How to learn Grade 5: Mathematics

Pennsylvania academic standards: Mathematics, Grade 5

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1. Use, describe, and develop procedures to solve problems involving volume.

Estimating answers (age 10+): Find the volume of right rectangular prisms by packing with unit cubes and show it equals l × w × h (or base area × height); apply V = l × w × h and V = B × h to solve real-world problems; calculate, estimate, and compare volumes of cubes and cuboids in standard units (cm³, m³)
Volume as additive: Recognise volume as additive; find volumes of composite solid figures made of two or more non-overlapping right rectangular prisms

Pictograms and tally charts (age 6+): Read, write, and use the vocabulary of data collection and display — data, tally, tally chart, frequency, frequency table, survey, pictogram, bar chart, axis/axes, scale, label, category, discrete data, continuous data, line graph, pie chart — and apply these terms when collecting, organising, and presenting data
Understanding fractions: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8); use operations on fractions to solve problems involving data in line plots (e.g. redistribute total equally)

Measurement Conversions: Solve problems involving the calculation and conversion of units of measure, using decimal notation and multi-step reasoning in real-world contexts

Classifying shapes by properties: Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties

Numbers on a number line: Understand a coordinate system defined by two perpendicular number lines (axes) with an origin at (0,0); know that an ordered pair (x, y) specifies a unique point where the first number gives horizontal distance and the second gives vertical distance from the origin
Plotting points in the first quadrant: Plot and read ordered pairs in the first quadrant of the coordinate plane; represent real-world and mathematical problems by graphing points and interpreting coordinate values in context

Number Pattern Relationships: Generate two numerical patterns using two given rules; identify relationships between corresponding terms; form ordered pairs and graph them on a coordinate plane

Substituting into Formulae: Substitute numerical values into formulae and expressions including scientific formulae; evaluate expressions by replacing variables with given values and computing the result using correct order of operations
Brackets in Expressions: Use parentheses, brackets, or braces in numerical expressions and evaluate expressions containing these grouping symbols

Dividing fractions (unit fractions): Interpret and compute division of a unit fraction by a non-zero whole number (e.g. 1/3 ÷ 4 = 1/12); use visual models and the relationship between multiplication and division to explain the result
Dividing unit fractions and whole numbers: Solve real-world problems involving division of unit fractions by whole numbers and whole numbers by unit fractions, using visual models and equations
Real-world fraction multiplication: Solve real-world problems involving multiplication of fractions and mixed numbers, using visual fraction models or equations
Multiplication as scaling: Interpret multiplication as scaling (resizing): compare the size of a product to a factor based on the size of the other factor without computing; explain the effect of multiplying by fractions greater than, equal to, or less than 1
Multiplying fractions: Understand a/b as a multiple of 1/b; multiply proper fractions and mixed numbers by whole numbers, supported by visual models (e.g. 3 × 2/5 = 6/5 = 1 1/5)

Adding and subtracting mixed numbers: Add and subtract mixed numbers with like denominators, including by converting to improper fractions or using properties of operations
Adding fractions (different denominators): Add and subtract fractions with denominators that are multiples of the same number by finding a common denominator
Adding Fractions (Unlike Denominators): Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce a common denominator

Division with Decimals: Use written division methods in cases where the answer has up to two decimal places; divide decimals to hundredths by whole numbers

Comparing Decimals: Compare two decimals to hundredths (or up to three decimal places) by reasoning about size using place-value understanding; record with >, =, <
Reading and writing numbers (age 10+): Read, write, and compare decimals to thousandths using base-ten numerals, number names, and expanded form; compare using >, =, < based on place-value meaning
Decimal place value: Round decimals to any place using place-value understanding; round any whole number to a required degree of accuracy
Reading Decimal Places: Identify the value of each digit in numbers given to three decimal places (e.g. in 4.378, the 7 represents 7 hundredths)
Place Value × 10 and ÷ 10: Recognise that in a multi-digit number, a digit in one place represents 10 times as much as in the place to its right and 1/10 of what it represents in the place to its left
Patterns with Powers of Ten: Explain patterns in zeros when multiplying by powers of 10 and in decimal-point placement when multiplying/dividing by a power of 10; use whole-number exponents to denote powers of 10 (e.g. 10³ = 1000)