Los Angeles Unified School District
District Mathematics Program
Instructional Guide
Harcourt
Fifth Grade
2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide
Rationale for Fifth Grade Instructional Guide
Organizational Highlights
In Fifth Grade students will apply the four basic operations to fractions and decimals and also learn how to add and subtract positive and
negative numbers. The Number Sense standards 1.2 and 1.5 are therefore critical for students to understand and master. (See page 138
Mathematics Framework for California Public Schools) Fifth graders will also use measuring units to determine length and area and know
and use formulas to determine the volume of simple geometric figures. Students will solve problems involving linear functions and be able
to write the equation and graph the resulting ordered pairs of integers on a grid.
The mathematics program in the fifth grade includes the use of manipulatives. All students should have access to concrete objects and/or
manipulatives as it enables student to create models which facilitate their understanding of the mathematical concepts being taught. The
Lesson Resources in Harcourt Math are identified in the Instructional Guide for each of the content standards. These resources include the
use of manipulatives to introduce lessons, math background for teachers, the Problem of the Day, home-school connection activities, and
strategies for English learners.
Quarter One
Number Sense is the focus of the first quarter of Fifth Grade in order to expand and enhance the understanding of whole number,
fraction and decimal concepts from fourth grade. This includes estimating and rounding very large and very small numbers, all
operations with decimals, and prime factors and exponents. The number of assessed standards in this quarter is purposely low to
allow sufficient time to develop these concepts.
Quarter Two
Building on quarter one, fractions, decimals and percents are the emphasized concepts for this quarter. Students will be able to apply
these concepts to representations on a number line and to use them to solve problems using all four operations. The Statistics, Data
and Probability standards are included in this quarter to facilitate the application of these fraction and percentage concepts. Graphing
of ordered pairs is introduced in this quarter in order to allow sufficient time for practice and review throughout the rest of the year.
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide
Quarter Three
Students will expand graphing from last quarter to include linear functions in this quarter. Operations with rational numbers are also
an area of emphasis. The balance of this quarter is devoted to the measurement and geometry standards with an emphasis on volume,
angles, and the use of appropriate tools such as the compass and protractor.
Quarter Four
Quarter Four is an opportunity for teachers to review key concepts that their students found difficult. The emphasis here will be
dictated by the needs of the students. In addition, students will preview the skills they will encounter during the first part of 6th grade,
with a particular emphasis on ratio, rates and probability.
LOS ANGELES UNIFIED SCHOOL DISTRICT
Grade Five
Mathematics Instructional Guide
Mathematical Reasoning
1.0
2.0
Students make decisions about how to approach problems:
1.1
Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and
prioritizing information, and observing patterns.
1.2
Determine when and how to break a problem into simpler parts.
Students use strategies, skills, and concepts in finding solutions:
2.1
Use estimation to verify the reasonableness of calculated results.
2.2
Apply strategies and results from simpler problems to more complex problems.
2.3
Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models to explain
mathematical reasoning.
2.4
Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language;
support solutions with evidence in both verbal and symbolic work.
2.5
Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of
accuracy.
2.6
Make precise calculations and check the validity of the results from the context of the problem.
LOS ANGELES UNIFIED SCHOOL DISTRICT
Grade Five
Mathematics Instructional Guide
3.0
Students move beyond a particular problem by generalizing to other situations:
3.1
Evaluate the reasonableness of the solution in the context of the original situation.
3.2
Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving
similar problems.
3.3
Develop generalizations of the results obtained and apply them in other circumstances.
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instructional Guide: Quarter One
NUMBER SENSE: WHOLE NUMBERS
DECIMALS
FRACTIONS
Standards:
* NS 1.1
* NS 1.3
Estimate, round, and manipulate very large (e.g. millions) and very small numbers (e.g. thousandths)
Understand and compute positive integer powers of non-negative integers; compute examples as repeated multiplication. (e.g. 24
=16)
* NS 1.4
Determine the prime factors of all numbers through 50 and write the numbers as the product of their prime factors b y using
exponents to show multiples of a factor(e.g. 24=2x2x2x3=23 x 3).
* NS 1.5
Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.
* NS 2.1
Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers;
and verify the reasonableness of the results.
* NS 2.2
Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors.
AF 1.3
Know and use the distributive property in equations and expressions with variables.
MR 1.0-3.3 See section prior to Quarter One for the mathematical reasoning standards. These standards are used to engage students in
making sense of the mathematics they are learning.
Dates**
Standard #
* NS 1.1
* NS 1.5
MR 2.3
Concepts
Place Value through
millions; Comparing and
ordering whole numbers
Harcourt
Lesson Resources 2A-10B
Lessons 1.1-1.5
Place value: tenths,
hundredths, thousandths;
Comparing and ordering
decimals
Lesson Resources 18A-26B
Lessons 2.1-2.5
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
Alternative Resources
Place Value Lessons from About Teaching Mathematics,
pp 173–182
The Place Value Game from 50 Problem-Solving Lessons
(by M. Burns), pp 65-67
Note: NS 1.5 not fully addressed
in textbook
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter One, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instructional Guide: Quarter One
NUMBER SENSE: WHOLE NUMBERS
Dates**
DECIMALS
FRACTIONS
Standard #
* NS 1.1
* NS 2.1
MR 2.5
Concepts
Rounding whole numbers
and decimals; Adding and
Subtracting greater whole
numbers and decimals
Harcourt
Lesson Resources 32A-38B
Lessons 3.1-3.4
Lesson Resources 46A-56B
Lessons 4.1- 4.5
Pages 64-65
Alternative Resources
Marilyn Burns, About Teaching Mathematics
“Palindromes” p. 127
“Cross Out Singles”, p 191
“101 and Out”, p. 192
* NS 1.1
AF 1.3
Estimating products;
Multiplication of Greater
Numbers;
Distributive Property
Lesson Resources 144A – 152B
Lessons 9.1 - 9.5
Multiplication Lessons from About Teaching Mathematics,
pp 194–203
* NS 2.1
MR 2.4
MR 3.1
Decimal Factors and
Products
Lesson Resources 160A-168B
Lessons 10.2-10.6
Decimals on Grids from About Teaching Mathematics, M.
Burns pp 240-241
* NS 1.1
* NS 2.2
MR 3.1
Estimate quotients;
3-digit Dividends
2-digit Divisors
Lesson Resources 182A-194B
Lessons 11.1-11.4; 11.6
Division Lessons from About Teaching Mathematics, M.
Burns pp 204–211
Lesson Resources 202A-210B
Lessons 12.2-12.5
Relating Multiplication and Division from About Teaching
Mathematics, pp 215–222
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
Extending Multiplication Concepts from About Teaching
Mathematics, pp 212–215
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter One, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instructional Guide: Quarter One
NUMBER SENSE: WHOLE NUMBERS
Dates**
Standard #
* NS 2.2
Concepts
Decimal Division
DECIMALS
Harcourt
Lesson Resources 218A-222B;
238A-238B
Lessons 13.1-13.3; 14.3
FRACTIONS
Alternative Resources
Decimals on Grids from About Teaching Mathematics, M.
Burns pp 240-241
Mental Calculation with Decimals from About Teaching
Mathematics, pp 241-244
* NS 2.2
Divisibility; Least Common
Multiple
Lesson Resources 254A- 256B
Lesson 15.1-15.2
* NS 1.3
* NS 1.4
MR 2.3
MR 2.4
Factors, Exponents,
Prime and Composite
Numbers
Lesson Resources 264A-272B
Lessons 15.5-15.8
A S S E S S M E N T
E N R I C H M E N T / R E T E A C H I N G
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter One, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instructional Guide: Quarter Two
FRACTIONS
DECIMALS
PERCENTS
GRAPHING
Standards:
* NS 1.2
Interpret percents as a part of a hundred; find decimal and percent equivalents for common fractions and explain why they
represent the same value; compute a given percent of a whole number.
* NS 1.5
Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers.
* NS 2.3
Solve simple problems including ones arising in concrete situations, involving the addition and subtraction of fractions and
mixed numbers (like and unlike denominators of 20 or less) and express answers in the simplest form.
* NS 2.4
Understand the concept of multiplication and division of fractions.
* NS 2.5
Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems.
* SDAP 1.1 Know the concepts of mean, median, and mode; compute and compare simple examples to show that they may differ.
* SDAP 1.2 Organize and display single-variable data in appropriate graphs and representations (e.g., histogram, circle graphs) and explain
which types of graphs are appropriate for various data sets.
SDAP 1.3 Use fractions and percentages to compare data sets of different sizes.
* SDAP 1.4 Identify ordered pairs of data from a graph and interpret the meaning of the data in terms of the situation depicted by the graph.
* SDAP 1.5 Know how to write ordered pairs correctly; for example (x, y).
MR 1.0-3.3 See section prior to Quarter One for the mathematical reasoning standards. These standards are used to engage students in
making sense of the mathematics they are learning.
Dates**
Standard #
* SDAP 1.1
* SDAP 1.2
Concepts
Analyze graphs;
Mean, Median, Mode
* SDAP 1.2
SDAP 1.3
* SDAP 1.4
* SDAP 1.5
Organizing Data; Writing Lesson Resources 122A-130B
and Graphing Ordered Pairs Lessons 8.3-8.6
Graphing in the Classroom from About
Teaching Mathematics, pp 75-78
* NS 1.5
Decimals and Fractions
Marilyn Burns, About Teaching Mathematics,
pp. 223 -237 Fractions
pp. 240-241 Decimals
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
Harcourt
Lesson Resources 98A-105B; 108AB
Lessons 7.1 - 7.3 and 7.5
Lesson Resources 258A-262B
Lessons 15.3-15.4
Lesson Resources 280A-294B
Lessons 16.1-16.5
Alternative Resources
Marilyn Burns, A Collection of Math Lessons
3-6, pp. 103-113
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Two, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instructional Guide: Quarter Two
FRACTIONS
Dates**
Standard #
* NS 1.2
DECIMALS
PERCENTS
Concepts
Relate fractions, decimals,
and percents
Harcourt
Lesson Resources 228A-228B
Lesson 13.5 pp 340-341
Lesson Resources 316A-324B
Lessons 18.1-18.5
Comparing Data sets
Lesson Resources 332AB
Lesson 18.7
* NS 2.3
Add and Subtract Fractions
and Mixed Numbers; LCD
Lesson Resources 344A-374B
Lessons 19.1-19.6; 20.1-20.5
p. 343
* NS 2.4
* NS 2.5
Multiplying and Dividing
Fractions and Mixed
Numbers
Lesson Resources 380A-400B
Lessons 21.1-21.4; 22.1-22.4
SDAP 1.3
GRAPHING
Alternative Resources
Introducing Percents from About Teaching
Mathematics, pp 238-252
A S S E S S M E N T
E N R I C H M E N T / R E T E A C H I N G
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Two, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Three
OPERATIONS WITH RATIONAL NUMBERS
ALGEBRA CONCEPTS
GEOMETRIC CONCEPTS
Standards:
* NS 1.5
* NS 2.1
Identify and represent on a number line decimals, fractions, mixed numbers, and positive and negative integers
Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers;
and verify the reasonableness of the results.
* AF 1.1
Use information taken from a graph or equation to answer questions about a problem situation.
* AF 1.2
Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by substitution.
* AF 1.4
Identify and graph ordered pairs in the four quadrants of the coordinate plane.
* AF 1.5
Solve problems involving linear functions with integer values; write the equations; and graph the resulting ordered pairs of
integers on a grid.
* MG 1.1
Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a
rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle
of the same area by pasting and cutting a right triangle on the parallelogram).
* MG 1.2
Construct a cube and rectangular box from two-dimensional patterns to compute the surface area for these objects.
* MG 1.3
Understand the concept of volume and use the appropriate units in common measuring systems ( ie cubic centimeter [cm 3] ) to
compute the volume of rectangular solids.
MG 1.4
Differentiate between and use appropriate units of measure for, two- and three-dimensional objects (i.e. find the perimeter, area,
and volume).
* MG 2.1
Measure, identify and draw angles, perpendicular and parallel lines, rectangles, and triangles, by using appropriate tools (e.g.
straightedge, ruler, compass, protractor, drawing software).
* MG 2.2
Know that the sum of the angles of any triangle is 180o and the sum of the angles of any quadrilateral is 360o and use this
information to solve problems.
MG 2.3
Visualize and draw two-dimensional views of three-dimensional objects made from rectangular solids.
MR 1.0-3.3 See section prior to Quarter One for the mathematical reasoning standards. These standards are used to engage students in
making sense of the mathematics they are learning.
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Three, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Three
OPERATIONS WITH RATIONAL NUMBERS
Dates**
ALGEBRA CONCEPTS
GEOMETRIC CONCEPTS
Standard #
* AF 1.1
* AF 1.2
Concepts
Expressions and Equations
Harcourt
Lesson Resources 68A-78B; 86A-92B
Lessons 5.1-5.4; 6.1-6.4
pp.190-193
* NS 2.1
* NS 1.5
Integers
Lesson Resources 416A-428B
Lessons 23.1-23.5
* AF 1.4
* AF 1.5
Graphing on a Coordinate Plane
Lesson Resources 438A-448B
Lessons 24.1-24.4
* MG 2.1
* MG 2.2
MR 1.1
MR 2.3
Lines and Angles; Polygons;
Circles
Lesson Resources 454A-490B
Lessons 25.1-26.5
pp. 500-501
M. Burns, Lessons for Algebraic Thinking,
pp. 43-87, pp. 168-169
* MG 1.1
Measurement
Lesson Resources 504A-508B
Lessons 27.1-27.3
M. Burns, About Teaching Mathematics,
“The Tangram Puzzle”, p. 83 and p. 272
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
Alternative Resources
M. Burns, Lessons for Algebraic Thinking,
pp. 276-294; About Teaching
Mathematics, “Patterns, Functions and
Algebra” pp. 112-124
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Three, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Three
OPERATIONS WITH RATIONAL NUMBERS
Dates**
ALGEBRA CONCEPTS
Standard #
* MG 1.1
MG 1.2
MG 1.3
MG 1.4
Concepts
Perimeter, Area. Circumference
Harcourt
Lesson Resources 524A-540B
Lessons 28.1-28.7
* MG 1.1
* MG 1.2
* MG 1.3
MG 2.3
Surface Area and Volume
Lesson Resources 546A-558B
Lessons 29.1-29.5
pp 566-567
GEOMETRIC CONCEPTS
Alternative Resources
About Teaching Mathematics, M. Burns,
Area on the Geoboard: p. 97-99, p. 283-4
The Banquet Table Problem p. 281
A S S E S S M E N T
E N R I C H M E N T / R E T E A C H I N G
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Three, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Four
RATIONAL NUMBERS
FRACTIONS
PROBABILITY
Standards:
NS 2.0
Students perform calculations and solve problems involving addition, subtraction, and simple multiplication of fractions and
decimals:
* NS 2.1
Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers;
and verify the reasonableness of the results.
* NS 2.2
Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors.
* NS 2.3
Solve simple problems including ones arising in concrete situations, involving the addition and subtraction of fractions and
mixed numbers (like and unlike denominators of 20 or less) and express answers in the simplest form.
NS 2.4
Understand the concept of multiplication and division of fractions.
* NS 2.5
Compute and perform simple multiplication and division of fractions and apply these procedures to solving problems
* AF 1.5
Solve problems involving linear functions with integer values; write the equations; and graph the resulting ordered pairs of
integers on a grid.
* MG 2.1
Measure, identify and draw angles, perpendicular and parallel lines, rectangles, and triangles, by using appropriate tools (e.g.
straightedge, ruler, compass, protractor, drawing software).
* MG 2.2
Know that the sum of the angles of any triangle is 180o and the sum of the angles of any quadrilateral is 360o and use this
information to solve problems.
* SDAP 1.1 Know the concept of mean, median, and mode; compute and compare simple examples to show that they may differ.
* SDAP 1.2 Organize and display single-variable data in appropriate graphs and representations (e.g. histogram, circle graphs) and explain
which types of graphs are appropriate for various data sets.
* SDAP 1.3 Use fractions and percentages to compare data sets of different sizes.
MR 1.0-3.3 See section prior to Quarter One for the mathematical reasoning standards. Mathematical Reasoning Standards are used to
engage students in making sense of the mathematics they are learning.
Dates**
Standard #
* SDAP 1.3
Concepts
Probability
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
Harcourt
Alternative Resources
Lesson Resources: 570A-578B
Lessons 30.1-30.5
pp 569, 586-587
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Four, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Four
RATIONAL NUMBERS
Dates**
Standard #
Concepts
FRACTIONS
Harcourt
NS 2.0
* NS 2.1
All Operations with
Decimals, Negative and
Positive Integers
Lesson Resources: 414 H–428B
Lessons 23.6
* NS 2.2
Division
Lesson Resources: 182A-242B
Lesson 12.6
* NS 2.3
NS 2.4
* NS 2.5
MR 1.1
MR 2.3
All Operations with
Fractions and Mixed
Numbers
Lesson Resources: 342A-400B
Lessons 16.6, 17.1-17.5
pp. 336-337, 412-413
* AF 1.5
Graphing Ordered Pairs
Lesson Resources: 438A-444B
* MG 2.1
* MG 2.2
Identify and Draw Angles;
Sums of Angles
Lesson Resources: 454A-484B
* SDAP 1.1
Mean, Median, Mode
Lesson Resources: 102A-104B
Lesson 7.4
MR 2.6
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
PROBABILITY
Alternative Resources
M. Burns, About Teaching Mathematics,
pp.212-222
M. Burns, About Teaching Mathematics, pp.
231-234
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Four, 2004-2005
LOS ANGELES UNIFIED SCHOOL DISTRICT
Fifth Grade
Mathematics Instruction Guide: Quarter Four
RATIONAL NUMBERS
Dates**
Standard #
* SDAP 1.2
* SDAP 1.3
NS 2.1
MR 1.1
MR 1.2
Concepts
FRACTIONS
Harcourt
Making a Graph;
Comparing Data Sets
Lesson Resources: 330A-332B
Lessons 18.6-18.7
Mathematical Reasoning in
Measurement
Lesson Resources 512A-516B
Lessons 27.4-27.7
PROBABILITY
Alternative Resources
M. Burns, About Teaching Mathematics, pp.
59-75
A S S E S S M E N T
E N R I C H M E N T / R E T E A C H I N G
Legend: Key Standard
* Assessed Standard
Strikethrough: Topic not taught or assessed this quarter
** Dates to be determined at school site
LAUSD Mathematics Program
Instructional Guide, Grade 5
Quarter Four, 2004-2005