Prioritized Categories All categories Not categorized Prioritized (G) Geometry (MD) Measurement and Data (NBT) Number and Operations in Base Ten (NF) Number and Operations - Fractions (OA) Operations and Algebraic Thinking

# Geometry

## Narrative for the (G) Geometry

By the end of Grade 5, competencies in shape composition and decomposition, and especially the special case of spatial structuring of rectangular arrays should be highly developed. Students need to develop these competencies because they form a foundation for understanding multiplication, area, volume, and the coordinate plane.

Students extend their spatial structuring in two ways. They learn to spatially structure in three dimensions; for example, they can decompose a right rectangular prism built from cubes into layers, seeing each layer as an array of cubes. They use this understanding to find the volumes of right rectangular prisms with edges whose lengths are whole numbers as the number of unit cubes that pack the prisms.  Second, students extend their knowledge of the coordinate plane, understanding the continuous nature of two-dimensional space and the role of fractions in specifying locations in that space.

Students in Grade 5 learn to analyze and relate categories of two-dimensional and three-dimensional shapes explicitly based on their properties. Based on analysis of properties, they classify two-dimensional figures in hierarchies.

Students also solve mathematical and real-world problems using coordinates. They graph connect ordered pairs of (whole number) coordinates to points on a grid, so that these coordinate pairs constitute numerical objects.  Students then interpret the results within the context of situation.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-05.G.02

(G) Geometry
Cluster: Graph points on the coordinate place to solve real­world and mathematical problems.

#### MAT-05.G.04

(G) Geometry
Cluster: Classify two ­dimensional figures into categories based on their properties.

# Measurement and Data

## Narrative for the (MD) Measurement and Data

In Grade 5, students extend their measurement abilities from Grade 4 to express measurements in larger or smaller units within a measurement system.  The major emphasis for measurement in Grade 5 is volume. Volume not only introduces a third dimension and thus a significant challenge to students’ spatial structuring, but also complexity in the nature of the materials measured.

As students work with data in Grades K–5, they build foundations for their study of statistics and probability in Grades 6 and beyond, and they strengthen and apply what they are learning in arithmetic. Kindergarten work with data uses counting and order relations.  In Grades 3–5, work with data is closely related to the number line, fraction concepts, fraction arithmetic, and solving problems that involve the four operations.

By the end of Grade 5, students should be comfortable making line plots for measurement data and analyzing data shown in the form of a line plot. As in earlier grades, students should work with data in science and other subjects. Grade 5 students working in these contexts should be able to give deeper interpretations of data than in earlier grades.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-05.MD.05

(MD) Measurement and Data
Cluster: Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

##### MAT-05.MD.05 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes. 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.
b. Represent threefold whole-number products as volumes to represent the associative property of multiplication.
c. 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.
d. 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 problem.

# Number and Operation in Base Ten

## Narrative for the (NBT) Number and Operation in Base Ten

In Grade 5, students extend their understanding of the base-ten system to the relationship between adjacent places, how numbers compare, and how numbers round for decimals to thousandths. New at Grade 5 is the use of whole number exponents to denote powers of 10. Students understand why multiplying by a power of 10 shifts the digits of a whole number or decimal that many places to the left. For example, multiplying by 104 is multiplying by 10 four times.

Students students fluently compute products of whole numbers using the standard algorithm for multiplication. In division, they reason about dividing whole numbers with two-digit divisors, and reason about adding, subtracting, multiplying, and dividing decimals to hundredths. Division strategies in Grade 5 involve breaking the dividend apart into like base-ten units and applying the distributive property to find the quotient place by place, starting from the highest place.

Place value understandings and general methods for computing are used to add, subtract, multiply, and divide decimals to hundredths. Students use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction to solve decimal problems. They also relate the strategy to a written method and explain the reasoning used.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-05.NBT.03

(NBT) Number and Operations in Base Ten
Cluster: Understand the place value system.

##### MAT-05.NBT.03 Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, word form, and expanded form.
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

#### MAT-05.NBT.05

(NBT) Number and Operations in Base Ten
Cluster: Perform operations with multi­digit whole numbers and with decimals to hundredths.

#### MAT-05.NBT.06

(NBT) Number and Operations in Base Ten
Cluster: Perform operations with multi­digit whole numbers and with decimals to hundredths.

#### MAT-05.NBT.07

(NBT) Number and Operations in Base Ten
Cluster: Perform operations with multi­digit whole numbers and with decimals to hundredths.

# Number and Operations - Fraction

## Narrative for (NF) Number and Operations - Fractions

In Grade 4, students have some experience calculating sums of fractions with different denominators.  They understand the process as expressing both quantities in terms of the same unit fraction so that they can be added. Grade 5 students extend this reasoning to situations where it is necessary to re-express both fractions in terms of a new denominator. Students make sense of fractional quantities when solving word problems, estimating answers mentally to see if they make sense.

Grade 4 students connected fractions with addition and multiplication. In Grade 5, students connect fractions with division, understanding a fraction as division of the numerator by the denominator.  Using the relationship between division and multiplication, students start working with simple fraction division problems. They 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.

Grade 5 work with multiplying by unit fractions, and interpreting fractions in terms of division, enables students to see that multiplying a quantity by a number smaller than 1 produces a smaller quantity. This work - interpreting multiplication as scaling (resizing), prepares students for Grade 6 work in ratios and proportional reasoning.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.

#### MAT-05.NF.01

(NF) Number and Operations - Fractions
Cluster: Use equivalent fractions as a strategy to add and subtract fractions.

#### MAT-05.NF.02

(NF) Number and Operations - Fractions
Cluster: Use equivalent fractions as a strategy to add and subtract fractions.

#### MAT-05.NF.06

(NF) Number and Operations - Fractions
Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

#### MAT-05.NF.07

(NF) Number and Operations - Fractions
Cluster: Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

##### MAT-05.NF.07 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.
b. Interpret division of a whole number by a unit fraction, and compute such quotients.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions using visual fraction models and equations to represent the problem.

# Operations and Algebraic Thinking

## Narrative for the (OA) Operations and Algebraic Thinking

Algebraic thinking is about generalizing arithmetic operations and operating on unknown quantities.  It involves recognizing and analyzing patterns and developing generalizations about these patterns.  In algebra, symbols can be used to represent generalizations. Operations and Algebraic Thinking deals with the basic operations - addition, subtraction, multiplication, and division - the relationships they model, the kinds of problems they can be used to solve, as well as, their mathematical properties and relationships.

As preparation for the Expressions and Equations domain in the middle grades, students in Grade 5 begin working more formally with expressions. They write expressions to express a calculation, e.g., writing 2 x (8+7) to express the calculation “add 8 and 7, then multiply by 2.” They also evaluate and interpret expressions, e.g., using their conceptual understanding of multiplication to interpret 3 x (18932+921) as being three times as large as 18932+921, without having to calculate the indicated sum or product. Students in Grade 5 begin to think about numerical expressions in ways that prepare their later work with variable expressions (e.g., three times an unknown length is 3 x L).

Students extend their Grade 4 pattern work by working briefly with two numerical patterns that can be related and examining these relationships within sequences of ordered pairs and in graphs. This work prepares students for studying proportional relationships and functions in middle school.

## Calculation Method for Domains

Domains are larger groups of related standards. The Domain Grade is a calculation of all the related standards. Click on the standard name below each Domain to access the learning targets and rubrics/ proficiency scales for individual standards within the domain.