One-Step Problems
Add To
Add To problems involve an addition expression but to solve them may require subtraction. When referring to the ‘Add to’ category in figure 1, it can be noted that there are three types of problems within this category: Result Unknown, Change Unknown, and Start Unknown. I have chosen one scenario and a fixed set of variables to demonstrate the relationship between each type of problem. The scenario describes a girl’s height and how much she grew in one summer. I have added an extra layer of difficulty by referring to some variables in one problem in inches and feet, and then using just inches in other problems. Students will have to recognize the relationship between the units in order to successfully solve these problems.
Result Unknown
Aleah was 39 inches tall at the end of the school year. Over the summer she grew 4 inches. How tall is she now?
In my classroom, I will start my unit by opening up a discussion of this type of problem, as students have typically encountered ‘Add To, Result Unknown’ problems by the fourth grade. All necessary information is given in this problem and I feel that my students will feel confident in their ability to solve this type. This particular problem is the most basic ‘Add to’ problem and can be solved by taking the initial height of ‘39’ inches and then adding the ‘4’ inches that Aleah grew over the summer. By combining the two known amounts, the ‘Result Unknown’ can be found: 39 + 4 = 43. After discussing a few problems of this type with students, I will facilitate a discussion wherein I ask students, “Can all word problems be solved by adding the numbers you find?” This is when I would move students into a slightly more difficult problem type: ‘Add to, Change Unknown’. Using the same scenario, I would ask students what they notice in the new problem and what information is needed to solve it.
Change Unknown
Aleah was 3'3" tall in the Spring. Now, in the Fall she is 43 inches tall. How much did she grow over the summer?
By slightly changing the ‘result unknown’ problem students are familiar with to a ‘change unknown’ problem, it is my hope that they will understand the relationship between the two word problems and readily identify the unknowns.
Initial Unknown
An ‘Add to, Initial Unknown’ problem can easily be created by changing the information given in the scenario and can be discussed much like the first two.
Example: Aleah is now 3’7” tall. and She grew by 4 inches over the summer. How tall was she in the spring?
Take From
After discussing ‘Add To’ problems with students, I would begin to present them with both ‘Add to’ and ‘Take from’ problems in order for them to explore the inverse relationship between the two operations. ‘Take from’ problems are expressed with a subtraction. When the ‘Take From’ category is studied in figure 1, it can be noted that there are three types. As with Add to problems, the three categories are: Result Unknown, Change Unknown, and Initial Unknown. To demonstrate the relationship among the three types of ‘Take From’ problems, I have chosen a new scenario with fixed variables to discuss. In this new scenario, a girl has a collection of desert plants that she wishes to share with her friend.
Result Unknown
There were 40 albums in Aniyah's collection. If she gave James 25, how many does she have left for herself?
This problem is the most basic of the ‘result unknown’ problems. The operation seems obvious enough to students that there will be little confusion regarding which should be used. To solve this problem and find the result unknown, the student will need to subtract 25 from 40. This should be expressed as 40 - 25 = 15, with the Result Unknown being 15.
Change Unknown
Aniyah now has 15 albums. She used to have 40 in her collection that she wanted to share with James. How many did she give to James?
By slightly changing the information given in this problem, a new problem can be examined and compared to the last. My goal is for students to understand the connection between the ‘Result Unknown’ and ‘Change Unknown’ through individual examination and whole class discussion.
Initial Unknown
Aniyah had a collection of albums. She gave 25 albums to James. Aniyah now has 15 albums left. How many albums did Aniyah have in her collection originally?
An initial unknown problem, although slightly more difficult for students, can easily be constructed by adjusting the given variables and feeding them into this third scenario problem. Students will be expected to compare this problem with other ‘Take From’ and ‘Add to’ problems through whole class discussion, and cooperative learning in pairs in groups. Ideas on how set up this cooperative learning and discussion will be discussed within the ‘Strategies’ heading.
Compare
Just as for change problems, compare problems can be divided into two main types and six subcategories in all. ‘Compare Greater’ refers to problems that use the term “more, higher, longer, greater, heavier, etc. ‘Compare Fewer’ refers to problems wherein the term “less, lower, smaller, lighter, shorter, etc.” is used and the known value is smaller than the missing value. They can be described as problems wherein one quantity or unit is compared to another. The subcategories of Comparison type problems will be described below.
I plan to spend a longer amount of time discussing ‘Compare’ problems because I notice that my students struggle with this type more frequently. I believe it may be helpful to paraphrase the first few problems as change problems to help students understand the meaning of comparison language before presenting them with the compare type. The scenario in the Compare problems I have constructed describe a boy and a girl who are comparing the number of donuts they have. They are constructed in a way that shows the relationship between the larger number, the smaller number, and the difference between each value As an extra layer of complexity, I have used two different units to describe donuts: base ten wherein an amount of donuts is referred to, and base 12 wherein donuts are described in terms of dozens. These problems can technically be described as two-step problems. It will be necessary for students to know how to convert between the two units (dozens and single donuts) to solve the problems.
Compare Greater-Smaller Unknown
If Eden gets 2 dozen more donuts, she will have 38, just like Quan'ye. How many does she have now?
In this problem students will need to understand that a dozen refers to 12. If I feel that converting between units will be too taxing on students, I will chose to leave the two-step aspect out of the problems. For example, instead of using the term ‘two-dozen’, I would stick to explicitly stating how many donuts Eden had ’24 more’. Given in the problem is how many more donuts Quan’ye has than Eden, which is 2 dozen more. We also know that Quan’ye has 38 donuts. In order to solve this problem the students will need to subtract rather than add to find the solution. This will be a key point in classroom discussion of this problem. Students who are conditioned to associate the word ‘more’ with addition will have a difficult time figuring out this problem. The solution to this problem can be expressed as 38 - 24 = 14.
Compare Greater- Difference Unknown
Quan'ye has 38 donuts and Eden has 14. How many more does Quan'ye have than Eden?
By changing the status of the three variables within the problem, a slightly different problem type can be created. Class discussion on the relationship between the ‘Smaller Unknown’ and ‘Difference Unknown’ will be held after introducing the two problems.
Compare Greater-Larger Unknown
Eden has 14 donuts. If Quan'ye has 2 dozen more, how many does Quan'ye have?
Changing the two variables whose values are given in the scenario created this third problem. Students will be asked to compare and contrast the three ‘Compare Greater’ problems and make a conclusion about the relationship among the three. By this point in the unit, it is expected that students will have a strong understanding of how class discussions work and will be expected to reason through their conclusion by using specific vocabulary and logical ‘language’. For example, their thoughts both verbally and on paper should be expressed something like this, “I know this is the answer because___________”, or “I know I need to subtract because__________.”
Compare Fewer-Smaller Unknown
Eden has 2 dozen fewer donuts than Quan'ye. If he has 38, how many does Eden have?
The given amounts in this problem are the difference (2 dozen fewer) and how many donuts Quan’ye has (38). In this case the word ‘fewer’ is consistent with the ‘key word’ method because it refers to the actual operation required: subtraction. For many of the problems presented, this is not the case. The solution to the problem can be represented numerically as 38 - 24 = 14
Compare Fewer- Difference Unknown
Eden has 14 donuts and Quan'ye has 38. How many fewer does Eden have?
By slightly changing the known variables from the ‘Smaller Unknown’ problem, a slightly more complex problem can be examined and discussed. Students will now be able to ‘compare and contrast’ both Compare Greater and Compare Fewer problems within the same scenario. Comparison of both types of ‘Compare’ problems will come after a very thorough discussion of both Change problems and ‘Compare Greater’ problems.
Compare Fewer- Larger Unknown
Eden has 2 dozen fewer donuts than Quan'ye. If she has 14 donuts how many does Quan’ye have?
Above is another example of how one scenario, with fixed variables, can be used to create and discuss many types of compare problems.
Put Together
This type of problem is often referred to as a Part-Part-Whole Problem. There are two types of ‘Put Together’ problems: Part Unknown and Whole Unknown. It should also be noted that the parts within the problem may be different. For example, if the problem is referring to a basket of apples, one part of the apples could be green and the rest red. Put Together problems are set up so that in one problem either the whole or one part will be missing. The order of the parts is irrelevant. I have created a new scenario with fixed variables wherein two boys are selling headphones. A boy is selling two different types of headphones for a fundraiser.
Part Unknown
Dylan is selling a total of 75 headphones for a fundraiser. If he has 19 'Beats' headphones to sell and the rest are iPhone brand, how many iPhone brand does he have to sell?
In this problem one part of the whole is missing, the whole being all headphones together. The parts we know are that there are a total of 75 headphones being sold, and that Dylan has 19 beats headphones to sell. In order to find the part unknown, students will need to subtract the part known (19) from the Whole (75). This can be expressed as 75 - 19 = 56. Possible aspects to discuss with students are the different parts of the problems, different units used, and if the units matter in finding a solution.
Whole Unknown
Dylan is selling beats and iPhone brand headphones for a fundraiser. Dylan has 19 'Beats' brand to sell and 56 iPhone brand to sell. How many headphones does he have to sell in all?
In this particular problem, both parts are known: Dylan’s Beats brand headphones (19) and Jeremiah’s 56 iPhone brand headphones. When added together, the whole can be found. This problem can now be expressed numerically as 19 + 56 = 75.
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