Processes and Problems
Unit Focus 1: Base Ten Operations
Key concepts I want to explore initially with the students are first of all writing numbers in expanded form. Following that we will move on to the concept of making and unmaking tens, also known as composing and decomposing tens. I particularly want to address the addition and subtraction process within 20 as the pivotal process or key understanding that students need in order to understand subtraction with regrouping. (A corollary understanding here is that a two-digit number is not two numbers, but one number with two digits: it is the sum of so many tens and so many ones, the many being specified by the digits. Although it is a subtle difference, in order to make and unmake tens it is important for children to comprehend numbers as being able to have more than one digit. Indeed, this is the whole notion behind place value.) The thinking here is that if students can compose and decompose numbers within 20, it should reasonably follow that, with instruction and guidance, they will be able to perform these operations with 100's, 1000's, etc. The goal is for students to understand that exchanges can take place between higher value units without changing the value of the number. I will introduce the concept of rounding as it relates implicitly to writing numbers in expanded form, and explicitly to higher value units of tens. Lastly, I will introduce a mental subtraction exercise that gets children thinking about place value unit exchanges that don't change the value of the minuend and subtrahend and that also facilitate a kind of mental gymnastics that children can build on and use to increase their arithmetic prowess.
Lesson 1: I begin teaching base ten operations by introducing a problem in expanded form. My reasoning is that this process gets children thinking about place value, it lays groundwork for adding, and later subtracting, across zeros and my experience has shown that children enjoy performing this operation, so they tend to think positively about their own abilities to handle place value with increasingly large numbers. Perhaps most importantly, this reasonably simple operation affords me the opportunity to lay the groundwork for the classroom format and group dynamics we will use for this unit.
I begin by writing a four digit number on the board, for example: 8,765. Employing the dialectic method (question and answer format) students identify the place values of each digit and I rewrite the number as 8,765=8,000+700+60+5. Then I introduce the Problem of the Day: Write the number 5,678 in expanded form. Students work individually, or in pairs if they are more comfortable and rewrite the problem in expanded form as per the paper-folded model.
As students work, I walk around the room and note who understands the process, who doesn't and why. A rule I will employ, borrowed from Magdalene Lampert as described in her wonderful narrative description of her own mathematics teaching process in her book, Teaching Problems and the Problems of Teaching, is that no student gets to ask the teacher a question unless she or he has asked everyone in the problem-solving group and nobody knows the answer (Lampert, 2001, p.82). However, at this time, I allow and encourage questions as I circumambulate the classroom.
Bringing the children back to whole-class attention we discuss student solutions and some are written on the board. We come to consensus as to which processes and solutions are reasonable. Here I begin modeling how to make conjectures as students discuss and continue to work the problem. For example a conjecture might be that a number in the thousands place, in expanded form, is a number from 1 to 9 with three zeroes behind it. What conjecture could we make about a number in the hundreds place, using the expanded form schema? I continue to ask questions about each place value unit until we get to ones. What conjecture can we make about the number in the ones' place? The answer I am seeking is that it will be a number from 0 to 9 that stands alone. Or some version of the above. The emphasis here is on how to formulate a conjecture or conjectures.
Next, I have students draw four columns on their papers and label the columns from right to left: ones, tens, hundreds and thousands as I do the same on the board. Then I plug in the numbers from the original number example, 8,765, and we discuss the place value positions of each digit and its place value name. Students repeat the exercise on their four-quadrant paper this time using their Problem of the Day, 5,678. Again we come together for a whole-class discussion, find reasonable solutions and come to an agreement about our conjectures.
After this I will break the children into groups of three or four and they will tackle some more challenging problems: Write 50,678 and 500,678 in expanded form. Students will store their work in their Word Problems Notebook, and I will check them for any misunderstandings that need to be cleared up the following day.
The next day I will follow up by having a conversation about the relative size of numbers. For example: What is the relationship between 8,000 and 800? How about that between 8,000 and 80? What about 8,000 and 8? The object is to have the children be thinking about tens and higher value units of tens.
After discussion and whole class practice with other examples, I will introduce the concept of rounding as a strategy to help determine the reasonableness of answers to whole number computations. Again, much practice will follow, in not only rounding of numbers, but using rounding as a computational strategy. A note here is that I am not strictly attached to a time frame, and if it takes students longer than two days to absorb these base-ten concepts, I will extend the modeling, discussion and practice as need be.
Lesson 2: This is a lesson in Adding Base Ten Numbers by Place Value Components. The objectives in this lesson are to teach the commutative rule of addition, to introduce the concept of decomposing numbers of the same place value without changing their value, and in the process, to learn a non-standard method of addition. This method of adding was taught to me by Professor Roger Howe of Yale University as a way to teach the commutative and associative rules for addition under the child-friendly name of the Any Which Way Rule. I name the procedure as I introduce and discuss the problem of the day. The procedure demonstrates to children how they can break up numbers into their place value components in order to perform an arithmetic operation.
I write a problem on the board such as 24+35 = (20+30) + (4+5) = 50+9 = 59. Again using a question and answer format we discuss the concepts presented. First of all, the Any Which Way Rule allows us to decompose and regroup the numbers into bundles of tens and ones, add the numbers in their respective place value positions together and finally recombine the place value parts into a familiar base ten number we all recognize, know and love. I will encourage the children to come up with their own versions of why this works, i.e. make conjectures. Ultimately, the conjecture I'm looking for is some version that recognizes that we are breaking the numbers up into tens and ones and then adding them up. Next I write the problem on the board vertically and we solve it using the standard algorithm and compare the two processes. Then I give students a number of problems to solve using both methods and they break into their working groups to solve them. We reassemble whole-class after they have worked the problems and discuss their processes and revise our conjectures if necessary. Here a conjecture might be a definition of decomposing numbers, for example. Again students store their work to be collated later.
Lesson 3 will be an exploration of a process called equal additions which is a method demonstrating the principle of maintaining the values of numbers by adding the same number to each number in a subtraction problem in order to mentally subtract it. The problem: 369 - 199. Working on the board, I add 1 to 369 and 1 to 199 giving me the new problem 370 -200, pointing out I haven't changed the value of the difference between the two numbers. This is called a method of compensation. Students check the answer, 170, by using "traditional' means. Some fourth graders will be able to do this and I will guide the regrouping process. After a number of demonstration problems, I assign problems for children to work on their own. Later we will make some conjectures about why this works and post some examples around the room.
By now students should be ready to tackle some word problems in base-ten operations, using the four-quadrant page format they have been practicing.
Sample Problems of the Day Reinforcing Operating in Base Ten
In this section of the unit I will lead and encourage students in an exploration of place value to 1000's, identifying and using even and odd numbers and rounding numbers to nearest ten, hundred or thousand in order to estimate the answer to an addition or subtraction problem. Please keep in mind that these are only sample problems. I will determine when to move on in light of the speed of comprehension of individual students and the class as a whole. (Note: I have not delineated a specific lesson teaching odd and even number recognition due to the fact it is out of the scope of this unit; however, this concept will need to be introduced and practiced before doing the following problems.)
The first three problems are a related suite of numerical word problems designed not only for their arithmetic content but to familiarize students with reading arithmetic problems and determining what to do. I will probably introduce all three in the same class period, depending on the level of comprehension. The problems would also work well as variants on a theme to be presented to different working groups who would then report out to the whole class at the end of the period. I will model the first one as a whole-class exercise. It is important to articulate the clues that are given in each problem; therefore, students draw a clue chart on their papers (in quadrant two), as I model it on the board. When working in their groups I again require them to make a clue chart for the second step of their procedure.
1A.The ones' digit is five. The number is greater than 640 and less than 650. What is the number?
1B. Escribe el número par lo más grande que tiene un seis en el lugar de centenas y un 4 en el lugar de decenas.
1C. ¿Cuáles son los cinco números pares que tienen un seis en el lugar de centenas y un cuatro en la posición de decenas?
In the next suite of problems students practice writing numbers in expanded form. Again I model one or two or more of the first type (2A), then students will tackle 2B as a Problem of the Day. More problems and practice follow as time permits and comprehension requires.
2A. Write the number one thousand, six hundred and fifty in expanded form.
2B. Se vendió Pablo 1000 billetes para el juego de fútbol. Se vendió Marta 900 billetes. Jaime se ha vendido 80 billetes y Hermanita Lupita se vendió 2. ¿Cuántos billetes en total se vendieron toda la familia López? Usa la forma extendida a resolver.
Parte dos: ¿Quien ha vendido más, Pablo solo, o Marta, Jaime y Lupita entre si? ¿Cuántos más?
The second part of the above problem can be an extension activity for those students who finish quickly and need more of a challenge, or it can be a required problem for all students.
The last problem in the Base Ten section is a sample problem in rounding. I will reintroduce the rounding operation as a whole class demonstration first. A key understanding is that a given number is sandwiched between two numbers that differ only in the last non-zero digit place.
We will be rounding to nearest thousands, so I'll begin by having students round off to nearest tens, hundreds and then thousands by representing the sample numbers I give them as numbers in expanded form: for example: write 278 in expanded form. If we want to round this number to nearest hundreds, what place value digit will we need to use to round off? What place value digit gives us our clue?
2C. There were 1,289 tickets sold for a baseball game. The following week 1, 982 tickets were sold to attend another game by the winning team. Round off the numbers to give an estimated number of how many more people attended the second game than the first. Hint: What place value will you use for rounding? What do you need to do after you round off the numbers in order to get an estimate of how many people attended the second game.
Unit Focus Two: Take Away/Finding Difference—Subtracting Across Zeroes
The next set of problems is designed to help students recall and reinforce what they have learned about addition and subtraction in third grade, i.e. problems that don't require regrouping. We will practice more two digit additions and subtraction using mental math. I will begin this subunit with a discussion of regrouping by composing and decomposing tens using simple one digit problems in order to formally introduce the concept and give them the new vocabulary. The objectives are for students to understand regrouping by making and unmaking tens; to identify and solve problems of finding the difference and to begin learning how to subtract across zeroes. I will conduct a review lesson on fact families, again to reinforce a concept the children should have learned in third grade, and give a diagnostic quiz on addition and subtraction facts to determine where the children are in memorization of the facts.
Lesson 1: I write the following problem on the board: 9 + 6 = 15. I lead students in an inquiry to discover that 9 is really nine ones, 6 represents six ones and the 15 means one ten and five ones. I'll use tokens to demonstrate this and bundle them into one bundle of tens with five ones. At the same time I tell them that I'm composing or making a ten. Then, as per the Commutative Rule of Addition, I'll switch the addends and have students verify that we get the same answer. From here I'll have students work with their own pile of tokens to bundle simple practice problems I give them into tens and ones.
Next I'll go to the inverse operation using the above problem as 15 - 6 = 9 and 15 - 9 =6. Again we will use tokens, this time to decompose or unmake a ten and five ones by taking the five ones and one token from the ten bundle away. And so forth. Again the students practice the inverse practice problems with their own pile of tokens.
It is important to give the students the new vocabulary, i.e. composing and decomposing so that they have a way to talk and write about the process. Also, it helps them to concretize the new concepts and to remember them. So the key learning here is that when we perform addition and subtraction we are really making and unmaking tens.
Lesson 2: Students meet in their groups to work a number of simple problems, arranged in quartets of two addition and two subtraction problems that I give them on a printed sheet. These quartets are fact families or cousins. A sample is: 5 + 4 = 9; 4 + 5 = 9; 9 - 5 = 4 and 9 - 4 = 5. The objective is for them to figure out what the pattern is and if it will always happen. They will create other quartets of fact families. When I bring them back to whole-class discussion, groups will report out and post their fact families on the board.
The following day we are ready to begin the word problem(s) of the day.
Sample Problems of the Day Demonstrating Inverse Relationships/Finding Difference—Subtracting Across Zeroes: An Introduction
The first two problems are a review. They are followed by a suite of two problems that move to three-digit addition and its inverse cousin, subtraction, and then two more that, by changing one digit, begin the process of adding and subtracting across zeroes. In this suite, our follow up discussion will talk about "successive composition and decomposition" as described by Liping Ma in her book Knowing and Teaching Elementary Mathematics. She points out that students need to learn that "when the next higher place in a summand or a minuend is zero, one has to compose or decompose a unit from further than the next higher place" (Ma, 1999, p.15). For example: 203 - 15 requires that we decompose 100 into 10 tens, and one ten into ten ones. (Ma, 1999, p.15). I will pay particular attention to students' conjectures before and after working these problems.
I teach each suite of the following as I have them grouped, i.e. 1A and 1B on one day and 2A, 2B, 2C on another.
1A. Sarah has 30 baseball cards. Tricia has 48 cards. How many cards do they have together?
This is a great mental math problem that I will model: If we round 48 to 50, we can simply add 30 and 50 to get 80 then subtract the two we rounded off and get 78. Another way to compute this problem mentally is to add 30 + 40 + 8 = 70 + 8 = 78. Now I'll have groups write two or three problems that can be solved the same way. I'd like them to solve 1B by writing it down and solving it in the traditional manner to judge whether or not they are able to regroup the tens, because the next trio will contain regrouping problems. I check individual work to determine how much additional instruction children need before moving to the next trio.
1B. Rogelio tiene 53 conchas. Juanita tiene 95 conchas. ¿Cuántas conchas tiene Juanita más que Rogelio?
In this trio of problems students move from solving a simple one column addition to regrouping with zeroes and then subtracting across zeroes.
2A. Maggie collected 543 beads last year. This year she collected 456 beads. How many beads did Maggie collect?
2B. Margarita recogió 544 cuentas de collar de su abuelita. Tambien recogió 456 cuentas de collar de su tía. ¿Cuántas cuentas de collar ha recogido en total?
2C. Margarita ha recogido 1000 cuentas de collar. Ha recogido 456 mas cuentas de collar que su amiga Catarina. ¿Cuántas cuentas de collar ha recogido Catarina?
After coming together whole class to post conjectures and discuss solutions, children will be given similar trios to practice the regrouping with zeros skill. Then the groups are asked to write and solve their own trios and present them to the class. This may take another day to complete.
Unit Focus Three: Building Skills in Composing and Decomposing in Base Ten—Introducing Subtraction of Decimals and Some Comparison Problems.
The next set of problems will expand on the composing and decomposing of tens while at the same time introducing addition and subtraction of decimals through the means of money by exploring comparison problems. The last problem in the set introduces students to the infamous missing addend problems.
Lesson 1: Each child receives a stack of ten dollar bills, one dollar bills and a few dollars change in play money. I ask them to keep the tens, ones and change in separate piles until they need them to work the following sample problem. How much money do you have if you add $7.69 and $9.67? Arrange your money to show the correct number of tens, ones and change you will have? Remember that 100 cents equals one dollar. How many cents do you have when you add up the change part of this problem? What are you going to do with that $1.36? What is the decimal point telling us in this problem? How do we read it? We will solve these problems together, one at a time, as I model them in front of the class. Children are working individually now, but they may work in pairs if they are more comfortable. I am circumambulating the room, guiding and answering questions. When we are finished children should have a ten, seven ones and thirty-six cents in change.
Again I will ask the question: What is the decimal point telling us in this problem? We discuss that it means less than one whole dollar; however, the relationship between the places is the same as for whole numbers. We will keep working problems until I'm satisfied that all children understand how to represent addition of money problems. Then, I'll switch to subtraction of money problems and model decomposing tens in order to solve the problem. Again, students will work with their own stacks of tens, ones and change in order to solve some decimal subtraction problems. Throughout the lesson I will introduce and employ the word hundredths, explaining the suffix "th" to signify less than one whole.
Using the money manipulatives we discover, and I articulate, that dimes are 1/10 of a dollar since a dollar equals ten dimes. Likewise, a penny is 1/100 of a dollar and 1/10 of a dime. I write the decimal representations on the board. We are ready for our problems of the day.
1A. Cuando fue al pueblo Juana tenía $14.87. Cuando regresaba, tenía $6. 39. ¿Cuánto dinero gastaba Juana?
I will use this problem to observe whether or not students know the rule: the largest decimal place rules. The next more difficult problem will point the way.
2A. Juana tiene $140.87. Su amiga Flora tiene $210.26. ¿Cuánto dinero más tiene Flora que Juana?
Problem 2A seems a simple comparison problem where the amount of money Juana has is compared to the amount Flora has. However; students must first be able to determine who has more money. Thus, I articulate the rule, largest decimal place rules, and we discuss it and explore more examples. Mathematically the problem is challenging, because it requires the student to decompose across every place value, while at the same time keeping track of the decimal points.
Lesson 2: I take some time to explain a missing addend problem as a subtraction problem that looks like addition and can, in fact, be solved by addition. Professor Roger Howe told me another way to teach this is to "think of a-b as the number you add to b to get a".
I introduce students to some terminology and patterns that will help them to identify a missing addend problem. At this point I will discuss the Carpenter et al taxonomy referred to in the Strategies section. Addition and subtraction problems are named "Join" and "Separate" problems. And these can be further identified as Join/Separate problems with "Result Unknown, Change Unknown or Start Unknown" (Carpenter et al, 1999, p.10). Thus, for Join problems: 5 + 7 = X, 5 + X = 12 and X + 7 = 12. Similarly, for Separate problems: 12 - 7 = X, 12 - X = 5 and X - 7 = 5. I model and discuss some sample problems that treat action situations which happen over time.
The Unknown, or what we want to find out, is X. I'll have students meet in their groups to solve some Join and Separate problems, identifying whether X represents Result Unknown, Change Unknown or Start Unknown. I also want them to identify the action in the problem and its time frame. When they report out to the class, we will review the new terms again, articulate the action and its time frame, check their problems and try a few more. Then, in groups, students will formulate their own set of four problems. This time the groups will post their new problems, and we will review them to discern if they are reasonable.
Lesson 3: A sample problem is X + 6 = 10. What is X? From what we learned yesterday, what kind of problem is this? When the Start or the Change is unknown, children often have difficulty. These are the missing addend problems. After the students identify the above as a "Start Unknown" problem, I give them a simple word problem that fits the formula pattern. For example, Alice found some marbles on the playground. After school Ahmad gave her 6 marbles. Then she had 10 marbles. How many marbles did Alice find?
Here is where I give them the term missing addend and explain how to identify the problem as such by setting it up as X + 6 = 10, to use the above example, and by identifying it as a Start Unknown problem. A missing addend can also be a Change Unknown problem, so I reverse the above problem to make it fit the pattern 4 + X = 10. I also discuss the time factor and the action (finding and receiving marbles) in the problem.
Sometimes, it's easier to ask a subtraction question than an addition question. So, could we ask the question, "What is 10 - 6 = X?" This is changing the Start Unknown problem to a Result Unknown problem. Let's try it. After I work the problem and we check it together, I'll give students a number of sample problems to explore to see if the algorithm works. Then I'll give them some time to practice the algorithm with other problems in their groups. Later, we'll come together to discuss solutions and make some conjectures. The goal is for the children to discover that these problems are most easily solved by subtraction. Now we are ready to continue our word problems.
1B. Habían muchos gatos en la calle al mediodía. Setenta y cinco gatos se fueron. Cuarenta y cinco gatos se quedan. ¿Cuántos gatos habían en la calle al mediodía?
2B. Mickey had 275 rocks in his rock collection. His little brother borrowed a number of them for his school science project. Now Mickey has only 170 rocks left. How many did his brother borrow?
3B. Mickey had some rocks in his rock collection. His little brother took 125 of them to build a fort. Now Mickey has only 375 rocks in his collection. How many did he have?
In the above problems although they could easily be mental math problems, the challenge is in identifying the problems as missing addend problems. I would like the children to identify first what component is missing in each one—the Start Unknown or the Change Unknown. Can we make a conjecture here that when either one is missing the problem is most easily solved by subtraction?
Now it is time to look at missing addend problems that are classified as Part-Part-Whole and Comparison Problems, according to the Carpenter et al taxonomy. Again whole class, I give students some sample problems that involve static relationships where there is no action and no change over time. This time I use two-digit numbers in the problems. For example, John has some oranges and Sally has 29 oranges. Together John and Sally have 54 oranges, how many oranges does John have? I articulate that there is no time frame here and no action. And I set up a formula: X + 29 = 54. What is X? Is it a Part Unknown problem or a Whole Unknown problem?
A Compare problem might be: John has 25 oranges. Sally has 4 more oranges than John. How many oranges does Sally have? In this case, students are choosing whether the unknown to be solved for is Compare Quantity [subtrahend Unknown, or Referent (minuend) Unknown]. These are fine, and to fourth-graders, subtle distinctions that will require lots of examples of each type and lots of practice. Again, I identify the problems as missing addend problems, and I make the connection that the Sowder taxonomy makes that missing addend problems can be Take-away (Separate) situations or Comparison situations.
After much practice of both types of problems over several days, we are ready to begin our Word Problems of the Day. Two examples follow.
1C. There were 4750 apples on the big tree. There were a total of 8295 apples on the big tree and the little tree together. How many apples were on the little tree?
This is a Change Unknown subtraction problem that also requires decomposing one thousand.
2C. The library has some books. In Ms. Wasser's fourth grade class of 21 students, each student checked out 2 books. Now the library has only 958 books. How many books does the library have altogether?
This is a multi-step, Start Unknown addition problem. Students need lots of practice with these types of problems to determine when to add and when to subtract in order to find the missing part.
3C.Flora tiene $2100.26 en el banco. Entre si, Flora y su amiga Juana tiene $3600.13 en el banco. ¿Cuánto dinero tiene Juana?
Problem 3C is a fairly substantial problem in that it deals with a number of mathematical dimensions, notably subtraction across zeroes and subtraction of decimals as well as finding the missing addend.
Please see the appendix for more problems of all the various types covered in this unit.
In sum, students will work individually and in small groups to solve word problems using their knowledge of base ten operations to explore subtraction problems involving finding difference, subtracting across zeros, using mental math, comparing numbers, finding the missing addend, subtracting with decimals and subtracting with fractions. They will learn to identify key elements of the problem, formulate conjectures and revise them if need be. It is to be hoped that the byproduct will be a lively spirit of inquiry and enthusiasm that will serve to inform student work and invigorate their mathematical thinking and problem-solving abilities.
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