- Add, subtract, multiply, and divide rational numbers.
- Create a game incorporating computation on rational numbers.
- Find the square and the cube of numbers.
- Use algebraic properties and apply a variety of computational methods and algorithms to evaluate expressions.
- Utilize the order of operations to correctly evaluate expressions.
- Work with a team to write and evaluate expressions.
- Calculate the rate of change caused by earned interest on investments.
- Use estimation to plan and budget for a trip to chosen location.
- Discovering Math: Computation video
- Creating a Game Directions (see below)
- Gameboard (see below)
- Number cubes
- Set of playing cards (numbers only, one per group of students)
- Calculators
- Number Cards (five 2 cards and five 3 cards), one set for each pair of students (see below)
- Order of Operations Poster (see below)
- Evaluating Expressions Practice Sheet (see below)
- Red, yellow, green, and blue chips
- Weekly circulars from local food stores
- Party Planning Directions (see below)
- Shopping List (see below)
- Rate of Change Activity Sheet (see below)
- Tell students they will be creating a game to practice operations on integers and rational numbers.
- Review addition, subtraction, multiplication, and division of integers and rational numbers by displaying practice problems on the board. Have students complete the practice problems and share and explain their answers.
- Divide the class into group of four students. Distribute copies of Creating a Game Directions and the gameboard to each group and discuss the directions. Students will create a game board by filling in operation symbols and numbers on the board. They should use at least three addition, three subtraction, three multiplication, and three division operations. They should also use positive numbers, negative numbers, decimals, and fractions (i.e., they may place multiply by ? in one box on the game board).
- When the gameboards are complete the students should play the game with their group.
- Each student should draw cards to create their starting value. Direct them as to the number of cards to draw and the type of number they should create. Drawing one card will create a one-digit number. Drawing two cards can create a two-digit number or a fraction. If students are to work with positive and negative numbers, red cards can be positive values and black cards can negative values.
- Students should roll number cubes to advance around the board. They must complete the computation on each space they land on, keeping track of their new value. For example, if a student started with 20 and landed on a space that directs them to add 5, they will then have a value of 25. Division calculations should be rounded to the nearest hundredth.
- Students should complete their computations using mental math, paper and pencil, or calculators.
- After each student has completed three trips around the board, the player with the highest value wins.
- Groups can switch gameboards if time allows.
- Display 4^{2}and 6^{3}on the board. Ask students to find the answers. Have them share and explain their work.
- Assign each student a partner. Distribute a number cube, Number Cards, and a calculator to each pair. Have one student roll the number cube and then pick a card. They must now find the square or the cube of the number they rolled (square if they picked a 2 number cared or cube if they picked 3 number card). The other student should check the work on the calculator. Have them take turn rolling the number cube and practicing squares and cubes.
- Display the phrase "Order of Operations." Ask students to describe the order of operations. They should recall the mnemonic, Please Excuse My Dear Aunt Sally, from the video. Review the order of operations (parentheses, exponents, multiplication, division, addition, and subtraction).
- Distribute copies of the Order of Operations Poster.
- Display the following expression:
3 [(11 - 1) + 8] x 5^{2}
Model how to evaluate the expression using the order of operations. Solve each step of the problem, using the appropriate color from the poster to show the step.
3 [(11 - 1) + 8] x 5^{2}
3 [10+ 8] x 5^{2}
3 x18x 5^{2}
3 x 18 x25
54x 25
1,350 - Ask students to identify any patterns they notice when using the color-coded order of operations system.
- When students are comfortable evaluating expressions, distribute the Evaluating Expressions Practice Sheet and have them complete it using the color-coded order of operations system.
- Assign each student a partner. Give each pair a bag containing red, yellow, blue, and green chips. Review the operations that each color represents from the Order of Operations Poster. Students will use the chips to write their own expressions.
- Have each student pull two chips from the bag.
- Ask them to write expressions that include the elements that their chips represent. For example, if they pull one yellow, one green, and two blue chips, they will write an expression that includes one addition or subtraction element, one exponential element, and two multiplication or division elements.
- Then have students evaluate their partner's expression. They can check their work using a calculator.
- Ask students to recall the algebraic properties they learned about in the video. Have them share and explain their ideas.
- Identity property of addition ? the sum of a number and zero is the number
- Identity property of multiplication ? the product of any number and one is the number
- Commutative property of addition — in a sum, you can add terms in any order
- Commutative property of multiplication — in a product, you can multiply factors in any order
- Associative property of addition — changing the grouping of terms in a sum does not change the sum
- Associative property of multiplication — changing the grouping of terms in a factor in a product does not change the product
- Distributive property of multiplication over addition — multiplication may be distributed across addition
- Ask students to identify situations in which they used an algebraic property when evaluating their expressions. Have each pair write two expressions that use an algebraic property. Ask them to share and explain their examples to the class or in writing.
- Assign each student a partner. Tell students they will be planning a party, making a menu, and determining how much money they will spend on food for the party. They will be using arithmetic to complete the party planning. Review the operations and their uses with students. Have students give examples of when they would use addition, subtraction, multiplication, and division. They may use examples from the video.
- Distribute copies of weekly circulars from local food stores and the Shopping List.
- Distribute copies of the Party Planning Directions and discuss with students.
- Allow students time to plan their menus and complete the Shopping list.
- Ask students to share their menus and Shopping Lists with the class. Have them explain the algorithms, operations, and strategies they used to find the costs of individual items, multiples items, and the total cost of the party.
- Discuss the benefits of using estimation and the importance of comparing estimates with actual figures.
- Review rate of change. Discuss how depositing money in an interest-earning bank account allows the value of the money to increase. Model an example by calculating how much interest $200 would earn at 3 percent in one year ($6). Next ask students how much the initial investment would be worth in two years ($206 + 3% = $212.18). Continue practicing rate of change until students are comfortable with concept.
- Distribute the Rate of Change Activity to students. Have them complete the calculations in the chart. Allow time for students to discuss the impact of earning interest on an initial investment (assume that no other deposits are made to avoid compounding and monthly interest).
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Use the following three-point rubric to evaluate students' work during this lesson.
- 3 points: Students clearly demonstrated the ability to add, subtract, multiply, and divide rational numbers; clearly demonstrated the ability to find the square and cube of given numbers; clearly demonstrated the ability to use algebraic properties, order of operations, and a variety of computational methods and algorithms to evaluate expressions; clearly demonstrated the ability to calculate the rate of change caused by an interest-earning bank account; and clearly identified the ability to use estimation in addition, subtraction, multiplication, and division.
- 2 points: Students satisfactorily demonstrated the ability to add, subtract, multiply, and divide rational numbers 80% of the time; satisfactorily demonstrated the ability to find the square and cube of given numbers 80% of the time; satisfactorily demonstrated the ability to use algebraic properties, order of operations, and a variety of computational methods and algorithms to evaluate expressions 80%of the time; satisfactorily demonstrated the ability to calculate the rate of change caused by an interest-earning bank account 80%of the time; and satisfactorily identified the ability to use estimation in addition, subtraction, multiplication, and division 80%of the time.
- 1 point: Students demonstrated the ability to add, subtract, multiply, and divide rational numbers less than 80%of the time; demonstrated the ability to find the square and cube of given numbers less than 80%of the time; demonstrated the ability to use algebraic properties, order of operations, and a variety of computational methods and algorithms to evaluate expressions less than 80%of the time; demonstrated the ability to calculate the rate of change caused by an interest-earning bank account less than 80%of the time; and identified the ability to use estimation in addition, subtraction, multiplication, and division less than 80%of the time.
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associative property of additionDefinition: changing the grouping of terms in a sum does not change the sum
Context: (9 + 4) + 3 = 9 + (4 + 3)
associative property of multiplication
Definition: changing the grouping of factors in a product does not change the product
Context: (7 x 3) x 2 = 7 x (3 x 2)
commutative property of additionDefinition: in a sum, you can add terms in any order
Context: 6 + 3 = 3 + 6
commutative property of addition
Definition: in a product, you can multiply factors in any order
Context: 3 x 9 = 9 x 3
distributive property of multiplication over additionDefinition: multiplication may be distributed across addition
Context: 5 x (35 + 45) = (5 x 35) + (5 x 45)
identity property of addition
Definition: the sum of any number and zero is the number
Context: 9 + 0 = 9
identity property of multiplicationDefinition: the product of any number and one is the number
Context: 7 x 1 = 7
order of operations
Definition: a set of rules for evaluating an expression with more than one operation
Context: The teacher told the students to observe the order of operations when solving expressions.
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Mid-continent Research for Education and Learning (McREL)McREL's Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education addresses 14 content areas. To view the standards and benchmarks, visit
www.mcrel.org/compendium/browse.asp.
This lesson plan addresses the following national standards:
- Adds, subtracts, multiplies, and divides integers, and rational numbers.
- Adds and subtracts fractions with unlike denominators; multiples and divides fractions.
- Understands exponentiation of rational numbers and root-extraction (e.g., squares and square roots, cubes and cube roots).
- Selects and uses appropriate computational methods (e.g., mental, paper and pencil, calculator, computer) for a given situation.
- Understands the correct order of operations for performing arithmetic computations.
- Uses proportional reasoning to solve mathematical and real-world problems (e.g., involving equivalent fractions, equal ratios, constant rate of change, proportions, and percents).
- Understands the properties of operations with rational numbers (e.g., distributive property, commutative and associative properties of addition and multiplication, inverse properties, identity properties).
- Knows when an estimate is more appropriate than an exact answer for a variety of problem situations.
- Understands how different algorithms work for arithmetic computations and operations.
National Council of Teachers of Mathematics (NCTM)The National Council of Teachers of Mathematics (NCTM) has developed national standards to provide guidelines for teaching mathematics. To view the standards online, go to
standards.nctm.org.
This lesson plan addresses the following national standards:
- Work flexibly with fractions, decimals, and percents to solve problems.
- Understand and use ratios and proportions to represent quantitative relationships.
- Develop meaning for integers and represent and compare quantities with them.
- Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.
- Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.
- Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
- Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.
- Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.
- Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
- Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
- Solve problems that arise in mathematics and in other contexts.
- Apply and adapt a variety of appropriate strategies to solve problems.
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