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6th Grade Unit 2: Expressions, Equations, and Inequalities

Page history last edited by Brigit Minden 10 months, 1 week ago

 

6th Grade Unit 2: Expressions, Equations, and Inequalities

Unit Driving Question

How do expressions, equations, and inequalities help to solve real-world problems?

 

Essential Questions

  1. How can patterns in real-world scenarios be explored using mathematics?

  2. How can properties help to simplify real-world problems?

  3. How can the order of operations be used to find solutions for real-world problems?

  4. How can equations be used to find solutions to real-world problems?

 

Big Ideas

  1. Mathematical relationships can be expressed using different representations.
  2. The order of operations is used to evaluate and compare expressions.
  3. The commutative, associative, and distributive properties are used to find equivalent expressions.
  4. Equations can be used to find an unknown value.

 

Useful Websites

The following apps, websites, and smartboard lessons can be used throughout the unit, as needed, during small groups, lessons, to reinforce standards.  They are also useful for students who may need reinforcement, remediation, or differentiation.

  1. Virtual Nerd: Virtual Nerd provides video tutorials as a supplemental resource for both students and teachers.

  2. Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard so that students may study at their own pace both in and outside of the classroom.

  3. Mr.Naussbaum.com: MrNussbaum.com offers interactive games specifically designed to pinpoint one or several essential concepts to make the learning process more interactive and enjoyable.

  4. Interactive Sites for Education: These interactive activities work great on your interactive whiteboard, computer, laptop, or Chromebook for whole group or small group instruction or use in the computer lab or at home for individual learning. Most of these activities are Flash-based.  This means that they will NOT work on iPads unless you are running an app that allows Flash to play such as Puffin.  

  5. Kahoot: Kahoots are fun, learning games best played in a group setting.  Players answer questions on individual devices (Ex: Chromebook, iPads) while games are displayed on a shared screen (Ex: Smartboard or TV).  Choose a Kahoot to match your desired skill or create your own.

  6. Zapzapmath: Zapzapmath has over 150 math lessons designed to incorporate higher order thinking skills in the fields of creation, evaluation, and analysis. This is combined into a game-based system of fun math learning.

  7. Mr.Anker Tests: Interactive activities and games for dozens of math skills.  Most of these activities run on Flash.

Launch Task

1 Lesson

  • Building Bridges (Illuminations): This task serves as a bridge between the number sense gained in Unit 1: Rational Numbers and the algebraic thinking needed for Unit 2: Expressions, Equations, and Inequalities. By posing the hamburger problem and the variation on the hamburger problem to students, they will begin to explore problem-solving strategies, such as creating expressions, equations, and inequalities and forming a table of values and graphs in order to investigate relationships between varying quantities.  The Valentine problem and its variations in this lesson would also be a good initial task.

Big Ideas for Development Lessons

4 - 5 Weeks (approximately 1 week per big idea)

Big Idea 1: Mathematical relationships can be expressed using different representations.

OAS-M: 6.A.1.1, 6.A.1.2, 6.A.3.16.N.2.3, 6.N.4.36.N.4.4 

Key Resources

 

  1. Algebraic Expressions (Open Up): The Warm Up provides students with the opportunity to review vocabulary used for mathematical operations.  In Activity 1, students investigate the meaning and use of a variable.  Activity 2 gives students the opportunity to connect algebraic expressions with real-world situations.  The Cool Down gives a task that connects what students learned in Activity 1 and 2.

  2. Which Goes with Which? (Illustrative Mathematics): Students match verbal statements to equations.  Teachers can extend this task by having students graph the solution to each equation on a number line.  At this point students, do not know how to formally solve an equation, but other problem-solving strategies can be used to find the solution such as guess and check or using a table of values.

  3. Inequalities on a Number line (Desmos): This Desmos activity allows students to explore inequalities using multiple representations including algebraic expressions, verbal statements, and number lines.

  4. Writing & Graphing Inequalities (Open Up): The Warm-Up has students compare the height of an adult to a child’s height using an inequality.  Activity 1 is a card sort in which students match real-world scenarios, verbal descriptions of inequalities, and graphs of inequalities.  Activity 2 has students estimate the height of a basketball goal by referencing the height of an adult standing next to a ladder in the same picture.  The Cool Down checks for student understanding of inequalities by asking if a given number satisfies the inequalities written by students in the first part of the Cool Down.

  5. Interpreting Points on a Coordinate Plane (Open Up): This lesson provides four tasks (Warm Up, Activity 1: Axes Drawing Decisions, Activity 2: Running Training, Cool Down) that allow students to explore points on a coordinate plane including ordered pairs with halves and fourths.  The Warm-Up and the Cool Down have students explore how a coordinate plane can be used to graph a table of values.

  6. Gym Membership Plans (Illustrative Mathematics):  In this task, students will represent a real-world scenario using a rule, a table, and a graph.  This task can be extended to include equations by asking students to solve an equation to determine how much Edwin would pay after so many months.

  7. Boat Coordinate Game (GeoGebra): This game gives students the opportunity to practice identifying points on a coordinate plane.
  8. Mini Golf Marbleslides (Desmos):  Students practice graphing coordinates in a game of mini-golf.

  9. Battle Boats (Desmos): Students build proficiency in the coordinate plane through a guess the location style game.

 

Big Idea Probe

 

 

Evidence of Understanding

 

Recognize real-world situations can be modeled using different representations.

  • Use expressions, equations, and inequalities to model real-world situations.

    • Define and use variables to represent unknown values in real-world situations.

    • Discuss the similarities and differences for expression, equations, and inequalities.

  • Use tables, graphs, and number lines to model real-world and mathematical situations.

    • Create a table of values for a pair of varying quantities.

    • Plot ordered pairs on a coordinate plane to create a graphical representation of a real-world scenario.

    • Analyze patterns using rules, tables, and graphs.

    • Discuss the similarities and differences for representing relationships using tables and graphs.

    • Recognize number lines can be used to graph solutions to equations and inequalities.

  • Translate between representations (rules, tables, graphs) for real-world and mathematical situations. 

 

 

Big Idea 2: The order of operations is used to evaluate and compare expressions.

OAS-M: 6.A.1.3, 6.A.2.16.N.4.36.N.4.4 

Key Resources

 

  1. Meaning of Exponents (Open Up): In Activity 2, students use the meaning of exponents to write and evaluate exponential expressions.  In Activity 3, students compare exponential expressions that have the same base.

  2. Order of Operations (Open Middle):  Students try to make the largest or smallest expression by inserting whole numbers 0-9.

  3. Exponential Expressions (Open Up): Students use the order of operations to evaluate numerical expressions in Activity 2 and algebraic expressions in Activity 3.

  4. Making it True (Open Up): The Warm-Up from Lesson 9: Writing and Solving Equations has students find values of x that make the equations and inequalities true.  The equations and inequalities were written specifically for students to investigate what happens to the product when you multiply by a number less than one, a number greater than 1, and a reciprocal.

 

Big Idea Probe

 

 

Evidence of Understanding 

 

Use and apply the order of operations to generate equivalent expressions.

  • Discuss the meaning and use of exponents in expressions.

 

Use order of operations to evaluate algebraic expressions, equations, and inequalities.

  • Recognize any rational number can replace a variable in an algebraic expression in order to evaluate the expression.

  • Compare expressions using inequality symbols (<, ≤,>, ≥, ).

  • Identify solutions and non-solutions for equations and inequalities.

  • Discuss the different results obtained when evaluating expressions, equations, and inequalities. 

 

Big Idea 3: The commutative, associative, and distributive properties are used to find equivalent expressions.

OAS-M: 6.A.2.16.N.4.36.N.4.4 

Key Resources 

 

  1. New Park - Commutative Property (Illustrative Mathematics): This task investigates the commutative property of multiplication using rational numbers by comparing the size of a new park with two design proposals.  This activity is accessible to 6th graders before Unit 6: Area because students learned about area of rectangles in 5th grade.

  2. Using Volume to Understand the Associative Property of Multiplication (Illustrative Mathematics): As fifth graders, students learned about volume of rectangular prisms.  This task uses that prior knowledge to investigates the associative property of multiplication.  To extend the task to apply the associative property to rational numbers, teachers can change the rectangular prism’s dimensions to rational numbers once students have identified the associative property of multiplication using the whole numbers given in this task.

  3. Length and Area Representations of Expressions - Distributive Property (Open Up): Students explore equivalent expressions in the Warm-Up.  In Activity 1, students investigate the distributive property using area of rectangles.  This activity is accessible to 6th graders before Unit 6: Area because students learned about area of rectangles in 5th grade.

  4. Exploring the Distributive Property (Open Middle): Students will insert whole numbers 0-9 to create equivalent expressions using the distributive property.  This task was presented in Unit 1 in order to investigate factors. Now with an understanding of the distributive property, challenge students to find different solutions than the ones they found the first time they tried the task.

  5. Using Equivalence (Open Up): The Warm-Up, Activity 1, and Activity 2 #1 all investigate equivalent expressions. Activity 2 #1 specifically explores the usefulness of using the commutative, associative, and distributive properties to create an equivalent expression that easier to evaluate than the other expressions.

 

Big Idea Probe

 

 

Evidence of Understanding

 

Use and apply the commutative, associative, and distributive property to numerical and algebraic expressions.

  • Recognize applying the commutative, associative, and distributive property to an expression generates an equivalent expression.

  • Use the commutative, associative, and distributive properties to show that different expressions for the same rule are equivalent.

  • Discuss how the commutative, associative, and distributive property are used to simplify expressions.

 

 

Big Idea 4: Equations can be used to find an unknown value.

OAS-M: 6.A.3.1, 6.A.3.26.N.2.3, 6.N.4.36.N.4.4 

Key Resources

 

  1. Visual Representations of Equations (Open Up): In the Warm-Up, students explore representing a relationship between varying using a tape diagram.  Activity 1 is a card sort in which students match stories, diagrams, and equations.  After matching the cards, students will solve the equation to answer the questions on the story cards.  To solve equations like x + 7x = 56 which appear on some cards, students can use the distributive property to show x + 7x is equivalent to 8x to get the one-step equation 8x = 56. If you choose to use the card sort, cards 7 & 8 can be set aside until students learn about ratios and percents in Unit 3  In Activity 2, students will use balanced weights of different shapes and equations to determine the weight of the square in each diagram.  The Cool Down asks students to determine which tape diagram and equation represents the given story and to use their selection to answer the question in the story.

  2. Structure of Equations (Open Up): In the Warm-Up, students learn the secret to a “mind-reading” trick by using equations to guess an unknown number.  Activity 2 gives students another opportunity to match an equation to a situation which they practiced in Big Idea 1.  Then, students are given equations and asked to write a situation to match the equation.

  3. Solving One-Step Equations (Open Middle): The first task has students use all of the whole numbers 1 through 9 one time to make each equation true.  The second task has students use whole numbers 1 through 9 that would make a true equation where x has the largest possible value.

  4. Writing & Solving Equations (Open Up): In Activity 2, students write an equation for the given scenario and solves the equation to answer the question for the scenario.  The Lesson Summary describes an activity in which the teacher calls on a student to share a situation describing a relationship, and another student is called on to create an equation for the situation. The teacher continues to call on students until the equation is solved. This would be a good activity to check for understanding.  The Cool Down has students find a missing side of a triangle by writing and solving an equation using the perimeter for a given triangle.  Every time a student is asked to find the solution to an equation they should graph this solution on a number line.

  5. Model Algebra (Math Playground): This virtual balance scale helps students visualize solving equations.

 

Big Idea Probe

 

 

Evidence of Understanding 

 

Recognize equations can be solved to find an unknown value using properties of operations and equality.

  • Model equations using tools, such as pan balances and algebra tiles.

  • Identify inverse operations needed to solve an equation.

  • Discuss the process used to solve equations.

 

Interpret solutions to equations that model real-world situations.  

  • Use a number line to graph solutions for equations. 

 

Unit Closure

1 Week (includes time for probes, re-engagement, and assessment)

  • Chocolate Bar Sales (Illustrative Mathematics): In this task, students will complete a table that represents a relationship between two varying quantities based on a real-world situation. Students then use the table to create an equation and a graph that can be used to solve the questions related to the situation.

 

OKMath Framework Introduction

6th Grade Introduction

 

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