# MA.912.LT.5.5

Explore relationships and patterns and make arguments about relationships between sets using Venn Diagrams.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Logic and Discrete Theory
Status: State Board Approved

## Benchmark Instructional Guide

### Vertical Alignment

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### Purpose and Instructional Strategies

In Math for College Liberal Arts students explore relationships between two sets using Venn Diagrams. In other classes students will extend this exploration to include relationships between three or more sets.
• Instruction includes usage of Venn Diagrams to represent relationships between sets. The universal set U is represented by a rectangle and the sets within the universe are represented by circles.

• In a Venn Diagram, the complement, A′, is represented by the shaded area.
• For example, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, A = {0, 1, 3, 4, 5, 7, 9} and B = {0, 1, 2, 4, 6, 8, 10}, then A′ = {2, 6, 8, 10, 11, 12, 13}. The Venn Diagram shows the shaded region for A′.
• In a Venn Diagram, the union of sets A and B, AB, is represented by the shaded area.
• For example, A = {0, 1, 3, 4, 5, 7, 9, 11} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then A ∪ B is {0, 1, 3, 4, 5, 7, 9, 11} ∪ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. The Venn Diagram of this is the following where the shaded region represents A ∪ B
• In a Venn Diagram, the intersection of sets A and B, AB, is represented by the shaded area.
• For example, A = {0, 1, 3, 4, 5, 7, 9, 11} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then AB is {0, 1, 3, 4, 5, 7, 9, 11} ∩ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 4}. The Venn Diagram of this is the following where the shaded region represents AB:
• Operations can be combined, following the order of operations.
• For example, the Venn Diagram below shows that the shaded region represents the complement of A ∪ B. A = {0, 1, 3, 4, 5, 7, 9} and B = {0, 1, 2, 4, 6, 8, 10, 12}, then AB is {0, 1, 3, 4, 5, 7, 9} ∪ {0, 1, 2, 4, 6, 8, 10, 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,12}. The (A ∪ B) ′ = {11, 13, 14}.
• Instruction includes finding the difference of two sets. Order matters when finding the difference of two sets.
• In a Venn Diagram, A − B is represented by the shaded area.
• In a Venn Diagram, B − A is represented by the shaded area.

### Common Misconceptions or Errors

• Students may repeat elements that are in both sets when writing the union.
• Students may confuse union and intersection.
• Students may incorrectly apply the word “and” when applying set operations.

• Find the following sets using the given Venn Diagram.
Part A. A
Part B. A′
Part C. B
Part D. B
Part E. AB
Part F. AB
Part G. (AB)′
Part H. AB
Part I. Describe a set of operations that would result in the set {w, y
Part J. Describe a set of operations that would result in the set {e, k, m, r, t
Part K. Describe a set of operations that would result in the set {a, b, c, d, f, g, h, j, n, p, w, y}

### Instructional Items

Instructional Item 1
• Find (A ∪ B)′.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1210305: Mathematics for College Statistics (Specifically in versions: 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1212300: Discrete Mathematics Honors (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Lesson Plans

Taxes using Venn Diagrams, Lesson 1:

Students will review constructing and solving Venn diagrams with two and three data sets. Students will then convert text about the collection of taxes from the local, state, and federal governments into a Venn diagram. This is lesson 1 of a three-part integrated mathematics and civics mini-unit.

Type: Lesson Plan

Taxes using Venn Diagrams, Lesson 2:

Students will discuss, recognize, and be challenged to list unions, intersections, and complements related to a Venn diagram created by three data sets. The data is the type of taxes assessed to citizens by the local, state, and federal governments. This is the second lesson in a 3-part integrated mathematics and civic mini-unit.

Type: Lesson Plan

Can You Walk in My Shoes?:

Students use real-life data to create dot-plots and two-way tables. Students will collect data at the beginning of the lesson and use that data to create double dot plots and frequency tables, finding and interpreting relative frequencies.

The assignment allows students to work collaboratively and cooperatively in groups. They will communicate within groups to compare shoes sizes and ages to acquire their data. From the collection of data they should be able to predict, analyze and organize the data into categories (two-way tables) or place on a number line (dot-plot).

As the class assignment concludes, a discussion of the final class display should take place about the purchasing of shoes versus ages and the relationship that either exists or doesn't exist.

Type: Lesson Plan

Human Venn Diagram:

Students will physically interact with Venn diagrams. The students will physically interact with Venn diagrams; the students in the class become the data to arrange. Students will physically move and see how and why elements belong in each section of the Venn diagram.

Type: Lesson Plan

## Perspectives Video: Expert

B.E.S.T. Journey:

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

## Student Resources

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## Parent Resources

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