7.1.1A Rational Numbers
Overview
Standard 7.1.1 Essential Understandings
Students have had prior experience with reading, writing and representing positive rational numbers as fractions, decimals, percents and ratios. They are fluent in writing positive integers as products of factors. They have developed a sense for the relative size of integers and can compare and order positive rational numbers. Students have been using positive rational numbers in a variety of ways: to label, measure, locate, compare and quantify. They are able to plot these values on a number line and also on a coordinate grid. Students will extend their understandings to all rational numbers, including negative values. Progression from just representation of value to comparison of values, both in the positive and now the negative, is strengthened in this standard. This standard extends the representation of equivalent representations of positive rational numbers to negative values. Students have also had prior experience plotting points in quadrant 1 on the coordinate grid. This standard extends the coordinate plane to all four quadrants.
In addition, students will deepen their understanding of the definition and meaning of a rational number. Students who master identifying examples of rational numbers in 7th grade will be prepared for the introduction to irrational numbers in 8th grade.
All Standard Benchmarks
7.1.1.1 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3.14.
7.1.1.2 Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator.
7.1.1.3 Locate positive and negative rational numbers on the number line, understand the concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate grid.
7.1.1.4 Compare positive and negative rational numbers expressed in various forms using the symbols <, >, ≤, ≥.
7.1.1.5 Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions.
7.1.1 Fraction & Decimals: Representations and Relationships
7.1.1.1 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3.14.
7.1.1.2 Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Understand the difference between a terminating (ending) and repeating decimal;
- Be able to give the definition of a rational number;
- Be able to identify rational versus non-rational numbers;
- Be able to pick the values that are/are not rational, given a set of numbers;
- Give examples of, explain and define an integer;
- Know that [math]\pi[/math] (pi) can be approximated by [math]\frac{22}{7}[/math] or 3.14.
Work from previous grades that supports this new learning includes:
- Know the calculator and non-calculator notation for repeating decimals (i.e. [math]4.\overline{3}[/math]);
- Develop an understanding of whole numbers;
- Understand positive integers;
- Understand fractions;
- Understand of decimals and decimal values;
- Be able to identify equivalent common benchmark fractions (1/4 = 0.25, ½ = 0.50, etc.).
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
- Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
- Develop meaning for integers and represent and compare quantities with them.
Common Core State Standards (CCSS)
6.NS (The Number System) Apply and extend previous understandings of numbers to the system of rational numbers.
- 6.NS.5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
- 6.NS.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.6.a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite.
6.NS.6.b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.6.c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
- 6.NS.7. Understand ordering and absolute value of rational numbers.
6.NS.7.a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right.
6.NS.7.b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3 degrees C > -7 degrees C to express the fact that -3 degrees C is warmer than -7 degrees C.
7NS (The Number System) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
- 7 NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.2.d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
8NS (The Number System) Know that there are numbers that are not rational, and approximate them by rational numbers.
- 8.NS.1. Know that numbers are not rational are called irrational. Understand informally that every number has a decimal expansion; rational numbers have decimal expansions that terminate in 0s or eventually repeat, and conversely.
Misconceptions
Student Misconceptions and Common Errors
- Students think that prime numbers are irrational because each prime number is only divisible by 1 and itself. Likewise, fractions such as [math]\frac{1}{17}[/math] will also be thought of as irrational because they are also primes.
- Students struggle with fractions like [math]\frac{1}{7}[/math] whose decimal equivalent is a long repeating decimal: [math]0.\overline{142857}[/math] Students who change into a decimal by dividing 1 by 7 on a calculator may not recognize the repeating decimal due to the rounding that occurs on the calculator. On most calculators [math]\frac{1}{7}[/math] results in a rounded decimal of 0.1428571429, and many students may not see the repeating pattern that is emerging.
- Students assume that since pi is often notated as 3.14 that it is a terminating and therefore rational number.
- Students think that all square roots (e.g.,[math]\sqrt{36}[/math]) are irrational numbers.
- Students struggle with defining integers as rational numbers. For example, students often struggle with converting 3 into the fraction [math]\frac{3}{1}[/math]. Students seem to be confused by a 1 in the denominator.
Vignette
In the Classroom
Mr. A. distributed grid paper to his class of 32 7th grade students. He asked them to work with their partners to draw a 4-quadrant grid on their paper. Next, he instructed them to number from the origin (0, 0) in the center of the paper to each of the 4 edges of the paper, numbering on the horizontal (X) axis with positive integers to the right of zero and negative integers to the left of zero. On the vertical (Y) axis, he instructed them to number positive integers above 0 and negative integers below zero.
Teacher: Class, what is the relationship between dividing one integer by another and all rational numbers?
Ty: What do you mean, Mr. A? Why would we divide a number by another number? What is a rational number?
Teacher: Good question, Ty. I want to help you understand the important group of numbers referred to as rational numbers. Let's use the coordinate grid to find out more about them.
Ty: So, we're just going to play around with the coordinate grid? Mr. A, you told us before, "The coordinate grid is the playground of algebra." Is today just playing around?
Teacher: I hope your "playing around" helps you see relationships in rational numbers you didn't know before. Let's get started. I want all of you and your partners to plot a fraction for me on the grid. Let's plot ½. The denominator 2 will be represented by the X-coordinate. The values on the X-axis will tell us what kind of fraction it is. For example, 2 on the X-axis is halves, 3 on the X axis is thirds, 4 on the X-axis is fourths and so on. The numerator 1 is our Y coordinate. The Y coordinate describes how many equal parts there are in our fractional value. So our fraction ½, from the origin, is up one, and over 2 units. Does that make sense? As an ordered pair, it would be (2, 1), but we are representing ½.
Kai: So this point (2,1) we just plotted represents the fraction ½, correct?
Teacher: Yes, every fraction has a "home" on the coordinate grid. Now plot another fraction that is equivalent to ½. Compare your new point to a point from another group.
Students compare plotted points. Many students plotted different points.
Teacher: What point did you plot that was equivalent to ½?
Alli: We plotted 3/6. From the origin we went up 3 and over 6.
Teacher: Is the fraction 3/6 equivalent to ½?
The class agreed.
Teacher: Everyone, plot this point on your grid. Did anyone plot a different point?
Sara: We plotted 4/8, up 4 and over 8 units. We know 4/8 is the same as ½.
Teacher: Sara, I agree. ½ and 4/8 are equivalent. Everyone should plot this point, too.
Students kept sharing until eight different points are plotted.
Teacher: Does anyone see a pattern on his or her grid?
Ty: These eight points form a line. The line starts at the origin and goes up 1, over 2, up 1, over 2, and continues like that.
Teacher: Does anyone else have this pattern on his or her graph?
All students agreed this pattern showed up on everyone's graph.
Teacher: Why did this happen? What caused this pattern?
Rachel: The fraction ½ is the same as 6/12. The numerator and denominator of 6/12 are both 6 times as much as ½. In 4/8, they are each 4 times greater than ½. Each point we plotted had the same number multiplied by the numerator and denominator of ½. These are all equivalent fractions.
Teacher: Are there any points between the points you plotted on this line? Did we skip any equivalent fractions to ½?.
Ross: Nope... there are no numbers between 3 and 4 so there cannot be any equivalent fractions between 3/6 and 4/8.
Rachel: Yes there is... what if you made 1 - 3½ times greater and made 2 - 3½ times greater. The equivalent fraction would be 3½/7. I could go up 3½ and over 7. It would fit perfectly between 3/6 and 4/8. There would be many equivalent fractions between the equivalent fractions we plotted.
Teacher: How many would there be?
Rachel: I don't know. There are so many numbers between 3 and 4 I couldn't count them all. Would this be infinite?
Jenna: I think it would be infinite. There are an infinite number of numbers between 3 and 4 on a number line, so this would be infinite too.
Teacher: Since there are an infinite number of equivalent fractions between the eight fractions we plotted, how can we graph all of them?
Rick: We can draw a line through them. That way we plot them all.
Teacher: Rick, should we draw a line, line segment, or a ray?
Rick: What's the difference... just make it a line.
Teacher: The difference is how many endpoints we need. Line segments have 2 endpoints, rays have 1 endpoint and lines have 0 endpoints. Which one is our best choice?
Sara: I think it's a ray. We need one endpoint at 0,0. The other end goes on forever to show there are an infinite number of equivalent fractions to ½.
The class agrees with Sara and they draw their ray.
Teacher: Remember when we started this lesson? I asked you about dividing one integer by another. What if we took those eight points you plotted and divided their numerators by the denominators. What happens? Try it.
Students work briefly...
Ty: Every time I divided a numerator by a denominator I got .5. Why did that happen?
Rachel: Simple... the numerator was half of the denominator. The .5 is an equivalent decimal for the fraction ½ or any other fraction on this line.
Teacher: Well... you told me it's not a line, but a ray. So... what if you extended your ray and made it a line. You included the points (-2, -1), (-6, -3), (-8, -4). What happens when you divide the numerator by the denominator?
Tom: Great... now we have to divide negative integers. What happens when you divide a negative by another negative number?
Sara: A negative divided by a negative is always a positive, but these are the same division problems we just did. We did 1 divided by 2, 3 divided by 6, and 4 divided by 8. The only difference now is they are all negative integers. The answers are all the same as before. It's always .5. Why does this happen?
Teacher: Does anyone see a relationship? If so, tell your partner. See if they agree with you.
The students discuss this question with their partner.
Teacher: What is the relationship between each fraction on this line?
Steve: If you take the numerator and divide it by the denominator, the answer is always .5. Every fraction, even the negative fractions, was equivalent to the original fraction ½. The decimal .5 is equivalent to ½. So every fraction on this line is equivalent to ½.
The teacher asks the class if they agree. The class thinks Steve is right about the equivalence of all the fractions on this line.
Teacher: So... this worked for ½, but does it work for any other fractions. Each group now must choose a different fraction and test this to see if it continues to work. You will be reporting out your results to the class.
The teacher works with his students so each group chooses a different fractional value. Each group plots the equivalences, graphs them and divides the numerators by denominators. Next, each group brings up their results, explains what they did and shows their findings with the document camera. Here's an example:
Rick and Sam: We chose the fraction 2/3. We plotted, from the origin, up 2, over 3; up 4, over 6; up 8, over 12; down 3, over 2 to the left; and down 9, over 6 to the left. They all fit on the same line. When we drew the line, we went off the grid going up to the right and down to the left. We think every point on that line is the same as the fraction 2/3. We think this is true because when we divided every point's numerator by the denominator we got .6666666 repeating. They are all equivalent to this repeating decimal value.
Teacher: Now that every group has reported out, they all found the same thing. Every time we divide a fraction integer numerator by its integer denominator we get an equivalent decimal value. Congratulations! You just found the property that explains what makes a number rational.
Ty: That's it? When you divide any fraction integer numerator by its integer denominator and you get a decimal that comes out even or repeats, that's a rational number? Oh... I get that idea.
Teacher: Good... now, how can we make comparisons between these rational numbers?
Ty: Decimals are easy to compare. We can just convert all the fractions to decimals.
Teacher: Sometimes decimals are needed, but other times we need to keep values in their fraction form. How can we make comparisons with these fractions?
Rachel: I noticed, as students were presenting, that the greater the value the fraction, the steeper the line. Is that how we can compare fractions?
Teacher: Good observation, Rachel. You are correct, but I would not want to draw this graph every time we needed to compare fractions.
Steve: We could just draw a number line and mark them out. That would be easier.
Teacher: True, that would be easier, but still that would be a lot of drawing and time to construct a diagram. Is there another way?
Sara: I noticed something on the coordinate grid. If we wanted to compare... say 2/3 to ¾, we could look for a vertical line where both graphs cross it on grid lines. The 2/3 line crosses at 8/12, and the ¾ line crosses at 9/12. So ¾ is greater than 2/3 by 1/12.
Teacher: Good observation. You found equivalent fractions with a common denominator. So we can compare rational numbers by steepness of lines on the coordinate grid, location on a number line, equivalent decimals, or equivalent fractions. There are many ways to compare rational number. Good work today, class!
Resources
Teacher Notes
- Seventh grade students need exposure to the square root symbol to understand what it means and represents.
- In order to fully develop an understanding of rational numbers, teachers will need to contrast rational numbers with non-rational (or irrational) numbers. Students should be introduced to a few examples of irrational numbers like pi ([math]\pi[/math]), [math]\sqrt3[/math], [math]\sqrt5[/math] . Teachers may want to talk about the square root of perfect squares as additional examples of rational numbers.
- The term "integer" is very important to define over and over again throughout 7th grade, but particularly with these benchmarks. Students' understanding of rational numbers is linked to a deeper understanding of the term "integer."
- Remind students that the all integers can also be written as fractions, such as 3 = 3/1. By seeing this representation, they can see that those are rational numbers, too.
- When students examine numbers such as [math]\frac{1}{7}[/math]which equals [math]0.\overline{142857}[/math], a misconception might result, in that students won't see the pattern on the calculator and might believe the number is not rational. If this happens, the student is not understanding the overall concept of a rational number being a number that can be expressed as the quotient of two numbers (ratio of two numbers). The original fraction being discussed, ([math]\frac{1}{7}[/math]), is clearly a rational number. They need to understand that the division of two integers will ALWAYS result in a rational number.
- When talking about numbers that are not rational, make sure students make the connection that pi is just approximated by 22/7 and 3.14. Pi is not equal to either of those two numbers. Pi is actually a number that goes on forever without repeating and cannot be expressed as a ratio of two integers and therefore is not rational (irrational).
- When we express a rational number as a decimal, then either the decimal will
Be exact, as [math]\frac{1}{4}[/math] = .25, or it will not, as [math]\frac{1}{3}[/math] ≈ .3333. Nevertheless, there will be a
predictable pattern of digits. But if we attempted to express an irrational number as an exact decimal, then, clearly we could not, because if we could, then the number would be rational. Moreover, there will not be a predictable pattern of digits. For example,
- [math]\sqrt2[/math] ≈ 1.4142135623730950488016887242097
Now, with rational numbers you sometimes see [math]\frac{1}{11}[/math] = 0.0909090909...
In this lesson, students use real-world models to develop an understanding of fractions, decimals, unit rates, proportions and problem solving. The three activities in this investigation center on situations involving rational numbers and proportions that students encounter at a bakery. These activities involve several important concepts of rational numbers and proportions, including partitioning a unit into equal parts, the quotient interpretation of fractions, the area model of fractions, determining fractional parts of a unit not cut into equal-sized pieces, equivalence, unit prices and multiplication of fractions.
Additional Instructional Resources
The Rational Number Project is an ongoing research project investigating student learning and teacher enhancement. RNP advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations.
The ancient Greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction), but one of his students, Hippasus, proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational. However, Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!
rational number: a number expressible in the form a/b or (- a/b) for some fraction a/b. The rational numbers include the integers and numbers that can be expressed as a quotient of two integers, where the divisor is not zero.
Examples:
[math]\frac{799}{1000}[/math], [math]\frac{1}{2}[/math], [math]\frac{9}{11}[/math], and [math]-\frac{7}{5}[/math] are rational numbers.
0.799 is a rational number, since 0.799=[math]\frac{799}{1000}[/math].
integer: the whole numbers and their opposites. 0 is an integer, but is neither positive nor negative. The integers from -4 to 4 are shown on the number line below.
terminating decimal: a decimal that contains a finite number of digits (0.5, 0.25, 0.4, 0.125, etc.)
repeating decimal: a decimal number in which a digit or group of digits repeats without end ([math]0.\overline{3}[/math], [math]0.\overline{66}[/math],
negative number: a number less than 0. On a number line, negative numbers are located to the left of 0; on a vertical number line, negative numbers are located below 0.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- What instructional strategies can be used to engage students with investigating the structure of rational numbers?
- What different representations can be used to illustrate relationships of rational numbers?
- What are the expected misconceptions students will have with rational numbers? What responses will help them develop understanding?
- Reflection questions to be used with all 7th grade standards:
How can the instruction be scaffolded for students?
Do the tasks that have been designed connect to underlying concepts or focus on memorization?
How can it be determined if students have reached this learning goal?
How did the lesson be differentiated?
Materials
- Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F.(Skip), Lewandowski, S., Trevino, E., and Zbiek, R.M. (2006). Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: National Council of Teachers of Mathematics.
- Linder-Scholer, W. (1998). SciMathMN: Minnesota K-12 Mathematics Frameworks. St. Paul, MN: SciMathMN.
- Lappan, G., Fey, J., Fitzgerald, W., Friel, S., and Philips, E. (2009). Accentuate the Negative, CMP2. Boston, Mass: Pearson.
- Introduction to negative numbers
- Number types
Numbers are classified according to type. The first type of number is the first type you ever learned about: the counting, or "natural" numbers. - Rational and irrational numbers
- Coordinate plane and coordinates
Assessment
1. On his mathematics test, Carlos had 20 correct out of 25 problems. Which of the following is NOT another way of expressing 20 out of 25?
A. [math]\frac{4}{5}[/math] B. 0.80 C. 80% D. [math]\frac{5}{4}[/math]
Answer: D
DOK Level: 2
Source: FCAT (Florida Comprehensive Assessment Test) grade 7, released August, 2006
2. An equation is shown.
n = 1 ÷ 17
Which describes n?
A. Integer
B. Irrational
C. Rational
D. Whole
Answer: C
DOK Level: 2
Source: Minnesota MCA Series III Mathematics' Item Sampler, Grade 7
3. Which one of the following numbers is not rational?
A. 7/16 B. 7/1 C. 7/0 D. 0.7
Answer: C
DOK Level: 2
Differentiation
Struggling Learners
- Provide struggling learners with a calculator to do division problems.
- Provide number lines for the students to use.
- Start with warm-ups for the day, such as writing different numbers; give them fractional values, decimal values, positives and negatives. Let them use a calculator to evaluate the numbers given (write the values in a table), then show them the square root (√) symbol and give them values under the square root symbol to have them evaluate on their calculators (adding them to their tables). Have them compare to the values they got earlier. Talk about the similarities and differences in any of the values they got.
- The use of scientific (preferably graphing) calculators would work best on this topic so the students can see what they have entered into their calculators.
- Review the non-mathematical meaning of the term "rational" - people can act in a rational or irrational manner. This term may be confusing due to the different contexts it can be used in.
- The word "negative" may confuse with 'no' or 'not' use of the term.
- Introduce the concept of square root/irrational numbers.
- Have students plot numbers, including square roots, on a number line, extending into the negative as well.
- Which natural numbers have rational square roots?
Only the square roots of the square numbers are rational; that is, the square roots of the perfect squares.
[math]\sqrt1[/math] = 1 Rational
[math]\sqrt2[/math] Irrational
[math]\sqrt3[/math] Irrational
[math]\sqrt4[/math] = 2 Rational
[math]\sqrt5[/math], [math]\sqrt6[/math], [math]\sqrt7[/math], [math]\sqrt8[/math] Irrational
[math]\sqrt9[/math] = 3 Rational
And so on.
Rational number concepts activity
This activity explores Egyptian achievements in mathematics and helps students understand how ancient Egyptians used hieroglyphics to write numerals and how to multiply, divide and write fractions using their methods.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
using calculators to see the decimal representations of different fractional values. | give examples of fractional values to try by hand first and then with a calculator. |
exploring situations where a number can or cannot be described as a ratio of two integers (such as the length of a square with an area of 5 sq. units). | using correct vocabulary terms: rational, terminating, repeating and integer. |
learning multiple ways of describing repeating decimals. | making sure students understand that a rational number is the ratio of two integers. |
dividing numbers to find the decimal value, first on paper and then using a calculator to see equivalent ones have same decimal value. | using multiple representations to engage students in different ways of understanding rational numbers. |
Parent Resources
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