## Introduction

What should students know about mathematical comparison? In this article, you'll learn about the progression across grades K-8, get tips for teaching ways to compare, and learn to incorporate algebraic thinking. Since most students are familiar with pumpkins, we will use pumpkins in the examples!

Generally, comparison means looking at two or more things to see how they are the same or different. For example, you might compare two pumpkins by saying that one is smoother or has a darker color than the other. However, mathematical comparisons are more specific and include numbers. To compare pumpkins mathematically, you consider measurable or countable attributes such as height, weight, or number of seeds.

To begin a discussion of ways to compare pumpkins mathematically, you may want to have students weigh and measure their own pumpkins and use those measurements to write and solve problems. A tape measure and inexpensive kitchen scale are very helpful tools for collecting real data. The actual weights of two pumpkins are shown in the image above.

Comparing with Numbers: Students in Kindergarten through Grade 2 should be able to compare by deciding which number is greater. For example, Pumpkin B is heavier than A because 36 ounces is greater than 20 ounces. Remind students that you can also conclude that Pumpkin A is lighter than B.

Using Inequality Symbols: Most first graders are expected to learn the basic inequality symbols for less and greater, < and >. So, students should be able to write 36 > 20 (36 is greater than 20), or 20 < 36 (20 is less than 36. Here are three ways to help students remember which symbol to use:
• Think of "<" as an arrow pointing to the smaller number.
• Think of "<" as being like a tilted capital L. Since "less" starts with L, the symbol < means "is less than."
• Think of "<" as a mouth opening toward the larger number.
Comparing By Subtracting: Many math problems ask for a comparison of two amounts as how much or how many more or less. This requires subtraction of the smaller number from the larger number. So, in comparing 20 and 36, students can write several different statements. Notice that in this case, the symbols < and > are not needed.
• The difference between 20 and 36 is 16. With symbols, write 36 – 20 = 16.
• 20 is 16 less than 36. With symbols, write 20 = 36 – 16.
• 36 is 16 more than 20. With symbols, 36 = 20 + 16
• The weight of Pumpkin A is 16 ounces less than B.
• The weight of Pumpkin B is 16 ounces more than A.
Tip for Algebra Readiness: Student may be confused by different statements including "less than" or "greater than." Consider these examples, with two numbers A and B, and the difference as a number N. (In these examples, A and B will be used as variables to represent the weight of Pumpkins A and B, respectively. If desired, you can use other data for A and B or other letters for variables.)
• "A is less than B" is written A < B.
• "A is N less than B" is written as A = B – N.
In the first sentence, we only know which number is greater. We can write an inequality. In the second sentence, we know that A is equal to the result of subtracting N from B. So, we can write an equation.

## Comparison Questions in Grades 3–5

Using Fractions or Decimals: At these grades, students are learning to add and subtract fractions, decimals, mixed numbers, and greater whole numbers. So, to increase the difficulty level of problems about comparing measurements by subtracting, include numbers that are similar to the numbers in computation problems for the specific grade level.

Multiplicative Comparison: This type of comparison involves statements such as "times as many" or "times as much." For example, consider how to compare a 10-pound pumpkin to a 2-pound pumpkin. You can say that the 10-pound pumpkin weighs 5 times as much as the 2-pound pumpkin. Explain to students that the factor (or multiplier) is found by thinking of a number sentence with a missing factor. 2 x ___ = 10. To find a missing factor, you can divide. For the pumpkins weighing 20 oz and 36 oz, you can divide 36 by 20 to get 1.8. The larger pumpkin weighs 1.8 times as much as the smaller one.

Tip: Some advertisements or commercials use expressions such as "times more than" instead of "times as many." This is a confusing phrase because "more than" usually relates to addition or subtraction. It's better to avoid saying "times more than." Here is a mathematical reason for the confusion: In later grades, the phrase "200% more than 5" means 5 + 200% of 5, or 15. So, because 2 = 200%, the phrase "2 times more than 5" could also mean 5 + 2 x 5 which is 3 x 5.

Connecting to Algebra: It's good to discuss the general procedure by using variables. If the problem is to find out how many times A is equal to B, you can write either of these equations with a blank for a missing factor:
x  ___ = B
___  x  A = B

In either case, you can divide B by A to get the missing factor. The missing factor is B ÷ A, which also can be expressed as a fraction B/A.

For example, suppose you compare a 6-pound pumpkin to a 4-pound pumpkin. How many "times as heavy is 6 pounds compared to 4 pounds?
x  ___ = 6
To find the missing factor, divide 6 by 4. The result is 1.5 or 3/2.

Consider how the answer changes if you compare the numbers in reverse order. That is, what is the missing factor if you want to find how many times B is equal to A?
x  ___ = A
In this case, you divide A by B to find the missing factor.
x  ___ = 4
The missing factor is the fraction 4/6, which is the same as 2/3. (NOTE: The decimal value is 0.666..., so an answer such as this is often left in fraction form.)

## Comparison Questions in Grades 6–8

Comparing with Ratios: At middle grades, students learn a new way to compare quantities or measurements. Ratio comparisons are related to "times as many" but not the same. Consider the two pumpkins, A and B, that weigh 20 oz and 36 oz. The ratio of A to B is 20 to 36. There are several ways to write the ratio.
• With the word "to", as 20 to 36
• With a colon, as 20:36
• As a fraction, 20/36 or simplified to 5/9
This ratio means that 20 is 5/9 of 36, which can be written as 20 = 5/9 x 36. So, 5/9 can be the missing factor if you are answering "How many times as much is 20 is compared to 36?" So, for the weights of the pumpkins, the smaller one weighs 5/9 as much as the larger one.

Tip: It is very common for students to mix up the order of the numbers. Remind students that the number being compared to belongs at the bottom of the fraction or ratio. Also, if students are finding a missing factor, remind them that multiplying by a factor greater than 1 yields a larger product, and multiplying by a factor less than 1 yields a smaller product.

Comparing with Percent: This type of comparison involves writing either a ratio or a missing factor as a percent. There are several ways to solve percent problems. Suppose you want to find what percent of 20 is 36? You are finding a missing factor, __%  x 20 = 36. So, you can divide 36 by 20 to get 1.8. As a percent, this is 1.8  x 100 = 180%. Or you can write and solve a proportion. In this case 36/20 = r/100, where r is the number of hundredths. Thus, 36 is 180% of 20. The larger pumpkin's weight is 180% of the smaller one's weight.

Percent of Increase or Decrease, Percent More or Less: This type combines aspects of comparison by subtraction and comparison by finding a missing factor. In middle school, change can be positive or negative. So, write the numeric change as a percent of the original number.
• Percent increase from 20 to 36: The change found by subtraction is 16. To write this as a percent increase, divide 16 by the original or starting number, 20. This is 0.8 or 80%. The answer is an 80% increase. Also notice that 36 was 180% of 20. And 100% of 20 is 20. So, another way to find the percent increase is to subtract 100% from 180%. So, the larger pumpkin's weight is 80% more than the smaller pumpkin's weight.
• Percent decrease from 36 to 20: The change is –16 or a decrease of 16. To write this as percent decrease, divide 16 by 36. This is approximately 0.44, or a 44% decrease. An alternate method is to change 20/36 to about 56%. So, if 100% of a number changes to 56% of the number, that is a 44% decrease.
Connecting to Algebra: Each of these situations can be generalized by using variables.
• The ratio of A to B can be written as A:B or A/B.
• To compare A to B as a percent, write a proportion A/B = r/100. Solve for r. The result can be written as a statement or equation: A is r% of B, or A = rx B.
• To find the percent of change from A to B, find B – A and divide the result by A. Multiply by 100. Or solve the proportion B/A = r/100, then subtract 100 from r.

## Conclusion

We have looked at many ways to compare two pumpkins, using weights of 20 oz and 36 oz. Here is a summary by difficulty level.
• Because 20 < 36, the 20-oz pumpkin weighs less.
• The smaller pumpkin weighs 16 ounces less than the larger one.
• 36 oz is 1.8 times as much as 20 oz, so the weight of the larger pumpkin is 1.8 times as much as the smaller one.
• The ratio of the pumpkin weights, larger to smaller, is 36 to 20 or 9/5.
• The ratio 36/20 equals 1.8 or 180%. So, the larger pumpkin's weight is 180% of the smaller one's weight.
• As percent more or less, the 36-oz pumpkin weighs 80% more than the 20-oz pumpkin.
Provide practice on these concepts by having students weigh several pumpkins and write comparison statements. The assignment can vary according to the ability level of a student or class. For 5th grade and up, you may want to download the free PDF activity, Pumpkin Math with Decimals. See the link below. This activity has several comparison problems including decimals.

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I hope your holidays are special!

Angie Seltzer
Math Curriculum Specialist
Owner of K8MathSense.com