Comparison Model Thinking Tool lessonSupports understanding of the structure of comparison subtraction problems

quan

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quan

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difference

Making Meaning of Operations

Background: Structure of Subtraction problemsThe comparison organizer supports is the “comparison” structures of subtraction. The difficulty students have with this structure is that the quantity total has nothing to do with responding to the request to find the difference between two quantities. A common error of students totaling the compared amounts is a sign that students do not understand the comparison model of subtraction. Concrete modeling of this understanding is a valuable step in helping students understand why subtraction is the operation needed to solve these type of problems. Connecting cubes could be used to stack two towers beside one another, one taller than the other, followed by the question “how many more cubes does the taller tower have than the shorter one?” Verify that students understand that they are looking for the number of cubes it would take to make the shorter tower equal in height to the taller tower. This would be finding the difference.

When using this tool with word problems, the understanding the difference between the two models of subtraction becomes very important when demonstrating understanding of the structure of the problem. Students need to read and comprehend the situation happening in the word problem to determine the facts given and the facts unknown (remember, the unknown can be any part of the equation). Confusion for students can be an issue when they are only looking for “key words” to determine what operation to use. For example, this can happen when the story in the word problem feels like a joining situation, but if one of the parts are missing, then subtracting may be the operation to use to find the unknown part. Teaching students “short-cut phrases” can only add to the confusion. We need to help our students understand the structures, so they have to tools to make sense of the situations they are solving for.

The answer being sought for in this lesson is the understanding of the structure that the situation presents in the word problems. The focus of this lesson is not finding the “unknown” of the word problem.

Initial instruction for the comparison model thinking tool:

1. Begin this lesson by asking what it means to subtract. There are two structures of subtraction students need to know. The “take away” structure is commonly understood as subtraction. The part-part-whole structure supports this understanding. The other meaning of subtraction, the “comparison” structure, is supported better with the comparison model organizer.

2. Next, discuss what symbols we use to communicate numerically what is taking place in a mathematical situation. Record their thinking. (numbers [any of the digits 0-9 and the combinations that represent numerical values]; operation symbols[+,-,x,÷, etc]; $, ., relationship symbols [=, <, >, etc] ) This student-generated list could be an anchor chart to add symbols to as situations arise. The number, operation, and relationship symbol =, will need to be symbols necessary for this lesson.

3. Now present the comparison thinking tool and discuss how it is used to represent comparison subtraction situations. Students may need to begin with concrete objects. Once the understanding is supported with concrete tools, then replace those objects with the symbols that represent the values in each section of the part-part-whole organizer.

4. Once the tool is understood, it is time to apply the tool with word problems. With each word problem, it is important to ask students with key questions that guide them to representing the structure of the problem with the tool. Here are some question stems to use:

What type of situation is happening in the story of this word problem? What information is given? Where does this fact belong on the comparison model thinking tool? What mathematical symbol(s) do I need to represent the situation in this problem? What is my unknown fact(s) in this problem? How am I going to represent the unknown? (variables are appropriate to model

in this situation, but a question mark could work just as well) Once I have my facts in place on the comparison thinking tool, then it is time to ask how to set up the number sentence(s)

that could be used to represent the situation for solving for the unknown.

5. Comparison situation example of #5 scenario Post the word problem ~

Lucy has two bracelets. Julies has five bracelets. How many fewer bracelets does Lucy have than Julie? “What type of situation is happening in this story?” (comparison subtraction) “How do you know?” (each person has their own

amount and am asked to compare those two amounts ~ fewer) “What information is given and how will it be represented on the part-part-whole organizer?” (“two” represented with “2”,

“five” with a “5”, “b” or “?” can represent the unknown…”2” and “5” represent the values that are being compared, so the ‘5” goes in the longer row to represent the greater quantity, the “2” goes in the shorter row to represent the lesser quantity, the unknown is the “difference”)

5

2 ? “What number sentence could I use to represent the situation in this story?” (5 - 2 = ?) “Why is “-” being used in this number

model? What does “=” mean?

6. Another comparison subtraction situation example: Post the word problem ~

Julie has three more shirts than Lucy. Lucy has two shirts. How many shirts does Julie have? “What type of situation is happening in this story?” (subtraction) “How do you know?” (asked to compare quantities)

“What information is given and how will it be represented on the part-part-whole organizer?” (“three” represented with “3”, “two” with a “2”, “s” or “?” can represent the unknown…”2” represents the lesser quantity because Julie three has more, so the number 3 goes in the “difference” and 2 goes in shorter row …the unknown is the “longer row” since it is the greater quantity)

?

2 3 “What number sentence could I use to represent the situation in this story?” (? - 3 = 2) “Why is the “-“ symbol being used in

this number model? Why might a person use addition to solve for this unknown? (addition/subtraction relationship understanding is a necessary part of discussion)

7. Once students have enough practice with the comparison model thinking tool, other number quantities can be used. It is important that students see the structure of the problem, no matter how large or small the quantities they are working with.

8. When finding other word problems that allow students the opportunity to apply this thinking tool, remember the type of situations that apply for this specific tool. Begin with one-step problems to get enough practice so that its application is easily applied in muliti-step problems (3rd grade up). Students will need to be flexible with several thinking tools (see illustration below), and will need to justify their thinking throughout the year as to why the tool they are using fits the structure of the problem they are working to solve.

Helpful hint: Student engagement will increase if each student has their own comparison model show-me mat. Post-it notes are nice for manipulating placement of the numbers/symbols as students are thinking. Laminating the mats and dry erase markers are another option to consider when asking students to demonstrate their thinking with this organizer.

Comparison Model Mat

Difference

Quantity

Quantity