2013 Math Framework, Grade 5 - Curriculum Frameworks (CA ...

Grade-Five Chapter

of the

Mathematics Framework

for California Public Schools: Kindergarten Through Grade Twelve

Adopted by the California State Board of Education, November 2013 Published by the California Department of Education Sacramento, 2015

8 Grade Five

7

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In the years prior to grade five, students learned strategies for multiplication and division, developed an understanding of the structure of the place-value

system, and applied understanding of fractions to addi-

tion and subtraction with like denominators and to mul-

5

tiplying a whole number times a fraction. They gained understanding that geometric figures can be analyzed

and classified based on the properties of the figures and

focused on different measurements, including angle mea-

4

sures. Students also learned to fluently add and subtract

whole numbers within 1,000,000 using the standard

algorithm (adapted from Charles A. Dana Center 2012).

3

Critical Areas of Instruction

In grade five, instructional time should focus on three

critical areas: (1) developing fluency with addition and

2

subtraction of fractions and developing understanding of

the multiplication of fractions and of division of fractions

in limited cases (unit fractions divided by whole numbers

and whole numbers divided by unit fractions); (2) extend-

1

ing division to two-digit divisors, integrating decimal

fractions into the place-value system, developing under-

standing of operations with decimals to hundredths,

K

and developing fluency with whole-number and decimal operations; and (3) developing understanding of volume

(National Governors Association Center for Best

Practices, Council of Chief State School Officers

[NGA/CCSSO] 2010l). Students also fluently multiply

multi-digit whole numbers using the standard algorithm.

Grade Five 233

Standards for Mathematical Content

The Standards for Mathematical Content emphasize key content, skills, and practices at each grade level and support three major principles: ? Focus--Instruction is focused on grade-level standards. ? Coherence--Instruction should be attentive to learning across grades and to linking major

topics within grades. ? Rigor--Instruction should develop conceptual understanding, procedural skill and fluency,

and application.

Grade-level examples of focus, coherence, and rigor are indicated throughout the chapter.

The standards do not give equal emphasis to all content for a particular grade level. Cluster headings can be viewed as the most effective way to communicate the focus and coherence of the standards. Some clusters of standards require a greater instructional emphasis than others based on the depth of the ideas, the time needed to master those clusters, and their importance to future mathematics or the later demands of preparing for college and careers.

Table 5-1 highlights the content emphases at the cluster level for the grade-five standards. The bulk of instructional time should be given to "Major" clusters and the standards within them, which are indicated throughout the text by a triangle symbol ( ). However, standards in the "Additional/Supporting" clusters should not be neglected; to do so would result in gaps in students' learning, including skills and understandings they may need in later grades. Instruction should reinforce topics in major clusters by using topics in the additional/ supporting clusters and including problems and activities that support natural connections between clusters.

Teachers and administrators alike should note that the standards are not topics to be checked off after being covered in isolated units of instruction; rather, they provide content to be developed throughout the school year through rich instructional experiences presented in a coherent manner (adapted from Partnership for Assessment of Readiness for College and Careers [PARCC] 2012).

Table 5-1. Grade Five Cluster-Level Emphases

Operations and Algebraic Thinking

5.OA

Additional/Supporting Clusters

? Write and interpret numerical expressions. (5.OA.1?2) ? Analyze patterns and relationships. (5.OA.3)

Number and Operations in Base Ten

5.NBT

Major Clusters

? Understand the place-value system. (5.NBT.1?4 ) ? Perform operations with multi-digit whole numbers and with decimals to hundredths. (5.NBT.5?7 )

Number and Operations--Fractions

5.NF

Major Clusters

? Use equivalent fractions as a strategy to add and subtract fractions. (5.NF.1?2 ) ? Apply and extend previous understandings of multiplication and division to multiply and divide

fractions. (5.NF.3?7 )

Measurement and Data

5.MD

Major Clusters

? Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. (5.MD.3?5 )

Additional/Supporting Clusters

? Convert like measurement units within a given measurement system. (5.MD.1) ? Represent and interpret data. (5.MD.2)

Geometry

5.G

Additional/Supporting Clusters

? Graph points on the coordinate plane to solve real-world and mathematical problems. (5.G.1?2) ? Classify two-dimensional figures into categories based on their properties. (5.G.3?4)

Explanations of Major and Additional/Supporting Cluster-Level Emphases

Major Clusters ( ) -- Areas of intensive focus where students need fluent understanding and application of the core concepts. These clusters require greater emphasis than others based on the depth of the ideas, the time needed to master them, and their importance to future mathematics or the demands of college and career readiness.

Additional Clusters -- Expose students to other subjects; may not connect tightly or explicitly to the major work of the grade.

Supporting Clusters -- Designed to support and strengthen areas of major emphasis.

Note of caution: Neglecting material, whether it is found in the major or additional/supporting clusters, will leave gaps in students' skills and understanding and will leave students unprepared for the challenges they face in later grades.

Adapted from Smarter Balanced Assessment Consortium 2011, 85.

Connecting Mathematical Practices and Content

The Standards for Mathematical Practice (MP) are developed throughout each grade and, together with the content standards, prescribe that students experience mathematics as a rigorous, coherent, useful, and logical subject. The MP standards represent a picture of what it looks like for students to understand and do mathematics in the classroom and should be integrated into every mathematics lesson for all students.

Although the description of the MP standards remains the same at all grades, the way these standards look as students engage with and master new and more advanced mathematical ideas does change. Table 5-2 presents examples of how the MP standards may be integrated into tasks appropriate for students in grade five. (Refer to the Overview of the Standards Chapters for a description of the MP standards.)

Table 5-2. Standards for Mathematical Practice--Explanation and Examples for Grade Five

Standards for Mathematical Practice

Explanation and Examples

MP.1

Make sense of problems and persevere in solving them.

In grade five, students solve problems by applying their understanding of operations with whole numbers, decimals, and fractions that include mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. For example, "Sonia had sticks of gum. She promised her brother that she would give him of a stick of gum. How much will she have left after she gives her brother the amount she promised?" Teachers can encourage students to check their thinking by having students ask themselves questions such as these: "What is the most efficient way to solve the problem?""Does this make sense?""Can I solve the problem in a different way?"

MP.2

Reason abstractly and quantitatively.

Students recognize that a number represents a specific quantity. They connect quantities

to written symbols and create logical representations of problems, considering appropriate

units and the meaning of quantities. They extend this understanding from whole numbers

to their work with fractions and decimals. Teachers can support student reasoning by asking

questions such as these: "What do the numbers in the problem represent?""What is the

relationship of the quantities?" Students write simple expressions that record calculations

with numbers and represent or round numbers using place-value concepts. For example,

students use abstract and quantitative thinking to recognize, without calculating the

quotient, that

is of

.

MP.3

Construct viable arguments and critique the reasoning of others.

In grade five, students may construct arguments by using visual models such as objects and drawings. They explain calculations based upon models, properties of operations, and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions such as "How did you get that?" and "Why is that true?" They explain their thinking to others and respond to others' thinking.

Students use various strategies to solve problems, and they defend and justify their work to others. For example: "Two after-school clubs are having pizza parties. The teacher will order 3 pizzas for every 5 students in the math club and 5 equally sized pizzas for every 8 students on the student council. How much pizza will each student get at the respective parties? If a student wants to attend the party where she will get the most pizza (assuming the pizza is divided equally among the students at the parties), which party should she attend?"

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