# 2.4.2: Navigating a Table of Equivalent Ratios - Mathematics

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## Lesson

Let's use a table of equivalent ratios like a pro.

Exercise (PageIndex{1}): Number Talk: Multiplying by a Unit Fraction

Find the product mentally.

(frac{1}{3}cdot 21)

(frac{1}{6}cdot 21)

((5.6)cdotfrac{1}{8})

(frac{1}{4}cdot (5.6))

Exercise (PageIndex{2}): Comparing Taco Prices

1. Noah bought 4 tacos and paid $6. At this rate, how many tacos could he buy for$15?
2. Jada’s family bought 50 tacos for a party and paid $72. Were Jada’s tacos the same price as Noah’s tacos? number of tacosprice in dollars Table (PageIndex{1}) Exercise (PageIndex{3}): Hourly Wages Lin is paid$90 for 5 hours of work. She used the table to calculate how much she would be paid at this rate for 8 hours of work.

1. What is the meaning of the 18 that appears in the table?
2. Why was the number (frac{1}{5}) used as a multiplier?
3. Explain how Lin used this table to solve the problem.
4. At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work?

Exercise (PageIndex{4}): Zeno's Memory Card

In 2016, 128 gigabytes (GB) of portable computer memory cost $32. 1. Here is a double number line that represents the situation: One set of tick marks has already been drawn to show the result of multiplying 128 and 32 each by (frac{1}{2}). Label the amount of memory and the cost for these tick marks. Next, keep multiplying by (frac{1}{2}) and drawing and labeling new tick marks, until you can no longer clearly label each new tick mark with a number. 1. Here is a table that represents the situation. Find the cost of 1 gigabyte. You can use as many rows as you need. memory (gigabytes)cost (dollars) (128)(32) Table (PageIndex{2}) 2. Did you prefer the double number line or the table for solving this problem? Why? Are you ready for more? A kilometer is 1,000 meters because kilo is a prefix that means 1,000. The prefix mega means 1,000,000 and giga (as in gigabyte) means 1,000,000,000. One byte is the amount of memory needed to store one letter of the alphabet. About how many of each of the following would fit on a 1-gigabyte flash drive? 1. letters 2. pages 3. books 4. movies 5. songs ### Summary Finding a row containing a “1” is often a good way to work with tables of equivalent ratios. For example, the price for 4 lbs of granola is$5. At that rate, what would be the price for 62 lbs of granola?

Here are tables showing two different approaches to solving this problem. Both of these approaches are correct. However, one approach is more efficient.

• Less efficient
• More efficient

Notice how the more efficient approach starts by finding the price for 1 lb of granola.

Remember that dividing by a whole number is the same as multiplying by a unit fraction. In this example, we can divide by 4 or multiply by (frac{1}{4}) to find the unit price.

### Glossary Entries

Definition: Table

A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.

For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.

pettail length (inches)
dog(22)
cat(12)
mouse(2)
Table (PageIndex{3})

## Practice

Exercise (PageIndex{5})

Priya collected 2,400 grams of pennies in a fundraiser. Each penny has a mass of 2.5 grams. How much money did Priya raise? If you get stuck, consider using the table.

number of penniesmass in grams
(1)(2.5)
(2,400)
Table (PageIndex{4})

Exercise (PageIndex{6})

Kiran reads 5 pages in 20 minutes. He spends the same amount of time per page. How long will it take him to read 11 pages? If you get stuck, consider using the table.

time in minutesnumber of pages
(20)(5)
(1)
(11)
Table (PageIndex{5})

Exercise (PageIndex{7})

Mai is making personal pizzas. For 4 pizzas, she uses 10 ounces of cheese.

number of pizzasounces of cheese
(4)(10)
Table (PageIndex{6})
1. How much cheese does Mai use per pizza?
2. b. At this rate, how much cheese will she need to make 15 pizzas?

Exercise (PageIndex{8})

Lin is paid $90 for 5 hours of work. She used the following table to calculate how much she would be paid at this rate for 8 hours of work. 1. What is the meaning of the 18 that appears in the table? 2. Why was the number 1/5 used as a multiplier? 3. Explain how Lin used this table to solve the problem. 4. At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work? #### 12.4 - Zeno’s Memory Card In 2016, 128 gigabytes (GB) of portable computer memory cost$32.

Here is a double number line that represents the situation:
Show Number Line

One set of tick marks has already been drawn to show the result of multiplying 128 and 32 each by 1/2. Label the amount of memory and the cost for these tick marks.

Next, keep multiplying by 1/2 and drawing and labeling new tick marks, until you can no longer clearly label each new tick mark with a number.

1. Here is a table that represents the situation. Find the cost of 1 gigabyte. You can use as many rows as you need.

#### Are you ready for more?

A kilometer is 1,000 meters because kilo is a prefix that means 1,000. The prefix mega means 1,000,000 and giga (as in gigabyte) means 1,000,000,000. One byte is the amount of memory needed to store one letter of the alphabet. About how many of each of the following would fit on a 1-gigabyte flash drive?

#### Lesson 12 Practice Problems

1. Priya collected 2,400 grams of pennies in a fundraiser. Each penny has a mass of 2.5 grams. How much money did Priya raise? If you get stuck, consider using the table.
1. Kiran reads 5 pages in 20 minutes. He spends the same amount of time per page. How long will it take him to read 11 pages? If you get stuck, consider using the table.
1. Mai is making personal pizzas. For 4 pizzas, she uses 10 ounces of cheese.

a. How much cheese does Mai use per pizza?
b. At this rate, how much cheese will she need to make 15 pizzas?

Lin is paid $90 for 5 hours of work. She used the table to calculate how much she would be paid at this rate for 8 hours of work. Expand Image Description: <p>A 2-column table with 5 rows of data. The first column is labeled "amount earned, in dollars" and the second column is labeled "time worked, in hours." Row 1: 90, 5 Row 2: 18, 1 Row 3: 144, 8 Row 4 and 5: blank. An arrow pointing from row 1 to row 2 is labeled "times one-fifth." Another arrow pointing from row 2 to row 3 is labeled "times 8."</p> 1. What is the meaning of the 18 that appears in the table? 2. Why was the number (frac15) used as a multiplier? 3. Explain how Lin used this table to solve the problem. At this rate, how much would Lin be paid for 3 hours of work? For 2.1 hours of work? ### 12.4: Zeno’s Memory Card In 2016, 128 gigabytes (GB) of portable computer memory cost$ 32.

Here is a double number line that represents the situation:

Expand Image

One set of tick marks has already been drawn to show the result of multiplying 128 and 32 each by (frac12) . Label the amount of memory and the cost for these tick marks.

Next, keep multiplying by (frac12) and drawing and labeling new tick marks, until you can no longer clearly label each new tick mark with a number.

Here is a table that represents the situation. Find the cost of 1 gigabyte. You can use as many rows as you need.

A kilometer is 1,000 meters because kilo is a prefix that means 1,000. The prefix mega means 1,000,000 and giga (as in gigabyte) means 1,000,000,000. One byte is the amount of memory needed to store one letter of the alphabet. About how many of each of the following would fit on a 1-gigabyte flash drive?

### Summary

Finding a row containing a “1” is often a good way to work with tables of equivalent ratios. For example, the price for 4 lbs of granola is $5. At that rate, what would be the price for 62 lbs of granola? Here are tables showing two different approaches to solving this problem. Both of these approaches are correct. However, one approach is more efficient. Expand Image Description: <p>A 2-column table with 6 rows of data. First column labeled "granola, in pounds," second column labeled "price, in dollars." The data is as follows: Row 1: 4, 5 Row 2: 8, 10 Row 3: 16, 20 Row 4: 32, 40 Row 5: 64, 80 Row 6: 62, 77.50. An arrow pointing from row 1 to row 2, then 2 to 3, then 3 to 4, then 4 to 5 is labeled "times 2". The last arrow from row 5 to 6 is labeled "subtract 2 pounds" on the left of the table and is labeled " subtract "$2.50" on the right.</p>

Expand Image

Description: <p>2-column table, 3 rows of data. First column labeled "granola, in pounds," second column labeled " price, in dollars." The data is as follows: Row 1: 4, 5 Row 2: 1, 1.25 Row 3: 62, 77.50. An arrow pointing from row 1 to row 2 is labeled "times one-fourth" and another arrow pointing from row 2 to row 3 is labeled "times 62."</p>

Notice how the more efficient approach starts by finding the price for 1 lb of granola.

Remember that dividing by a whole number is the same as multiplying by a unit fraction. In this example, we can divide by 4 or multiply by (frac14) to find the unit price.

### Glossary Entries

A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.

For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

## Finding Missing Values in a Table of Equivalent Ratios

You can find equivalent ratios by multiplying or dividing both terms of a ratio by the same number. This is similar to finding equivalent fractions of a given fraction. All the ratios in the tables below are equivalent.

The table below represent the equivalent ratios 1:3, 2:6, 3:9

The table below represent the equivalent ratios 1:4, 3:12, 5:20

Such tables of equivalent ratios can be used to find missing values as follows.

Find the missing values in the following table of equivalent ratios:

### Solution

Find the missing values in the following table of equivalent ratios :

$frac <6>= frac<10><3> x = frac<10> <3> imes 6 = frac<10> <3> imes frac<6> <1>= 20$

$frac <40>= frac<3><10> y = frac<3> <10> imes 40 = frac<3> <10> imes frac<40> <1>= 12$

Find the missing values in the following table of equivalent ratios:

### Solution

Since the table gives values of equivalent ratios

$frac <6>= frac<3><2> x = frac<3> <2> imes frac<6> <1>= frac<3> <2> imes frac<6> <1>= 9$

$frac <12>= frac<2><3> y = frac<2> <3> imes 12 = frac<2> <3> imes frac<12> <1>= 8$

## Tables of Equivalent Ratios

Video Solutions to help Grade 6 students learn how to use table ratios to solve problems.

### New York State Common Core Math Grade 6, Module 1, Lesson 9

Lesson 9 Student Outcomes

&bull Students understand that a ratio is often used to describe the relationship between the amount of one quantity and the amount of another quantity as in the cases of mixtures or constant rates.
&bull Students understand that a ratio table is a table of equivalent ratios. Students use ratio tables to solve problems.

A ratio table is a table of pairs of numbers that form equivalent ratios.

Example 1
To make Paper Mache, the art teacher mixes water and flour. For every two cups of water, she needs to mix in three cups of flour to make the paste.
Find equivalent ratios for the ratio relationship 2 cups of water to 3 cups of flour. Represent the equivalent ratios in the table below:

Example 2
Javier has a new job designing websites. He is paid at a rate of $700 for every 3 pages of web content that he builds. Create a ratio table to show the total amount of money Javier has earned in ratio to the number of pages he has built. Javier is saving up to purchase a used car that costs$4,300. How many web pages will Javier need to build before he can pay for the car?

Exercise 1
Spraying plants with “cornmeal juice” is a natural way to prevent fungal growth on the plants. It is made by soaking cornmeal in water, using two cups of cornmeal for every nine gallons of water. Complete the ratio table to answer the questions that follow.
a. How many cups of cornmeal should be added to 45 gallons of water?
b. Paul has only 8 cups of cornmeal. How many gallons of water should he add if he wants to make as much cornmeal juice as he can?
c. What can you say about the values of the ratios in the table?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

## Lesson Numbering for Learning Targets

In some printed copies of the student workbooks, we erroneously printed a lesson number instead of the unit and lesson number. This table provides a key to match the printed lesson number with the unit and lesson number.

Lesson Number Unit and Lesson Lesson Title
1 1.1 Tiling the Plane
2 1.2 Finding Area by Decomposing and Rearranging
3 1.3 Reasoning to Find Area
4 1.4 Parallelograms
5 1.5 Bases and Heights of Parallelograms
6 1.6 Area of Parallelograms
7 1.7 From Parallelograms to Triangles
8 1.8 Area of Triangles
9 1.9 Formula for the Area of a Triangle
10 1.10 Bases and Heights of Triangles
11 1.11 Polygons
12 1.12 What is Surface Area?
13 1.13 Polyhedra
14 1.14 Nets and Surface Area
15 1.15 More Nets, More Surface Area
16 1.16 Distinguishing Between Surface Area and Volume
17 1.17 Squares and Cubes
18 1.18 Surface Area of a Cube
19 1.19 Designing a Tent
20 2.1 Introducing Ratios and Ratio Language
21 2.2 Representing Ratios with Diagrams
22 2.3 Recipes
23 2.4 Color Mixtures
24 2.5 Defining Equivalent Ratios
25 2.6 Introducing Double Number Line Diagrams
26 2.7 Creating Double Number Line Diagrams
27 2.8 How Much for One?
28 2.9 Constant Speed
29 2.10 Comparing Situations by Examining Ratios
30 2.11 Representing Ratios with Tables
31 2.12 Navigating a Table of Equivalent Ratios
32 2.13 Tables and Double Number Line Diagrams
33 2.14 Solving Equivalent Ratio Problems
34 2.15 Part-Part-Whole Ratios
35 2.16 Solving More Ratio Problems
36 2.17 A Fermi Problem
37 3.1 The Burj Khalifa
38 3.2 Anchoring Units of Measurement
39 3.3 Measuring with Different-Sized Units
40 3.4 Converting Units
41 3.5 Comparing Speeds and Prices
42 3.6 Interpreting Rates
43 3.7 Equivalent Ratios Have the Same Unit Rates
44 3.8 More about Constant Speed
45 3.9 Solving Rate Problems
46 3.10 What Are Percentages?
47 3.11 Percentages and Double Number Lines
48 3.12 Percentages and Tape Diagrams
49 3.13 Benchmark Percentages
50 3.14 Solving Percentage Problems
51 3.15 Finding This Percent of That
52 3.16 Finding the Percentage
53 3.17 Painting a Room
54 4.1 Size of Divisor and Size of Quotient
55 4.2 Meanings of Division
56 4.3 Interpreting Division Situations
57 4.4 How Many Groups? (Part 1)
58 4.5 How Many Groups? (Part 2)
59 4.6 Using Diagrams to Find the Number of Groups
60 4.7 What Fraction of a Group?
61 4.8 How Much in Each Group? (Part 1)
62 4.9 How Much in Each Group? (Part 2)
63 4.10 Dividing by Unit and Non-Unit Fractions
64 4.11 Using an Algorithm to Divide Fractions
65 4.12 Fractional Lengths
66 4.13 Rectangles with Fractional Side Lengths
67 4.14 Fractional Lengths in Triangles and Prisms
68 4.15 Volume of Prisms
69 4.16 Solving Problems Involving Fractions
70 4.17 Fitting Boxes into Boxes
71 5.1 Using Decimals in a Shopping Context
72 5.2 Using Diagrams to Represent Addition and Subtraction
73 5.3 Adding and Subtracting Decimals with Few Non-Zero Digits
74 5.4 Adding and Subtracting Decimals with Many Non-Zero Digits
75 5.5 Decimal Points in Products
76 5.6 Methods for Multiplying Decimals
77 5.7 Using Diagrams to Represent Multiplication
78 5.8 Calculating Products of Decimals
79 5.9 Using the Partial Quotients Method
80 5.10 Using Long Division
81 5.11 Dividing Numbers that Result in Decimals
82 5.12 Dividing Decimals by Whole Numbers
83 5.13 Dividing Decimals by Decimals
84 5.14 Using Operations on Decimals to Solve Problems
85 5.15 Making and Measuring Boxes
86 6.1 Tape Diagrams and Equations
87 6.2 Truth and Equations
88 6.3 Staying in Balance
89 6.4 Practice Solving Equations and Representing Situations with Equations
90 6.5 A New Way to Interpret $a$ over $b$
91 6.6 Write Expressions Where Letters Stand for Numbers
92 6.7 Revisit Percentages
93 6.8 Equal and Equivalent
94 6.9 The Distributive Property, Part 1
95 6.10 The Distributive Property, Part 2
96 6.11 The Distributive Property, Part 3
97 6.12 Meaning of Exponents
98 6.13 Expressions with Exponents
99 6.14 Evaluating Expressions with Exponents
100 6.15 Equivalent Exponential Expressions
101 6.16 Two Related Quantities, Part 1
102 6.17 Two Related Quantities, Part 2
103 6.18 More Relationships
104 6.19 Tables, Equations, and Graphs, Oh My!
105 7.1 Positive and Negative Numbers
106 7.2 Points on the Number Line
107 7.3 Comparing Positive and Negative Numbers
108 7.4 Ordering Rational Numbers
109 7.5 Using Negative Numbers to Make Sense of Contexts
110 7.6 Absolute Value of Numbers
111 7.7 Comparing Numbers and Distance from Zero
112 7.8 Writing and Graphing Inequalities
113 7.9 Solutions of Inequalities
114 7.10 Interpreting Inequalities
115 7.11 Points on the Coordinate Plane
116 7.12 Constructing the Coordinate Plane
117 7.13 Interpreting Points on a Coordinate Plane
118 7.14 Distances on a Coordinate Plane
119 7.15 Shapes on the Coordinate Plane
120 7.16 Common Factors
121 7.17 Common Multiples
122 7.18 Using Common Multiples and Common Factors
123 7.19 Drawing on the Coordinate Plane
124 8.1 Got Data?
125 8.2 Statistical Questions
126 8.3 Representing Data Graphically
127 8.4 Dot Plots
128 8.5 Using Dot Plots to Answer Statistical Questions
129 8.6 Interpreting Histograms
130 8.7 Using Histograms to Answer Statistical Questions
131 8.8 Describing Distributions on Histograms
132 8.9 Mean
133 8.10 Finding and Interpreting the Mean as the Balance Point
135 8.12 Using Mean and MAD to Make Comparisons
136 8.13 Median
137 8.14 Comparing Mean and Median
138 8.15 Quartiles and Interquartile Range
139 8.16 Box Plots
140 8.17 Using Box Plots
141 8.18 Using Data to Solve Problems

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R the most common are " a

b " and " ab ", which are used when R is implicit, and variations of " a

on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X:

b if and only if b

X together with the relation

### Equivalence relations Edit

The following relations are all equivalence relations:

• "Is equal to" on the set of numbers. For example, 1 2 <2>>> is equal to 4 8 <8>>> . [3]
• "Has the same birthday as" on the set of all people.
• "Is similar to" on the set of all triangles.
• "Is congruent to" on the set of all triangles.
• "Is congruent to, modulon" on the integers. [3]
• "Has the same image under a function" on the elements of the domain of the function.
• "Has the same absolute value" on the set of real numbers
• "Has the same cosine" on the set of all angles.

### Relations that are not equivalences Edit

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 does not imply that 5 ≥ 7. It is, however, a total order.
• The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.
• The empty relationR (defined so that aRb is never true) on a non-empty set X is vacuously symmetric and transitive, but not reflexive. (If X is also empty then Ris reflexive.)
• The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point, then this defines an equivalence relation.
• A partial order is a relation that is reflexive, antisymmetric, and transitive. is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class.
• A strict partial order is irreflexive, transitive, and asymmetric.
• A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is serial, that is, if ∀aba

is an equivalence relation on X, and P(x) is a property of elements of X, such that whenever x

y, P(x) is true if P(y) is true, then the property P is said to be well-defined or a class invariant under the relation

A frequent particular case occurs when f is a function from X to another set Y if x1

x2 implies f(x1) = f(x2) then f is said to be a morphism for

, a class invariant under

, or simply invariant under

. This occurs, e.g. in the character theory of finite groups. The latter case with the function f can be expressed by a commutative triangle. See also invariant. Some authors use "compatible with

More generally, a function may map equivalent arguments (under an equivalence relation

A) to equivalent values (under an equivalence relation

B). Such a function is known as a morphism from

### Equivalence class Edit

A subset Y of X such that a

b holds for all a and b in Y, and never for a in Y and b outside Y, is called an equivalence class of X by

### Quotient set Edit

The set of all equivalence classes of X by

. If X is a topological space, there is a natural way of transforming X/

into a topological space see quotient space for the details.

### Projection Edit

Theorem on projections: [5] Let the function f: XB be such that a

bf(a) = f(b). Then there is a unique function g : X/

B, such that f = gπ. If f is a surjection and a

bf(a) = f(b), then g is a bijection.

### Equivalence kernel Edit

The equivalence kernel of a function f is the equivalence relation

### Partition Edit

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.

#### Counting partitions Edit

Let X be a finite set with n elements. Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number Bn:

A key result links equivalence relations and partitions: [6] [7] [8]

In both cases, the cells of the partition of X are the equivalence classes of X by

. Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by

, each element of X belongs to a unique equivalence class of X by

. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X.

and ≈ are two equivalence relations on the same set S, and a

b implies ab for all a,bS, then ≈ is said to be a coarser relation than

is a finer relation than ≈. Equivalently,

is finer than ≈ if every equivalence class of

is a subset of an equivalence class of ≈, and thus every equivalence class of ≈ is a union of equivalence classes of

is finer than ≈ if the partition created by

The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.

is finer than ≈" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice. [9]

• Given any set X, an equivalence relation over the set [XX] of all functions XX can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
• An equivalence relation

on X is the equivalence kernel of its surjective projection π : XX/

b if and only if there exist elements x1, x2, . xn in X such that a = x1, b = xn, and (xi, xi+1) ∈ R or (xi+1, xi) ∈ R, i = 1, . n−1.

be the equivalence relation on X defined by (a, 0)

(a, 1) for all a ∈ [0, 1] and (0, b)

(1, b) for all b ∈ [0, 1]. Then the quotient space X/

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

### Group theory Edit

Just as order relations are grounded in ordered sets, sets closed under pairwise supremum and infimum, equivalence relations are grounded in partitioned sets, which are sets closed under bijections that preserve partition structure. Since all such bijections map an equivalence class onto itself, such bijections are also known as permutations. Hence permutation groups (also known as transformation groups) and the related notion of orbit shed light on the mathematical structure of equivalence relations.

' denote an equivalence relation over some nonempty set A, called the universe or underlying set. Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀xAgG (g(x) ∈ [x]). Then the following three connected theorems hold: [11]

In sum, given an equivalence relation

over A, there exists a transformation group G over A whose orbits are the equivalence classes of A under

This transformation group characterisation of equivalence relations differs fundamentally from the way lattices characterize order relations. The arguments of the lattice theory operations meet and join are elements of some universe A. Meanwhile, the arguments of the transformation group operations composition and inverse are elements of a set of bijections, AA.

Moving to groups in general, let H be a subgroup of some group G. Let

be an equivalence relation on G, such that a

b ↔ (ab −1 ∈ H). The equivalence classes of

—also called the orbits of the action of H on G—are the right cosets of H in G. Interchanging a and b yields the left cosets.

Related thinking can be found in Rosen (2008: chpt. 10).

### Categories and groupoids Edit

Let G be a set and let "

" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows. The objects are the elements of G, and for any two elements x and y of G, there exists a unique morphism from x to y if and only if x

The advantages of regarding an equivalence relation as a special case of a groupoid include:

• Whereas the notion of "free equivalence relation" does not exist, that of a free groupoid on a directed graph does. Thus it is meaningful to speak of a "presentation of an equivalence relation," i.e., a presentation of the corresponding groupoid
• Bundles of groups, group actions, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies
• In many contexts "quotienting," and hence the appropriate equivalence relations often called congruences, are important. This leads to the notion of an internal groupoid in a category. [18]

### Lattices Edit

The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker: X^XCon X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: XX to its kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples:

• Reflexive and transitive: The relation ≤ on N. Or any preorder
• Symmetric and transitive: The relation R on N, defined as aRbab ≠ 0. Or any partial equivalence relation
• Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "ab is divisible by at least one of 2 or 3." Or any dependency relation.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

• The number of equivalence classes is finite or infinite
• The number of equivalence classes equals the (finite) natural number n
• All equivalence classes have infinite cardinality
• The number of elements in each equivalence class is the natural number n.

Euclid's The Elements includes the following "Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). By "relation" is meant a binary relation, in which aRb is generally distinct from bRa. A Euclidean relation thus comes in two forms:

(aRcbRc) → aRb (Left-Euclidean relation) (cRacRb) → aRb (Right-Euclidean relation)

The following theorem connects Euclidean relations and equivalence relations:

Theorem If a relation is (left or right) Euclidean and reflexive, it is also symmetric and transitive. Proof for a left-Euclidean relation (aRcbRc) → aRb [a/c] = (aRabRa) → aRb [reflexive erase T∧] = bRaaRb. Hence R is symmetric. (aRcbRc) → aRb [symmetry] = (aRccRb) → aRb. Hence R is transitive. ◻ >

with an analogous proof for a right-Euclidean relation. Hence an equivalence relation is a relation that is Euclidean and reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention.

1. ^"Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25 . Retrieved 2020-08-30 .
2. ^
3. Weisstein, Eric W. "Equivalence Class". mathworld.wolfram.com . Retrieved 2020-08-30 .
4. ^ abc
5. "7.3: Equivalence Classes". Mathematics LibreTexts. 2017-09-20 . Retrieved 2020-08-30 .
6. ^If: Given a, let a

b hold using seriality, then b

a by symmetry, hence a

a by transitivity. — Only if: Given a, choose b=a, then a

## RightStart™ Mathematics, second edition

RightStartMathematics is an intriguing program for kindergarten through junior high. It is difficult to convey within this review the thoroughness with which concepts are taught using multi-sensory learning approaches. The author, Dr. Joan Cotter, integrates concepts more than you find in most programs, and her development of conceptual understanding is outstanding. Her goal is to lay a much stronger foundation in mathematical thinking and understanding, and I think she accomplishes this very well.

RightStart Levels A through F are considered the elementary grade program. RightStart Levels G and H target students in grades six through eight. (These last two courses share a common set of manipulatives, and I review them separately after Levels A through F.)

While first editions of Levels A through E are still available, Levels A through F are now available as RightStart second editions, referred to by the publisher as RS2. I recommend the second editions for most situations since they are more user-friendly than the first editions. The scope and sequence, as well as the presentation, have been adjusted to reflect new research and discoveries about how children learn and to better align with the more advanced scope and sequence of newer math programs. This is particularly evident in Level A where the first edition has 77 lessons while the second edition has 132. The additional lessons address topics such as subtraction and fractions.

The scope and sequence of RightStart Mathematics is a bit different than other programs, and children should move through it at their own pace rather than treating each book as equivalent to a particular grade level. Because it covers so much and even teaches some concepts typically taught to older students, you might take longer than one school year to complete a level. The scope and sequence is very advanced in some areas, so you might find that your child needs to work in a lower level book. Remedial students should definitely start at a lower level. A free placement test is available on the publisher's website. I think that a child completing Level H in junior high will be functioning at a level equivalent to or higher than that of most other programs for the same level.

The second edition courses build in review lessons at the beginning of each course that will familiarize those new to the program with its unique methodology. (Math Set Bridges remain available for those using a first edition course who are moving into an RS2 course. See the publisher's website for information if you need one of these.)

For each level, you will need both the book bundle plus the set of manipulatives. Each book bundle has a lessons book and a separate worksheet book (either print or digital). There is significantly less worksheet activity compared to most programs. Much of the practice and drill takes place through learning activities and games. Because of this, RightStart should work especially well for students who don’t like to do a lot of writing.

Levels A through F use the same set of manipulatives called the RightStart™ Math Set. The basic manipulative set includes the AL Abacus, place value cards, base 10 picture cards, six special card decks for math games, fraction charts, Drawing Board Geometry Set, geoboards, color cubes, colored tiles, calculator, geared clock, math balance, tangrams, centimeter cubes, 4-in-1 ruler, folding meter stick, goniometer (angle measurer), a set of wooden geometry solids, and a set of plastic coins. (Note that the calculator is used infrequently—not as a substitute for mastering computation skills.) All of these components are non-consumable. You can customize this set to reflect a different monetary system at the time you order the set, selecting from U.S., Euro, Canadian, New Zealand, and Australian versions.

You might be able to save almost half the cost of the manipulatives by purchasing the RightStart Mathematics Super Saver Set. This includes only the hard-to-find items from the larger set and leaves it to you to supply items such as a geared clock, a geoboard, and a meter stick. Downloadable files on the publisher's website provide substitutes for some items such as tangrams and abacus tiles.

Whether you use the complete RightStart Math Set or the RightStart Mathematics Super Saver Set, you will need to supply some household/school items in addition to the items from RightStart. These include such things as scissors, colored pencils, a thermometer, a digital clock, and a measuring tape.

RightStart instructs parents to frequently use the math card games that come with the manipulatives set. Think of the card games as you think of phonics readers for a reading program. The card games are used up through Level H rather than just for beginners. The RightStart math Set includes a Math Card Games book with instructions for more than 300 games plus a DVD with videos demonstrating some of the games.

Periodic assessments are included, but these are often interactive using manipulatives as well as written responses. They are not at all like typical tests.

As you might have guessed, this program requires a great deal of one-on-one or group presentation through the primary grades. The parent or teacher must familiarize him or herself with the methodology and the concepts to be taught in each lesson beforehand. Because the methodology is unique, this will take more prep time when you start into the program, then less as you move along. Students gradually become more independent, and Levels G and H are written directly to students who are expected to work on their own.

I’ve had discussions about the prep time with a few people who have used the program. One mom told me that she thought it unnecessary to really try to understand the methodology before starting. She thinks it works fine if you just prepare and present lesson by lesson. She’s comfortable with picking it up as she goes. Another parent felt the opposite. She wanted to grasp the “big picture” before she was comfortable starting to teach. I’m in the latter camp. I think this teaching style preference is something to consider if you are concerned about prep time. If you can work with it, learning as you go, then your up-front prep time drops considerably.

You can purchase the book bundle for each level with either one coil-bound, printed workbook or a PDF file of downloadable worksheets. The workbook is consumable and cannot be reproduced, but the PDF worksheets may be printed for your family. While the initial cost might seem a little high, keep in mind that the investment in the manipulatives is spread out over six years. In addition, if you plan to use the course for more than one child and purchase the version with downloadable worksheets, there will be no additional costs in the future for that same course.

### Levels A through F

RightStart Math for Levels A through F uses the AL Abacus, a specially designed abacus used throughout these levels. This particular abacus highlights a key feature of the program: the technique of teaching children to visualize numbers rather than counting. Children learn to quickly spot groups of five and think in terms of "five plus." This same sort of visualization (called subitizing) is used in other ways throughout the program.

The abacus and visualizing are not the only things unique to this program. RightStart Mathematics incorporates methods based onDr. Cotter's research about how children learn. Like some other programs, it is multisensory—using manipulatives, teacher-directed conversation, experiential learning, oral responses, games, and written work. However, the variety of manipulatives and the ways in which they are used, coupled with an unusual scope and sequence, set this program apart from others.

RightStart uses a variety of approaches for teaching almost every concept. For example, in one lesson, children learn to solve simple equations like 3 + 4 with tally sticks, then with the abacus, then on the worksheet. They might use the math balance and the abacus in another lesson, then the balance, tiles, and a geared clock in another lesson. Simple card games provide drill and reinforcement.

RightStart covers the Common Core State Standards as a minimum but goes beyond the standards with advanced mathematical thinking and pacing such as in Singapore Math. Many concepts are introduced earlier and taught in more depth than in other programs. It's an ambitious program!

For example, Level A (RS2) teaches addition and subtraction facts through 18, place value up to the thousands, mental addition, fractional units up to 1/10, telling time to the half hour, money (pennies, nickels, and dimes), measurement in both inches and centimeters, and geometry concepts such as cubes, cylinders, parallel lines, and perpendicular lines, This level includes the Yellow is the Sun book and CD that use three songs and a number of activities to teach counting, visual number recognition (subitizing), and working with units of five. This ties in directly to instruction using the AL Abacus.

Level B (RS2) spends a significant amount of time on addition and subtraction, and it introduces multiplication as arrays. As early as Lesson 27, students are adding numbers such as 40 + 10. They also learn about topics such as even and odd numbers, skip counting, the concepts of hundreds and thousands, parallel and diagonal lines, rectangles, right triangles, equilateral triangles, symmetry, addition with carrying, patterns, transformations, values of coins, perimeter, measurement, time telling, fractions (up through writing equations such as ½ of 12 = 6), and creating bar graphs.

Level C (RS2) begins with review then introduces an Addition Table as a tool for helping children understand relationships between addition facts. It continues with topics such as evens and odds, Roman numerals, trading, adding several two-digit numbers, adding four-digit numbers, arrays, multiplication, area, perimeter, subtraction with two-digit numbers, geometry related concepts (e.g., drawing horizontal lines and dividing equilateral triangles into fourths), telling time to the minute, money (including making change), measurement, line plots, area plots, working with tangrams, introductory division, fractions, negative numbers, and algebraic thinking.

Level D (RS2) spends a good deal of time on skills that will be needed as students move into more complex math. For example, they become very familiar with multiplication facts as well as patterns of multiples for each number. Then they begin work on factoring. Division with remainders is taught, but less time than I expected is devoted specifically to division. Other topics covered include fractions, measurement in both the metric and U.S. systems, geometry (including work with drawing tools), time, money, charts, graphs, and problem-solving skills.

Level E (RS2) teaches the multiplication of multiple-digit numbers by two-digit numbers, the division of multiple-digit numbers by a single-digit number, and both equivalent and mixed number fractions. It also covers prime numbers, factors, decimals to the hundredths, and percents. Algebraic concepts are introduced and problem-solving is emphasized throughout the course. For geometry concepts, students study the classification of triangles and polygons, symmetry, reflections, angle measurement, and work with three-dimensional figures. Other topics explored are measurement, elapsed time, distance, capacity problems, and money. This level also teaches students how to use a calculator. However, the calculator is used strategically rather than as a crutch. For example, students learn the order of operations on a calculator, how to show dollars and cents, and how to calculate the cost of diesel fuel. Three lessons use the calculator for advanced work with prime numbers.

Level F (RS2) continues to reinforce and then build upon earlier levels. Students work with the four basic operations with whole numbers as well as with fractions and decimals. The concept of percentages is taught in relation to both fractions and decimals. Early in the course, students learn about exponents and complex order of operations. For example, students evaluate the expression (5 2 - 4 2 ) - [(5 - 4) • (5 + 4)] = on Worksheet 21-A. Students learn to convert units of measurement using both the U.S. and metric systems. Geometry concepts such as the area and volume of various geometric figures are taught, and students learn to make some geometric constructions. The Cartesian coordinate system is introduced, and students learn to construct lines based on a series of coordinate points. Even at this level, card games are continually used as key tools for practice and review.

### Levels G and H

The first edition of Level G was a single course that took more than a year to complete. With the second edition, that course has been split into two courses that should each be completed in one year: Levels G and H. Students can begin the RightStart program at Level G without having used other RightStart courses. The publisher's choice to split the first edition of Level G into two courses and to expand explanations and review are all significant improvements.

Levels G and H both focus heavily upon geometry, and the manipulative kit covering both levels is called the Geometry Set. While these courses primarily teach geometry, as they do so, students apply and expand upon the range of arithmetic skills and mathematical knowledge they have learned previously. Levels G and H address the Common Core math standards for the middle grades (grades six through eight), and they also introduce many algebra and geometry concepts that are generally taught in high school. To keep arithmetic skills sharp, students continue to play card games as in the lower levels. Frequent review is built into the lessons.

These courses use a discovery approach that I consider very effective for learning geometry. Throughout the course, students make constructions (drawings), compare results, and discover mathematical principles.

The Geometry Set uses many items from the manipulative set for the younger levels. If you already have the younger level set, you need only two items which are sold together as the Geometry Set Short. These items are a scientific calculator and an mmArc™ (a combination protractor and compass). The complete Geometry Set includes those two items plus a small drawing board, a T-square, two triangles, a goniometer (for measuring angles), a 4-in-1 ruler, geometry panels (flat geometric shapes that can be connected to create three-dimensional shapes), the math card decks, and the card deck instruction book. An optional deluxe drawing board is available.

In an explanatory document, "RightStart™ Mathematics: Level G Lessons," the publisher says:

RightStart™ Mathematics Level G Second Edition also incorporates other branches of mathematics as well as arithmetic and algebra. Some lessons have an art flavor, for example, constructing Gothic arches or circle designs. Other lessons have an engineering focus, like creating designs for car wheels. Even some history of mathematics is woven throughout the lessons.

For each course, there are three books: Lessons, Worksheets, and Solutions. Students will use the first two books, and the parent or teacher will use the Solutions books as answer keys and to help if a student needs assistance. Worksheets is a set of three-hole-punched pages that are inserted in a binder, while the other two books have plastic coil bindings. Three assessments are included in Worksheets, so parents or teachers might want to remove those assessments for storage elsewhere.

The Lessons books lay out the lessons with illustrations and instructions about when to use worksheets, manipulatives, and games. These layouts are easy to follow, so students should be able to work independently.

Among concepts covered in Level G are fractions, ratios, area, perimeter, square roots, geometric constructions, triangle properties, angles, triangle congruency, the Pythagorean theorem, circle properties, pi, tangents, bisectors, arcs, rotations, transformations, and symmetry. Many basic postulates and theorems for geometry are presented although they are not named as such. There are no formal proofs, but solutions in the answer key are written out sequentially, similar to the way you might do it for a proof.

Level H is a continuation of Level G. I have not personally reviewed Level H, but the website description reads:

Using a drawing board, T-square, triangles, compass, and goniometer, the student continues to work with fractions and decimals while investigating volume, tessellations, fractals, ratios, angles, and other geometry concepts. Trigonometry is introduced along with platonic solids, 3-dimensional figures, surface area, patterning, and plane symmetry. Connections between various aspects and branches of mathematics are explored. Pre-algebraic concepts are included. The history of mathematics is woven throughout the lessons. Daily card games are included in the lessons.

RightStart recommends VideoText's Interactive Algebra as the next step after Level H because the approaches of both programs are very compatible.

I have a minor concern about the overall progression of the program. Since Levels G and H duplicate a significant amount of what will be covered in high school geometry that means most students will end up repeating topics in high school. While repetition will be helpful for some students, it might be redundant for others. However, I can see where RightStart's geometric approach might work very well for a student who continues geometry study with a rigorous, proofs-based course like Geometry: Seeing, Doing, Understanding.

### Summary

RightStart Mathematics second edition courses make a good program even better. They should work well for a wide range of students because of the variety of learning strategies.

### Pricing Information

When comparison prices appear, please keep in mind that they are subject to change. Click on links where available to verify price accuracy.

You might want to check out the premade lesson plans from Homeschool Planet that are available for RightStart Mathematics.

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## Equivalent Ratios

Recall that ratios are comparisons. They are used to compare two quantities.

Sometimes it is useful to write a ratio in another way. For example, there are 8 cups of strawberries added to a fruit salad for every 16 cups of other fruits. This is an 8:16 ratio of strawberries to other fruits. However, it can also be represented as 1 :2. So if we add one cup of strawberries we can add two cups of other fruits and keep the same mixture in a smaller amount.

We can get equivalent ratios by scaling up or scaling down.

1.) This is an example of scaling up by a scale factor of 3. The ratio 2 : 3 is the same as the ratio of 6 : 15.

2.) This is an example of scaling down by a scale factor of 5. The ratio 8 to 11 is the same as the ratio 40 : 55.

3.) Here, we scaled up by a factor of 8 to create two equivalent ratios.

We can make a string of equivalent ratios by continuing to scale up or scale down.

1 : 3 = 2 : 6 = 3 : 9 = 4 : 12 = 5 : 15 = 6 : 18 = 7 : 21.

All of these ratios show the same relationship.

Simplifying ratios to the simplest form can be helpful when solving problems that deal with ratios. You can simplify a ratio the same way that you simplify a fraction. This is basically just scaling down until you can no longer scale down anymore.

Let's take a look at how scaling up and scaling down can be helpful with a couple of word problems.

1. At East Ridge Middle School, the ratio of boys to girls is 3 to 2. There are 600 students at the school. How many are boys?

Solution: We have been given the whole. So let's write a ratio that compares the boys to the whole. 3 : 5 is the part-to-whole relationship. Now we can scale up from 5 to 600. 5 x 120 = 600, so 3 x 120 = 360. There are 360 boys at the school.