HMH Unit 1: The Number System Progression Before: 6 th grade During the years from kindergarten to eighth grade, students must repeatedly extend their conception of number. At first, number means counting number : 1, 2, 3... Soon after that, 0 is used to represent none and the whole numbers are formed by the counting numbers together with zero. The next extension is fractions. At first, fractions are barely numbers and tied strongly to pictorial representations. Yet by the time students understand division of fractions, they have a strong concept of fractions as numbers and have connected them, via their decimal representations, with the base-ten system used to represent the whole numbers. During middle school, fractions are augmented by negative fractions to form the rational numbers. During: 7 th grade In seventh grade students consolidate their understanding of operations on signed numbers. They add, subtract, multiply, and divide rational numbers, gaining fluency in one of the parts of the middle school curriculum that is most difficult to conceptualize. Operations on signed numbers is particularly difficult for students because of a conflation between the symbol indicating the operation of subtraction and the symbol indicating that a number is negative. Another aspect that leads to difficulty in learning about operations on signed numbers is the proliferation of rules that students are expected to remember, too often without recourse to methods for reasoning out how the numbers behave. Collections of rules that students memorize without understanding are likely to break down under the pressure of difficult problems or after a period of disuse. Grade 7 students understand that a rational number is the ratio of two integers, and are given the opportunity to convert a rational number to a decimal using long division. They also know that the decimal form of a rational number terminates in 0s or eventually repeats. After: 8 th grade In Grade 8 students are given the opportunity to refine their understanding of terminating and repeating expansions of rational numbers. In Grade 8 students are given the opportunity to understand that terminating decimals are actually repeating because it is the number zero that repeats. In Grade 8 students are given the opportunity to convert a rational number to a fraction. In Grade 8 also, students extend their notation of the number system. This time they augment the rational numbers with the irrational numbers to form the real numbers. After having worked with rational approximations of π, many students mistakenly believe that is an irrational number, and this betrays the lack of opportunity to study irrational numbers and the idea of using rational approximations of irrational numbers. With regard to irrational numbers students are given the opportunity to learn how to write rational approximations of irrational numbers and use these to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.
UNIT 1: The Number System Planning Organizer TIME: 7 weeks UNIT NARRATIVE: This unit builds upon the students understanding of rational numbers that was developed in 6 th grade. In Grade 7, learning now moves to exploring and ultimately formalizing rules for operations (addition, subtraction, multiplication and division) with integers. Using both contextual and numerical problems, students should explore what happens when negative numbers and positive numbers are combined. Repeated opportunities over time will allow students to compare the results of adding, subtracting, multiplying and dividing pairs of numbers, leading to the generalization of the rules. Fractional rational numbers and whole numbers should be used in computations and explorations. Students will be able to give contextual examples of integer operations, write and solve equations for real-world problems and explain how the properties of operations apply. Real-world situations could include: profit/loss, money, weight, sea level, debit/credit, football yardage, etc. Textbook Correlations: Additional Resources HMH GOMATH: Unit 1: Modules 1,2,3 Personal Math Trainer, Math on the Spot, Formative Assessment Lessons, and Real Player Activator videos ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: proportion, rational number, ratio, 1. How can previous mathematical understanding of number operations apply to reciprocal, least common multiple, inverse operation adding and subtracting rational numbers? 2. How can previous mathematical understanding of number operations apply to multiplying and dividing rational numbers? 3. What strategies can be used to accurate solve real world and mathematical problems involving rational numbers? CLUSTER HEADING & STANDARDS: Apply and extend previous understandings of operations with fractions 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7. NS.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7. NS.1b Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit. Those receiving special attention in this unit are bolded. MP 1 Make sense of problems and persevere when solving them. 7.NS.3 MP 2 Reason quantitatively and abstractly. 7.NS.1, 7.NS.2, 7.NS.3 MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. 7.NS.1, 7.NS.2 MP 5 Use appropriate tools strategically. 7.NS.3 MP 6 Attend to precision. 7.NS.3 MP 7 Look for and make use of structure.
7. NS.1c Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 7. NS.1d Apply properties of operations as strategies to add and subtract rational numbers. 7. NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7. NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then - (p/q) = (-p)/q = p/ (-q). Interpret quotients of rational numbers by describing real-world contexts. 7. NS.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.1, 7.NS.2, 7.NS.3 MP 8 Look for and express regularity in repeating reasoning 7.NS.8 7. NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7. NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers. Learning Outcomes: Interpret sums of rational numbers by describing real-world contexts. Explain and justify why the sum of p + q is located a distance of q in the positive or negative direction from p on a number line. Apply and extend previous understanding to represent addition and subtraction problems of rational numbers with a horizontal or vertical number line. Represent the distance between two rational numbers on a number line is the absolute value of their difference and apply this principle in real-world contexts. Apply the principle of subtracting rational numbers in real-world contexts. Apply properties of operations as strategies to add and subtract rational numbers. Apply the properties of operations, particularly distributive property, to multiply rational numbers. Interpret the products of rational numbers by describing realworld contexts. Interpret the quotient of rational numbers by describing real-world contexts. Apply properties of operations as strategies to multiply and divide rational numbers. Solve real-world mathematical problems by adding, subtracting, multiplying, and dividing rational numbers, including complex fractions. End of the Unit Assessment: AC Driven
Balancing equations using Hands-On Algebra. Revisiting the basic operations of fractions Scaffolding 6 th grade Common Core regarding one step equations and inverse properties. Skill base lessons regarding properties and integers. Number lines Number Talks Visual charts using T-Tables DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Virtual Technology of real-world ELD Literacy Standards situations. Graphic organizers Use of Rational expressions Highlighting : cloze activities Multiple step with multiple SIOP strategies simplifications. Real-world visuals Multi standard processes Group collaboration Talk Moves Number talks Presentations Small group instruction One on one peer support Smaller size quantities Number Talks Revisiting the basic operations of fractions Scaffolding 6 th grade Common Core regarding one step equations and inverse properties. Skill base lessons regarding properties and integers. Number lines
HMH Unit 2 Ratios and Proportional Relationships This unit builds on the students knowledge and understandings of rate and unit concepts that were developed in Grade 6. This includes the need to develop proportional relationships through the analysis of graphs, tables, equations, and diagrams. Grade 7 will push for the students to develop a deep understanding of the characteristics of a proportional relationship. Mathematics should be represented in as many ways as possible in this unit by using graphs, tables, pictures, symbols and words. Some examples of providing the students with this opportunity are the following: researching newspaper ads, constructing their own questions, keeping a log of prices (particularly sales) and determining savings by purchasing items on sale. Before: 6 th grade Students study of ratio in grade 6 underpins their work with proportional and all linear relationships in later grades. When students find the ratio of one number, a, to another number, b, by dividing a by b, they compare the quantities using division. The CCSS asks students do this in a variety of situations. Students learn that the same ratio can be represented in different ways (for example,.75, 75%, 3:4, 3 to 4, or 3/4 ) and that a ratio can show the relationship between quantities measured with the same units or with different units. Further, for some situations, the ratio of one quantity to another may change depending on the circumstances (the ratio of the number of miles a car travels for each gallon of gas changes depending on where and how the car is driven), while for others (the relationship between any circle s circumference and diameter), the ratio remains constant. It is important to notice how the CCSS shifts emphasis away from setting up a proportion and solving it by cross-multiplication. A major emphasis of the CCSS is finding the unit rate. Also the CCSS does not expect-students to be troubled by such things as distinguishing a ratio from a fraction or from a rate. During: 7 th grade In the CCSS the treatment of Ratio and Proportional Relationships represents a fundamental departure from traditional seventh grade standards. By design, there is no mention of solving a proportion or of cross-multiplication. Instead, the emphasis is on identifying the specific nature of the relationship between varying quantities. The CCSS calls on students to learn to recognize proportional relationships, identify the constant of proportionality, represent proportional relationships with tables, graphs, and formulas, and understand the unit rates contained in proportional relationships. This broader treatment forges a conceptual link among ratio, proportional relationships, and functions. Students will learn that in any proportional relationship the ratio between the varying quantities is constant. When such a ratio is constant, the quantities are in direct proportion to each other. In a proportional relationship, when one quantity is multiplied by a certain factor, the other quantity will be multiplied by that same factor. The constant, which appears in division as the constant ratio and in multiplication as the constant multiple, is called a constant of proportionality. A useful form for a constant of proportionality is the unit rate. In other words, the ratio y/x = k thus there is a constant ratio between the varying quantities. Student will understand that the constant of proportionality is also the scale factor in scale drawings. After: 8 th grade In Grade 7 students build out their understanding of proportional relationships in the Constructions and Angles Unit when they solve problems involving scale drawings. One very exciting idea is that in the Measurement students could put their understanding of proportional relationships to work by recognizing and representing proportional relationships in the context of measurement. For example, students could determine if the diameter is proportional to the circumference or area enclosed by the circle and then wonder about the constant of proportionality. They could also determine if the height of a right rectangular prism is proportional to volume and then determine the constant of proportionality. The understanding of proportional relationships that Grade 7 students gain, lays a firm foundation for learning about functions, particularly linear functions.
UNIT 2: Ratios and Proportional Relationships Planning Organizer TIME: 6 weeks UNIT NARRATIVE: This unit builds on the students knowledge and understandings of rate and unit concepts that were developed in Grade 6. This includes the need to develop proportional relationships through the analysis of graphs, tables, equations, and diagrams. Grade 7 will push for the students to develop a deep understanding of the characteristics of a proportional relationship. Mathematics should be represented in as many ways as possible in this unit by using graphs, tables, pictures, symbols and words. Some examples of providing the students with this opportunity are the following: researching newspaper ads, constructing their own questions, keeping a log of prices (particularly sales) and determining savings by purchasing items on sale. Textbook Correlations: Additional Resources HMH Go Math Unit 2: Modules 4 and 5 Personal Math Trainer, Math on the Spot, Formative Assessment Lessons, and Real Player Activator videos ESSENTIAL QUESTIONS: ACADEMIC VOCABULARY: unit rates, ratios, proportional 1. How can ratio and rate reasoning be used to efficiently solve real world relationships, proportions, constant of proportionality, complex problems? fractions, 2. What mathematical language and representation can be used when examining proportional relationships? 3. How can proportional reasoning be used to efficiently solve real world problems involving rational numbers? CLUSTER HEADING & STANDARDS: Analyze proportional relationships and use them to solve real-world and mathematical problems. 7. RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour. 7. RP.2 Recognize and represent proportional relationships between quantities. 7. RP.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7. RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. MATHEMATICAL PRACTICE: All mathematical practice standards are addressed in every unit. Those receiving special attention in this unit are bolded. MP 1 Make sense of problems and persevere when solving them. MP 2 Reason quantitatively and abstractly. 7.RP.1, MP 3 Construct viable arguments and critique the reasoning of others. MP 4 Model with mathematics. MP 5 Use appropriate tools strategically.
7. RP.2c Represent proportional relationships by equations. For example, if total cost tt is proportional to the number nn of items purchased at a constant price pp, the relationship between the total cost and the number of items can be expressed as tt = pppp. 7. RP.2d Explain what a point (xx,) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (00,) and (11,) where rr is the unit rate. 7. RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. MP 6 Attend to precision. 7.RP.1, MP 7 Look for and make use of structure. MP 8 Look for and express regularity in repeating reasoning Learning outcomes: By the end of this unit, students should be able to extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships. End of Unit Assessment: AC Driven DIFFERENTIATION REMEDIATION ACCELERATION ENGLISH LEARNERS SPECIAL EDUCATION Review simple cross-multiplication and progressing to simple word problems at a slower rate. Small group instruction and activities to scaffold the 4th & 5th foundational concepts: Content: Concept of equivalent fractions Rates and ratios reinforced Identify two quantities Sketch out the problem visually Coordinate plane Have students explore actual markup and commission tables and solve tasks based on best ways to save money. Content: Slope of a line Direct variation Input/output data Real world problems Geometrical Shapes Application/Strategies: Technology Projects/presentation ELD Literacy Standards Graphic organizers Cloze activities SIOP strategies Real-world visuals Group collaboration Number talks One on one peer support Student s IEP s and /or 504 s should be consulted. With the intent of meeting individual student needs, instructional groups can be formed accordingly. Concept of equivalent fractions Rates and ratios reinforced Identify two quantities Sketch out the problem visually Coordinate plane