Rational numbers are a subset of the number system, including and beyond whole numbers. ADD and SUBTRACT rational numbers

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1 Course: Grade 7 Mathematics Year: Teachers: Cheryl Flanagan, Morgan O Brien, Nicole Brustman Unit 1: Operating with Rational Numbers (add/sub) Approximate Time Frame: 7 weeks 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. a. 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. b. 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. c. 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. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.3. Solve real world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) When am I going to use positive and negative numbers? How are positive and negative numbers related? How can I model negatives numbers, including how I operate with them? How is it possible to add two quantities and get a sum that is less than what you started with? Rational numbers are a subset of the number system, including and beyond whole numbers. Properties of whole number operations can be applied to solving real world and mathematical problems involving rational numbers, including integers. Subtraction of a rational number is thought of as adding an opposite because the negative sign reverses your direction on the number line, p q = p + ( q). Estimation is a means for predicting & assessing the reasonableness of a solution. ADD and SUBTRACT rational numbers REPRESENT addition and subtraction on a horizontal or vertical number line DESCRIBE the relationship of opposite quantities UNDERSTAND positive or negative direction on a number line SHOW additive inverses APPLY absolute value principle in context APPLY properties of operations as strategies to solve real world problems CONVERT between equivalent forms of numerical expressions USE mental computation and estimation strategies to ASSESS reasonableness of answers Strategies to represent and solve problems involving operations with rational numbers (including decimals, fractions, integers) The sum of two opposites is zero A negative number can also be interpreted as the opposite of the positive number Absolute value of a rational number is its distance from zero on a number line Computation with integers is an extension of computation with fractions and decimals: Models: Number line, chip model, area model, arrays, bar model, fraction circles, picture/visual Rational Numbers Number Line Opposite Quantities Additive Inverse Absolute Value Unit 2: Operating with Rational Numbers (mult/div) Approximate Time Frame: 4 weeks

2 When am I going to use positive and negative numbers? 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. 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. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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 realworld contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. 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. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.3 Solve multi step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools How can I use what I already know about multiplying and dividing positive numbers to multiply and divide with negative numbers? What rules can we find to generalize patterns when multiplying and dividing positive and negative numbers? How can the use of mental math and estimation strategies help determine the reasonableness of answers? Rational numbers are a subset of the number system including & beyond whole numbers. Properties of whole number operations can be applied to solving real world and mathematical problems involving rational numbers, including integers. Integers can be divided, provided that the divisor is not zero, and every quotient of integers is a rational number. Estimation and mental math are more complex with rational numbers. Multiply and Divide rational numbers Develop and apply rules for multiplying & dividing signed numbers Apply properties of operations as strategies Solve multi step problems in context Convert between equivalent forms of rational numbers Understand the relationship between equivalent forms of an expression Use mental computation and estimation strategies Assess reasonableness of answers. Rules for multiplication and division of positive and negative numbers Properties of operations (Distributive Property, Commutative, Associative, and Identity Properties of Addition and Multiplication, Additive Inverse Property) Strategies for converting a rational number into a decimal The decimal form of a rational number terminates in zeros or eventually repeats Opposites and absolute value of rational numbers A negative number can also be interpreted as the opposite of the positive number. (Ex: 5 can be interpreted as the opposite of 5.) Computation with integers is an extension of computation with fractions and decimals Essential Terms: Absolute Value Integer Multiplicative Inverse/Reciprocal Rational Number Expression Equation Properties of Operations (Commutative Property, Distributive Property, Additive Inverse, Multiplicative Identity, Multiplicative Property of Zero) Order of Operations Other terms students should know: Income, Profit Expense, Loss Deposit Withdrawal Product Quotient

3 strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. Models: Number line, chip model, area model, arrays, bar model, fraction circles, picture/visual Estimation as a means for predicting & assessing the reasonableness of a solution. Mental math and estimation strategies (benchmarks, clustering, rounding, front end estimation, compatible numbers) Rational numbers are a set of numbers that includes the whole numbers, integers, as well as numbers that can be written as quotient of two integers, a b, where b 0. Rational numbers can be represented as fractions, decimals, & percents in infinitely many equivalent forms. Expressions can be rewritten in different forms

4 Unit 3: Algebraic Reasoning II Approximate Time Frame: 5 weeks 7.EE.1. Apply properties of 1.How do you USE variables Variables can be used operations as strategies to add, Expressions can be Variable subtract, factor, and expand differentiate between to represent numbers manipulated to suit a Numerical linear expressions with rational a situation that can be REPRESENT quantities whose exact values coefficients particular purpose and Expression represented with an in/out of context are not yet specified. solving problems Algebraic 7.EE.2. Understand that rewriting an expression in equation and one that efficiently. CONSTRUCT simple Expressions can be Expression different forms in a problem can be represented context can shed light on the Mathematical equations and manipulated to Term problem and how the with an expression? expressions, equations, generate equivalent inequalities Coefficient quantities in it are related. expressions to and inequalities are SOLVE problems in When would we use simplify the problem. Constant used to represent and context Equivalent variables to represent solve real world and o Simple Expressions can be Equation real world situations? mathematical equations decomposed and Inequality problems. o Simple recomposed in How can we use inequalities Linear different ways to inequalities to generate equivalent represent real world REASON about quantities forms. situations? 7.EE.4 (emphasis on a) Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. b. Solve word problems leading to equations of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Why are the properties of operations important? How do you translate real world problems to algebraic expressions? How do you differentiate between a situation that can be represented by an arithmetic solution and one that can be represented by an algebraic solution? COMPARE algebraic and arithmetic solutions IDENTIFY sequence of operations GRAPH inequality INTERPRET graphed inequality using context APPLY properties of operations FACTOR Linear expressions with rational coefficients EXPAND Linear expressions with rational coefficients Flexibility with the equivalent forms of an expression (expanded form, factored form, etc) allows for efficient problem solving. Properties of Operations and Order of Operations are used to simplify, evaluate, or find equivalent expressions. The equal sign demonstrates equivalence. Ex: 2x + x = 3x (equivalent expressions) 2x + x and 3x + 4 are not equivalent expressions

5 WRITE an expression in different forms INTERPRET and REPRESENT expressions in different forms to make meaning of context Rational numbers can be represented in equivalent forms to solve problems efficiently (25% can be represented as 1/4 or 0.25). Estimation as a means for predicting & assessing the reasonableness of a solution. Fluency with mental math and estimation facilitates efficient problem solving. Inverse operations are used to solve equations and inequalities. Solutions to an equation/ inequality are the values of the variables that make the equation/ inequality true. There are some inequalities that have infinitely many solutions (those in the form of x>c or x<c). Solutions to an inequality are represented symbolically or using a number line.

6 Unit 4: Proportional Relationships Approximate Time Frame: 5 weeks 7.RP.1. Compute unit rates Where do we use We can use Compute unit rates Dividing the analyze associated with ratios of fractions, including ratios of proportions and ratios proportions to find associated with ratios denominator into the area lengths, areas and other in everyday life? equivalent values. of fractions, including numerator results in quantities measured in like or coordinate plane different units. ratios of lengths, areas finding the unit rate. equivalent How can I model unit Ratios compare two and other quantities fraction rate and proportional items. measured in like or Graphing the gratuities or relationships? different units. coordinates of ratios commissions Unit rate takes the will result in seeing if mark downs How can I use what I ratio of two items Decide whether two the ratios are mark ups already know about being compared any quantities are in a proportional origin percentages and gives the equivalent proportional demonstrated by a percent error proportions to find amount if there is relationship. straight line. percent solutions to real world only one unit being increase/decrease situations such as tax, measured. Identify the constant Proportional point commissions, percent of proportionality (unit representations can proportion increase/decrease or If two comparisons rate) in tables, graphs, be written as an ratio interest? are proportional then equations, diagrams, equation by the total their ratios are equal. and verbal descriptions = the constant times simple interest How can I use algebra of proportional the quantity. unit rates to find the solution to When we graph ratios relationships. proportional we can recognize if it situations? is proportional. 7.RP.2. Recognize and represent proportional relationships between quantities. a. 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. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r 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. 7.G.1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. How can the use of proportions help in finding measurements in geometric situations? I can make a ratio chart and plot the ratios. I can use proportions to solve interest, sales tax, tipping, and other percentages problems. I can use proportions to make a map correct in its dimensions. Represent proportional relationships by equations. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. In creating a graphic display of proportional relationships, when the x coordinate is 1, the y coordinate is my unit rate. If I multiply the decimal equivalent of a percentage by the whole, I can find the percent of a number. If set up a proportion, I can find the percent by putting the

7 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. part/whole = x/100. Key s Actual Distance/Key s Scaling Distance = Actual Distance/Map Distance Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Unit 5: Two and Three Dimensional Geometry Approximate Time Frame: 4 weeks 7.G.4. Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.6. Solve real world and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on How are the area and circumference of a circle related How can we apply surface area and volume of solids to solve real world problems? How are cross sections of three dimensional objects formed? Parallelograms and rectangles can be used to derive the formula for the area of a circle. Coordinate geometry can be a useful tool for understanding geometric shapes and transformations. SOLVE problems using formulas DERIVE informally the relationship between circumference and area of a circle DRAW/CONSTRUCT geometric shapes with given conditions Formulas o area of two dimensional figures (circles, rectangles, triangles) o circumference of a circle o volume of threedimensional figures Adjacent Angle Circumference Complementary Angle Congruent Cross section Plane Irregular Polygon Parallel Lines Pi Regular Polygon Supplementary Angle

8 constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 7.G.3. Describe the twodimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 7.G.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi step problem to write and solve simple equations for an unknown angle in a figure. How are algebra and geometry related Cross sections of three dimensional objects can be formed in a variety of ways, depending on the angle of the cut with the base of the object. The area of irregular and regular polygons can be found by decomposing the polygon into triangles, squares, and rectangles. Approximate volumes and surface area of simple geometric solids may be found using estimation. Manipulatives and the construction of nets may be used in computing the surface area of right rectangular prisms. Pi (π) is the relationship between a circle s circumference and diameter. o USE rulers, protractors, technology IDENTIFY unique triangles DECOMPOSE threedimensional shapes into two dimensional shapes (i.e. nets) DESCRIBE twodimensional figures that result from plane sections of threedimensional figures WRITE and SOLVE problems using equations to find an unknown angle in a figure o surface area of right rectangular prisms Surface area is the sum of the area of the faces Nets can be used to evaluate surface area calculations Properties of special shapes o right triangles o isosceles triangles o squares o equilateral shapes Slicing a threedimensional figure results in various twodimensional figures (created by the faces of the slice) Angles (supplementary, complementary, vertical, adjacent) Vertical Angles Algebraic equations can be used to find unknown angles of geometric figures.

9 Unit 6: Probability Approximate Time Frame: 3 weeks 7.SP.5. Understand that the 1. Does the outcome Probability gives a Determine the The probability of an Certain probability of a chance event is a number between 0 and 1 that of one event have an quantitative likelihood of an event. event is a ratio of the Compound event expresses the likelihood of the event occurring. Larger impact on the description of the number of favorable Dependent events numbers indicate greater likelihood. A probability near 0 outcome of likelihood of an Approximate the outcomes to the Experimental indicates an unlikely event, a event. probability of a chance number of possible subsequent events? Probability probability around 1/2 event by collecting outcomes. indicates an event that is Impossible neither unlikely nor likely, and a Probability ranges data on the chance probability near 1 indicates a 2. How do you Independent from 0 to 1. process that produces The probability of an likely event. events describe the it and observing its event, E, is given by Outcome 7.SP.6. Approximate the probability of events; probability of a chance event by If the probability is long run relative the formula: Sample Space collecting data on the chance What does it mean known, I can predict frequency, and predict P(E)= number process that produces it and Simple event observing its long run relative for a probability to with relative accuracy the approximate of favorable Theoretical frequency, and predict the be 0 or 1? the number of times relative frequency outcomes approximate relative frequency Probability given the probability. an event will occur. given the probability. Number Tree Diagram 3. What is the of possible difference between Develop a probability outcomes model and use it to experimental and find probabilities of theoretical events. probability? 7.SP.7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound event is the 4. Why do the results of experiments sometimes differ from the theoretical probability? 5. How can I use probability to help me predict what can happen in real life? Just because the probability suggests an event will occur x number of times, doesn t mean it will unfold in real life that way. I can use a tree diagram to find compound probability. Compound probability is the probability of numerous events. I can represent probability as a fraction, decimal, or percentage. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. P(E)=0, impossible; 0<P(E)<1/2 is unlikely; P(E)=½ as likely as not; ½ <P(E)<1 likely; P(E)=1 certain When predicting number of occurrences, take the probability and multiply it by the number of events to occur. Theoretical Probability as compares to Experimental Probability.

10 fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the events. c. Design and use a simulation to generate frequencies for compound events. 7.SP.3. Informally assess the degree of visual overlap of two numerical data distributions Compound Probability to find multiply the probability of each event. A tree diagram is a visual depiction of compound probability. Unit 7: Inferences about Population Approximate Time Frame: 3 weeks 7.SP.1. Understand that 1. How do you explain Formulating UNDERSTAND/USE Random sampling Box and Whisker statistics can be used to gain information about a population real world problems questions, designing statistics tends to produce Plot by examining a sample of the using statistics? studies, and representative population; generalizations Frequency about a population from a collecting data about EXAMINE a sample of a samples. Grouped sample are valid only if the 2. How do you sample is representative of that a population through population Frequency Table population. Understand that interpret data from random sampling Finding a valid, Histogram random sampling tends to produce representative samples statistical allow us to make GENERALIZE representative sample Inter Quartile and support valid inferences. representations? inferences and information about a will enable valid Range (IQR) compare data. population inferences to be made 7.SP.2. Use data from a random Mean Absolute sample to draw inferences 3. How is sampling about a population. Deviation about a population with an used to make DETERMINE if a unknown characteristic of Mean interest. Generate multiple predictions? sample is What it means to samples (or simulated samples) Median of the same size to gauge the representative/valid have a valid, random Mode variation in estimates or predictions. 4. How do you draw sample representative Outlier inferences from USE measures of of a population(s). 7.SP.4. Use measures of center Range and measures of variability for random samples? center and measures Sample numerical data from random of variability for Inferences about a samples to draw informal Simple Random numerical data from population are only comparative inferences about 5. Why is data Sampling two populations. random samples valid if the sample is collected and random and Stem and Leaf analyzed? representative. Plot 6. How do people use data to influence others? DRAW informal comparative inferences USE data from a random sample Proportional reasoning is used to make estimates or predictions about a population.

11 o DRAW inferences about a population GENERATE multiple samples of the same size o GAUGE the variation in estimates or predictions EXPRESS/CALCULATE the difference between the centers of two numerical data distributions as a multiple of a measure of variability mean absolute deviation Having multiple samples for the same population allows for gauging the variation of estimates or predictions. Measures of center can be used to compare data and measure variability between data sets. Data displays are used to visually compare data sets and draw informal comparative inferences. Box plots are way to show measures of variability such as the range (Other data displays may highlight other measures of variability). Formulating questions, designing studies, and collecting data about a population through random sampling allow us to make inferences and compare data.