Standards for Mathematical Content Grade 5 #

Operations and Algebraic Thinking #

Write and interpret numerical expressions. #

• 5.OA.1. Use parentheses to construct numerical expressions, and evaluate numerical expressions with these symbols.

• 5.OA.2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 x (8 + 7). Recognizing that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Analyze patterns and relationships. #

• 5.OA.3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Numbers and Operations in Base Ten #

Understand the place value system. #

• 5.NBT.1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

• 5.NBT.2. Explain and extend the patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain and extend the patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

• 5.NBT.3. Read, write, and compare decimals to thousandths.

  • a) Read and write decimals to thousandths using base-ten numerals, number names, and expanded form [e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 (1/10) + 9 (1/100) + 2 (1/1000)].
  • b) b. Compare two decimals to thousandths place based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

• 5.NBT.4. Use place values understanding to round decimals to any place.

Perform operations with multi-digit whole numbers and with decimals to hundredths. #

• 5.NBT.5. Fluently multiply multi-digit whole numbers using a standard algorithm.

• 5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, number lines, real life situations, and/or area models.

• 5NBT.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between the operations. Related the strategy to a written method and explain their reasoning in getting their answers.

Number and Operations - Fractions #

Use equivalent fractions as a strategy to add and subtract fractions. #

• 5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.

• 5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and check the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Apply previous understandings of multiplication and division to multiply and divide fractions. #

• 5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers (e.g., by using visual fraction models or equations to represent the problem). For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

• 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

  • a) Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
  • b) Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

• 5.NF.5 Interpret multiplication as scaling (resizing), by:

  • a) Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
  • b) Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. (Division of a fraction by a fraction is not a requirement at this grade.)

• 5.NF.6. Solve real world problems involving multiplication of fractions and mixed numbers (e.g., by using visual fraction models or equations to represent the problem).

• 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

  a) Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

  b) Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

  c) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions (e.g., by using visual fraction models and equations to represent the problem). For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Measurement and Data #

Convert like measurement units within a given measurement system and solve problems involving time. #

• 5. MD.1. Identify, estimate measure, and convert equivalent measures within systems English length (inches, feet, yards, miles) weight (ounces, pounds, tons) volume (fluid ounces, cups, pints, quarts, gallons) temperature (Fahrenheit) Metric length (millimeters, centimeters, meters, kilometers) volume (milliliters, liters), temperature (Celsius), (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems using appropriate tools.

• 5. MD.2. Solve problems involving elapsed time between world time zones. (L)

Represent and interpret data. #

• 5.MD.3. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

• 5.MD.4. Explain the classification of data from real-world problems shown in graphical representations including the use of terms mean and median with a given set of data. (L)

Geometric measurement: understand concepts of volume and relate to multiplication and addition. #

• 5.MD.5. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a) A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

b) A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

• 5.MD.6. Estimate and measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standard units.

• 5.MD7. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

  • a) Estimate and find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Demonstrate the associative property of multiplication by using the product of three whole-numbers to find volumes (length x width x height).
  • b) Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
  • c) Recognize volume as additive. Find volumes of solid figures composed of two, non-overlapping, right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Geometry #

Graph points on the coordinate plane to solve real-world and mathematical problems. #

• 5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

• 5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

Classify two-dimensional (plane) figures into categories based on their properties. #

• 5.G.3. Understand that attributes belonging to a category of two dimensional (plane) figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

• 5.G.4. Classify two-dimensional (plane) figures in a hierarchy based on attributes and properties.

Standards for Mathematical Practice #

Instruction around the Standards of Mathematical Practices is delivered across all grades K-12. These eight standards define experiences that build understanding of mathematics and ways of thinking through which students develop, apply, and assess their knowledge.

Make sense of problems and persevere in solving them. #

• explain correspondences between equations, verbal descriptions, tables, and graphs

• draw diagrams of important features and relationships, graph data, and search for regularity or trends

• use concrete objects or pictures to help conceptualize and solve a problem

• understand the approaches of others to solving complex problems

• identify correspondences between different approaches

• check if the solution makes sense

Reason abstractly and quantitatively. #

• represent a situation symbolically

• create a coherent representation of the problem

• have the ability to show how problem has a realistic meaning

• reflect during the manipulation process in order to probe into the meanings for the symbols involved

• use units consistently

Construct viable arguments and critique the reasoning of others. #

• construct arguments using concrete referents such as objects, drawings, diagrams, and actions

• justify conclusions, communicate conclusions, listen and respond to arguments, decide whether the argument makes sense, and ask questions to clarify the argument

• reason inductively about data, making plausible arguments that take into account the context from which the data arose

Model with Mathematics. #

• apply mathematics to solve problems arising in everyday life

• identify important quantities in a practical situation and model the situation using such tools as manipulatives, diagrams, two-way tables, graphs or pictures

• interpret mathematical results in the context of the situation and reflect on whether the results make sense

• apply mathematical knowledge, make assumptions and approximations to simplify a complicated situation

Use appropriate tools strategically. #

• select the available tools (such as pencil and paper, manipulatives, rulers, calculators, a spreadsheet, and available technology) when solving a mathematical problem

• be familiar with tools appropriate for their grade level to make sound decisions about when each of these tools might be helpful

• identify relevant external mathematical resources and use them to pose or solve problems

• use technological tools to explore and deepen their understanding of concepts

• detect possible errors by strategically using estimation and other mathematical knowledge

• know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data

Attend to precision. #

• give carefully formulated explanations to each other

• use clear definitions and reasoning in discussion with others

• state the meaning of symbols, including using the equal sign consistently and appropriately

• specify units of measure, and label axes to clarify the correspondence with quantities in a problem

• calculate accurately and efficiently

• express numerical answers with a degree of precision appropriate for the problem context

Look for and make use of structure. #

• discern a pattern or structure

• understand complex structures as single objects or as being composed of several objects

• check if the answer is reasonable

Look for and express regularity in repeated reasoning. #

• identify if calculations or processes are repeated

• use alternative and traditional methods to solve problems

• evaluate the reasonableness of their intermediate results, while attending to the details