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7th Grade Unit 1 Resource Grade Level: 7th Grade Subject Area: 7th Grade Math Lesson Title: Unit 1 Number and Lesson Length: 9 days Operations THE TEACHING PROCESS Lesson Overview This unit bundles student expectations that address sets and subsets of rational numbers, operations with rational numbers, and personal financial literacy standards regarding sales tax, income tax, financial assets and liabilities records, and net worth statements. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. During this unit, students use a visual representation to organize and display the relationship of the sets and subsets of rational numbers, which include counting (natural) numbers, whole numbers, integers, and rational numbers. Students also apply and extend operations with rational numbers to include negative fractions and decimals. Grade 7 students are expected to fluently add, subtract, multiply, and divide various forms of positive and negative rational numbers which include integers, decimals, fractions, and percents converted to equivalent decimals or fractions for multiplying or dividing. Students also create and organize a financial assets and liabilities record, construct a net worth statement, calculate sales tax for a given purchase, and calculate income tax for earned wages. Unit Objectives: Students will extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers. Students will add, subtract, multiply, and divide rational numbers fluently. Students will apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers. Students will calculate the sales tax for a given purchase and calculate income tax for earned wages. Students will create and organize a financial assets and liabilities record and construct a net worth statement. Standards addressed: TEKS: 7.1A Apply mathematics to problems arising in everyday life, society, and the workplace. 7.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. 7.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. 7.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. 7.1E Create and use representations to organize, record, and communicate mathematical ideas. 7.1F Analyze mathematical relationships to connect and communicate mathematical ideas. 7.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. 7.2A Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers. 7.3A Add, subtract, multiply, and divide rational numbers fluently 7.3B Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers. 7.13A Calculate the sales tax for a given purchase and calculate income tax for earned wages 7.13C Create and organize a financial assets and liabilities record and construct a net worth statement ELPS: The student will ELPS.c.1A: use prior knowledge and experiences to understand meanings in English ELPS.c.1C: use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary ELPS.c.1D: speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known) ELPS.c.1E: internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment ELPS.c.2D: monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed ELPS.c.2E: use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language ELPS.c.3D: speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency ELPS.c.3H: narrate, describe, and explain with increasing specificity and detail as more English is acquired ELPS.c.4D: use pre-reading supports such as graphic organizers, illustrations, and pre-taught topic-related vocabulary and other pre-reading activities to enhance comprehension of written text ELPS.c.4F: use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language ELPS.c.4H: read silently with increasing ease and comprehension for longer periods ELPS.c.5B: write using newly acquired basic vocabulary and content-based grade-level vocabulary ELPS.c.5F: write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired ELPS.c.5G: narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired. Misconceptions: Some students may think the sum of any two rational numbers is always greater than the two addends. Some students may think the difference of any two rational numbers is always less than the greater. Some students may think the product of any two rational numbers is always greater than the factors. Some students may think the quotient of any two rational numbers is always less than the dividend. Some students may think the value of a property or home is a liability, rather than an asset, if there is an outstanding mortgage on the property or home. Some students may think the sales tax is the total cost, rather than the amount added to the price to determine the total cost. Some students may think that a percent may not exceed 100%. Some students may think that a percent may not be less than 1%. Some students may multiply a decimal by 100 moving the decimal two places to the right when trying to convert it to a percent rather than dividing by 100 and moving the decimal two places to the left. Some students may think the value of 43% of 35 is the same value of 43% of 45 because the percents are the same rather than considering that the wholes of 35 and 45 are different, so 43% of each quantity will be different. Some students may attempt to perform computations with percents without converting them to equivalent decimals or fractions for multiplying or dividing. Vocabulary: Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n} Earned wages – the amount an individual earns over given period of time Financial asset – an object or item of value that one owns Financial liability – an unpaid or outstanding debt Fluency– efficient application of procedures with accuracy Income tax – a percentage of money paid on the earned wages of an individual or business for the federal and/or state governments as required by law Integers – the set of counting (natural numbers), their opposites, and zero {-n, …, -3, -2, -1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z. Net worth – the total assets of an individual after their liabilities have been settled Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are whole numbers and b ≠ 0, which includes the subsets of whole numbers and counting (natural) numbers (e.g., 0, 2, etc.) Rational numbers – – the set of numbers that can be expressed as a , where a and b are integers and b≠ 0, which includes the subsets of fraction integers, whole numbers, and counting (natural) numbers (e.g., -3, 0, 2, , etc.). The set of rational numbers is denoted by the symbol Q. Sales tax – a percentage of money collected by a store (retailer), in addition to a good or service that was purchased, for the local government as required by law Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n} INSTRUCTIONAL SEQUENCE Phase: Engage List of Materials: Engage Day 1 cards (one set per student, cut and bagged) Activity: Common Characteristics Prior to the lesson, the teacher has the following numbers written on the board (in a scattered manner). There is a page of cards with these numbers in the lesson (Engage Day 1 cards). 24 2 1 2 13 , 0, 50%, 4, 0.75, −5, 1 5, 32 , −0.25, 120%, 100, 3 2 Ask the students to group the numbers that have common characteristics and to be ready to justify how they grouped the numbers. What’s the teacher doing? What are the students doing? Walking around, answering questions and monitoring students’ participation. Sorting and grouping the numbers, discussing the attributes with a partner. Questions to ask after students have grouped the numbers: Why did you group these numbers together? Why are other numbers not included in that group? How would you classify those numbers? Fractions? Whole numbers? Integers? Percents? Mixed numbers? Proper fractions? Improper fractions? Decimals? Exponents? A neutral number? Non-positive? Non-negative? Do any of these numbers simplify to another number? If so, which ones? Is there more than one way to represent these numbers? Can you give me another number (from the list or one you create) that would also be in the group? Why? Phase: Explore List of Materials: Engage Day 1 cards Explore 1 Grouping Mat Activity: Representing Rational Numbers The teachers provides students with the Explore 1 Grouping Mat and ask students to place the numbers from the Engage and the board (includes numbers students added) on the mat in their appropriate location. What’s the teacher doing? What are the student’s doing? The teacher is monitoring as students work and is asking questions: The students are placing the numbers onto the mat and asking the teacher and a partner questions about the placement. How can you identify a natural or counting number? It is a number that I learned when I first started counting on my fingers: 1, 2, 3, 4, … How can you identify a whole number? A whole number is 0, 1, 2, 3, 4, … The set of whole numbers contains all of the counting numbers plus the number 0. How can you identify an integer? I think of an integer as the positive and negative “whole” numbers and 0. How do you identify a rational number? It can be written as a fraction of integers, with the denominator not equal to 0. How can a decimal be a rational number? A terminating or repeating decimal can be written as a fraction of integers, so it is a rational number. Phase: Explain (Part I & II) List of Materials: Explore 1: Day 1 Grouping Mat ( one per student) Notes on Rational Numbers (one per student) Notes on Rational Numbers (one per student) Activity Part I: Justify The students present their mats and justify where they placed the numbers. The teacher asks the students the following questions: How can you identify a natural or counting number? It is a number that I learned when I first started counting on my fingers: 1, 2, 3, 4, … How can you identify a whole number? A whole number is 0, 1, 2, 3, 4, … The set of whole numbers contains all of the counting numbers plus the number 0. How can you identify an integer? I think of an integer as the positive and negative “whole” numbers and 0. How do you identify a rational number? It can be written as a fraction of integers, with the denominator not equal to 0. How can a decimal be a rational number? A terminating or repeating decimal can be written as a fraction of integers, so it is a rational number. In which set does a percent belong? A percent is a rational number because it can be written as a ratio (fraction) of the percent to 100. Which set of number contains all of the other numbers? Why? The set of rational numbers contains all of the other numbers (natural, whole, integers) and also contains numbers that can be written as fractions of integers (with the denominator not equal to zero). Which set is a subset of the set of whole numbers? Why? The set of natural numbers is the subset of the set of whole numbers because all of the natural numbers are contained in the whole numbers. What does the Venn diagram tell us about how the sets are related to each other? The Venn diagram The Venn diagram shows us that the natural numbers are contained in the whole numbers, the whole numbers are contained in the integers, and the integers are contained in the rational numbers. Activity Part II: Notes on Rational Numbers Then the teacher provides each student with a copy of Notes on Rational Numbers for students to add to their journals. The students will determine if the statements are true or false. Also, have the students create a diagram of the rational number set and its subsets then the students can write an explanation in their journals of where they placed three numbers in the mat. What’s the teacher doing? What are the students doing? The teacher is monitoring the students as they work. The students glue the Notes on Rational Numbers in their journals. Have students make vocabulary flashcards from index cards, with the definition on one side and the vocabulary word on the other. Students will make will make vocabulary flashcards for : Counting numbers (natural numbers), integers, whole numbers, rational numbers, positive rational numbers, and keep them for future use. Phase: Evaluate List of Materials: Paper Activity: Performance Assessment 1: Analyze the situation(s) described below. Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each mathematical process. 1) Number sets are interrelated a) Create a visual representation to organize and display the relationship of the sets and subsets of numbers: counting (natural) numbers integers rational numbers whole numbers b) Write a description describing the relationships between sets of rational numbers. What’s the teacher doing? What are the students doing? The teacher is monitoring the students as they work, asking questions as needed. The students are completing the Performance Assessment individually. Questions to check or understanding: How are counting (natural) numbers, whole numbers, integers, and rational numbers related? What types of visual representations can be used to represent the relationships between sets and subsets of numbers? How can a number belong to the same set of numbers but not necessarily the same subset of numbers? Phase: Explore List of Materials: Another Look at Adding and Subtracting Fractions Activity: Another Look at Adding and Subtracting Fractions. Students will add and subtract fractions using Another Look at Adding and Subtracting Fractions. What’s the teacher doing? Monitoring students as they work and assisting students as needed. What are the students doing? Working in groups of two on the assignment. How can you rename the fractions so that they have the same denominator? I can write equivalent fractions with the least common denominator Phase: Explain List of Materials: Another Look at Adding and Subtracting Fractions Activity: Teacher and Student Justifications Teacher clarifies steps to work the problems and students will explain how they worked the problems. What’s the teacher doing? What are the students doing? The teacher is clarifying the steps to working the problems as needed. The students are explaining how they worked each problem. Which words in problems 9-14 indicated the operations of addition or subtraction? In problem 9, the words “how much does she have left” indicate subtraction. In problem 10, the words, “how much in 5 minutes,” indicates addition, because the other amounts were for 2 and 3 minutes. In problem 11, the operation is subtraction because part of the distance is already traveled. In problem 13, the operation is addition for perimeter (the sum of the side lengths). In problems 12 and 14 both addition and subtraction are used. Phase: Explore/Explain List of Materials: Mac’s Brownies Activity: Mac's Brownies Display Mac’s Brownies, keeping the directions covered, and only displaying the problem. Instruct students to read the problem and use mental math to quickly select the range of numbers they believe represents the solution. Survey students and tally the ranges students have estimated, as this will be used in a later activity. Facilitate a class discussion for students to justify their range of numbers selection. What is the teacher doing? 1. Display teacher resource: Mac’s Brownies, keeping the directions covered, and only displaying the problem. Instruct students to read the problem and use mental math to quickly select the range of numbers they believe represents the solution. Survey students and tally the ranges students have estimated, as this will be used in a later activity. Facilitate a class discussion for students to justify their range of numbers selection. Ask: What method did you use to determine the estimate? Possible answer: I chose the 24 – 32 range because know 1 pan yields 8 servings and 1 serving is 1 brownie and there are 3 1 pans with 8 2 What are the students doing? Students to read the problem and use mental math to quickly select the range of numbers they believe represents the solution. Students justify their range of numbers selection. brownies in each pan. Since 3 1 is 2 between 3 and 4, did 3 x 8 = 24 and 4 x 8 = 32; etc. Why did you select this method to give an estimate? . I chose to multiply the 2 whole numbers 3 1 is 2 between; etc. Is there another method that may be used to determine an estimate? Answers may vary. I chose 3 x 8 = 24 because 3 1 comes 2 after 3. I know the answer will be more than 24; etc. How are you going to determine how many brownies Mac has made using pictorial models? A whole square can represent 1 pan, which can be divided into 8 equal parts – each one-eighth fraction piece that represents 1 brownie. Since Mac made 3 1 2 pans of brownies, 3 whole squares plus half a whole square will need to be the model for the brownies. How is the mathematical operation of multiplication being demonstrated as you cover the whole square with the model for brownie pieces? The operation of multiplication is being demonstrated because repeatedly creating multiple groups of 8 brownies until 3 1 pans 2 have been filled with 8 brownies on each whole pan. What multiplication number sentence can be written to record the relationship among the numbers in the problem and how these numbers relate to the model created with the fraction squares? The multiplication number sentence is 3 1 pans x 8 brownies per pan = 2 28 total brownies. What does the 3 1 represent in 2 this problem and the in multiplication number sentence you wrote? The number of pans (groups) of brownies Mac has baked. How are the 8 brownies used in this problem and the in multiplication number sentence you wrote? The 8 represents the number of brownies (size of each group – the factor, also known as the multiplicand) placed on each pan. You only have half a pan. What does this mean? Only going to create a partial group - 1 2 pan, which means you only have 4 brownies from the half a pan because 1 pan yields 8 brownies. How many brownies has Mac made? The total number of brownies Mac baked using 3 1 pans 2 that yield 8 brownies per pan is 28 brownies. ***Facilitate a class discussion, using the previously tallied results gathered from students about their estimate of the range of numbers (20 to 30 brownies, 24 to 30 brownies, 24 to 32 brownies) from the displayed teacher resource: Mac’s Brownies. Demonstrate how to determine the fraction of student responses for each range of numbers. Ask: What fraction of the class chose this estimated range? Answers may vary. How did you choose your range of numbers? Answers may vary. I chose the estimated range of 24 – 32 brownies: knew 1 pan yielded 8 brownies so, I did 3 x 8 = 24 and 4 x 8 = 32 because 3 1 is between 3 2 and 4; etc. What fraction of the class did not choose this estimated range? Answers may vary. What is the sum of the fraction of Students demonstrate how to determine the fraction of student responses for each range of numbers. Students discuss a class discussion about the meaning and purpose of the multiplication number sentence to make a connection to the diagram. Students record their actions on the fraction diagrams. students who did choose the estimated range of 24 to 30 brownies and the fraction of students who did not choose the estimated range of 24 to 30 brownies? (The sum of the 2 fractions, the chosen estimated range and the not chosen estimated range should be 1 or very close to 1.) 2. Display the multiplication number sentence 3 1 x 8 = 28 for the class 2 to see. 3. Using the displayed teacher resource: Mac’s Brownies, facilitate a class discussion about student solutions to the problem. Without using the formal algorithm, sketch a picture to model the problem. Facilitate a class discussion about the meaning and purpose of the multiplication number sentence to make a connection to the diagram. Ask: What is the multiplication number sentence as a whole communicating? (The number sentence: 3 1 x 8 = 28 represents 3 2 1 2 pans (groups) of 8 brownies in each pan (number of objects in each group) which produces 28 total brownies.) How is the addition number sentence 8 + 8 + 8 + 4 = 24 related to the multiplication number sentence 3 1 x 8 = 28? (Both 2 number sentences are representing 3 1 groups of 8. The addition 2 number sentence shows the number in each group (8) being repeatedly added. The multiplication number sentence is communicating how many groups of 8 are needed.) What did you do in this problem? (Combined multiple groups of a given number.) What is the mathematical term for combining multiple groups of a given number? (multiplication) 4. Explain to students that there are various terms used in relation to multiplication; the process of combining multiple groups (factor, also known as the multiplier) of a given number (factor, also known as the multiplicand) to produce a product (the total quantity after multiplying the factors). Ask: What do the 3 1 pans represent in 2 the multiplication number sentence: 3 1 x 8 = 28? (The 2 number of groups – first factor multiplier.) Explain to students that in mathematics this is called a “factor." What is the purpose of this factor? (This first factor, also referred to as the multiplier, indicates the number of groups.) What do the 8 brownies in the multiplication number sentence: 3 1 x 8 = 28 2 represent? (The number of brownies that 1 pan yields - the number in each group – the second factor.) Explain to students that in mathematics this is also called a “factor." What is the purpose of this factor? (This factor, also referred to as the multiplicand, indicates the number or size of each group.) What do the 28 brownies in the multiplication number sentence: 3 1 x 8 = 28 2 represent? (The total number of brownies there are from 3 1 pans 2 that yield 8 brownies per pan.) Explain to students that in mathematics this is called a “product." What is the purpose of the product? (It communicates the total quantity after multiplying the 2 factors.) For the multiplication number sentence 2 1 x 6 = 14, what 3 word problem can be written that would fit this number sentence? Answers may vary. Mac baked some of his famous “Choco-Loco-Mint” brownies, but decided to cut larger size brownies so that 1 pan yields 6 servings. (Note: 1 brownie represents 1 serving.) If Mac baked 2 1 pans of “Choco-Loco3 Mint” brownies, how many brownies has he made?; etc. How would you draw a diagram that communicates the meaning of the multiplication number sentence and shows the relationship between the number sentence and the numbers in the problem? (The diagram would be 3 whole squares to represent the pans with 1 whole square divided into 3 equal parts to show 1 3 pan. Each pan would show 6 equal size sections drawn to represent the 6 brownies/pan.) Example of the diagram: 1 pan 1 pan 1 3 pan 6 brownies 6 brownies 2 brownies 21 3 x 6 = 14 1st factor x 2nd factor = product multiplier multiplicand # of pans brownies/pan How would you label each factor and the product in the diagram to identify what each factor and the product represents? (In the multiplication number sentence 2 1 3 x 6 = 14, the 2 1 is the first 3 factor, multiplier, and represents the number of pans (groups). The second factor is 6, multiplicand, and represents the number of brownies per pan (size of each group). The product 14 represents the total number of brownies in the 2 1 pans with 6 3 brownies per pan, total quantity after multiplying the 2 factors.) Phase: Explore/Explain List of Materials: Dividing Trim Activity: Dividing Trim What’s the teacher doing? What are the students doing? The teacher is monitoring students as they work, assisting as needed and asks the following guiding questions: Students are completing the Dividing Trim activity, recording their actions on the fraction diagrams. How many lengths of one-fourth yard are in a length of threefourths yard? (There are 3 lengths of one-fourth yard in a length of three-fourths yard.) What does the number line represent in this problem? (Number of lengths of yards.) How can you show lengths of one-fourth yard on this number line? (Begin at 0 and circle lengths of one-fourth.) What on the number line shows how many lengths of one-fourth yard are in a length of three- Students discover the pattern for dividing fractions with like denominators. fourths yard? (The number of circled lengths of one-fourth yard.) What represents the size of each group in this problem? (The length of one-fourth yard.) What represents the beginning quantity, the dividend, on the number line? (The mark and label for three-fourths yard.) What division number sentence can be written to represent the model? ( 3 4 yd. ÷ 1 4 yd. = 3 lengths) What is the dividend in the number sentence and what is its purpose? (Three-fourths, it indicates the beginning quantityamount of ribbon available to by cut.) What is the divisor in the number sentence and what is its purpose? (One-fourth, it indicates the size of each group which will be successively removed from the dividend.) What is the quotient in the number sentence and what is its purpose? (3, it indicates the number of groups of a given size that were successively removed from the dividend.) ***Facilitate a class discussion to encourage the discovery of the pattern for dividing fractions with like denominators. Ask: What do you notice about the denominators of the dividend and divisor? (They are like denominators.) What do you notice about the numerators of the dividend and the divisor? (If you divide the numerators from the dividend and the divisor, you get the quotient.) What do you notice about the value of the quotient? (It is not always smaller than the dividend and the divisor.) What rule can you write for the pattern in the problems? (If the fractions have common denominators, then divide the numerators of the dividend and the divisor to calculate the quotient.) Phase: Explore/Explain List of Materials: Extending Integers Activity: Extending Integers Distribute Extending Integers to pairs of students. Ask student to work together to complete the handout. What’s the teacher doing? What are the students doing? The teacher is monitoring students as they work. The students are working in pairs to complete the assignment. How do you show the sum of the integers on the number line? To sum -2, -4, and 4, start at 0 and draw a ray from 0 to -2. Then draw a ray 4 units to the left to represent -4. From the end of that arrow draw a ray 4 units to the right to represent 4. Phase: Explore/Explain List of Materials: Basic Operations Notes (one per student) Basic Operations Notes and Skill Drills Part I (one per student) Basic Operations Notes and Skill Drills Part II (one per student) Activity: Basic Operations Notes and Skill Drills Part I Distribute Basic Operations Notes to each student Talk through the notes briefly, discussing any processes that are particularly difficult for your students. Distribute Basic Operations Skill Drills Part I to each student. Ask students to work independently to complete the handout. What’s the teacher doing? What are the students doing? Monitor students as they work, assisting as needed. Use the facilitation questions below as needed. What would you do to solve ( -3 + 2)? Subtract 2 from 3 and keep the sign of the larger one; etc. How would you solve an addition problem with two negative numbers? Add the two numbers together and keep the negative sign. What would you do to solve (-6) (-2)? Add the opposite of the second number. So, in this problem it would become (-6) + 2, which would then require you to follow the addition rule of subtracting 2 from 6 and keeping the negative sign; etc. How would you solve a subtraction problem with one negative number? Add the opposite of the second number and follow the addition rules to solve. How is multiplication and division different from addition and subtraction of integers? In multiplication and division, you will multiply or divide the numbers as they appear. If they are both negative numbers, the answer is positive. If they are both positive numbers, the answer is still positive. If one number is positive and the other is negative, the answer is negative. What is the process you use to add fractions? Find a common denominator and then add the numerators. If the problem has mixed numbers, add the whole numbers as well. What is the process you use to subtract fractions? I will find a common denominator and then subtract the numerators. If the After working several problems, students can check their work with a calculator. Students complete the handout. problem has mixed numbers, I subtract the whole numbers as well. How do you solve the problem 5 1 3 2 ?. You will get a common 4 8 denominator of 8, so the new mixed 2 8 3 8 number will be 5 2 . Since you cannot subtract 2 - 3, you will have to borrow from the 5, making it a 4 and add the whole it 8 2 to the making 8 8 10 . Now, you subtract the 8 7 8 numerators 10 and 3, getting . The whole numbers are now 4 and 2, so 7 8 your final answer is 2 ; etc. What is the process you use to multiply fractions? Multiply the numerators by numerators and the denominators by denominators. If the factors are mixed numbers, you will need to first change them to an improper fraction. What is the process you use to divide fractions? Take the reciprocal of the second fraction and then follow the multiplication process. 3 5 How would you solve ( ) 1 Change 1 9 ? 10 9 to an improper fraction 10 19 . Then, take the reciprocal by 10 10 flipping the fraction over . Then, 19 of multiply 3 by 10 and 5 by 19 to get 30 , and keep the negative sign 95 because of the integer rules. Then, simplify the fraction to 6 ; etc. 19 Activity: Basic Operations Skill Drills Part II Distribute Basic Operations Skill Drills Part II to each student. Ask students to work independently to complete the handout. What’s the teacher doing? What are the students doing? Monitor students as they work, assisting After working several problems, as needed. students can check their work with a Use the facilitation questions below as calculator. needed. How do you solve 2.56 + 3.293? (Line up the decimals, add the numbers and move the decimal point straight down.) What is the process for subtracting decimals? (Line up the decimals, subtract the numbers and move the decimal point straight down.) What is the process for multiplying decimals? (Perform the multiplication as you would for whole numbers. Then, count the number of digits to the right of the decimal point in each factor. Place the decimal point in your answer so that your answer has the same number of digits after the decimal point.) How do you solve 55.04 ÷ 3.2? (First, you move the decimal out of the divisor, and then move the decimal point the same number of places in the dividend. Since we have to move it one place in 3.2, we will move it one place for 55.04. So, we will divide 550.4 by 32. You get 172, but need to bring the decimal point straight up, so that your answer is 17.2.) List of Materials: Fractions without Distractions (one per student, in color preferably) Activity: Fractions without Distractions From the following website, http://store.lonestarlearning.com/everything-else/conference-handouts/ select “Fractions” and print the second page “Fractions without Distractions and provide one per student.” Ask students to fold the notes into a little book and glue the notes into their journal. What is the teacher doing? What are the students doing? The teacher is monitoring students as Students are reviewing the process for they work. adding, subtracting, multiplying, and Then the teacher calls on different dividing rational numbers. students and has them explain a process from the Fractions without Distractions notes. The teacher then assigns several different problems from the Basic Skills Review Part I for students to explain their processes using the notes from Fractions without Distractions. Phase: Evaluate List of Materials: Paper Activity: Performance Assessment: For each expression below, determine the solution and complete each generalization. 1) When adding two rational numbers, if a pair of addends has the same sign, then the sum will have the __________ of both addends. 2) When adding two rational numbers, if a pair of addends has the opposite signs, then the sum will have the sign of the addend with the _________ ____________ _____________. 3) When subtracting two non-zero positive rational numbers (with the subtrahend larger than the minuend), the difference will always be ____________ than the minuend and be ____________. 4) When multiplying two non-zero positive rational numbers, where at least one of the factors is greater than one, the product always will be __________ than the smallest factor. 5) When dividing two non-zero positive rational numbers, where the dividend is greater than the divisor, the quotient will always be ___________ than one. 6) When multiplying two or more rational numbers with a(n) __________ number of negative signs, then the product is ___________________. What’s the teacher doing? What are the students doing? The teacher is monitoring students as they work. The students are working individually to complete the assignment. Phase: Explore/Explain List of Materials: Buyer’s Bonanza Activity: Buyer’s Bonanza Distribute Buyer’s Bonanza to each student. Ask them to complete the tables. What the teacher is doing What the students are doing The teacher is monitoring students as they work, assisting as needed. Students are working in pairs to complete the handout. How did you find the amount of discount? How is that like finding the sales tax? What did you do to find the amount of sales tax? How did you find the total cost of the curtains? How is that different from finding the amount of the sale price? What effect does the purchase price of a given item have on sales tax? What process is used to calculate sales tax on a given purchase? Phase: Explore/Explain Materials: You Can’t’ Hide from Taxes (one per student) Activity: You Can’t Hide from Taxes From the Texas Council on Economic Education lessons for Personal and Financial Literacy for Grades 7 & 8, (free to all Texas educators on the internet http://economicstexas.org/?page_id=5497 ), please distribute Lesson 1 “You Can’t Hide from Taxes” and follow the lesson and teacher notes. Phase: Elaborate Materials: Know Your Worth (one per student) Activity: From the Texas Council on Economic Education lessons for Personal and Financial Literacy for Grades 7 & 8, (free to all Texas educators on the internet http://economicstexas.org/?page_id=5497 ), please distribute Lesson 4 “Know Your Worth” and follow the lesson notes. Phase: Evaluate List of Materials: Paper Activity: Performance Assessment: Analyze the problem situation(s) described below. Organize and record your work for each of the following tasks. Using precise mathematical language, justify and explain each solution process. Drew has decided it is time to purchase a new car. He has shopped around and decided which car he would like to purchase. 1) The price of the car Drew wants to purchase is $17,998. This price includes all fees associated with the purchase of a vehicle, such as title and registration fees; however, the price does not include the required sales tax. a) In Texas, the sales tax rate for automobiles is 6 %. Calculate the sales tax and final price of the new car. 2) Drew will have to finance his car purchase through his bank with a car loan. The financial institution requires him to provide a net worth statement to determine if he will qualify for the loan. Drew’s current financial information is below: a) Create and organize a financial assets and liabilities record and include a net worth statement. 3) Drew’s car loan was approved for the purchase of a vehicle with monthly payments that cannot exceed 12% of his current net worth. a) Determine if the loan Drew was approved for will allow him to purchase his new car, if his monthly car payment is projected to be $352.18. 4) Drew is going to file his 2014 federal income tax return and would like to use his refund, if any, towards a down payment for the new car. a) Using the taxable income brackets and rates from the Internal Revenue Service below, calculate the income tax Drew is required to pay if: his taxable income for 2014 was $48,750 he plans on filing as Head of Household 5) If the amount of paid income tax exceeds the required income tax, Drew will receive a refund of the difference when he files his 2014 federal income tax return. If the amount paid in income tax is less than the required income tax, Drew will owe the difference when he files his federal income tax return. Drew paid a total of $9,000 in income tax in 2014 through monthly deductions in his paycheck. a) Determine if Drew will receive an income tax refund that he can use as a down payment towards the purchase of his car and how this down payment will affect the amount Drew will need to finance. What is the teacher doing? What are the students doing? The teacher is monitoring students as they work, allowing for individual prompting, support and encouragement through the task. The student are working individually to complete the task. Engage Day 1 24 2 0 0.75 120% −5 32 1 4 2 1 5 13 100 −0.25 3 2 50% Explore 1: Day 1 Grouping Mat Rational Numbers Integers Whole Numbers Natural or Counting Numbers Notes on Rational Numbers Sets of Numbers Examples Natural or Counting Numbers 1, 2, 3, 4, 5, …. Whole Numbers 0, 1, 2, 3, 4, 5, … Integers …-5, -4, -3, -2, -1, 0, 1,2 ,3 ,4 ,5 … Rational Numbers All natural or counting numbers plus the whole numbers plus the integers plus the fractions where the numerator and denominator are integers (positive and negative) plus the terminating and repeating decimals plus the mixed numbers and decimals Statement All natural numbers are whole numbers. All whole numbers are integers. All integers are rational numbers. All whole numbers are rational numbers. All rational numbers are integers. All integers are whole numbers. All natural numbers are rational numbers. The set of integers is a subset of the set of rational numbers. The set of natural numbers is a subset of the set of whole numbers. True or False? Notes on Rational Numbers Key Statement True or False? All natural numbers are whole numbers. True All whole numbers are integers. True All integers are rational numbers. True All whole numbers are rational numbers. True All rational numbers are integers. False All natural numbers are rational numbers. True All integers are whole numbers. False The set of integers is a subset of the set of rational numbers. The set of natural numbers is a subset of the set of whole numbers. True True Another Look at Adding and Subtracting Fractions KEY Problem 1. 2. 3. 1 3 8 8 1 3 4 8 8 44 1 84 2 3 1 4 3 3 3 1 4 9 4 4 3 3 4 12 12 94 5 12 12 3 2 5 3 3 3 2 5 9 10 5 3 3 5 15 15 9 10 19 15 15 19 4 1 15 15 4. 5 5 3 5 3 5 Description of Process Sample descriptions are given Already has a common denominator Add the numerators Keep the common denominator Simplify the resulting fraction Rename each fraction with a common denominator of 12 Subtract the numerators Keep the common denominator Rename each fraction with a common denominator of 15 Add the numerators Keep the common denominator Write the improper fraction as a mixed number Does not need a common denominator, since there is only one fraction. Combine the fraction with the whole number to make a mixed number. Another Look at Adding and Subtracting Fractions KEY Problem 7 1 5. 2 8 2 7 1 4 7 4 2 2 8 24 8 8 74 3 2 2 8 8 Description of Process Sample descriptions are given Rename each fraction with a common denominator of 8 Combine the fractions by subtracting the numerators and keeping the common denominator Keep the whole number and combine with the resulting fraction to form a mixed number 3 8 3 8 3 2 1 8 8 8 83 5 1 1 8 8 3 3 7. 1 8 4 3 32 3 6 1 1 8 42 8 8 36 9 1 1 8 8 9 1 1 1 1 1 2 8 8 8 1 3 8. 2 5 10 1 2 3 2 3 2 2 5 2 10 10 10 12 3 10 2 3 1 1 10 10 10 10 10 12 3 9 1 1 10 10 6. 2 Does not need a common denominator Rename the two as one and eighteighths so that three-eighths can be subtracted. Subtract the numerators Keep the common denominator Combine the fraction with the whole number to make a mixed number. Rename each fraction with a common denominator of 8 Add the numerators Keep the common denominator Change the improper fraction to a mixed number and combine it with the whole number Rename each fraction with a common denominator of 10 Rename two and two-tenths as one and twelve-tenths so that three-tenths can be subtracted Subtract the numerators Keep the common denominator Another Look at Adding and Subtracting Fractions KEY 9. Alice has two and one-fourth pies to eat. In one minute she ate six-eighths of a pie. How much pie does Alice have left to eat? 1 6 2 6 10 6 4 1 2 2 1 1 1 pies 4 8 8 8 8 8 8 2 10. Charles ate one and one-half pies in 3 minutes and two-thirds of a pie in 2 minutes. How much pie has Charles eaten in 5 minutes? 1 2 3 4 7 1 1 1 1 1 1 1 2 pies 2 3 6 6 6 6 6 11. Rachel has walked three-fifths of a mile. She has ten minutes to reach the two mile marker. How far must Rachel walk to reach the two mile marker? 3 5 3 2 2 1 1 miles 5 5 5 5 12. What is the perimeter of the rectangle shown? 1 1 1 1 5 5 4 4 4 4 3 3 4 4 3 3 4 4 5 5 20 20 20 20 18 9 14 14 feet 20 10 4 feet 3 feet 3 hours a day outside her home traveling to and from work and working 4 1 2 at her job. Kelly spends hour one way traveling to work. Kelly is allowed hour total 2 3 1 for her breaks. She is allowed 1 hour for lunch. The rest of the time Kelly spends 4 working in her office. How much time does Kelly spend working in her office? 3 1 1 2 1 9 11 21 11 10 5 9 1 9 2 8 2 6 6 hours 4 2 2 3 4 12 12 12 12 12 6 13. Kelly spends 9 5 cm, how long is the base? 6 5 3 3 5 1 5 3 2 1 17 6 6 17 13 17 13 4 4 cm 6 4 4 6 2 6 6 6 3 14. If the perimeter of the isosceles triangle below is 17 cm Another Look at Adding and Subtracting Fractions Problem 1. 1 3 8 8 2. 3 1 4 3 3. 3 2 5 3 4. 5 3 5 Description of Process Another Look at Adding and Subtracting Fractions Problem 7 1 5. 2 8 2 6. 2 3 8 3 3 7. 1 8 4 1 3 8. 2 5 10 Description of Process Another Look at Adding and Subtracting Fractions 9. Alice has two and one-fourth pies to eat. In one minute she ate six-eighths of a pie. How much pie does Alice have left to eat? 10. Charles ate one and one-half pies in 3 minutes and two-thirds of a pie in 2 minutes. How much pie has Charles eaten in 5 minutes? 11. Rachel has walked three-fifths of a mile. She has ten minutes to reach the two mile marker. How far must Rachel walk to reach the two mile marker? 12. What is the perimeter of the rectangle shown? 4 feet 3 feet 3 hours a day outside her home traveling to and from work and working 4 1 2 at her job. Kelly spends hour one way traveling to work. Kelly is allowed hour total 2 3 1 for her breaks. She is allowed 1 hour for lunch. The rest of the time Kelly spends 4 working in her office. How much time does Kelly spend working in her office? 13. Kelly spends 9 14. If the perimeter of the isosceles triangle below is 17 5 cm, how long is the base? 6 cm Mac’s Brownies Mac has created a new brownie, “Choco-Loco-Mint." One pan of brownies yields 8 servings. (Note: 1 brownie represents 1 serving.) If Mac baked 3 1 2 pans of “Choco-Loco-Mint” brownies, how many brownies has he made? Select one of the ranges of numbers below that could represent the solution: (A) (B) (C) 20 to 30 brownies 24 to 30 brownies 24 to 32 brownies Directions Let the whole fraction square represent one pan. Use the fraction squares to model the problem. Draw a diagram of the fraction square model. Label all parts of the diagram to communicate the solution. Write an addition number sentence and a multiplication number sentence to show the relationship between the numbers in the problem. Mac’s Brownies KEY Mac has created a new brownie, “Choco-Loco-Mint." One pan of brownies yields 8 1 servings. (Note: 1 brownie represents 1 serving.) If Mac baked 3 pans of “Choco2 Loco-Mint” brownies, how many brownies has he made? Select one of the ranges of numbers below that could represent the solution. (A) (B) (C) 20 to 30 brownies 24 to 30 brownies 24 to 32 brownies Have students explain their selection. Keep a tally of all the student responses. This tally will be used to compare to the reasonableness of the solution. Directions Let the whole fraction square represent one pan. Use the fraction squares to model the problem. Draw a diagram of the fraction square model. Label all parts of the diagram to communicate the solution. Write an addition number sentence and a multiplication number sentence to show the relationship between the numbers in the problem. 1 We need to determine how many brownies are made with 3 pans (number of groups) 2 Each pan yields 8 brownies. (The number in each group) I will lay 8 equal size pieces on each pan to represent the number of brownies each pan will yield. 1 pan yields 8 brownies 3 1 2 + 1 pan yields 8 brownies + pans of 8 brownies per pan = 3 pans of 8 brownies per pan + 1 pan yields 8 brownies 1 2 + half pan yields 4 brownies pan of 8 brownies = 24 brownies + 4 brownies = 28 brownies Addition Number Sentence: 8 brownies + 8 brownies + 8 brownies + 4 brownies = 28 brownies Multiplication Number Sentence: 3 1 2 pans x 8 brownies per/pan = 28 brownies 1 2 The multiplier (the first factor 3 ) indicates the number of times the multiplicand (the second factor 8) is used as an addend when using repeated addition. Dividing Trim Use the number line to model each given situation. Translate each situation to a division number sentence. 1. What is the maximum number of 1 yard lengths of ribbon trim that can be cut from a 4 3 yard length of ribbon trim? 4 3 yd. 4 0 yd. 1 yd. Number Sentence: 2. What is the maximum number of an 1 yard lengths of ribbon trim that can be cut from 12 8 yard length of ribbon trim? 12 0 yd. 8 yd. 12 1 yd. Number Sentence: 3. What is the maximum number of 2 yard lengths of ribbon trim that can be cut from a 6 4 yard length of ribbon trim? 6 0 yd. Number Sentence: 4 yd. 6 1 yd. Dividing Trim 4. What is the maximum number of a 3 yard lengths of ribbon trim that can be cut from 12 9 yard length of ribbon trim? 12 0 yd. 9 yd. 12 1 yd. Number Sentence: 5. What is the maximum number of 2 yard lengths of ribbon trim that can be cut from a 6 5 yard length of ribbon trim? 6 5 yd. 6 0 yd. 1 yd. Number Sentence: 6. What is the maximum number of an 5 yard lengths of ribbon trim that can be cut from 12 8 yard length of ribbon trim? 12 0 yd. Number Sentence: 8 yd. 12 1 yd. Dividing Trim KEY Use the number line to model each given situation. Translate each situation to a division number sentence. 1. What is the maximum number of 1 yard lengths of ribbon trim that can be cut from a 4 3 yard length of ribbon trim? 4 1 yd 4 1 yd 4 1 yd 4 1 length 1 length 1 length 3 yd. 4 0 yd. Number Sentence: 3 1 yd. yd. = 3 lengths 4 4 2. What is the maximum number of an 1 yd. 1 yard lengths of ribbon trim that can be cut from 12 8 yard length of ribbon trim? 12 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 1 yd 12 1 length 8 yd. 12 0 yd. Number Sentence: 1 yd. 8 1 yd. yd. = 8 lengths 12 12 3. What is the maximum number of 2 yard lengths of ribbon trim that can be cut from a 6 4 yard length of ribbon trim? 6 1 yd. 6 1 yd. 6 1 length 1 yd. 6 1 length 4 yd. 6 0 yd. Number Sentence: 1 yd. 6 4 2 yd. yd. = 2 lengths 6 6 1 yd. Dividing Trim KEY 4. What is the maximum number of a 3 yard lengths of ribbon trim that can be cut from 12 9 yard length of ribbon trim? 12 1 yd 12 1 yd 12 1 yd 12 1 yd 12 1 length 1 yd 12 1 yd 12 1 yd 12 1 length 1 yd 12 1 yd 12 1 length 9 yd. 12 0 yd. Number Sentence: 1 yd. 9 3 yd. yd. = 3 lengths 12 12 5. What is the maximum number of 2 yard lengths of ribbon trim that can be cut from a 6 5 yard length of ribbon trim? 6 1 yd. 6 1 yd. 6 1 yd. 6 1 length 1 yd. 6 1 yd. 6 1 2 1 length 5 yd. 6 0 yd. Number Sentence: 1 yd. 5 2 1 yd. yd. = 2 lengths 6 6 2 6. What is the maximum number of an length 5 yard lengths of ribbon trim that can be cut from 12 8 yard length of ribbon trim? 12 1 yd 12 1 yd 12 1 yd 12 1 length 1 yd 12 1 yd 12 1 yd 12 1 yd 12 3 5 length 8 yd. 12 0 yd. Number Sentence: 1 yd 12 8 5 3 yd. yd. = 1 lengths 12 12 5 1 yd. Extending Integers 1. Use the given clues to identify the three integers. Write an equation to show the sum of the three integers and use a number line to model the equation. Clue 1: The first integer is 2 units to the left of 0. Clue 2: The second integer is 2 units less than the first integer. Clue 3: The third integer is 6 units more than the first integer. 2. Use the given clues to identify the three integers. Write an equation to show the difference of the three integers and use a number line to model the equation. Clue 1: The first integer is 2 more than negative 7. Clue 2: The second integer is 4 less than 3. Clue 3: The third integer is 6 units more than the first integer. 3. Use the given clues to identify the three integers. Write an equation to show the product of the three integers. Clue 1: The first integer is 4 less than negative 6. Clue 2: The second integer is 4 less than 23. Clue 3: The third integer is 8 units more than the first integer. 4. Use the given clues to identify the two integers. Write an equation to show the quotient of the two integers and use positive and negative tiles to model the equation. Clue 1: The first integer is 14 more than negative 6. Clue 2: The second integer is 4 less than 2. Extending Integers Write an equation and use a vertical number line to match the given situation for problems 5 through 8. 5. An elevator was on the third floor and went down four floors. Where is the elevator? 3 2 7. The elevator was on the first floor and was needed on the second floor down in the basement. How many floors did it need to move and in what direction? 6. The elevator was two floors down in the basement and went up five floors. Where is the elevator? 3 2 1 1 0 0 –1 –1 –2 –2 –3 –3 3 2 1 8. The elevator was 1 floor down in the basement and was needed on the second floor. How many floors did it need to move and in what direction? 3 2 1 0 0 –1 –1 –2 –2 –3 –3 Extending Integers 9. Gary was reviewing his monthly bank statement. He had three withdrawals of $75, $150, and $425. He had two deposits of $500 and $206. If his balance at the beginning of the month was $29, what is his current balance after the withdrawals and deposits? Write a number sentence to match the situation and solve the problem. Extending Integers KEY 1. Use the given clues to identify the three integers. Write an equation to show the sum of the three integers and use a number line to model the equation. Clue 1: The first integer is 2 units to the left of 0. Clue 2: The second integer is 2 units less than the first integer. Clue 3: The third integer is 6 units more than the first integer. Answer: first integer is (−2); second integer is (−4) = (−2) + (−2); third integer is (−2) + 6 = 4 Equation: (−2) + (−4) + 4 = (−2) –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 2. Use the given clues to identify the three integers. Write an equation to show the difference of the three integers and use a number line to model the equation. Clue 1: The first integer is 2 more than negative 7. Clue 2: The second integer is 4 less than 3. Clue 3: The third integer is 6 units more than the first integer. Answer: first integer is (−5) = 2 + (−7); second integer is (−1) = 3 + (−4); third integer is (−5) + 6 = 1 Equation: (−5) ─ (−1) ─ 1 = (−5); Since 1 and (−1) create a zero pair, they do not count as an overlap on the number line. opposite of 1 opposite of (−1) –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 3. Use the given clues to identify the three integers. Write an equation to show the product of the three integers. Clue 1: The first integer is 4 less than negative 6. Clue 2: The second integer is 4 less than 23. Clue 3: The third integer is 8 units more than the first integer. Answer: first: (−10) = (−6) + (−4); second: 19 = 23 + (−4); third: (−2) = (−10) + 8 Equation: (−10)(19)(−2) = 380 4. Use the given clues to identify the two integers. Write an equation to show the quotient of the two integers and use positive and negative tiles to model the equation. Clue 1: The first integer is 14 more than negative 6. Clue 2: The second integer is 4 less than 2. Answer: first: 8 = (− 6) + 14 second: (−2) = 2 + (−4) Equation: 8 ÷ (−2) = (−4) Divide 8 positive tiles equally into 2 rows and then take the opposite of the tiles in each row. (+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (–) (–) (–) (–) (–) (–) (–) (–) (+) (+) (+) (+) (+) (+) Extending Integers KEY Write an equation and use a vertical number line to match the given situation for problems 5 through 8. 5. An elevator was on the third floor and went down four floors. Where is the elevator? Answer: 3 + (−4) = (−1) 7. The elevator was on the first floor and was needed on the second floor down in the basement. How many floors did it need to move and in what direction? You want the distance between (−2) and 1, so you subtract. Answer: (−2) ─ 1 = (−3) (−2) + (−1) = (−3) 3 floors down. 3 2 6. The elevator was two floors down in the basement and went up five floors. Where is the elevator? Answer: (−2) + 5 = 3 3 2 1 1 0 0 –1 –1 –2 –2 –3 –3 3 2 1 0 –1 –2 –3 8. The elevator was 1 floor down in the basement and was needed on the second floor. How many floors did it need to move and in what direction? You want the distance between 2 and (−1), so you subtract. Answer: 2 ─ (−1) = 3 2+1=3 3 floors up. 3 2 1 0 –1 –2 –3 Extending Integers KEY 9. Gary was reviewing his monthly bank statement. He had three withdrawals of $75, $150, and $425. He had two deposits of $500 and $206. If his balance at the beginning of the month was $29, what is his current balance after the withdrawals and deposits? Write an equation to match the situation and solve the problem. Answer: $29 ─ 75 ─ 150 ─ 425 + 500 + 206 = $85 Basic Operations Notes Fractions Multiplication: To multiply fractions, you multiply straight across. 1 3 3 = Example: 8 8 64 In some cases you might need to simplify the answer, so make sure you double check your answer. You can simplify the product after multiplying Example: 2 1 2 2 1 = ( is simplified to ) 3 2 6 6 3 Or you can simplify the factors before multiplying Example: 4 1 2 1 4 2 2 1 2 becomes because ( is simplified to ), so = 6 5 3 5 6 3 3 5 15 If one of the factors is a whole number, you write the whole number as an improper fraction by writing the whole number as the numerator and by placing a “1” in the denominator. 1 3 1 3 becomes = Example: 3 5 1 5 5 If one of the factors is a mixed number, you write the mixed number as an improper fraction. 2 1 17 1 17 2 becomes = =1 Example: 3 5 3 5 3 15 15 (To write a mixed number as an improper fraction, you multiply the whole number by the denominator, and then add the numerator. In the problem above, multiply 5 times 3 then add 2, keeping the original denominator.) Division: To divide fractions, you keep the first term the same, change the division to multiplication, then flip the second term. Another way of saying the same thing is to multiply by the reciprocal of the second term. If either term is a mixed number or whole number, you will use the procedures given above in the section on multiplication. Example: 5 1 5 2 10 5 1 = = . This answer simplifies to =1 . 8 2 8 1 8 4 4 Example: 4 2 1 22 3 66 66 1 becomes = and = 13 . 5 3 5 1 5 5 5 Basic Operations Notes Addition and Subtraction (fractions): To add or subtract fractions you must have a common denominator. 5 1 1 2 + , the least common demoninator is 8. can be written as . 8 2 2 8 Example: 5 2 7 So, + = . 8 8 8 For 13 5 13 39 , the least common demoninator is 9. can be written as . 3 9 3 9 Example: 39 5 34 34 7 So, = . This solution may be written as or 3 . 9 9 9 9 9 For Decimals Addition and Subtraction: The decimals in the problem must have the place values aligned to add or subtract. If the problem is written across the page, it is recommended you rewrite the problem vertically to perform the addition or subtraction. Example: 2.345 + 17.4 would be written as: 2.345 +17.4 Example: 62.34 +51.4 0.0135 + 3.09 21.897 4.5060 – 0.645 –1.3450 Multiplication: Complete the multiplication, then place the decimal in the answer based on the number of digits to the right of the decimal in each factor. The decimals in the problem do not have to line up. Example: 1.234 (there are 3 digits to the right of the decimal) X 5.67 (there are 2 digits to the right of the decimal) -------6.99678 (there are 5 digits to the right of the decimal) Example: 5.4(3.02) = 16.308 (There is one digit to the right of the decimal in the first factor plus two digits to the right of the decimal in the second factor, which makes three digits to the right of the decimal in the product.) Decimal Division: When dividing a decimal by a whole number, the decimal in the product must be lined up with the 1 .4 decimal in the quotient. Example: 4.2 ÷ 3 = 1.4 3 4 .2 When dividing a decimal by a decimal, you must move the decimals in the dividend and divisor the number of places required to make the divisor a whole number. Example: 24.9 ÷ 0.3 you will need to move the decimals in each number exactly one digit. The problem becomes 249 ÷ 3. 0.3 24.9 3 249 0.03 24.9 3 2490 It might be necessary to add one or more zeros as you are moving decimals. Example: 24.9 ÷ 0.03. Each decimal must be moved two digits. The problem becomes 2490 ÷ 3. Basic Operations Notes Integers Multiplication and Division: If the signs are the same, the answer is positive. If the signs are different, the answer is negative. Example: (24)(3) = 72 (−108)(−4) = 432 (24)(−3) = (−72) (−108)(4)= (−432) If the problem has more than two integers, you begin with the first two integers and perform the multiplication or division using the rule above. Then, using the sign of that product perform the next operation. Example: (−2)(3)(4) (−2)(3) = (−6) (−6)(4) = (−24) Addition: When adding integers that have the same sign, just add the integers and keep the sign. Example: 4 + 9 = 13 or (−4) + (−9) = (−13) If the signs are NOT the same, find the difference between the two numbers and use the sign of the number that is the greatest distance from zero. Example: (−5) + 4 = (−1) and 12 + (−3) = 9 Subtraction: To subtract an integer you will need to add the opposite of the integer being subtracted. Example: 7 – (−3) first change the sign from subtraction to addition, then change the (−3) to (+3). The problem becomes 7+ (+3) = 10. Example: (−12) – 8 would become (−12) + (−8). It is important to remember that the first term does not change its sign! Basic Operations Skill Drills Part 1 KEY Integers Simplify: 1) -102+ 465 = 363 2) 365 + (-241) = 124 3) (-18) + (-16) = (-34) 4) 3657 – (-1245) = 4902 5) -357 – (-291) = (− 66) 6) (-487) -1357 = (-1844) 7) (13)(15) = 195 8) (-144)(12) = (-1728) 9) (- 60)(- 50) = 3000 10) (21)(-17) = (-357) 11) 468 ÷ (-12) = (-39) 12) ( -678) ÷ (-226) = 3 13) -575 1 = (- ) 1150 2 14) (15)(-27) ÷(-5) = 81 Basic Operations Skill Drills Part 1 KEY Fractions Simplify: 1. 5 1 2 1 = = 6 2 6 3 2. 2 4 8 = 5 5 25 3. 7 2 11 1 = or 1 10 5 10 10 4. 1 1 2= 2 4 4 3 12 6 5. ( )( ) = or 5 14 70 35 6. 3 1 13 = 8 6 24 3 4 15 7. (- ) (- ) = 4 5 16 8. 3 5 3 = 4 4 5 8 2 14 5 9. (- ) = (- ) or -1 9 3 9 9 10. 4 2 8 4 = or 6 9 18 9 2 3 6 2 11. (- ) = (- ) or (- ) 3 5 15 5 12. 5 3 5 19 3 13. -4 (-1 ) = (- ) or (-2 ) 8 8 8 1 2 14. 2 = or 1 1 2 2 1 7 3 = or 1 4 4 4 Basic Operations Skill Drills Part 1 Integers Simplify: 15) -102+ 465 = 16) 365 + (-241) = 17) (-18) + (-16) = 18) 3657 – (-1245) = 19) -357 – (-291) = 20) (-487) -1357 = 21) (13)(15) = 22) (-144)(12) = 23) (-60)(-50) = 24) (21)(-17) = 25) 468 ÷ (-12) = 26) ( -678) ÷ (-226) = -575 = 1150 28) (15)(-27) ÷(-5) = 27) Basic Operations Skill Drills Part 1 Fractions Simplify: 15. 5 1 = 6 2 16. 2 4 = 5 5 17. 7 2 = 10 5 18. 1 2= 2 4 3 19. ( )( ) = 5 14 20. 3 1 = 8 6 3 4 21. (- ) (- ) = 4 5 22. 3 5 = 4 4 8 2 = 23. (- ) 9 3 24. 4 2 = 6 9 2 3 = 25. (- ) 3 5 26. 5 3 5 27. -4 (-1 ) = 8 1 28. 2 = 1 2 1 = 4 Basic Operations Skill Drills Part 2 KEY Decimals Simplify: 29) 8.3 + 0.24 + 6 = 14.54 30) 46 – 18.9 = 27.1 31) (0.21)(0.3) = 0.063 32) 5 – 2.4 – 1.38 + 0.62 = 1.84 33) 33.7 – 98.68 = (-64.98) 34) (1.3)(0.005) = 0.0065 35) (-2.5)(0.012) = (-0.03) 36) 7.026 ÷ (-0.03) = (-234.2) 37) -8.34 = 4,170 -0.002 38) 13.05 ÷ 0.4 = 32.625 39) (-2.5) – 3.4 + 2 = (-3.9) 40) 5 – (-2.5) – 9 = (-1.5) 41) 4 + 3.08 – 0.99 = 6.09 42) 3.2 ÷ 0.002 = 1,600 Basic Operations Skill Drills Part 2 KEY Mixed Review Simplify: 1) 7 1 35 3 1 = or 1 8 4 32 32 3) 4 3 3 437 45 3 = or 7 8 7 56 56 2) 5 3 20 5 = or 8 4 24 6 4) 5 3 1 25 1 2 = or 3 8 4 8 8 5) 8% sales tax for $24.58 (0.08)(24.58) = 1.9664 6) 152.5 ÷ 0.05 = 3050 7) 5 – 2.57 + 8.623 = 11.053 8) Half off $189.98 1 189.98 • = 94.99 2 9) (-50) (-4) = 200 10) 11) (-18) + (-7) = (-25) 12) (-202) – 81 = (-283) 1 13) (4.03)( (- ) ) = (-2.015) 2 14) -52.31 – 48.224 = (-100.534) -305 = 61 -5 Basic Operations Skill Drills Part 2 Decimals Simplify: 43) 8.3 + 0.24 + 6 = 44) 46 – 18.9 = 45) (0.21)(0.3) = 46) 5 – 2.4 – 1.38 + 0.62 = 47) 33.7 – 98.68 = 48) (1.3)(0.005) = 49) (-2.5)(0.012) = 50) 7.026 ÷ (-0.03) = 51) -8.34 = -0.002 52) 13.05 ÷ 0.4 = 53) (-2.5) – 3.4 + 2 = 54) 5 – (-2.5) – 9 = 55) 4 + 3.08 – 0.99 = 56) 3.2 ÷ 0.002 = Basic Operations Skill Drills Part 2 Mixed Review Simplify: 15) 7 1 1 = 8 4 17) 4 3 3 3 = 8 7 16) 5 3 = 8 4 18) 5 3 1 2 = 8 4 19) 8% sales tax for $24.58 (0.08)(24.58) = 20) 152.5 ÷ 0.05 = 21) 5 – 2.57 + 8.623 = 22) Half off $189.98 1 189.98 • = 2 23) (-50)(-4) = 24) 25) (-18) + (-7) = 26) (-202) – 81 = 1 27) (4.03)( (- ) ) = 2 28) -52.31 – 48.224 = -305 = -5 Buyer’s Bonanza Bonnie’s mother bought curtains on sale. The regular price of the curtains was $76. The curtains were marked 25% off. Use the percent bar model below to show the amount of the discount and the sale price of the curtains. Dollars 0% 25% 50% 75% 100% Percent Regular Price of Curtains Amount of 25% Discount Sale Price How Sales Tax Works Bonnie’s mother bought curtains. The sales tax on the price of the curtains was 5%. Use the percent bar model below to show the amount of the sales and the total cost of the curtains. Dollars 0% 25% 50% 75% 100% Percent Price of Curtains Amount of Sales Tax Total Cost of Curtains Buyer’s Bonanza Key Bonnie’s mother bought curtains on sale. The regular price of the curtains was $76. The curtains were marked 25% off. Use the percent bar model below to show the amount of the discount and the sale price of the curtains. Dollars $0 $19 $57 $38 Amount of discount 0% 25% $76 Sale price 50% 75% 100% Percent Regular Price of Curtains Amount of 25% Discount Sale Price $76 $19 $57 How Sales Tax Works Bonnie’s mother bought curtains. The sales tax on the price of the curtains was 5%. Use the percent bar model below to show the amount of the sales and the total cost of the curtains. Dollars $0 $2.85 $57 Tax 0% 25% 5% 50% 75% 100% $59.85 $0 $2.85 $57 Tax 0% 5% 25% 50% 75% 100% 105% Percent Price of Curtains Amount of Sales Tax Total Cost of Curtains $57 $2.85 $59.85