Grade Level: Grade 5

Algebraic Reasoning

Concepts
Patterns and Functions

Enduring Understandings
Algebraic representations generalize patterns and relationships.
Numerical and geometric patterns can be extended to determine an outcome.
Changing one variable in an equation results in a corresponding change in a second variable in the same equation.
Number patterns and relationships can be represented using variables in equations and inequalities.

Skills
Identify trends and make predictions based upon patterns and data displayed in different formats.
   1.)       Extend and compare arithmetic and geometric sequences.
   2.)       Represent geometric and numeric patterns using words, tables, graphs and equations.
   3.)       Analyze patterns and data to make generalizations and predictions.  
Recognize that a change in one variable may relate to a change in another variable.
   1.)       Describe how a change in one variable relates to a change in a second variable in context. 
Describe the general relationship between two sets of data using an equation or inequality.
   1.)       Represent mathematical relationships using variable in expressions, equations and inequalities, including a variable to represent an unknown number.
   2.)       Model and solve one stop equations using materials that model equivalence.
   3.)       Replace variables with a numeric value and simplify expression.

Numerical & Proportional Reasoning

Concepts
Numeric Relationships, Ratios, Proportion and Number Sense

Enduring Understandings
Place value patterns are repeated in large numbers.
Fractions, percents and decimals can be compared using a variety of models such as money, pictures, numerical expressions, etc.
Ratios can be used to compare two proportional quantities.

Skills
Extend whole number place value patterns, models, and notations to include decimals, which are fractions that have denominators that are multiples of ten.
   1.)       Identify, round, order and compare whole numbers to 1,000,000 using place value models, diagrams and number lines.
   2.)       Express numbers in expanded and regrouped forms and use the numbers to support computational strategies.
   3.)       Solve problems involving finding 10, 100 and 1000 more and less.
   4.)       Estimate products and missing factor using multiples of 10, 100 and 1000 or 10,000.
   5.)       Use models to extend whole number place value concepts and patterns to decimals.
   6.)       Explore numbers less than zero and extend the number line to introduce the concept of integers within practical applications.  
   7.)       Order and compare decimals to the hundredths.
   8.)       Name whole numbers using exponents.
Classify numbers by their factors.
   1.)       Use rectangular arrays to identify factor pairs and to classify numbers as prime, composite, and perfect squares.
   2.)       Explore divisibility rules and patterns with remainders.  
Demonstrate an understanding of numbers as equivalent fractions, decimals or percents.
   1.)       Represent a rational number in its equivalent fraction, decimal, ratio and percent forms with models, number patterns and common factors.
   2.)       Construct and use models and pictures to add and subtract fractions, decimals and mixed numbers with like and unlike denominators.
   3.)       Use equivalence and substitution with like and unlike denominators when adding and subtracting.
   4.)       Construct and use models and pictorial representations to multiply and divide common fractions and mixed numbers.  
   5.)       Order and compare decimals, fractions, mixed numbers and percents.
   6.)       Rename fractions in lowest terms.
Represent ratios and proportions and solve problems using models and pictures.
   1.)       Build models to identify and compare ratios and describe quantitative relationships using fraction and decimal equivalents.
   2.)       Write division problems in fraction form and round the fraction form to estimate an answer to a division problem.
   3.)       Use ratios and proportions to solve practical problems such as interpreting maps and scale drawings or identifying probability.  
Estimate and compute using models, pictures and numbers.
   1.)       Choose and use benchmarks to approximate locations on number lines and coordinate grids.
   2.)       Estimate and use counting, grouping of objects, number patterns, equivalent ratios and division to find fractional parts of a set of objects.
   3.)       Develop strategies, using place value relationships, inverse operations and commutative, associative and distributive properties, to simplify computations with 2-, 3-, and 4-digit numbers and money amounts.
   4.)       Use estimation to predict results and to recognize when an answer is or is not reasonable when adding, subtracting, multiplying and dividing.
   5.)       Explain when an estimation strategy will result in an over- or underestimate.
   6.)        Create and solve multi-step problems and explore order of operations in the context of practical situations.
   7.)       Add, subtract and use estimation up to 6-digit numbers to solve problems.
   8.)       Multiply and divide 2 and 3-digit numbers by 1 or 2-digit numbers.

Geometry & Measurement

Concepts
Shapes; Geometric Comparisons; Measurement

Enduring Understandings
Changing the position of an object does not affect its attributes.
Lines and angles can be described using geometric attributes.
The perimeters, areas, and volumes of objects depend on their dimensions.
The perimeter, area, and volume of a geometric figure may change as angles, sides, and lengths change

Skills
Use geometric relationships to describe polygons and solids.
   1.)       Use geometric relationships such as parallel, perpendicular and congruent to describe the attributes of sets and subsets of shapes and solids.
   2.)       Make and test conjectures about geometric relationships.  
Recognize that changes in the perimeter of a polygon may affect its area, and changes in area may affect the perimeter.
   1.)       Explore the relationship between area and perimeter when the dimensions of a polygon change.
   2.)       Develop formulas to find the perimeter and area of squares, rectangles and triangles. Identify, describe and build nets for solid figures and objects.
   1.)       Represent the surface of 3-dimensional objects through the use of 2-dimensional nets.
   2.)       Investigate and develop strategies to determine the volume of rectangular solids.  
   3.)       Analyze, compare and classify solids using attributes such as corners, edges and faces.  
Determine geometric relationships through spatial visualization.
  1.)       Plot points on the rectangular coordinate system and estimate and determine the distance between points.  
Solve problems in the measure of time and in converting units of length in the customary and metric systems using specific ratios.
  1.)       Solve length problems involving conversion of measures within customary and metric systems.
  2.)       Solve problems involving the conversion of measure so f time and elapsed time (days, hours, minutes and seconds).
  3.)       Estimate and choose appropriate units and tools to measure and solve a variety of problems involving length, perimeter, area, volume, capacity, mass, time, angle and temperature.

Working with Data: Probability & Statistics

Concepts
Data Relationships; Prediction; Numeric Communication

Enduring Understandings
Data can be used to see trends and make predictions and interpretations.
Measures of central tendency (mean, median, mode) can be used to compare data.

Skills
Differentiate between numerical and categorical data and their appropriate representation. (*Grade 4 - also in Unit 1 - Data About Us)
  1.)       Construct and interpret broken line graphs, line plots, bar graphs, picture graphs, simple circle graphs, and stem and leaf plots and evaluate how well each kind of display represents the features of the data.   
  2.)       Draw conclusions from data by determining range, mean, median and mode of a set of data.
Examine different data collection methods and their affect.
  1.)       Design and conduct surveys and samplings to collect data that represent a general population.
  2.)       Explore how a change in an outlier can change the measures of central tendency.  
Relate the likelihood of an event to a numerical value.
  1.)       Identify possible outcomes and express the likelihood of events as a fraction.
  2.)       Design and conduct probability experiments and games of chance.
  3.)       Make and test predictions of probability and fairness.