Strategies
In Children's Mathematics, it is stated that of the various ways that word problems can be distinguished from one another, one of the most useful ways of classifying them is by focusing on the types of action or relationships described in the problem. Within addition and subtraction problems, there are four classes of problems that can be identified, and ultimately described and taught to children. The four basic classes of problems are join, separate, part-part-whole, and compare. The size of the numbers can vary, as well as the theme or context of the problems; however, the basic structure involving the actions and relationships remains the same. Helping my students to understand this "four class" framework will be essential in assisting them in becoming better word problem solvers. As we progress through addition and subtraction word problems, I will name these four classes, I will have students identify the four-classes, and I will work to help them understand the relationships between the classes.
Joining - Class One
Join problems involve action; they take place over time. In Join problems, elements are added to a given set. They involve a direct or implied action in which a set in increased by a given amount. The following are examples of join problems with the result unknown. I think that it will be important for students to realize the differences within the Join class because it will better equip them with the cognitive tools that they need in order to solve the word problems.
Join Problems with Result Unknown
- Todd cooks 26 hotdogs. Will cooks 14 hotdogs after Todd. How many hotdogs did the boys cook altogether?
- Mrs. Kelsey baked 13 pies last week. She baked 3 more pies this week. How many pies did Mrs. Kelsey bake in all?
The word problems all require students to find the unknown result. This is probably the most basic type of addition word problem, although there are a few subtleties that may prove to be important based on the achievement level of the students that are in the class. One possible variation is to use number words rather than numerals. I have included such problems in my problem set because my students have had difficulty with reading word problems. I believe that the more that they see certain words, the easier it will be for them to recognize them. This will take away the stress that they feel with reading the problems and allow them an opportunity to concentrate on the math component of the problems.
Another pair of similar type problems is the differentiation between problems that require the students to regroup and those that do not. Problem numbers one, three, and four in the appendix do not require students to regroup. I may introduce these to my students first. I say this because concurrently, I may have to teach students adding and subtracting, with and without regrouping. Though this will not be included in the discussion of this curriculum unit, I plan to utilize the strategies discussed by Liping Ma in her book Knowing and Teaching Elementary Mathematics (7). The problems in the appendix that require regrouping are numbers two and five.
An added dimension to this problem set is extraneous information. Problem number five in the appendix shows a number of cones being produced in a certain amount of time - specifically, five minutes. Although this problem seems very simple, I believe that this is a type of problem that would cause several of my students to focus on the terms "five minutes" instead of the numbers of cones being produced, which is what the problem asks for. I will include many types of problems with extraneous information in my problem sets because there are always several of them on the standardized test that we take in Georgia, the Criterion Referenced Competency Test (CRCT).
Join Problems with Change Unknown
Although the problems presented in this set may seem familiar or even identical to the problem set in the previous section, they are very different. Each of these problems represents a different problem to young children because children use different strategies to solve them. Changes in the wording of the problems and the situations they depict can make a problem more or less difficult for children to solve.
- Todd cooks 26 hotdogs. Will cooks some more hotdogs. How many hotdogs did Will cook if the boys ended up with 38 hotdogs altogether?
- Mrs. Kelsey baked 13 pies last week. She baked some more pies this week. How many pies did she bake this week if she ended up with a total of 16 pies?
Since I know that these problems will challenge many of my children. As they work to solve problems, I will keep in mind that problems are easier for children to solve when the action or the relationships in the problems are as clear as possible. This will be something that I check for as I create sets of word problems for my students. It is evident that the largest group of these problems includes word problems one through seven in the appendix because they are all Join problems in which the change is unknown. A second dimension of this set is problems that do not require regrouping to find the answer. Problem numbers one, three, and six in the appendix are representative of this group. A final similar set of this group is problems that require regrouping to find the answer. These types of problems include numbers two, four, and seven in the appendix.
As I created several of the problems in this word problem set, I realize that these problems are difficult for my students to solve as addition problems. I realize that I have to find alternate ways for my students to "set up" the problems. For example, if I use problem number one to demonstrate, it will look like this:
26 + ___ = 38
One of the important practices in the Singapore Math(5)collection is to make sure that students are knowledgeable about adding numbers to ten. If I had students organize the problem differently, I believe that my students would see things much differently (See below.)
26
+___
38
I may even have my students draw a line all the way down the middle of the problem to focus on the ones and tens column separately. When students are able to look at the problem in this manner, it becomes easier for them to see that in the ones column, six plus a number equals eight. They are also able to see that in the tens column, two plus a number equals three. If they are not able to mentally use what they have learned about adding to ten, then they can use the Joining To method as described in Children's Mathematics: Cognitive Guided Instruction. In this method, a child would use six counters (to show the number in the ones column) and "count on" from six in order to find how many they need to get to eight. In this case, the child would add two more counters to the six. This would help the student to know that six plus two is eight. They would do the same for the number two in the tens column. One should make sure that the student realizes that s/he is adding tens when dealing with the tens column.
Join Problems with Start Unknown
The third type of Join problem is created when the start (minuend) of the problem is unknown. Several of these types of problems are listed below.
- Todd had some hotdogs. Will gave him 12 more hotdogs. Now Todd has 27 hotdogs. How many hotdogs did Todd start with?
- Mrs. Kelsey some pies on Monday. She baked 12 pies on Wednesday. She ended up with 21 pies in all. How many pies did she bake on Monday?
The dimensions of this problem include the overall dimension of being problems that all have the first addend, or the start, missing. Another dimension includes the problems that call for students to utilize regrouping. Examples of problems in this dimension include numbers one, three, and five. Problem numbers two and four are examples of problems that do not require regrouping to solve.
These types of problems will likely cause students the same trouble as those in the "Join with Change Unknown" section of this unit. I would use this opportunity to use the prior knowledge that the students have acquired with setting the problems up differently, utilizing their knowledge of place value, and adding to ten to solve these in a way similar to the previous ones. Using problem number four as an example, I would initially like to see the students set up problem number three in the following way:
___ + 14 = 28
After this, I would have the students set the problem up again vertically. It would look like this:
___
+ 14
28
In a way that is not very different from before, I would probably have the students draw a line down the middle of the problem in a way that would divide the ones column from the tens column. If necessary, I would encourage them to use the Joining To method in order to solve the problem. There is an additional issue with this problem that allows me to incorporate the students' prior knowledge of fact families. The fact that the students need to start "counting on" with the bottom number will allow me to have the conversation with them that shows them more about the commutative property. It will be interesting to see if students will explore the possibilities of listing the addend on the bottom first in order to find the addend on the top. Some students will need to review commutative basics such as:
23 + 15 = 38 and 15 + 23 = 38
Students could even write the problems; once again, in a vertical form so that they can see that they are completing the same type of problem as word problem number four.
Separating - Class Two
Separate problems are closely related to Join problems. Very similar to Join, Separate problems require an action that takes place over time. The difference is that the initial quantity is decreased over time versus increased with Join problems. Again, very similar to Join problems, Separate problems involve three quantities. There is a starting quantity (minuend), a change quantity, or the amount taken away (subtrahend), and the result (difference). Any of these quantities can be the unknown.
Separate Problems with Result Unknown
- Todd cooks 27 hotdogs. Will took 14 of Todd's hotdogs. How many hotdogs does Todd have left?
- Mrs. Kelsey baked 20 pies. She gave 8 pies to her neighbor. How many pies did Mrs. Kelsey have left?
The above word problem set include the overall dimension of being subtraction problems that require the students to find the result. Another dimension includes problem number one solely because it uses number words instead of numbers in the problem. Problem numbers one and two are of the dimension that requires subtraction without regrouping. There is also a dimension that requires regrouping with subtraction and those include problems number three, four, and five.
Separate Problems with Change Unknown
- Todd cooks 27 hotdogs. Todd then gives some hotdogs to his neighbor Will. Todd is left with 13 hotdogs. How many hotdogs did Todd give to Will?
- Mrs. Kelsey had 20 pies. She gave some pies to Ms. Kelsey. If Mrs. Kelsey was left with 12 pies, how many pies did she give to Ms. Kelsey?
There is a dimension that includes both of the above word problems because they are subtraction problems that include a missing subtrahend. There is also another dimension of this problem that does not require regrouping in order to find the answer. Ironically enough, all five of the problems are a part of this category. I would encourage the students to find the answer to these types of word problems in a way that is very similar to the way in which we solve the Join with Change Unknown problems. Initially, I would have the students set up the word problem as follows:
27 - ___ = 15
Again, students would need to realize that if they are not able to rely on their mental math skills, they will need to recall what was learned about the commutative property and how it can be utilized in this circumstance. They would also need to rely upon what they have learned about fact families. I would guide the students to arrange the problem vertically as follows:
27
-__
15
As done earlier in the unit, students requiring more assistance than others would need to draw a line down the middle of the problem which would divide the tens column from the ones column. From there, if necessary, they would need to use the Join To method and figure out "five plus what number equals 7" or "what number plus five equals 7", depending on the direction that they chose to work.
Separate Problems with Start Unknown
The last type of separation problem that students will need to understand is one that is missing the first number, or the minuend. Some examples of those types of problems follow:
- Todd had some hotdogs. He then gave Will12 hotdogs. Now Todd has 27 hotdogs. How many hotdogs did Todd start with?
- Mrs. Kelsey baked some pies. She gave 8 pies to her neighbor. She was left with 24 pies. How many pies did Mrs. Kelsey start with?
The largest dimension of these problems includes all of them and that is the fact that they are all missing the starting number, or the minuend. Another dimension is when a problem requires regrouping and those are problems one and three. A third dimension includes problem numbers two, four, and five. This dimension includes the problems that do not require regrouping. The last dimension includes problems numbers one and four. These problems utilize number words instead of numbers in the problems. As stated before, this is important for my students who experience literacy issues with the problem solving activities. In other words, they are not reading on grade level and I want to incorporate some type of practice every chance I get.
Part-Part-Whole Problems - Class Three
With Part-Part-Whole problems, there is no direct or implied action and no change over a period of time. Children's Mathematics: Cognitively Guided Instruction defines these types of problems as "static relationships among a particular set and its two disjoint (separate) subsets." Unlike the three types of Join and Separate problems, there are only two types of Part-Part-Whole problems.
Whole Unknown
This type of problem gives the two disjointed parts and asks the problem solver to give the size of the whole. A few of these problems are listed below.
- There are 18 girls in the class and 15 boys in the class. How many students are in the class?
- Four blue fish are in the tank with eighteen green fish. How many blue and green fish are in the tank?
The overall dimension which includes all five of the aforementioned problems is that they are all requiring the problem solver to give the size of the whole. The next dimension (although in no particular order) includes the problems where you are required to regroup. These problems include numbers one, two, three, and five. Another dimension does not require regrouping and problem number four is the sole problem in this dimension. A further look into the dimensions of these problems shows us a group of problems that use number words instead of numbers. This dimension consists of problems number two and five.
Part Unknown
In these types of word problems, the students are given one of the parts along with the whole, and they are to find the size of the other part. Unlike the Join and Separate problems, it is not obviously clear to the students as to how they are to go about solving these problems. For example, if you look at number one below, you can either subtract 15 - 7 to solve the answer, or you can say, "7 plus a number equals 15" and this will cause you to use the count on or Join To strategies. For the purpose of this curriculum unit, I will guide my students to use subtraction to solve these problems. As an aside, if students used an alternative (addition) method to solve this type of problem, I would not count it against them or tell them that they were wrong. Several of the problems in this category are identified below.
- 15 children were playing baseball. 7 were boys and the rest were girls. How many girls were playing baseball?
- Deedre has 28 marbles. 17 are pink and the rest are green. How many marbles are green?
The most obvious of the dimensions is that they are all giving the whole and one part and they are asking for the missing part. There is a dimension that asks for students to regroup with the subtraction and those are question numbers one and four. A dimension that does not require regrouping with subtraction in the word problems are numbers two, three, and five. An added dimension to this set of word problems is a problem that contains extraneous information. This is problem number four. Students will need to realize that the eight bones that T.J. lost is not important as we work to solve this problem.
Compare Problems - Class Four
Compare problems are more like Part-Part-Whole problems versus Join and Separate problems. Where the Part-Part-Whole problems "involve relationships between quantities rather than a joining or a separating action", Compare problems "involve the comparison of two distinct, disjoint sets rather than the relationship between a set and its subsets." There are three elements of a Compare problem. The chart below describes them.
Mark has 3 mice. | Referent set
Joy has 7 mice. | Compared set
Joy has 4 more mice than Mark. | Difference | Amount by which one set exceeds the other
Very similar to the Part-Part-Whole problem set, it is difficult to absolutely specify how I would prefer my students to solve Compare problems. As you will see in the set of problems that follow, students will be able to draw on their prior knowledge of Join To or counting on skills as well as Separate strategies in order to find the answers.
Difference Unknown
Some of the most common Compare problems are listed below.
- Mrs. Kelsey had 19 pies. Mrs. Polite had 7 pies. How many more pies did Mrs. Kelsey have than Mrs. Polite?
- If Veronica has 17 sandwiches and Jill has 11 sandwiches, how many more sandwiches does Veronica have than Jill?
In order to identify the dimensions of this problem, you have to realize the most evident dimension of all five of these problems. This dimension is that all of the questions are missing the difference. Another dimension is one that includes the problems that require regrouping. The regrouping problems in this set are numbers three and five. An added dimension is the set of problems that do not require regrouping with subtraction and those problems are numbers one, two, and four. The last dimension for this set includes a problem with extraneous information. The only problem in this category is problem number five.
Compared Set Unknown
These word problems are a little more difficult than the previous problems and call for students to pay more attention to the way in which the problems are worded. Although the students are comparing, these are not subtraction problems. They are actually problems in which they have to add in order to solve. Samples of Compared Set Unknown problems are listed below.
- Lacie had 9 marbles. Laila has 8 more marbles than Lacie. How many marbles does Laila have?
- Langston has 24 gumballs. Lamar has 19 more gumballs than Langston. How many gumballs does Lamar have?
One dimension in the problem set listed above requires students to regroup when they add. These problems are two, three, four, and five. There is only one problem in the dimension that does not require regrouping when adding and that is problem number one. Another dimension that I may explore when I continue to create word problems for my student is increasing complexity. Problems number one and two are basic addition problems and should not be very difficult for students to solve. Problem number one - although it is a two-digit plus one-digit problem - is simple because there is no regrouping required. Problem number two should be very easy to solve as well because it consists of two numbers that students are exposed to early on when they learn "adding doubles" and "adding doubles plus one." A final dimension is word problems that use number words as opposed to numbers within the context of the sentence. In this set, problem number four is the only one that is in this dimension.
Referent Unknown
In this Compare problem category, students are at a point where they need to subtract to solve the problems. Examples of Referent Unknown problems are listed below.
- Joy has 9 toy cars. She has 3 more cars than Mike. How many toy cars does Mike have?
- Tanya has 27 pens. She has 16 more than Travis. How many pens does Travis have?
As stated before, students must know that they have to subtract to solve these word problems. One dimension of these problems is that they all require subtraction to find the answer. Another dimension includes the problems that require regrouping. These problems are numbers three and five. An added dimension which include problems that do not require regrouping are numbers one, two, and four.
Although the wording in the problems in the preceding "four classes" of problems are similar, the basic structure of each problem is unique. The structures of the problems are related to how the students solve the problems.
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