Math

= = = New Common Core Standards = = =

= = = Mathematical Reasoning = = =

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1. Make sense of problems and persevere in solving them.
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2. Reason abstractly and quantitatively.
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3. Construct viable arguments and critique the reasoning of others.
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4. Model with mathematics.
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5. Use appropriate tools strategically.
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6. Attend to precision.
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7. Look for and make use of structure.
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8. Look for and express regularity in repeated reasoning.
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= = = Strands (need to make new pages and links) = = = Number Sense = = Algebra and Functions = = Measurement and Geometry = = Statistics, Data Analysis, and Probability = =

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= = = Mathematics Year Overview (break down into Teaching Points/clear lesson objectives--1 per lesson) =

From: http://commoncore.org/_docs/math/6-8_curriculum_overview.pdf
 * **1st Trimester ** || **2nd Trimester ** || **3rd Trimester ** ||
 * MODULE 1: RATIOS AND UNIT RATES (35 DAYS/7 WEEKS)


 * * Understand ratio concepts and use ratio reasoning to solve problems. **

RP 1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratioof wings to beaks in the bird house at the zoo was 2:1, because forevery 2 wings there was 1 beak.” “For every vote candidate A received,candidate C received nearly three votes.”

RP 2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

RP 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, thenat that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

<span style="font-family: Arial,Helvetica,sans-serif;">MODULE 2: ARITHMETIC OPERATIONS INCLUDING DIVIDING BY A FRACTION (25 DAYS/5 WEEKS)


 * <span style="font-family: Arial,Helvetica,sans-serif;">* Apply and extend previous understandings of multiplication and division to divide fractions by fractions. **

<span style="font-family: Arial,Helvetica,sans-serif;">NS 1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fractionmodel to show the quotient; use the relationship between multiplicationand division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3.(In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each personget if 3 people share 1/2 lb of chocolate equally? How many 3/4-cupservings are in 2/3 of a cup of yogurt? How wide is a rectangular strip ofland with length 3/4 mi and area 1/2 square mi?


 * Compute fluently with multi‐digit numbers and find common factors and multiples.**

NS.2 Fluently divide multi‐digit numbers using the standard algorithm.

NS.3 Fluently add,subtract,multiply, and divide multi‐digit decimals using the standard algorithm for each operation.

NS.4 Find the greatest common factor of two whole numberslessthan or equal to 100 and the least common multiple oftwo whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as amultiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). || <span style="font-family: Arial,Helvetica,sans-serif;">MODULE 3: RATIONAL NUMBERS (25 DAYS/5 WEEKS)

<span style="font-family: Arial,Helvetica,sans-serif;">* **<span style="font-family: Arial,Helvetica,sans-serif;"> Apply and extend previous understandings of numbers to the system of rational numbers. **

<span style="font-family: Arial,Helvetica,sans-serif;">NS 5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

<span style="font-family: Arial,Helvetica,sans-serif;">NS 6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

<span style="font-family: Arial,Helvetica,sans-serif;">a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

<span style="font-family: Arial,Helvetica,sans-serif;">b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

<span style="font-family: Arial,Helvetica,sans-serif;">c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

<span style="font-family: Arial,Helvetica,sans-serif;">NS 7. Understand ordering and absolute value of rational numbers.

<span style="font-family: Arial,Helvetica,sans-serif;">a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example,interpret –3 > –7 as a statement that –3 is located to the right of –7 ona number line oriented from left to right.

<span style="font-family: Arial,Helvetica,sans-serif;">b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3°C > –7°C toexpress the fact that –3°C is warmer than –7°C.

<span style="font-family: Arial,Helvetica,sans-serif;">c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. Forexample, for an account balance of –30 dollars, write |–30| = 30 todescribe the size of the debt in dollars.

<span style="font-family: Arial,Helvetica,sans-serif;">d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30dollars represents a debt greater than 30 dollars.

<span style="font-family: Arial,Helvetica,sans-serif;">NS 8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

<span style="font-family: Arial,Helvetica,sans-serif;">MODULE 4: EXPRESSIONS AND EQUATIONS (45 DAYS/9 WEEKS)


 * <span style="font-family: Arial,Helvetica,sans-serif;">* Apply and extend previous understandings of arithmetic to algebraic expressions. **

<span style="font-family: Arial,Helvetica,sans-serif;">EE 1. Write and evaluate numerical expressions involving whole-number exponents.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 2. Write, read, and evaluate expressions in which letters stand for numbers.

<span style="font-family: Arial,Helvetica,sans-serif;">a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation“Subtract y from 5” as 5 – y.

<span style="font-family: Arial,Helvetica,sans-serif;">b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe theexpression 2 (8 + 7) as a product of two factors; view (8 + 7) as botha single entity and a sum of two terms.

<span style="font-family: Arial,Helvetica,sans-serif;">c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volumeand surface area of a cube with sides of length s = 1/2.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) toproduce the equivalent expression 6 + 3x; apply the distributive propertyto the expression 24x + 18y to produce the equivalent expression6 (4x + 3y); apply properties of operations to y + y + y to produce theequivalent expression 3y.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3yare equivalent because they name the same number regardless of whichnumber y stands for.


 * <span style="font-family: Arial,Helvetica,sans-serif;">* Reason about and solve one-variable equations and inequalities. **

<span style="font-family: Arial,Helvetica,sans-serif;">EE 5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. || <span style="font-family: Arial,Helvetica,sans-serif;">MODULE 4 CONT.

<span style="font-family: Arial,Helvetica,sans-serif;">EE 8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.


 * <span style="font-family: Arial,Helvetica,sans-serif;">* Represent and analyze quantitative relationships between dependent and independent variables. **

<span style="font-family: Arial,Helvetica,sans-serif;">EE 9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in aproblem involving motion at constant speed, list and graph ordered pairsof distances and times, and write the equation d = 65t to represent therelationship between distance and time.

<span style="font-family: Arial,Helvetica,sans-serif;">MODULE 5: AREA, SURFACE AREA, AND VOLUME PROBLEMS (25 DAYS/5 WEEKS)

<span style="font-family: Arial,Helvetica,sans-serif;">MODULE 6: STATISTICS (25 DAYS/5 WEEKS) || = =

= (Old CA Standards) = = Mathematical Reasoning =

//<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">1.0 Students make decisions about how to **approach** problems: //

 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">1.1 Analyze problems by:
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">identifying //relationships// (how are these things related?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">distinguishing //relevant// from //irrelevant// information (what information do we need to solve this?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">identifying //missing// information (")
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">//sequencing// and //prioritizing// information (what should I do first?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">observing //patterns// (what happens when we try things out?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">( <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">[|Function Tables] <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">, <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">[|Function Tables 2] <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">1.2 //<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">Formulate //<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;"> and //<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">justify //<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;"> mathematical conjectures based on a general description of the mathematical question or problem posed. (what do you think we should do to solve this, and why?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">1.3 Determine when and how to break a problem into //<span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">simpler parts //<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">. (should we break this down into separate steps? how?)


 * //<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.0 Students use strategies, skills, and concepts in finding solutions: //**
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.1 Use estimation to verify the //reasonableness// of calculated results. (do I seem to be getting closer or farther away? does my answer make sense in the context?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.2 //Apply// strategies and results from simpler problems to more complex problems. ([|Train Problems]) (what have I learned that might work here?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.3 Estimate unknown quantities //graphically// and solve for them by using logical //reasoning// and arithmetic and algebraic //techniques//. (how can I make a visual representation of this?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.4 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. (can I explain my process in pictures and words and other ways of representing data?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.5 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. (can I show my work in a clear mathematical way?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.6 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. (can I say when to use an estimate and when to be exact?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">2.7 Make precise calculations and check the validity of the results from the context of the problem. (can I get accurate answers and make sure they are correct?)

//**<span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">3.0 Students move beyond a particular problem by generalizing to other situations: **//
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">3.1 Evaluate the reasonableness of the solution in the context of the original situation. (does my answer fit back into the original problem?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. (can I use this approach on other problems?)
 * <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px;">3.3 Develop generalizations of the results obtained and the strategies used and apply them in new problem situations. (can I think of other problems that would use this same strategy?)

= Strands = <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif;">Number Sense <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif;">Algebra and Functions <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif;">Measurement and Geometry <span style="background-color: #ffffff; font-family: Verdana,Arial,Helvetica,sans-serif;">Statistics, Data Analysis, and Probability

= Mathematics Year Overview =
 * ===1st Trimester=== || ===2nd Trimester=== || ===3rd Trimester=== ||
 * * NS 2.3 – Solve addition, subtraction, multiplication, and division problems including those arising in concrete situations that use positive and negative integers and combinations of these operations.


 * NS 2.1 – Solve problems involving addition, subtraction, multiplication and division of positive fractions and mixed numbers and explain why a particular operation was used for a given situation. || * AF 1.1 – Write and solve one-step linear equations in one variable.


 * NS 1.3 – Use proportions to solve problems (e.g., find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.


 * NS 1.4 – Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned and tips. || * NS 1.4 cont. – Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned and tips.


 * MG 2.2 – Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.


 * MG 1.1 – Understand the concept of a constant such as pi; know the formulas for the circumference and area of a circle.

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 * SDAP 3.3 – Represent probabilities as ratios, proportions, decimals between 0 and 1, and percentages between 0 and 100 and verify that the probabilities computed are reasonable; know that if P is the probability of an event, 1-P is the probability of an event not occurring. ||