6.1.1A Locate & Compare Numbers

Grade: 
6
Subject:
Math
Strand:
Number & Operation
Standard 6.1.1

Read, write, represent and compare positive rational numbers expressed as fractions, decimals, percents and ratios; write positive integers as products of factors; use these representations in real-world and mathematical situations.

Benchmark: 6.1.1.1 Number Line & Coordinate Grid

Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.

Benchmark: 6.1.1.2 Compare Numbers

Compare positive rational numbers represented in various forms. Use the symbols  < ,  =  and  >.

For example: [math]\frac{1}{2}[/math] > 0.36.

Overview

Big Ideas and Essential Understandings 

Standard 6.1.1 Essential Understandings

Relationships of equivalence with different forms of rational numbers can be illustrated in a variety of representations. Students use fractions, decimals, and percents to describe equivalent positive rational numbers. Clement (2004) suggests five different kinds of representations for teaching students the concepts of fractions, decimals and percents. These five representations are pictures, manipulatives, spoken language, written symbols, and relevant situations. Conceptual understanding of these equivalencies is developed as students describe these rational numbers in concrete representational forms (fraction strips, Cuisenaire rods, pattern blocks, etc.); visual representational forms (grids, diagrams, pictures, etc.); and abstract symbolic form. Using these same structures, students expand their understanding of equivalence with rational numbers to making comparisons between them. Students' learning experiences with these different forms of representation guide them in identifying and selecting appropriate forms for making comparisons and conversions in a particular situation.

The skills of prime factorization, least common multiple, and greatest common factor become tools for students in their formation of equivalent fractional numbers. Student understanding of representing whole numbers as a product of factors with exponents is aided by their previous work with whole numbers, multiples, factors and exponents.

 All Standard Benchmarks

●      6.1.1.1 Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.

●      6.1.1.2 Compare positive rational numbers represented in various forms. Use the symbols <, =, and >.

●      6.1.1.3 Understand that percent represents parts out of 100 and ratios to 100.

●      6.1.1.4 Determine equivalences among fractions, decimals, and percents: select among these representations to solve problems.

●      6.1.1.5 Factor whole numbers; express a whole number as a product of prime factors with exponents.

●      6.1.1.6 Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions.

●      6.1.1.7 Convert between equivalent representations of positive rational numbers.

Benchmark Cluster 

Benchmark Group A

  • 6.1.1.1 Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid.
  • 6.1.1.2 Compare positive rational numbers represented in various forms. Use the symbols <, =, and >.

 What students should know and be able to do [at a mastery level] related to these benchmarks.

 ●      Use a number line to locate and compare rational numbers, including fractions, decimals, and percents;

●      Convert between fractions, decimals, and percents;

●      Represent the relationship between two positive rational numbers using <, =, or >;

●      Identify the origin, horizontal (x), and vertical (y) axes on a coordinate grid;

●      Plot ordered pairs of rational numbers on a coordinate grid.

Work from previous grades that supports this new learning includes:

●      Read and write decimals using place value to describe decimals in terms of groups from millionths to millions; recognize the numbers to the left are less than the numbers to the right on a number line;

●      Find 0.1 more than a number and 0.1 less than a number.  Find 0.01 more than a number and 0.01 less than a number.  Find 0.001 more than a number a 0.001 less than a number;

●      Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line;

●      Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts;

●      Round numbers to the nearest 0.1, 0.01, and 0.001;

●      Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system.

Correlations 

NCTM Standards

 Understand numbers, ways of representing numbers, relationships among numbers, and number systems

  • work flexibly with fractions, decimals, and percents to solve problems;
  • compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
  • develop meaning for percents greater than 100 and less than 1;
  • understand and use ratios and proportions to represent quantitative relationships;
  • use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

 

Common Core State Standards

4NF (NUMBER AND OPERATIONS -  FRACTIONS) Extend understanding of fraction equivalence and ordering.

  • 4NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

4OA. (OPERATIONS AND ALGEBRAIC THINKING) Gain familiarity with factors and multiples.

  • 4OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

5NBT (NUMBER AND OPERATIONS IN BASE TEN) Understand the place value system.

  • 5NBT.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

6NS (NUMBER SYSTEM) Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

  • 6NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
  • 6NS.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.
  • 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.
  • 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.
  • 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.
  • 6NS.7 Understand ordering and absolute value of rational numbers.
  • 6NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -30C > -70C to express the fact that -30C is warmer than -70C.

6RP (RATIOS AND PROPORTIONAL RELATIONSHIPS) Understand ratio concepts and use ratio reasoning to solve problems.

  • 6RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6RP3.c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Misconceptions

Student Misconceptions 

Student Misconceptions and Common Errors

  • When partitioning a whole on a number line to locate fractional parts, students may count partition marks rather than pieces resulting in an answer of fifths rather than sixths; or count the beginning and end marks, resulting in sevenths rather than sixths.
  • When comparing fractions, students may focus only on the size of the denominator.

Example: [math]\frac{3}{8}[/math] is NOT larger than [math]\frac{3}{4}[/math] even though the number 8 is larger than the number 4.

  • Overgeneralizing can be a problem when comparing same size of fraction with differing sized wholes.

Example: [math]\frac{1}{4}[/math] cup as compared to [math]\frac{1}{4}[/math] gallon

  • When comparing decimals, students may not have an understanding of place value to the right of the decimal point. Example: Students may believe that 5.155 is greater than 5.3 because there are more digits.
  • Students may believe that adding "0" to the end of a decimal makes it 10 times larger, since 530 is 10 times larger than 53.When considering adding a "0" to the end of a whole number, the value changes. Students sometimes do not recognize that value of a decimal does not change when a "0" is added to the end of a decimal.
  • Students confuse the symbols for "less than" and "greater than;"
  • Students may not understand the order of the coordinates when locating points on a coordinate grid;
  • Students struggle identifying the coordinates of points located on the horizontal or vertical axes.

Vignette

In the Classroom

In this vignette, students discover a mystery word by graphing ordered pairs on a coordinate grid.

Teacher:  At your desks, each of you has a piece of graph paper and a set of directions that will lead you to discover a mystery word.  It's important to follow the directions carefully. Sometimes you'll be asked to connect the points in order and sometimes the points will not get connected. 

Directions:

Use your graph paper to draw a coordinate grid.  Clearly label the x- and y- axes.

Plot the following ordered pairs and connect them in the order given.

Start (8, 3), (8, -3) Stop

Start (13, 3), (13, -3) Stop

Start (8, 0), (13, 0) Stop

 

Start (-6, 3), (-6, 0), (-3.5, 3), (-1, 0), (-1, -3) Stop

Start (-1, 0), (-6, 0) Stop

 

Start (1, 3), (6, 3) Stop

Start (3.5, -3), (3.5, 3) Stop

 

Start (-13, -3), (-13, 3), (-10.5, 0), (-8, 3), (-8, -3) Stop

 

Teacher: Let's do the first step together. The directions say, "Start (8, 3), (8, -3) Stop." What do you think that means?

Student: I think it means to start by graphing a point at (8, 3), then draw a line to connect that point to a point at (8, -3), and then stop.

Teacher: So how do you know where to graph the point (8, 3)? Student: Well, you start at the origin, move 8 units to the right, then up 3 units and plot the point.

Teacher: Tell me about the 'origin'.

Student: The origin is the point at (0, 0). It's the place where you always start when you locate points.

Teacher: That makes sense, because 'origin' means starting point. Why do you think the origin (0, 0), has two numbers?

Student: All points on the coordinate grid have two numbers. They're an ordered pair.

Teacher: Why did you describe the numbers as an 'ordered' pair?

Student: Because the order of the two numbers matters. The first number tells you how far to move right or left, and the second number tells you how far to move up or down.

Teacher: So what happens if you move up or down first, then move left or right?

Student: You end up in a different place.

Teacher: Show me an example.

Student: Sure. Here's where (8, 3) and (3, 8) are located.

Teacher: We can see that reversing the order of the numbers results in a different location, so the order certainly matters. It's very important to know that the first number tells you how far to move left or right, and the second number tells how far to move up or down. What strategies do you use to help you remember that?

Student: I think about the coordinate axes as two number lines - one horizontal and the other vertical. In elementary school, we often used horizontal number lines. We used them to learn about all sorts of things, like adding, subtracting, multiplying, and dividing. I think about learning math when I was younger and that helps me remember to begin with the horizontal number line and move right or left first.

Student: I have another way to remember the order. My teacher last year told me to think of a ladder. You must walk to the ladder first and then go up or down.

Teacher: Those are two good strategies. Thank you for sharing. Yes, the coordinate axes are really two number lines that intersect at right angles. We refer to the horizontal axis as the x-axis and the vertical axis as the y-axis. The origin (0, 0), is the point where the two axes intersect.

Student: I use another strategy to help me remember the order of the numbers.

Teacher: What's that?

Student: Well, the ordered pair is in alphabetical order. First is the x-coordinate, then the y-coordinate. The x-coordinate tells you how far to move on the x-axis and the y-coordinate tells you how far to move on the y-axis.  I draw an imaginary line at those two places, and I know that the place where those lines intersect is where the point is located.

Teacher: Show us how you would use that strategy to locate (8, 3).

Student: Like this.

Teacher: That's the same location as we found using the other strategy. How do you remember that the horizontal axis is the x-axis and the vertical axis is the y-axis?

Student: Easy. I think about drawing an outdoor picture. I would start by drawing the horizon, or horizontal axis. Because the ordered pair is in alphabetical order and x comes before y, I know that the horizontal axis is the x-axis. Then the vertical axis must be the y-axis.

Teacher: Clever. Sounds like you have some strategies to help you discover the mystery word.  Go ahead and begin your investigation.

(Students discover the Mystery Word is "MATH.")

Student:  "Math" isn't much of a mystery word in math class.

Teacher:  Good point. Perhaps you have other ideas for words that could be used as mystery words.  That can be tomorrow's homework assignment.  I want each of you to choose a Mystery Word and write the directions necessary to discover it using a coordinate grid.  Tomorrow you'll be given time in class for your partners to discover your Mystery Word.

 

Resources

Instructional Notes 

Teacher Notes

  • Determining whether a student understands partitioning a whole on a number line can be assessed informally by observation. When misconceptions are observed, it is helpful to highlight divided sections to aid in the correct identification of numbered parts.
  • Many students understand [math]\frac{4}{5}[/math] to be "4 parts out of 5." These students are confused when they encounter [math]\frac{7}{5}[/math] as they do not understand how you can have "7 parts out of 5." Be aware of the language used when talking about fractions. Emphasize the relationship as part-to-whole, rather than part "out of" whole. Students can visualize [math]\frac{7}{5}[/math] when they think of having 7 leftover brownies, with a whole pan being cut into 5 parts.
  • Comparisons among fractions with different denominators are difficult for most students. Draw pictures with students to visualize the fractions. For situations when the pictures aren't sufficient to make comparisons, emphasize the need to have the whole cut into the same number of parts and find equivalents;
  • Instead of always encouraging students to simplify every time, remind students to look for the equivalent form that best expresses what they want or need to say.
  • Students will need a classroom "museum" filled with opportunities to work with specific fractions of various sizes.

Example: [math]\frac{1}{4}[/math] cup and [math]\frac{1}{4}[/math] gallon vary greatly in size even though both use the measure of [math]\frac{1}{4}[/math].

  • It will be helpful to review converting between equivalent representations of rational numbers before asking students to make comparisons between fractions, decimals, and percents.
  • Demonstration of the addition of a "0" to both a whole number and a decimal number may need to be given. For example: If we add a "0" to the back of 54, we have 540; but, if we add a "0" to the end of 5.4 to make 5.40, we still have 5.4 as a value.
  • A quick review with the symbols used for comparing numbers may be needed as a refresher for students.
  • Students may try to round numbers that have repeating decimals and incorrectly order the numbers. Remind them that even though you often round repeating decimals, the decimal is actually less than or greater than the rounded decimal.

Example: [math]0.\bar{3}[/math] is greater than 0.3 and less than 0.33.

  • When asking students to plot points on a coordinate grid, it will be helpful to refer to the coordinates as ordered pairs, to help students make the connection that order matters.

 

Instructional Resources 

Students practice naming and graphing ordered pairs, and use the coordinate plane to create geometric figures.

Students create equivalent fractions by dividing and shading squares or circles, and match each fraction to its location on the number line.

Author: Francine Nettesheim

2009© Wisc-Online.com

Students use a coordinate grid to plot points in all four quadrants.

Students explore different representations for fractions, including improper fractions, mixed numbers, decimals, and percents.

This applet allows students to work with relationships among fractions and ways of combining fractions.

 Additional Instructional Resources

 

New Vocabulary 

coordinate grid: a grid for locating points in a plane by using ordered pairs of numbers. It is formed by two number lines that intersect at right angles at their zero points. Example:

horizontal axis: positioned in a left-to-right position, parallel to the line of the horizon; referred to as the x-axis.  Example:

 

integers: a number in the set {... -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.  All integers are rational numbers, but not all rational numbers are integers.  Example: [math]-\frac{1}{4}[/math] is a rational number but not an integer.) 

            All whole numbers are integers, but not all integers are whole numbers. 

            Example: -2 is an integer, but not a whole number.)

            Examples: 3 and -3

 is greater than (>): a number "is greater than" another number if its value is larger than the compared number.  When two numbers are plotted on the number line, the number with the larger value will be on the right.  Example:

is less than (<): a number "is less than" another number if its value is smaller than the compared number.  When two numbers are plotted on the number line, the number with the smaller value will be on the left.  Example:

ordered pair (x,y): a pair of numbers used to locate a point on a coordinate grid.  The first number corresponds to a position along the horizontal (x) axis, and the second number corresponds to a position along the vertical (y) axis.  Example:

per: for each, or for every.  Example:  If apples cost $1.99 per pound, then each pound of apples costs $1.99.

percent (%):  per hundred, or out of 100

Example:  "12% of sixth grade students are left-handed" means that out every 100 sixth grade students, 12 are left-handed.

positive integer: an integer greater than zero   Example:

ratio:  comparison of two quantities by division.  This relationship can be expressed as a fraction, decimal, or percent, as well as with a colon or in words. Note: Fractions can be used to express both part-to-part and part-to-whole relationships. It is recommended to avoid writing part-to-part relationships as [math]\frac{a}{b}[/math] to avoid confusion with part-to-whole relationships. Example:  If a team wins 3 games out of 5 games played, the ratio of wins to total games played can be written as [math]\frac{3}{5}[/math], 0.6, 60%, 3 to 5, 3:5  or "three to five."

rational number: any number that can be expressed in the form [math]\frac{a}{b}[/math] where a and b are integers and b≠0.  A rational number can always be represented by either a terminating or a repeating decimal.  Examples:  [math]\frac{2}{3}[/math], 4 (which can be expressed as [math]\frac{4}{1}[/math]);  2.25 (which can be expressed as [math]\frac{225}{100}[/math]).

simplify (fraction): to express in simplest form, or lowest terms.  The numerator and denominator of proper fractions in simplest form have no common factor other than 1. Improper fractions and mixed numbers are in simplest form when the fraction part is proper and in simplest form.  Examples: The numerator and denominator of [math]\frac{4}{8}[/math] share the common factor 4, so must be rewritten as [math]\frac{1}{2}[/math] to be in simplest form; [math]\frac{10}{4}[/math] written in simplest form is [math]\frac{5}{2}[/math] or [math]2\frac{1}{2}[/math].

x-axis: the horizontal number line in the coordinate planeExample:

vertical axis: the vertical number line in the coordinate plane.  Example:

 y-axis: the vertical axis.  Example:

Professional Learning Communities 

Reflection - Critical questions regarding the teaching and learning of these benchmarks

  • Beyond procedural knowledge, what does a student need to understand about place value and the number system to make conversions?
  • How can we help students compare numbers when they struggle to translate among fractions, decimals, and percents?
  • How can we reinforce the relationship among decimals, percents, and fractions?
  • How can we connect locating points on a number line to locating pairs of points on the coordinate grid?
  • What connections can be made to real life to motivate students?

 

Materials

  • Cramer, K., Behr, M., Post T., Lesh, R. (2009). The Rational Number Project  (choose RNP: Initial Fraction Ideas)The Rational Number Project (RNP) advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations.
  • Clarke, D. M., Roche, A., & Mitchell, A. Ten practical tips for making fractions come alive and make sense.Article discusses key ideas and concepts involved in understanding fractions and offers a range of hints to the classroom teacher on how to support students to develop a connected understanding of this important topic. Includes a fractions activity sheet. Located at:


 

References 

Clement, L. L. (2004). A model for understanding, using, and connecting representations. Teaching children mathematics, 11 (2), 97 - 102.

Keeley, P., & Rose, C. (2006). Mathematics curriculum topic study. Thousand Oaks, CA: Corwin Press.

Kilpatrick, J., Martin, W., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

Mathematics Curriculum Framework. (2000). Malden, MA: Massachusetts Department of Education.

Minnesota's K-12 Mathematics Frameworks. (1998). St. Paul, MN:  SciMathMN.

Mathematics Framework for the 2009 National Assessment of Educational Progress. (2009).   Washington, D.C.: National Assessment Governing Board U.S. Department of Education.

Mathematics 6-8 GaDOE: Georgia Department of Education, n.d. Web. 29 Mar. 2011.

Cramer, K., Behr, M., Post T., Lesh, R. (2009).The Rational Number Project   (choose RNP: Initial Fraction Ideas) The Rational Number Project (RNP) advocates teaching fractions using a model that emphasizes multiple representations and connections among different representations.

Behr, M., & Post, T. (1992). Teaching rational number and decimal concepts.  In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201-248). Boston: Allyn and Bacon.

Assessment

(DOK: Level 1)

1.      Find the value of x


Answer:  x = 0.7

 

(DOK: Level 1)

2.      Place the ordered pairs on the coordinate grid shown below.

(9, 0)                      (2, 6)                (8, 5)                (0, 7)

Answer:

 

(DOK: Level 2)

3..      Place each of the numbers on the number line shown below.

  [math]1\frac{1}{4}[/math]                  0.7                   2.3                   [math]\frac{3}{2}[/math]

Answer: 

 

(DOK: Level 2)

4.      Tell whether the following statements are true or false.

____ a. [math]\frac{3}{5} < 0.6[/math]

____ b. [math]0.3 = 0.33[/math]

____ c. [math]\frac{2}{3} > \frac{2}{5}[/math]

____ d. [math]5.4 < \frac{5}{4}[/math]

____ e. [math]\frac{2}{5} > 25%[/math]

Answers:

_F__ a. [math]\frac{3}{5} < 0.6[/math]

_F__ b. [math]0.3 = 0.33[/math]

_T__ c.  [math]\frac{2}{3} > \frac{2}{5}[/math]

_F__ d. [math]5.4 < \frac{5}{4}[/math]

_F__ e. [math]\frac{2}{5} > 25%[/math]

   

(DOK: Level 3)

5.      Marta and Jeff each ordered pizzas - the same size pizza.  Marta cut her pizza into 6 pieces, and ate 3 pieces.  Jeff cut his pizza into 7 pieces and also ate 3 pieces.  Who ate more pizza?  Explain how you know.

Sample Answer:  Marta ate more pizza.  Although Marta and Jeff both ate 3 pieces of pizza, each of Jeff's pieces were smaller since he cut his pizza into 7 pieces rather than 6.

 

(DOK: Level 3)

6.      Reaching the Goal:  The goal of a community fundraiser was to earn $10,000.  The picture below shows the results after one week.  What part of the goal was reached after one week?  Express your answer as a fraction, decimal, and percent.  Explain how you found your answers.

Sample Answer:  [math]\frac{12}{25}[/math]; 0.48, or 48%; there are 5 pieces between 0 and 10,000 and each piece is broken into 5 parts.  Since 5 x 5 = 25, there are 25 parts total parts.  The thermometer shows that 12 of the 25 are colored in. To find a decimal, the denominator needs to be a power of 10.  Multiplying [math]\frac{12}{25}[/math]  by [math]\frac{4}{4}[/math] or 1, tells me that [math]\frac{48}{100}[/math] has the same value. [math]\frac{48}{100}[/math] written as a decimal is 0.48.  Since percent means out of 100, that is the same as 48%.

 

(DOK Level 4)

7.  

A)     Identify three numbers between 0.5 and 0.6. Express one as a decimal, one as a fraction, and one as a percent.

B)     Locate and label the numbers on the number line shown below.

C)     Write an expression using < to show their relationship.

D)     Tell how you could make the smallest number equivalent to the largest number.

 

Sample Answers:

A)  0.52, 57%, [math]\frac{59}{100}[/math]

B) 

C)  0.52 < 57% < [math]\frac{59}{100}[/math]

D)  Adding 0.07 to 0.52 results in 0.59, which is equivalent to [math]\frac{59}{100}[/math].

Differentiation

Struggling Learners 

Struggling Students

●      Provide constant use of the number line to make comparisons. Remind students that values increase as you move to the right. Include decimal and percent equivalencies for benchmarks.

●      Create a fraction "tool-kit." Have several strips that represent candy bars cut to the same dimensions. Spend several days folding the candy bars to represent common fraction denominators such as fourths, eighths, halves, thirds, sixths, twelfths, fifths, and tenths. Ask students to "compare" or "show" various fractions.  Example: "Which would you rather have? 3 fifths of a candy bar or 5 sevenths of a candy bar?"

●      Provide a rich and varied supply of manipulatives and tools such as geoboards, 100 grids, base-ten blocks, and number lines to represent rational numbers and make comparisons.  Example: Base-100 blocks can be used to compare 0.23 and 0.17, or geoboards can be used to compare [math]\frac{1}{4}[/math] to [math]\frac{3}{8}[/math].

  • Review fraction equivalency concepts before connecting to decimals or percents.

  • To prevent misconceptions and increase the understanding of decimals, fractions, and percents among special education students, it may help to develop students' understanding of rational numbers through percents first. Research suggests that humans easily see objects in proportional terms (full, half full, nearly empty, etc.) Students can estimate the "fullness" of beakers filled with water by assigning a numerical value from 1 to 100. Frequent exposure to percents can strengthen equivalency comprehension. Another benefit of beginning with percents is that every percentage has a corresponding fractional or decimal equivalent that is simple to determine. For example, 77% can easily be written as [math]\frac{77}{100}[/math] or 0.77. This conversion process is effortless and allows students to develop their own procedures rather than memorizing algorithms.
  • Provide pocket-sized fact charts to aid in equivalency.
  • Have students verbalize as they translate from one form to another.
  • Help students see relevancy of fractions, decimals, and percents by using real-world examples; e.g. discounts, sports statistics, sales tax.
  • Connect learning to real-world experiences. Example: Library materials are organized using the Dewey decimal system. To locate a book numbered 392.134, they need to understand that it comes between 392.13 and 392. 14, even though it appears "longer."

 

English Language Learners 
  • Commas and periods are used differently to denote place value in other countries. Example, 2,3141.5 in the United States is written as 2.341,5 in some South American countries. For these students, it is necessary to explicitly connect their prior experiences of notation to notation in the United States.
  • Use fractional language when describing decimals. Example: Read 2.75 as "2 and 75 hundredths" rather than "2 point 75." This reinforces place value and helps build a more conceptual understanding of decimals.
  • Encourage Think-Alouds. Example: Partner up an ELL student with another student with good English language skills to explain his/ her thinking.
  • Identify that the fraction [math]\frac{1}{4}[/math] can have many meanings. For example:
  • A part of a whole: [math]\frac{1}{4}[/math] of a pizza;
  • A part of a collection: [math]\frac{1}{4}[/math] of a group of students;
  • A measurement: [math]\frac{1}{4}[/math] of a mile;
  • A probability: 1 chance in 4;
  • A number: the number halfway between 0 and [math]\frac{1}{2}[/math]
  • A division: [math]{1}\div{4}[/math]

●      Make vocabulary connections explicit in your teaching. Example: In addition to identifying the denominator as the "number on the bottom," include that the denominator represents the number of equal-sized pieces into which the whole is cut.

  • Provide a picture dictionary.
  • Use graphic organizers such as the Frayer model shown below, for vocabulary development.

           

 

Extending the Learning 
  •     Ask students to use a city map (or create their own coordinate grid) to communicate their path to school using given map coordinates.
  • Have students play chess and record their moves using chess notation.

Chess Notation: This website can be used as a resource for teachers who are interested in teaching chess notation.

                       

Example: Ng1tof3 means moving the Knight from space g1 to space f3.

  • Chesskids This website provides a tutorial for learning about chess.

●      Have students create a deck of cards with fractions, decimals, and percents that can be used to play "War," a game where two players battle by comparing two cards and the player with the card showing the larger value "wins" the battle and takes the "loser's" card.  When the cards show equivalencies, the battle continues until a winner is declared.

●      Students can create pairs of cards showing unfamiliar equivalencies and use them to play Concentration, a memory game. Negative rational numbers could be included as a challenge.

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

Students are:  (descriptive list)         

Teachers are:  (descriptive list)        

using concrete and visual tools like number lines, fraction bars, geoboards, base-ten blocks, drawings, and drawings to compare fractions, percents, and decimals.

designing lessons that move students from the necessary concrete hands-on experiences and visual representations toward the abstract when making comparisons with fractions, percents, and decimals.

locating and comparing positive rational numbers using a number line.

assisting students with meta-cognition by encouraging self-reflective thinking. Example: how would you explain to a second grader that 1.8 is greater than [math]1\frac{3}{4}[/math] using a number line?  

comparing numbers and showing their relationships using <, >, and =.

asking students to compare numbers that require translation between forms, represent the relationships symbolically, and justify their thinking.

plotting ordered pairs of rational numbers on coordinate grid.

explaining that the coordinate axes are two intersecting number lines that form right angles and intersect at point 0, and connecting the location of points on a coordinate gird to previous experiences of locating points on a number line.

using a calculator for tedious computations necessary for comparing rational number.

ensuring students are utilizing their calculators to determine equivalencies.

discussing and writing about their mathematical reasoning.

requiring students to communicate their thinking.

 

 

Parents 

Parent Resources

This interactive website provides practice for locating points on a coordinate grid.

This website provides brief written instructions, a video tutorial, practice problems, and a quick quiz on converting fractions to decimals and percents.

Students identify equivalent fractions through various levels of difficulty to win points.

  • Ordering Dominoes

Use dominoes to represent fractions, as shown in the example below. Ask students to put 5 dominoes in ascending order and justify the order.  A calculator may be helpful for converting fractions

= [math]\frac{3}{5}[/math]

  • Students are asked to order 5 dominoes in ascending order and justify the order.
    Sample dominoes that work well when teaching basic equivalency include: [math]\frac{1}{3}, \frac{2}{5}, \frac{3}{5}, \frac{2}{4}, \frac{2}{3}, \frac{1}{4}, \frac{1}{5}[/math].