Big Idea #4: Comparison

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*Big Ideas:*

A “big idea” is a statement of ideas that are vital to the comprehension and development of mathematical knowledge. Each big idea is linked to countless mathematical understandings that come together into a coherent whole.

The twenty-one big ideas, by Randall Charles in the NCSM Journal, were developed through the analysis of mathematical concepts and skills. As content was studied, they looked for connections and common characteristics that ran across all grade levels and themes.

As teachers, the responsibility of an educator is to know what skills and concepts are taught at each grade. With this information, teachers can build on previous information and prepare their students for what they will learn in the future. By connecting these concepts to the past and the future, students will learn these key concepts that will be fundamental to their comprehension and mastery of mathematics.

The big ideas can and should be used to revise and improve district and state curriculum standards. Through these statements, the mathematic standards and skills will be developed to full understanding. Effective curriculums embrace the mathematical skills, ideas, and understandings to offer students the coherent whole.

*The Big Idea- Comparison: *

Comparisons: Numbers, expressions, and measures can be compared by their relative values.

**Numbers and Expressions**

- One-to-one correspondence can be used to compare sets.
- A number to the right of another on the number line is the greater number.
- Numbers can be compared using greater than, less than, or equal.
- Three or more numbers can be ordered by repeatedly doing pair-wise comparisons.
- Whole numbers and decimals can be compared by analyzing corresponding place values.
- Numerical and algebraic expressions can be compared using greater than, less than, or equal.

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**Fractions, Ratios, & Percent**

- A comparison of a part to the whole can be represented using a fraction
- A ratio is a multiplicative comparison of quantities; there are different types of comparisons that can be represented as ratios.
- Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes.
- Rates are special types of ratios where unlike quantities are being compared.
- A percent is a special type of ratio where a part is compared to a whole and the whole is 100.
- The probability of an event is a special type of ratio.

**Geometry and Measurement**

- Lengths can be compared using ideas such as longer, shorter, and equal.
- Mass/weights can be compared using ideas such as heavier, lighter, and equal.
- Measures of area, volume, capacity and temperature can each be compared using ideas such a greater than, less than, and equal.
- Time duration for events can be compared using ideas such as longer, shorter, and equal.
- Angles can be compared using ideas such as greater than, less than, and equal.

Charles, R.I. (2005).Big ideas and understandings as the foundation for elementary and middle school mathematics. *Journal of Mathematics Education Leadership*. *7*, 1-16.

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*Comparisons in the Early Grades:*

Comparison occurs when something is evaluated against another standard. When children are first exposed to this concept, they need to be exposed to the proper language and vocabulary. Such words include: shorter than, longer than, more than, and less than. As children explore length, weight, and quantity, they will be able to use the proper terminology to effectively compare the objects.

Initially when teaching these concepts it is important that children learn how to compare objects without using numbers. The concept of “big” and “small,” when compared to other objects, needs to be explicitly taught (IE A giraffe and bus are big, but a bus is bigger). Quoting Baratta-Lorton in *Elementary Mathematics Is Anything But Elementary*,

“At this early stage, numbers interfere rather than enhance the development of the concept. At this stage, we want children to experience the whole, and the numbers focus the child’s attention on the particular difference in measurement…. Using standard measurements of centimeters or inches is a much later step and should only be used after the children have had an opportunity to create their own standard and fully explore the idea of measurement” (Baratta-Lorton, p. 195, 139).

When teaching, it is important that children are given a solid foundation in comparisons so that they can continue to use this knowledge as they grow into more challenging areas of study.

*Links to Comparison Websites (K-2 Grade):*

http://www.uen.org/Lessonplan/preview?LPid=21363

This website has a list of center activities that can be used in the kindergarten classroom to teach the concept of more, fewer, or same. This site has book suggestions, worksheets, lesson plans, and information needed to teach in the classroom.

http://www.uen.org/Lessonplan/preview?LPid=16226

This lesson plan, for kindergarten, is a wonderful activity that teaches children about both measurement and comparisons. Throughout the story children can measure and compare the distance that the ladybug travels.

http://www.uen.org/Lessonplan/preview?LPid=16227

This lesson plan allows kindergarten students to compare and order objects according to length. Students can compare using ideas such as longer, shorter, and equal.

http://www.apples4theteacher.com/greater-than-less-than.html

Working with greater than and less than, students will try to guess the number that the computer has chosen. Appropriate for 1^{st} to second grade.

http://www.eduplace.com/rdg/gen_act/large/whatbig.html

Students will learn to compare objects between big and bigger/small and smallest. Appropriate for 1^{st} grade.

http://www.uen.org/Lessonplan/preview?LPid=21445

Using the game tic-tac-toe, students will learn the concepts of more and less. Appropriate for 1^{st} grade.

http://www.harcourtschool.com/activity/lets_compare/

This site has an interactive game that deals with greater than, less than, and equal to. In this game, students will identify the amount of two groups of coins and then identify the greatest amount. Appropriate for 2^{nd} grade.

http://pbskids.org/cyberchase/games/dataclusters/dataclusters.html

This is a fun, interactive site that has students comparing the height of trolls. Students predict which troll has the average height and then order the trolls from tallest to shortest. This activity helps students with comparisons of length. Appropriate for 2^{nd} grade.

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*References:*

Bahr, D.L., de Garcia, L.A. (2009). Elementary mathematics is anything but elementary. Belmont, Ca: Wadsworth Cengage Learning.

Carpenter, T. P. (1999). Children’s mathematics: Cognitively guided instruction. New

Hampshire: Heinemann.

Charles, R.I. (2005).Big ideas and understandings as the foundation for elementary and middle school mathematics. *Journal of Mathematics Education Leadership*. *7*, 1-16.

Anna Pugliano