# Montana City School GRADE 5

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1 Montana City School GRADE 5 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 1. Formulate and solve multi-step and nonroutine problems using a variety of strategies. Generalize methods to new problem situations. Grade 5 Solve problems by using the strategy work backward (1.7) Solve problems using the strategy find a pattern (2.5) Solve problems by using the skill interpret the remainder (3.3) Solve problems by using the skill relevant or irrelevant information (3.10) Solve problems by using the strategy predict and test (4.9) Solve problems by using the skill draw conclusions (5.4) Solve problems by using the skill estimate or find exact answers (6.5) Solve problems by using the skill evaluate answer for reasonableness (8.4) Solve problems by using the strategy use logical reasoning (9.5) Solve problems by using the strategy make a graph (10.6) Solve problems by using the strategy make an organized list (11.5) Solve problems by using the strategy make a model (12.6) Compare different strategies to solve problems (13.6) Solve problems by multiplying fractions (15.2) and whole numbers (15.3) Solve problems by using the skill choose the operation (15.8) Solve the problem by using the strategy make a table (16.5) Solve problems by using the skill estimate or actual measurement ( 22.6) Solve problems by using the skill identify relationships (18.4) Solve problems by using the strategy write an equation (21.3)

2 Montana City School GRADE 5 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 1. Formulate and solve multi-step and nonroutine problems using a variety of strategies. Generalize methods to new problem situations. 2. Select and apply appropriate estimation strategies throughout the problem-solving process. Grade 5 Solve problems by using the strategy solve a simpler problem (24.6) Solve problems by using the skill make generalizations Solve problems by comparing different strategies (24.10) Apply estimation strategies when utilizing a variety of problem solving strategies 3. Interpret and communicate mathematical ideas and logical arguments using correct mathematical terms and notations. Read and write whole numbers through millions (1.1) to billions (1.2) Read and write decimals through thousandths (5.1) Read, write, and represent fractions as a whole or as a group (12.1) Model, read, and write percents (16.6)

3 Montana City School GRADE 5 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 4. Recognize and investigate the relevance and usefulness of mathematics through applications, both in and out of school. Grade 5 Evaluate the need for mathematical skills in a variety of careers and real world situations 5. Select and use appropriate technology to enhance mathematical understanding. Appropriate technology may include, but is not limited to, paper and pencil, calculator, computer, and data collection devices. Choose mental math, paper and pencil, or a calculator to multiply whole numbers (2.6) Choose mental math, paper and pencil, or a calculator to find decimal sums and differences (6.4) Choose mental math, paper and pencil, or a calculator to add and subtract fractions (13.7)

4 Montana City School GRADE 6 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 1. Formulate and solve multi-step and nonroutine problems using a variety of strategies. Generalize methods to new problem situations. Grade 6 Solve problems by using the strategy work backward Solve problems using the strategy find a pattern Solve problems by interpreting the remainder and quotient (3.8) Solve problems by using the strategy guess and test Solve problems by using the strategy use logical reasoning Solve problems by using the strategy make an organized list Solve problems by using the strategy make a model Solve the problem by using the strategy make a table Solve problems by using the strategy solve a simpler problem Solve problems by using the strategy Act It Out Solve problems by using the strategy draw a diagram 2. Select and apply appropriate estimations strategies throughout the problem-solving process. Apply estimation strategies when utilizing a variety of problem solving strategies

5 Montana City School GRADE 6 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 3. Interpret and communicate mathematical ideas and logical arguments using correct mathematical terms and notations Grade 6 Learn to translate words and math (2.2) Describe figures by using the terms of geometry (8.1) 4. Recognize and investigate the relevance and usefulness of mathematics through applications, both in and out of school. Evaluate the need for mathematical skills in a variety of careers and real world situations 5. Select and use appropriate technology to enhance mathematical understanding. Appropriate technology may include, but is not limited to, paper and pencil, calculator, computer, and data collection devices. Choose appropriate methods and materials to understand and solve a variety of problems Learn to use an appropriate method of computation and justify your choice (1.6)

6 Montana City School GRADE 7 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 1. Formulate and solve multi-step and nonroutine problems using a variety of strategies. Generalize methods to new problem situations. 2. Select and apply appropriate estimation strategies throughout the problem-solving process. Grade 7 Solve problems by using the strategy work backward Solve problems using the strategy find a pattern Solve problems by interpreting the remainder and quotient (3.8) Solve problems by using the strategy guess and test Solve problems by using the strategy use logical reasoning Solve problems by using the strategy make an organized list Apply estimation strategies when utilizing a variety of problem solving strategies 3. Interpret and communicate mathematical ideas and logical arguments using correct mathematical terms and notations. Translate words into numbers, variables, and operations (1.8)

7 Montana City School GRADE 7 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 4. Recognize and investigate the relevance and usefulness of mathematics through applications, both in and out of school. Grade 7 Evaluate the need for mathematical skills in a variety of careers and real world situations Relate graphs to situations (4.3) 5. Select and use appropriate technology to enhance mathematical understanding. Appropriate technology may include, but is not limited to, paper and pencil, calculator, computer, and data collection devices. Choose appropriate methods and materials to understand and solve a variety of problems

8 Montana City School GRADE 8 Montana Standard 1: Students engage in the mathematical processes of problem solving and reasoning, estimation, communication, connections and applications, and using appropriate technology. 1. Formulate and solve multi-step and nonroutine problems using a variety of strategies. Generalize methods to new problem situations. 2. Select and apply appropriate estimation strategies throughout the problem-solving process. Grade 8 Solve problems by using the strategy work backward Solve problems using the strategy find a pattern Solve problems by interpreting the remainder and quotient (3.8) Solve problems by using the strategy guess and test Solve problems by using the strategy use Apply estimation strategies when utilizing a variety of problem solving strategies 3. Interpret and communicate mathematical ideas and logical arguments using correct mathematical terms and notations Translate between algebraic expressions and word phrases (1.2) 4. Recognize and investigate the relevance and usefulness of mathematics through applications, both in and out of school. Evaluate the need for mathematical skills in a variety of careers and real world situations 5. Select and use appropriate technology to enhance mathematical understanding. Appropriate technology may include, but is not limited to, paper and pencil, calculator, computer, and data collection devices. Choose appropriate methods and materials to understand and solve a variety of problems

9 Montana City School GRADE 5 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 1. Use four basic operations with whole numbers, fractions, decimals, and integers. Grade 5 Add and subtract whole numbers (1.6) Multiply basic facts using mental math and patterns of zeros (2.1) Multiply by a 1-digit number (2.3) a 2- digit number (2.4) Divide 3-digit and 4-digit dividends by 1- digit divisors (3.2) Use zeros in division (3.4) Divide by 2-digit numbers (3.7) Practice division by 1 and 2 divisors (3.9) Write and solve equations (2.1) Find whole number solutions to inequalities (4.8) Find sums and differences of decimals (6.2) Use models to multiply whole numbers and decimals (7.1) Multiply a decimal by a whole number (7.3,7.7) Model multiplication of decimals (7.4) Place decimal point in decimal multiplication. Multiply decimals with zeros in the product (7.6) Use models to divide decimals by whole numbers (8.1) Divide decimals by whole numbers (8.3) Identify multiples and least common multiple of a set of whole numbers (11.1) Identify numbers that are divisible by 2,3,5,6,9, and 10, and learn the divisibility rules (11.2) Write and evaluate powers of 10 (11.6)

10 Montana City School GRADE 5 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 1. Use four basic operations with whole numbers, fractions, decimals, and integers. Grade 5 cont. Add and subtract like fractions (13.1) unlike fractions (13.2) Subtract unlike fractions using fraction bars (13.3) Use common denominator to add and subtract unlike fractions (13.5) Hands On: Model addition (14.1) and subtraction (14.2) of mixed numbers Record addition and subtraction (14.3) Solve subtraction with renaming (14.4) Add and subtract mixed numbers (14.5) Use models to multiply fractions (15.1) Multiply two mixed numbers (15.4) Model fraction division (15.5) Divide whole numbers by fractions (15.6) Divide with fractions (15.7)

11 Montana City School GRADE 5 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 2. Use mental mathematics and number sense in using order of operations, and order relations for whole numbers, fractions, decimals, and integers. Grade 5: Use place value and number lines to compare and order whole numbers (1.3) Round whole numbers to a given place value (1.4) Estimate sums and differences of whole numbers (1.5) Estimate products by rounding and using the expanded form of numbers (2.2) Estimate the quotients by using compatible numbers and rounding (3.1) Estimate quotients when dividing by 2-digit numbers (3.6) Use order of operations to evaluate expressions (4.3) Use the properties and mental math to solve problems (4.4) Identify and write equivalent decimals (5.2) Use models and place value to compare and order decimals (5.3) Round decimals to a given place value (6.1) Use rounding to estimate the sums and differences of decimals (6.3) Estimate products of decimals (7.5) Estimate decimals quotients (8.2) Identify and write equivalent fractions (12.2) Write fractions in simplest form (12.3) Rename fractions greater than 1 as mixed numbers and mixed numbers as fractions greater than 1 (12.4) Compare and order fractions and mixed numbers (12.5) Name fraction and decimal equivalents (12.7) Estimate the sum and differences of fractions (13.4) Write equivalent forms of fractions, decimals, and percents (16.7) Identify, compare, and order integers (21.5)

12 Montana City School GRADE 5 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 3. Use the relationships and applications of ratio, proportion, percent, and scientific notation. Grade 5: Write and evaluate repeated factors in exponent form and identify square numbers (11.7) Write ratios as part-to-part, part-to-whole, and whole-to-part (16.1) Identify and write equivalent ratios and use proportional reasoning to solve problems (16.2) Use a model to find the percent of a number (16.8) 4. Develop and apply number theory concepts (e.g. primes, factors and multiples) in real world and mathematical problem situations. Identify and represent integers, their opposites, and their absolute value (21.4) Identify factors and tell whether a number is prime or composite (11.4) Find and write the prime factorization of a number using exponents (11.8) Identify factors and the greatest common factor of a set of whole numbers (11.3)

13 Montana City School GRADE 6 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 1. Use four basic operations with whole numbers, fractions, decimals, and integers. Grade 6 Solve equations involving decimals (3.9) Learn to add and subtract decimals (3.3) Multiply decimals by whole numbers and by decimals (3.5) Learn to divide whole numbers and decimals by decimals (3.7) Divide decimals by whole numbers (3.6) Use divisibility rules (4.1) Add and subtract fractions with like denominators (4.8) Add and subtract fraction with unlike denominators (5.2) Solve equations by adding and subtracting fractions (5.5) Learn to regroup mixed numbers to subtract (5.4) Add and subtract mixed numbers with unlike denominators (5.3) Multiply fractions (5.7) Multiply fractions by whole numbers (5.6) Multiply mixed numbers (5.9) Solve equations by multiplying and dividing fractions (5.10) Add (11.4), Subtract (11.5), Multiply (11.6), and Divide (11.7) integers

14 Montana City School GRADE 6 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 2. Use mental mathematics and number sense in using order of operations, and order relations for whole numbers, fractions, decimals, and integers. Grade 6 Compare and order whole numbers using place values or a number line (1.1) Estimate with whole numbers (1.2) Use order of operations (1.4) Write, compare, and order decimals using place value and number lines (3.1) Learn to estimate decimal sums, differences, products, and quotients (3.2) Write equivalent fractions (4.5) Estimate sums and differences of fractions and mixed numbers (4.9) Convert between mixed numbers and improper fractions (4.6) Use pictures and number lines to compare and order fractions (4.7) Compare and order integers (11.2)

15 Montana City School GRADE 6 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 3. Use the relationships and applications of ratio, proportion, percent, and scientific notation. Represent numbers by using exponents (1.3) Learn to write large numbers in scientific notation (3.4) Write ratios and rates to find unit rates (7.1) Write and solve proportions (7.3) Use rations to identify similar figures (7.4) Use proportions and similar figures to find unknown measures (7.5) Write percents as decimals and as fractions (7.7) Write decimals and fractions as percents (7.8) 4. Develop and apply number theory concepts (e.g. primes, factors and multiples) in real world and mathematical problem situations. Write prime factorizations of composite numbers (4.2) Find the greatest common factor (GCF) of a set of numbers (4.3) Convert between decimals and fractions (4.4) Find the least common multiple (LCM) of a group of numbers (5.1)

16 Montana City School GRADE 7 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 1. Use four basic operations with whole numbers, fractions, decimals, and integers. Grade 7 Learn to add and subtract decimals (3.2) Multiply decimals (3.3) Divide decimals by integers (3.5) Add and subtract fractions (3.8) Add and subtract mixed numbers (3.9) Multiply fractions and mixed numbers (3.11) Add (2.2), Subtract (2.3), Multiply (2.4), and Divide (2.4) integers 2. Use mental mathematics and number sense in using order of operations, and order relations for whole numbers, fractions, decimals, and integers. Use order of operations to simplify numerical expressions (1.5) Estimate decimal sums, differences, products, and quotients (3.1) Estimate sums and differences of fractions and mixed numbers (3.7) Identify, write, and convert between equivalent fractions and mixed numbers (2.9) Write fractions as decimals, and visa versa, and determine whether a decimal is termination or repeating (2.10) Compare and order fractions and decimals (2.11)

17 Montana City School GRADE 7 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 3. Use the relationships and applications of ratio, proportion, percent, and scientific notation. Represent numbers by using exponents (1.2) Identify, write, and compare ratios (5.1) Find equivalent ratios and identify proportions (5.4) Express large numbers in scientific notation (1.4) Use ratios to determine if two figures are similar (5.7) Use similar figures to find unknown lengths (5.8) Understand ratios and proportions in scale drawing Use ratios and proportions with scale (5.9) Model percents and write percents as equivalent fractions and decimals (6.1) Write decimals and fractions as percents (6.2) Estimate percents (6.3) Find the percent of a number (6.4) Solve problems involving percents (6.5) Solve problems involving percent of change (6.6) Solve problems involving simple interest (6.7) Express and evaluate numbers using powers and roots (9.7) 4. Develop and apply number theory concepts (e.g. primes, factors and multiples) in real world and mathematical problem situations. Find the prime factorization of composite numbers (2.6) Find the greatest common factor of two or more numbers (2.7) Find the least common multiple to two or more whole numbers (2.8)

18 Montana City School GRADE 8 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 1. Use four basic operations with whole numbers, fractions, decimals, and integers. Grade 8 Add and subtract rational numbers with like denominators (2.3) Multiply fractions, mixed numbers, and decimals (2.4) Divide fractions and decimals (2.5) Add and subtract fractions with unlike denominators (2.6) 2. Use mental mathematics and number sense in using order of operations, and order relations for whole numbers, fractions, decimals, and integers. Write rational numbers in equivalent forms (2.1) Compare and order positive and negative rational numbers written as fractions, decimals, and integers (2.2) Compare and order fractions and percents (6.1)

19 Montana City School GRADE 8 Montana Standard 2: Students demonstrate understanding of and an ability to use numbers and operations. 3. Use the relationships and applications of ratio, proportion, percent, and scientific notation. Evaluate expressions with exponents (4.1) Evaluate expressions with negative exponents and to evaluate the zero exponent (4.2) Find equivalent ratios to create proportions (5.1) Express large and small numbers in scientific notation and to compare two numbers written in scientific notation (4.4) Solve proportions (5.4) Estimate percents (6.2) Find percents (6.3) Find a number when a percent is known (6.4) Find commission, sales tax, and percentage of earnings (6.6) Find percent increase and decrease (6.5) Compute simple interest (6.7) Find square roots (4.5) Estimate square roots to a given number of decimal places and solve problems using square roots (4.6) Use rates and ratios (5.2) 4. Develop and apply number theory concepts (e.g. primes, factors and multiples) in real world and mathematical problem situations. Determine if a number is rational or irrational (4.7) Find commission, sales tax, and percent of earnings (6.6) Compute simple interest (6.7)

20 Montana City School GRADE 5 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 1. Understand the concepts of variable, expression and equation. Grade 5 Write and evaluate numerical and algebraic expressions (4.1) Evaluate numerical and algebraic expressions using order of operations (4.2) Write and solve equations (4.6) 2. Represent situations and number patterns using tables, graphs, verbal rules, equations, and models. Use function tables and equations to graph equations on a grid (21.2) Find whole number solutions to inequalities (4.8) 3. Recognize and use the general properties of operations (e.g., the distributive property). Identify the properties of addition and multiplication (4.3)

21 Montana City School GRADE 5 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 4. Solve linear equations using concrete, numerical, and algebraic methods. Grade 5 Write(4.1) and Evaluate (4.2) numerical and algebraic expressions 5. Investigate inequalities and nonlinear relationships informally. Explore relationships of numbers and equations during math lessons

22 Montana City School GRADE 6 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 1. Understand the concepts of variable, expression and equation. Grade 6 Learn to determine whether a number is a solution of an equation (2.4) Learn how to solve whole-number addition equations (2.5) subtraction equations (2.6) multiplication equations (2.7) division equations (2.8) 2. Represent situations and number patterns using tables, graphs, verbal rules, equations, and models. Use data in a table to write an equation for a function and to use the equation to find the missing value (11.9) Identify and graph integers, find opposites, and find the absolute value of an integer (11.1) 3. Recognize and use the general properties of operations (e.g., the distributive property). Learn to use number properties to compute mentally (1.5)

23 Montana City School GRADE 6 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 4. Solve linear equations using concrete, numerical, and algebraic methods. Grade 6 Identify and evaluate expressions (2.1) Find the missing value in a percent problem (7.9) Solve percent problems that involve discounts, tips, and sales tax (7.10) Solve equations containing integers (11.6) 5. Investigate inequalities and nonlinear relationships informally. Make an organized list to find all possible outcomes (12.3)

24 Montana City School GRADE 7 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 1. Understand the concepts of variable, expression and equation. Grade 7 Determine whether a number is a solution of an equation (1.10) Solve one-step equations by using addition and subtraction (1.11) and multiplication and division (1.12) Evaluate algebraic expressions (1.7) Simplify algebraic expressions (1.9) Solve one-step equations that contain decimals (3.6) Solve one-step equations that contain fractions (3.12) Identify and graph linear equations (4.6) 2. Represent situations and number patterns using tables, graphs, verbal rules, equations, and models. Compare and order integers and determine absolute value (2.1) Solve two-step equations (12.1) Solve multi-step equations (12.2) Solve equations that have variable on both sides (12.3) Read and write inequalities and graph them on a number line (12.4) Solve one-step inequalities by adding or subtracting (12.5) Solve one-step inequalities by multiplying or dividing (12.6) Solve simple two-step inequalities (12.7)

25 Montana City School GRADE 7 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 3. Recognize and use the general properties of operations (e.g., the distributive property). Grade 7 Identify properties of rational numbers and use them to simplify numerical expressions (1.6) Solve proportions by using cross products (5.5) 4. Solve linear equations using concrete, numerical, and algebraic methods. Solve one-step equations with integers (2.5) 5. Investigate inequalities and nonlinear relationships informally. Explore relationships of numbers and equations during math lessons

26 Montana City School GRADE 8 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 1. Understand the concepts of variable, expression and equation. Grade 8 Evaluate algebraic expressions (1.1) Combine like terms in an expression (11.1) Solve equations using addition and subtraction (1.7) Solve equations using multiplication and division (1.8) Classify polynomials by degree and by the number of terms (14.1) 2. Represent situations and number patterns using tables, graphs, verbal rules, equations, and models. Compare and order integers and evaluate expressions containing absolute values (1.3) Solve and graph inequalities (1.9) Graph points and lines on the coordinate plane (3.2) Solve multi-step equations (11.2) Solve equations with variables on both sides of the equal sign. (11.3) Solve and graph inequalities by using multiplication or division (11.4) Solve two-step inequalities and graph the solution of an inequality on a number line (11.5) Identify and graph linear equations (12.1) Find the slope of a line and use slope to understand and draw graphs (12.2) Use slopes and intercepts to graph linear equations (12.3) Find the equation of a line given one point and the slope (12.4) Recognize direct variation by graphing tables of data and checking for constant ratios (12.5) Recognize relationships in data and find the equation of a line of best fit (12.7)

27 Montana City School GRADE 8 Montana Standard 3: Students use algebraic concepts, processes, and language to model and solve a variety of real-world and mathematical problems. 3. Recognize and use the general properties of operations (e.g., the distributive property). Grade 8: Apply the properties of exponents (4.3) Determine if a number is rational or irrational (4.7) Solve systems of equations (11.6) 4. Solve linear equations using concrete, numerical, and algebraic methods. Solve two-step equations (2.8) Learn to add integers (1.4) Learn to subtract integers (1.5) Learn to multiply and divide integers (1.6) Write solutions of equations in two variables as ordered pairs (3.1) 5. Investigate inequalities and nonlinear relationships informally. Graph inequalities on the coordinate plane (12.6)

28 Montana City School GRADE 5 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 1. Identify, describe, construct, and compare plane and solid geometric figures. 2. Understand and apply geometric properties and relationships (e.g., the Pythagorean Theorem). Grade 5 Identify and use the concepts of point, line, and angle and classify angles as acute, right or obtuse (18.1) Estimate, measure, and draw angles (18.2) Identify, classify, and draw polygons (18.3) Draw a circle and its parts, relate radius and diameter, and find the measure of angles surrounding a point (18.5) Identify, describe, and classify types of triangles (19.1), quadrilaterals (19.2) solid figures (19.4) Draw and identify triangles and quadrilaterals (19.3) 3. Represent geometric figures on a coordinate grid. Identify and make nets for solid figures (19.6)

29 Montana City School GRADE 5 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 4. Explore properties and transformations of geometric figures. Grade 5 Identify, describe, predict, and extend transformations (20.1), tessellations (20.2) numeric patterns (20.4) Use transformations to create and extend geometric patterns (20.3) Identify congruent and similar figures (18.6) Identify and describe line and rotational symmetry (18.7) 5. Use geometry as a means of describing the physical world. Visualize and draw two-dimensional views of three-dimensional objects (19.7) Identify congruent and similar figures (18.6)

30 Montana City School GRADE 6 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 1. Identify, describe, construct, and compare plane and solid geometric figures. Grade 6 Name measure, draw, and classify angles (8.2) Classify the different types of lines (8.4) Understand the relationship of angles (8.3) Identify regular and not regular polygons and find the angle measure of regular polygons (8.7) Identify, classify, and compare quadrilaterals (8.6) Name three-dimensional figures (10.6) 2. Understand and apply geometric properties and relationships (e.g., the Pythagorean Theorem). Classify triangles and solve problems involving angle and side measures of triangles (8.5) 3. Represent geometric figures on a coordinate grid. Locate and graph points on the coordinate plane (11.3)

31 Montana City School GRADE 6 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 4. Explore properties and transformations of geometric figures. Grade 6 Use translations, reflections, and rotations to transform geometric shapes (8.10) Identify line symmetry (8.11) 5. Use geometry as a means of describing the physical world. Recognize, describe, and extend geometric patterns (8.8) Identify congruent figures and use congruence to solve problems (8.9)

32 Montana City School GRADE 7 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 1. Identify, describe, construct, and compare plane and solid geometric figures. Understand and apply geometric properties and relationships (e.g., the Pythagorean Theorem). Grade 7 Identify angles and angle pairs (8.2) Identify and name polygons (8.5) Find the measure of angles in polygons (8.8) Identify and describe geometric figures (8.1) Identify various three dimensional shapes (10.1) Identify parallel, perpendicular, and skew lines and angles formed by transversal (8.3) Identify parts of a circle and find central angle measures (8.4) Classify types of triangles by their side lengths and angle measures (8.6) Name, identify, and draw types of quadrilaterals (8.7) Use the Pythagorean Theorem to find the length of a side of a right triangle (9.8) 3. Represent geometric figures on a coordinate grid. Locate and graph points on the coordinate plane (11.3)

33 Montana City School GRADE 7 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 4. Explore properties and transformations of geometric figures. Grade 7 Recognize, describe, and show transformations (8.10) Identify congruent figures and use congruence to solve problems (8.9) Identify symmetry in figures (8.11) 5. Use geometry as a means of describing the physical world. Identify congruent figures and use congruence to solve problems (8.9)

34 Montana City School GRADE 8 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 1. Identify, describe, construct, and compare plane and solid geometric figures. Grade 8 Classify and name figures (7.1) Identify and create dilations of plane figures (5.6) 2. Understand and apply geometric properties and relationships (e.g., the Pythagorean Theorem). Use the Pythagorean Theorem to solve problems (4.8) Use the properties of the congruent figures to solve problems (7.6) 3. Represent geometric figures on a coordinate grid. Identify polygons in the coordinate plane (7.5)

35 Montana City School GRADE 8 Montana Standard 4: Students demonstrate understanding of shape and an ability to use geometry. 4. Explore properties and transformations of geometric figures. Grade 8 Create tessellations (7.9) Transform plane figures using translations, rotations, and reflections (7.7) Identify symmetry in figures (7.8) Find measures indirectly by applying properties of similar figures (5.7) 5. Use geometry as a means of describing the physical world. Draw and identify parts of threedimensional figures (8.4) Determine whether figures are similar, to use scale factors, and to find missing dimensions in similar figures (5.5) Make comparisons between and find dimensions of scale drawings, models, and actual objects (5.8)

36 Montana City School GRADE 5 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 1. Estimate, make, and use measurements to describe, compare, and/or contrast objects in real-world situations. 2. Select and use appropriate units and tools to measure to a level of accuracy required in a particular setting. Grade 5 Estimate and measure using appropriate customary units (22.1) and metric units (22.2) and determine which measurement is more precise Identify and convert linear measurements of time (22.3) Identify and convert measurements involving elapsed time (22.7) Identify and convert measurements involving temperature (22.8) Identify the appropriate units of measure for perimeter, area, and volume (5.2) 3. Apply the concepts of perimeter, area, volume and capacity, weight, and mass, angle measure, time, and temperature. Find perimeter (23.1) Find perimeter of polygons (5.3) Find the area of triangles (5.3) Find the area of parallelograms (24.5)

37 Montana City School GRADE 5 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 4. Demonstrate understanding of the structure and use of systems of measurement, including English and metric. Grade 5 Convert customary (22.4) and metric (22.5) capacity and weight units 5. Use the concepts of rates and other derived and indirect measurements. Find rates and unit rates (16.3) Read and interpret a scale on a map (16.4) 6. Demonstrate relationships between formulas and procedures for determining area and volume. Find the perimeter of polygons by using formulas (23.3) Find the circumference of a circle (23.5) Estimate the area of regular and irregular figures (24.1) Find the area of squares and rectangles by using formulas (24.2) Identify how perimeter and area are related (24.3) Estimate and find the volume of a rectangular prism (24.8)

38 Montana City School GRADE 6 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 1. Estimate, make, and use measurements to describe, compare, and/or contrast objects in real-world situations. Grade 6 Make a model to explore how area and perimeter are affected by changes in the dimensions of a figure (10.4) 2. Select and use appropriate units and tools to measure to a level of accuracy required in a particular setting. Compare the precision of measurements and determine acceptable levels of accuracy 3. Apply the concepts of perimeter, area, volume and capacity, weight, and mass, angle measure, time, and temperature. Find perimeter and missing side lengths of a polygon (9.7) Find the area of triangles and trapezoids (10.2) Estimate the area of irregular figures and find the area of rectangles and parallelograms (10.1) Find measures of time and temperature ( 9.5) Break a polygon into simpler parts to find its area (10.3) Find the area of a circle (10.5)

39 Montana City School GRADE 6 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 4. Demonstrate understanding of the structure and use of systems of measurement, including English and metric. Grade 6 Convert customary units of measure (9.3) metric (9.4) Understand and select appropriate customary units of measure (9.1) Understand and select appropriate metric units of measure (9.2) 5. Use the concepts of rates and other derived and indirect measurements. Lean to read and use map scales and scale drawings (7.6) 6. Demonstrate relationships between formulas and procedures for determining area and volume. Identify parts of a circle and find the circumference of a circle (9.8) Estimate and find the volumes of rectangular prisms and triangular prisms (10.7) Find volumes of cylinders (10.8) Find the surface area of prisms, pyramids, and cylinders (10.9)

40 Montana City School GRADE 7 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 1. Estimate, make, and use measurements to describe, compare, and/or contrast objects in real-world situations. Grade 7 Identify and make models of solid figures (9.7) 2. Select and use appropriate units and tools to measure to a level of accuracy required in a particular setting. Compare the precision of measurements and determine acceptable levels of accuracy (9.1) 3. Apply the concepts of perimeter, area, volume and capacity, weight, and mass, angle measure, time, and temperature. Find perimeter of a polygon and the circumference of a circle (9.2) Find the area of triangles and trapezoids (9.4) Find the area of rectangles and other parallelograms (9.3) Find the area of irregular figures (9.6) Find the area of a circle (9.5)

41 Montana City School GRADE 7 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 4. Demonstrate understanding of the structure and use of systems of measurement, including English and metric. Grade 7 Identify, convert, and compare metric units (1.3) customary units (5.6) of measure 5. Use the concepts of rates and other derived and indirect measurements. Find unit and compare unit rates, such as average speed and unit price (5.2) Determine the slope of a line and to recognize constant and variable rates of change (5.3) 6. Demonstrate relationships between formulas and procedures for determining area and volume. Find the volume of prisms and cylinders (10.2) Find volume of pyramids and cones (10.3) Find the surface area of prisms and cylinders (10.4) Find the volume and surface area of similar three dimensional figures (10.5)

42 Montana City School GRADE 8 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 1. Estimate, make, and use measurements to describe, compare, and/or contrast objects in real-world situations. Grade 8 Make scale models of solid figures (8.10) 2. Select and use appropriate units and tools to measure to a level of accuracy required in a particular setting. Compare the precision of measurements and determine acceptable levels of accuracy (8.10) 3. Apply the concepts of perimeter, area, volume and capacity, weight, and mass, angle measure, time, and temperature. Find the perimeter and area of triangles and trapezoids (8.2) Find the perimeter and area of rectangles and parallelograms (8.1) Classify and find angles in polygons (7.4) Find circumference and area of circles (8.3) Identify parallel and perpendicular lines and the angles formed by a transversal (7.2) Find unknown angles in triangles (7.3)

43 Montana City School GRADE 8 Montana Standard 5: Students demonstrate understanding of measurable attributes and an ability to use measurement processes. 4. Demonstrate understanding of the structure and use of systems of measurement, including English and metric. Grade 8 Use measurement for a variety of different reasons (8.0) 5. Use the concepts of rates and other derived and indirect measurements. Use one or more conversion factors to solve rate problems (5.3) 6. Demonstrate relationships between formulas and procedures for determining area and volume. Find the volume of prisms and cylinders (8.5) Find the volume of pyramids and cones (8.6) Find the surface area of prisms and cylinders (8.7) Find surface area of pyramids and cones (8.8) Find volume and surface area of spheres (8.9)

44 Montana City School GRADE 5 Montana Standard 6: The students demonstrate understanding of and an ability to use data analysis, probability, and statistics. 1. Systematically collect, organize, and describe data. Grade 5 Collect data by using surveys and organize data in tables and line plots (9.1) Compare the types of graphs that can be used for categorical data and numerical data, and select an appropriate graph (10.7) 2. Construct, read, and interpret tables, charts, and graphs. Read, interpret, and analyze the data in graphs (9.4) Represent data by making a bar graph, double-bar graph, and pictograph (10.1) Represent data by making a pictograph or a bar graph (24.7) Represent data by making a histogram (10.2), line graph (10.4), circle graph (10.5) Graph and name points on a coordinate grid using ordered pairs (10.3) 3. Draw inferences, construct and evaluate arguments based on data analysis and measures of central tendency. Describe data using the mean, medium, and mode (9.2) Compare two or more sets of data (9.3) 4. Construct sample spaces and determine the theoretical and experimental probabilities of events. Find all possible outcomes of an event (17.5) Find combinations and arrangements of sets of objects (17.6) 5. Make predictions based on experimental results or probabilities. Predict outcomes of probability experiments (17.1, 17.2, 17.3)

45 Montana City School GRADE 6 Montana Standard 6: The students demonstrate understanding of and an ability to use data analysis, probability, and statistics. 1. Systematically collect, organize, and describe data. 2. Construct, read, and interpret tables, charts, and graphs. 3. Draw inferences, construct and evaluate arguments based on data analysis and measures of central tendency. 4. Construct sample spaces and determine the theoretical and experimental probabilities of events. 5. Make predictions based on experimental results or probabilities. Grade 6 Use tables to record and organize data (6.1) Choose appropriate ways to display data (6.10) Display and analyze data in bar graphs (6.4) Organize data in frequency tables and histograms (6.5) Graph ordered pairs on a coordinate grid (6.6) Represent linear functions using ordered pairs and graphs (11.10) Display and analyze data in line graphs (6.7) Make and analyze stem-and-leaf plots (6.9) Find the range, mean, median, and mode of a data sheet (6.2) Identify the effect of additional data and outliers (6.3) Recognize misleading graphs (6.8) Make an organized list to find all possible outcomes (12.3) Estimate the likelihood of an event write and compare probabilities (12.1) List all the outcomes and find the theoretical probability of a compound event (12.5) Find the experimental probability of an event (12.2) Find the theoretical probability and complement of an event (12.4) Use probability to predict events (12.6)

46 Montana City School GRADE 7 Montana Standard 6: The students demonstrate understanding of and an ability to use data analysis, probability, and statistics. 1. Systematically collect, organize, and describe data. 2. Construct, read, and interpret tables, charts, and graphs. Grade 7 Select and use appropriate representations for displaying data (7.7) Display and analyze data in bar graphs and histograms (7.3) Plot and identify ordered pairs on a coordinate plane (4.1) Identify and graph ordered pairs from a table of values (4.2) Use function tables to generate a graph ordered pairs (4.4) Read and interpret data presented in boxand-whisker plots (7.5) Display and analyze data in line graphs (7.6) Organize and interpret data in frequency tables, stem-and-leaf plots, and line plots (7.1) Read and interpret data presented in circle graphs (7.4) Display and analyze data in scatter plots (7.9) 3. Draw inferences, construct and evaluate arguments based on data analysis and measures of central tendency. Find the mean, median, mode, and range of a data set (7.2) Identify and analyze misleading graphs (7.10)

47 Montana City School GRADE 7 Montana Standard 6: The students demonstrate understanding of and an ability to use data analysis, probability, and statistics. 4. Construct sample spaces and determine the theoretical and experimental probabilities of events. 5. Make predictions based on experimental results or probabilities. Grade 7 Find the theoretical probability of an event (11.4) Compare and analyze sampling methods (7.8) Use informal measures of probability (11.1) Use counting methods to determine possible outcomes (11.3) Find the probability of independent and dependent events (11.5) Find the number of possible combinations (11.6) Find experimental probability (11.2) Find the number of possible permutations (11.7)

48 Montana City School GRADE 8 Montana Standard 6: The students demonstrate understanding of and an ability to use data analysis, probability, and statistics. 1. Systematically collect, organize, and describe data. 2. Construct, read, and interpret tables, charts, and graphs. 3. Draw inferences, construct and evaluate arguments based on data analysis and measures of central tendency. 4. Construct sample spaces and determine the theoretical and experimental probabilities of events. 5. Make predictions based on experimental results or probabilities. Grade 8 Select the best representation for a set of data (9.8) Identify sampling methods and recognize biased samples (9.1) Display data in bar graphs, histograms, and line graphs (9.5) Organize data in tables and stem-and-leaf plots (9.2) Create and interpret scatter plots (9.7) Recognize misleading graphs and statistics (9.6) Find measures of variability (9.4) Find appropriate measures of central tendency (9.3) Convert between probabilities and odds (10.7) Find the probabilities of independent and dependent events (10.5) Find the number of possible outcomes in an experiment (10.9) Estimate probability using experimental methods (10.2) Estimate probability using theoretical methods (10.4)

49 Montana City School GRADE 5 Montana Standard 7: Students demonstrate understanding of and an ability to use patterns, relations and functions. 1. Describe, extend, analyze, and create a variety of patterns and functions. 2. Describe and represent relationships with tables, graphs, and rules. 3. Analyze functional relationships to explain how a change in one quantity results in a change in another. 4. Use patterns and functions to represent and solve problems. 5. Describe functions using graphical, numerical, physical, algebraic, and verbal models or representations. Grade 5 Represent a function with a table and an equation (4.7) Learn to find patterns and to recognize, describe, and extend patterns in sequences (1.7) Graph relationships from input-output tables (20.5) Use function tables and equations to graph equations on a grid ( 21.2) Investigation of algebraic concepts Use patterns to divide (3.5) Use patterns in decimal factors to find products (7.2) Read, interpret, and analyze the data in graphs (9.4) Write(4.1) and Evaluate (4.2) numerical and algebraic expressions

50 Montana City School GRADE 6 Montana Standard 7: Students demonstrate understanding of and an ability to use patterns, relations and functions. 1. Describe, extend, analyze, and create a variety of patterns and functions. Grade 6 Identify and extend patterns (1.1) 2. Describe and represent relationships with tables, graphs, and rules. Write expressions for tables and sequences (2.3) 3. Analyze functional relationships to explain how a change in one quantity results in a change in another. 4. Use patterns and functions to represent and solve problems. Use a table to find equivalent ratios and rates (7.2) Solve percent problems that involve discounts, tips, and sales tax (7.10) Make an organized list to find all possible outcomes (12.3) Use data in a table to write an equation for a function and to use the equation to find the missing value (11.9) 5. Describe functions using graphical, numerical, physical, algebraic, and verbal models or representations. Choose appropriate ways to display data (6.10)

51 Montana City School GRADE 7 Montana Standard 7: Students demonstrate understanding of and an ability to use patterns, relations and functions. 1. Describe, extend, analyze, and create a variety of patterns and functions. Grade 7 Find the mean, median, mode, and range of a data set (7.2) 2. Describe and represent relationships with tables, graphs, and rules. Read and write inequalities and graph them on a number line (12.4) 3. Analyze functional relationships to explain how a change in one quantity results in a change in another. Solve one-step equations with integers (2.5) Solve equations that have variable on both sides (12.3) 4. Use patterns and functions to represent and solve problems. Relate graphs to situations (4.3) Use order of operations to simplify numerical expressions (1.5) 5. Describe functions using graphical, numerical, physical, algebraic, and verbal models or representations. Find patterns to complete sequences using function tables (4.5)

52 Montana City School GRADE 8 Montana Standard 7: Students demonstrate understanding of and an ability to use patterns, relations and functions. 1. Describe, extend, analyze, and create a variety of patterns and functions. 2. Describe and represent relationships with tables, graphs, and rules. 3. Analyze functional relationships to explain how a change in one quantity results in a change in another. Grade 8 Find terms in an arithmetic sequence (13.1) Find terms in a geometric sequence (13.2) Find patterns in sequences (13.3) Identify linear functions (13.4) Identify and graph exponential functions (13.5) Identify and graph quadratic functions (13.6) Represent functions with tables, graphs, or equations (3.4) Generate different representations of the same data (3.5) Recognize inverse variation by graphing tables of data (13.7) Solve two-step equations (2.8) Solve equations with variables on both sides of the equal sign. (11.3) 4. Use patterns and functions to represent and solve problems. Identify and evaluate arithmetic sequences (3.6) 5. Describe functions using graphical, numerical, physical, algebraic, and verbal models or representations. Interpret information given in a graph or table and to make a graph to solve problems (3.3)

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