Instructional Sequence for Teaching Students Place Value, Addition and Subtraction to 1000.
Main Idea
Place Value is a means by which students understand the value of the numbers in a sequence, so when they are added or subtracted, the quantity is understood. One benefit of understanding numbers as quantity is that students will have a sense of the approximate value of the solution to the problem before performing the algorithm. Once the students have completed the calculation, they’ll know if the answer makes sense.
The goal of this instructional sequence is to move students to the point where they can perform the standard algorithm but also have a sense of the numbers and the quantity they represent. During this sequence, students will also learn to utilize this sense of quantity so that the techniques of borrowing and regrouping become less mechanical and attain meaning.
Starting Point
The introduction of this sequence is intended for a third grade curriculum in which the students have already developed a sense of patterns and partitioning. This would include the development of flexible finger patterns, imagined spatial patterns and conceptual partitioning of collections up to 20 items. Through prior lessons involving arithmetic racks, students should have moved beyond counting by one to the use of thinking strategies like doubling and going through ten.
The social norms for whole-class discussions should have already been established to include explaining and justifying solutions, attempting to make sense of explanations given by others, indicating agreement and disagreement, and questioning alternatives in situations where a conflict in interpretations or solutions are apparent.
Phase One: Counting & Creating Arrangements of Thousands, Hundreds, Tens and Ones into Cases, Boxes and Rolls
During this phase, students are introduced to the scenario of a Candy Factory that presently keeps track of the large number of candies it produces by counting each candy individually. Students are encouraged to think of easier ways of counting the factory’s inventory through tasks geared to grouping by tens, hundreds and thousands. The goal of this phase is for students to be able to understand the quantity each number represents in a base-ten place value system.
Task 1
The first task will be done as a whole class. The purpose is to develop the Candy Factory scenario and introduce the idea of rolls candy (10 in each) by engaging students in imagining an easier way to count single candies, thus supporting the sociomathematical norm of efficiency.
Students are encourage to share their ideas of an easier way of counting (i.e. 2,5,10) in order to introduce the idea of rolls as the method by which the Candy Factory decided to begin counting.
Possible Discussion Questions
Pairs of students are given bags of Unifix cubes as substitutes for candies and asked to estimate how many rolls of candy they can make. Estimates are posted on the board for later comparison. The counting of each bag of candy is then done by putting them into rolls of ten and results are discussed in light of predictions.
Possible Discussion Questions
Once students are able to envision the results of packing, a notation system is introduced involving drawing squares to signify boxes, short vertical lines to signify rolls and small circles to signify individual candies.
Anticipated Student Thinking/Common Misconceptions
During this task the students may initially suggest counting in pairs or by fives, since they are smaller numbers and probably more familiar to them. Upon questioning whether anyone else could think of an easier way to count more, expect that students would eventually suggest the idea of ten, perhaps even 100. Upon hearing 10, it should be confirmed that the Candy Factory did indeed decide to package candies in groups of ten. Many will suggest the ease with which to count by 10, while others may claim that they are not all that familiar or comfortable counting by 10. Re-teaching of counting by tens may need to take place to enable students to understanding the later tasks.
Task 2
This discussion/simulation is designed to support student’s development of imagery for packing and unpacking in the setting of the candy factory and understanding the quantity different packaging represents.
The idea is introduced that if the Candy Factory had to count all the rolls the class had counted, it would take a real long time. Discussion should lead to an easier way of counting rolls — packing 10 rolls into one box and the number of candies that would represent and counting cases — packing 10 boxes into one case and the number of candies that would represent. Cases, boxes, rolls need to be visualized as composite units that could be counted and composites composed of ten smaller units. For example, a box is a countable unit that was itself composed of one hundred candies. Actual rolls, boxes and cases of Unifix cubes are used to illustrate the packing of candies into these units. The composite units and their values are discussed.
Possible Discussion Questions
To expand upon the idea of what each unit represents in terms of value, students would participate by packing up the candies (Unifix cubes) they have (all like quantities), any way they like (into rolls, boxes, cases) and class discussion would center around the number of rolls or candies they each had. Teacher notation about the discussion should utilize the previously introduced squares, lines, and circles.
Possible Discussion Questions
Anticipated Student Thinking/Common Misconceptions
When students are sharing like quantities packaged differently, some students may argue the fact that different packaging means different quantities. By asking the students to verify the number of candies represented in a packaging example, it will help to clear up any misconceptions.
Task 3
This task is designed to move students beyond thinking concretely (use of Unifix cubes) with the solitary use of the notation system of squares, lines and circles. It also provides additional practice in counting by groups and understanding the quantity involved.
The teacher flashes a variety of configurations of single candies ( multiple rows and columns) on the overhead, indicating that if each circle represents a candy, students
should estimate how many rolls they see. Class discussion should center around how students went about estimating and how hard it was to do in such a short period of time, given the way the configurations were laid out. The overhead images are then expanded to include pictures of rolls. Students are asked to estimate the groups of candies now represented and through this manner learn to count groups through the counting of lines or rolls. Discussion centers around how a greater number of candies were counted simply because of the way in which they were packaged. During discussion, all overheads and notations made by the teacher are visible.
Possible Discussion Questions
Anticipated Student Thinking/Common Misconceptions
During this task, students may not assign the proper quantities to the pictures on the overhead. They may not be entirely comfortable yet with the notation system to be able to estimate from a quick flash on the overhead. During class discussion, when the overhead is in full view should serve to clarify how students reasoned their estimates. Established classroom norms of indicating agreement or disagreement and questioning alternative solutions should serve to pinpoint any students having difficulty with this task.
Assessment
Much of the assessment of understanding during this phase can be done through observations of class discussions. It is recommended that students be given a series of problems relating to the task that they either work on with a partner or individually to reinforce any learning that took place. Problems should also be assigned to determine if that particular student understands the concepts being discussed.
The key understandings that need to be learned are built up through each new task. Instruction should not move to a new task until student understanding has taken place at the current task. These understandings are as follows:
Task 1
Understanding of the Candy Factory scenario and efficient ways of counting through the method of packaging candies into larger quantities.
Task 2
Understanding of packing and unpacking within the scenario of the Candy Factory and the quantity represented.
Task 3
Understanding of the notation system (squares, lines, circles) and the quantity represented by such notation.
Phase Two: Transforming Quantities by Packing and Unpacking
During this phase, students find different arrangements that contain that same number of candies, but they are also able to create new arrangements by transforming previous ones without the need to explicitly justify why the new arrangement contained the same number of candies. Here, the understanding is generated that transformations conserve quantity. Students are moved further beyond the concrete through the introduction of an inventory form showing the number of boxes, rolls and pieces in the storeroom.
Task 1
During this task, students gain a greater understanding of quantity by finding different arrangements (packaging) that contain the same number of candies and/or take a single arrangement and find different ways of packaging it. A series of overheads are used, with pictures of squares, rolls and circles, from which the students are first asked to draw additional ways they could have the same number of candies and then later asked to identify like quantities among a series of differently arranged/packaged pictures. Class discussion should help them understand that the number of candies are the same, they are simply packaged differently.
It is during this task that a new rule should be introduced: All candies should always be packed up (if they count to 10) and cases, boxes or rolls cannot be partially unpacked (once they are opened, they need to be completely unpacked and counted).
This is to discourage students from borrowing without reducing the number, when they eventually get to subtraction problems.
Possible Questions
Anticipated Student Thinking/Common Misconceptions
Students may not see at first the fact that the different pictures actually represent the same number of candies. In order to clarify, it would be necessary to introduce the rule of packing and unpacking completely.
Task 2
During this task, students are exposed to the inventory form representing numerical quantities for boxes, rolls and pieces instead of the notation system of squares, lines, and circles. It is explained to the students that the Candy Factory found it very time consuming to continue to draw the inventory in the storeroom, so they moved to the form in order to make it easier.
Students are taken through a series of exercises where they are asked to discuss different ways that the candies indicated on the inventory form could be arranged. The problems present scenarios in which the students will pack up or unpack arrangement to determine if like inventory forms exist. This serves to create a taken-as-shared view that packing and unpacking candies are ways of creating different numerically equivalent arrangements of candies.
Possible Questions
Anticipated Student Thinking/Common Misconceptions
Students may not see at first, the relationship between the notation system and the columns in the inventory form. It would be necessary for the teacher to ensure the transition from pictures to the form is understood. If needed, pictures could continue to be drawn alongside the numbers to illustrate the representation. Also, students, having been used to unpacking or packing a single representation at a time, may need assistance in making the connection to the effect of unpacking or packing on all three columns in succession.
Assessment
Much of the assessment of understanding during this phase as well can be done through observations of class discussions. It is recommended that students be given a series of problems relating to the task that they either work on with a partner or individually to reinforce any learning that took place. Problems should also be assigned to determine if that particular student understands the concepts being discussed.
The key understandings that need to be learned are built up through each new task. Instruction should not move to a new task until student understanding has taken place at the current task. These understandings are as follows:
Task 1
Understanding of quantity. What each arrangement of pictures represents and that different arrangements can be created that represent the same quantity.
Task 2
Understanding of a taken-as-shared view that packing and unpacking candies are ways of creating different numerically equivalent arrangements of candies.
Phase Three: Adding and Subtracting by Packing and Unpacking
The major step taken from Phase 2 to this Phase is that the students move from dealing with a single collection of candies, to dealing with multiple collections of candies. In earlier Phases, students compared different collections of candies which were of equal quantity and different appearance. Beginning in this phase, students will begin to manipulate collections of candies which are of different quantity in order to simulate transactions being filled.
Task 1
This task involves the teacher introducing the idea that transactions involving quantities of candies are occurring and that candies are being brought to the storeroom and also being delivered to customers. In this task, students engage in role-playing scenarios in which candies are either brought to the storeroom or are sent to customers. The initial transaction problems are stated as follows:
There are 4 boxes, 3rolls, and 5 pieces in the storeroom. A customer places an order for 1 box, 4 rolls, and 3 pieces. How many candies are left in the storeroom?
or for addition/packing
There are 4 boxes, 3rolls, and 5 pieces in the storeroom. 1 box, 4 rolls, and 3 pieces are brought to the storeroom. How many candies are now in the storeroom?
The students will rely on skills, learned in Phase 2, involving transforming quantities by packing and unpacking to solve these problems. The students may, during this Task, draw pictures or use other notations to aid in solving problems and the teacher may initially present problems to the students with accompanying pictures of the storeroom. One result of the students’ drawing pictures may be that they continue to discuss the transactions in terms of actions performed on boxes, rolls, and pieces.
Good problems for this Task would include any which require the students to unpack (for subtraction) or pack (for addition) candies in order to complete the transactions.
Possible Discussion Questions
As the students are solving problems like the ones above, the teacher has to focus attention on their reasoning for each steps that they make. Ask questions about why they are unpacking boxes and rolls, packing candies into rolls, or rolls into boxes. During this task, keep students focused on the situation of the Candy Store and also the terminology which is used with that scenario. Students should explain their answers using terms such as packing/unpacking, boxes, rolls, and candies.
Anticipated Student Thinking/Common Misconceptions
There are two ways in which students will solve problems like the one above and they depend greatly upon the students’ interpretation of the problem. In the first way, students interpret the customer’s order as a quantity of 143 individual candies. Students that interpret the order in this way may be viewing 1 box, 4 rolls, and 3 pieces as three separate quantities which have to be taken as individual candies in order to be manipulated and understood as a single quantity. In other words, they will interpret the customer’s order as 100 candies plus 40 candies plus 3 candies. These students will unpack 2 of the boxes in the storeroom in order to send these 143 candies to the customer because they see the boxes in the storeroom as 100 candies.
The other way students might interpret the customer’s order is as a quantity of 143 candies structured into units of one hundred (boxes), ten (rolls), and three (individual candies). It is important that the teacher guide the students toward the second interpretation so that their methods for solving such problems will be more easily adapted to similar problems once standard notation is adopted. Note, however, that the first interpretation is not incorrect, simply different — as valid as the second interpretation.
The issue of multiple interpretations is not likely to surface when dealing with addition/packing problems. In these problems, students should understand by now that when there are more than ten individual pieces in the storeroom, a roll is packed and when there are more than ten rolls in the storeroom, a box is packed.
Assessment
Assessment should be continuous and ongoing throughout this Task. Assessment should be focused on the process by which the students solve the given problems. Special attention should be given to determining which students interpret the boxes, rolls, and candies as separate quantities and which students interpret them as a single, structured quantity.
Task 2
The delineation between Tasks 1 and 2 is subtle, and these tasks could conceivably be taken as two pieces of a whole. Task 2 is nearly identical to Task 1, but contains one important change in process. During this Task, students participate in discussions of problems of the sort used in Task 1, but with the focus being on finding the most efficient way of completing transactions. The problems are rephrased during this Phase so that the example problem from Task 1 would be written as:
There are 435 candies in the storeroom. A customer places an order for 143 candies. How many candies are in the storeroom now?
or for addition/packing
There are 435 candies in the storeroom. 143 candies are brought to the storeroom. How many candies are now in the storeroom?
Also during this Task, the teacher reintroduces the Inventory form from Phase 2 with the problems the students must solve, written on the forms. Using the same sample problem as in Task 1:
By focusing on the most efficient method for completing the transactions, students should discover the necessity of unpacking only the necessary boxes or rolls, as opposed to the first interpretation given in Task 1, since the inventory form makes clear the structured units of boxes, rolls, and pieces. At the same time, students should develop a better sense of when they will need to unpack boxes or rolls in order to solve the problems. In this way, students should begin to anticipate the necessity of unpacking.
Possible Discussion Questions
Questions posed during this Task should be of the sort found in Task 1. Also in this Task, the teacher should emphasize the use of the inventory form and begin to wean students from using pictures of the storeroom to solve problems. Some students may attempt to use more calculational terms when explaining their solutions. This is acceptable so long as the students are clear in their explanations and both the teacher and the class can follow the calculational explanation. If some of the other students become confused, a reversion to conceptual explanations, using terms from the Candy Shop scenario, may be required. One technique which might be effective at checking students’ understanding would be to have one student solve a problem on the board, and then have another student explain the solution to the class. The first student can then confirm the explanation or re-explain it if the first explanation was not accurate.
Another technique, useful for checking to see if students can anticipate the need to unpack rolls and/or boxes might be to flash a problem on the overhead projector for a few seconds only.
For example:
Then ask students if they believe that unpacking will be necessary in order to solve the problem. The same can be done for addition/packing problems.
Anticipated Student Thinking/Common Misconceptions
First, it is possible and likely that some students will try to resort to using pictures of the storeroom and the candies moving in and out in order to solve these problems. Teachers must try to get the students to use only the inventory form. In Task 1, it was noted that some students would interpret a customer’s order as being separate quantities. In the example given in Task 1, these students would see 1 box, 4 rolls, and 3 candies as separate quantities of 100, 40, and 3 candies respectfully. These students may have difficulty making the transition from pictures to the inventory form and numbers. When using the pictures, the students see their activity as actions involving individual collections of candies; the numbers on the inventory form are just a means of keeping track of those actions. Other students — those that were able to see the boxes, rolls, and candies as structured units in Task 1 will be able to see the numbers on the inventory form as a means of eliminating the need for the pictures.
Assessment
Assessment should be a continuous process during Task 2. Teachers should look to make sure that students understand the mechanics of partitioning and recombining collections of candies using the inventory form. Teachers should also look for students’ anticipation of the need to unpack boxes and/or rolls during the course of a solution. Teachers should also check students to determine if they are able to view the collections of candies in terms of structured units or not.
Phase Four: Adding and Subtracting Structured Quantities
There are two significant conceptual transitions made between Phases 3 and 4. The first is that students will no longer be using pictures and numbers to describe a sequence of actions and transactions involving boxes, rolls, and candies. The students will come to use numbers to act symbolically on quantities structured into units of one hundred, ten, and one. The other shift that will be noticed involves the discussions which take place. The focus of class discourse should shift away from the topic of how to enact a transaction to that of how to notate transactions more effectively. This second shift should occur naturally as students distance themselves from the use of pictures and rely more on simple numbers. It should also be noted that some students may not shift away from the use of pictures as readily as others. This is to be expected. These students will need more time and practice to make this shift.
Task 1
In this Task, students will begin to use standard mathematical notation as well as more calculational explanations for their solutions. Continuing with the same sample problem, what had been read as
There are 4 boxes, 3rolls, and 5 pieces in the storeroom. A customer places an
order for 1 box, 4 rolls, and 3 pieces. How many candies are left in the storeroom?
and then
There are 435 candies in the storeroom. A customer places an order for 143 candies. How many candies are in the storeroom now?
is now read and written in the standard form of
The same is true for the addition/packing problems. What was read as
There are 4 boxes, 3rolls, and 5 pieces in the storeroom. 1 box, 4 rolls, and 3 pieces are brought to the storeroom. How many candies are now in the storeroom?
and then
There are 435 candies in the storeroom. 143 candies are brought to the storeroom. How many candies are now in the storeroom?
is now read and written in the standard form of
For this Task then, typical textbook style problems can be used.
Possible Discussion Questions
It is very important that teachers have a clear picture of their students’ reasoning as they solve problems. In this final phase, the students reach the point where the problems they solve could have come from any math textbook that a student sees in school. The difference is that these students should have developed a sense of quantity through their work with the Candy Shop sequence. It is imperative that during this final Phase, teachers check for their students’ clear reasoning and number sense.
At this point teachers should try to engage students in discussion about their notation as opposed to the way in which they went about enacting transactions. The discussion should, as often as possible, be calculational in nature. If some students need conceptual explanations in order to keep up with the discussion, then they should be provided with, or allowed to work through those explanations.
Anticipated Student Thinking/Common Misconceptions
The biggest surprise for teachers using the Candy Shop sequence might be that as students are nearing the end of the sequence (or after it is finished) they begin to talk about the math involved during that sequence and math typically practiced in the classroom as two different things. Some students will make the transition from this sequence on talking about the candy metaphor and pictures as a means of justifying their solutions easily. These students will have gained a sense of quantity that will allow them to understand that the standard notation and the metaphor of the Candy shop involve the same "imagery of partitioning and recombining composite collections." Other students will continue to refer to the pictures to perform the equations given to them — counting the pictures and using the numbers to make a record of that counting.
Assessment Activities
Again, assessment should be formative. The teacher should be looking for clear and logical reasoning to support a student’s answers and not whether or not the answer is correct. As noted above, some students will be able to transcend the Candy Shop metaphor and utilize their understanding of quantity to understand place value and the mechanics of addition and subtraction in other situations.
** All information based on the following articles:
The Evolution of Mathematical Practices: A Case Study by Janet Bowers, Paul Cobb, and Kay McClain
Mathematizing and Symbolizing: The Emergence of Chains of
Signification in One First Grade Classroom by Paul Cobb, Koeno
Gravemeijer, Erna Yackel, Kay McClain, and Joy Whitenack