addition, the opposite effects of adding and taking away, and how to undo a problem by doing the opposite operation. Such activities can be done again and again and varied to fit children's needs and interests.

## Bibliography

Carlyle, Ann, and Barbara Moses. Session chaired at the NCTM annual meeting on brainstorming ideas for the Teaching Children Mathematics algebra focus issue. Boston, April 1995.

Tierney, Cornelia, Amy Shulman Weinberg, and Ricardo Nemirovsky. Up and Down the Number Line. Investigations in Number, Data, and Space Series. Palo Alto, Calif.: Dale Seymour Publications, 1995.

Tierney, Cornelia, Ricardo Nemirovsky, and Amy Shulman Weinberg. Changes Over Time. Investigations in Number, Data, and Space Series. Palo Alto, Calif.: Dale Seymour Publications, 1995.

Author Info:

The authors work at TERC, Cambridge, MA 02140, doing research on the learning of mathematics and the development of innovative curriculum materials. A classroom teacher for many years, Cornelia Tierney, cornelia_tierney@terc.edu, also works with teachers on implementing new curriculur materials. Ricardo Nemirovsky's other interest include the design of interactive devices that promote student's understanding of the mathematics of change.

Other Articles by Cornelia C. Tierney:

Other Articles by Ricardo Nemirovsky:

Children’s Spontaneous Representations of Changing Situations