## Grade 5 – CA Common Core Standards and Learning Objectives

### 5.5.OA Operations and Algebraic Thinking

#### 5.5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Perform multiple operations with whole numbers (Fifth grade – O.4)

#### 5.5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

Write numerical expressions (Fifth grade – O.3)

#### 5.5.OA.2.1 Express a whole number in the range 2-50 as a product of its prime factors.

Prime factorization (Fifth grade – F.2)

#### 5.5.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.

Complete a table for a two-variable relationship (Fifth grade – U.8)

Complete a table from a graph (Fifth grade – U.9)

Graph a two-variable relationship (Fifth grade – U.10)

### 5.5.NBT Number and Operations in Base Ten

#### 5.5.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.

Place values (Fifth grade – A.1)

Convert between place values (Fifth grade – A.2)

Place values in decimal numbers (Fifth grade – G.4)

#### 5.5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain 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.

Scientific notation (Fifth grade – A.11)

Multiplication patterns over increasing place values (Fifth grade – C.3)

Multiply numbers ending in zeroes (Fifth grade – C.4)

Multiply numbers ending in zeroes: word problems (Fifth grade – C.5)

Division patterns over increasing place values (Fifth grade – D.7)

Multiply a decimal by a power of ten (Fifth grade – I.2)

Divide by powers of ten (Fifth grade – J.1)

Decimal division patterns over increasing place values (Fifth grade – J.2)

#### 5.5.NBT.3.a Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

What decimal number is illustrated? (Fifth grade – G.1)

Model decimals and fractions (Fifth grade – G.2)

Understanding decimals expressed in words (Fifth grade – G.3)

Place values in decimal numbers (Fifth grade – G.4)

Convert decimals between standard and expanded form (Fifth grade – G.5)

Convert decimals between standard and expanded form using fractions (Fifth grade – G.14)

#### 5.5.NBT.3.b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Equivalent decimals (Fifth grade – G.6)

Decimal number lines (Fifth grade – G.8)

Compare decimals on number lines (Fifth grade – G.9)

Compare decimal numbers (Fifth grade – G.10)

Put decimal numbers in order (Fifth grade – G.11)

Compare decimals and fractions on number lines (Fifth grade – G.15)

Inequalities with decimal multiplication (Fifth grade – I.10)

#### 5.5.NBT.4 Use place value understanding to round decimals to any place.

Round decimals (Fifth grade – G.7)

Estimate sums and differences of decimals (Fifth grade – H.8)

#### 5.5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

Multiply by 2-digit numbers: complete the missing steps (Fifth grade – C.12)

Multiply 2-digit numbers by 2-digit numbers (Fifth grade – C.13)

Multiply 2-digit numbers by 3-digit numbers (Fifth grade – C.14)

Multiply 2-digit numbers by larger numbers (Fifth grade – C.15)

Multiply by 2-digit numbers: word problems (Fifth grade – C.16)

Multiply three or more numbers up to 2 digits each (Fifth grade – C.17)

Multiply by 3-digit numbers (Fifth grade – C.18)

Multiply three numbers up to 3 digits each (Fifth grade – C.19)

Multiply three or more numbers: word problems (Fifth grade – C.20)

#### 5.5.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, and/or area models.

Properties of multiplication (Fifth grade – C.6)

Division facts to 12 (Fifth grade – D.1)

Division facts to 12: word problems (Fifth grade – D.2)

Divide multi-digit numbers by 1-digit numbers (Fifth grade – D.3)

Divide multi-digit numbers by 1-digit numbers: word problems (Fifth grade – D.4)

Divide numbers ending in zeroes (Fifth grade – D.8)

Divide numbers ending in zeroes: word problems (Fifth grade – D.9)

Divide 2-digit and 3-digit numbers by 2-digit numbers (Fifth grade – D.10)

Divide 2-digit and 3-digit numbers by 2-digit numbers: word problems (Fifth grade – D.11)

Divide larger numbers by 2-digit numbers (Fifth grade – D.12)

Divide larger numbers by 2-digit numbers: word problems (Fifth grade – D.13)

Choose numbers with a particular quotient (Fifth grade – D.16)

#### 5.5.NBT.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 addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Divide money amounts: word problems (Fifth grade – D.14)

Subtract decimal numbers (Fifth grade – H.2)

Choose decimals with a particular sum or difference (Fifth grade – H.5)

Multiply a decimal by a one-digit whole number (Fifth grade – I.3)

Multiply a decimal by a multi-digit whole number (Fifth grade – I.4)

Multiply decimals and whole numbers: word problems (Fifth grade – I.5)

Multiply money amounts: word problems (Fifth grade – I.6)

Multiply three or more numbers, one of which is a decimal (Fifth grade – I.7)

Multiply two decimals using grids (Fifth grade – I.8)

Multiply two decimals (Fifth grade – I.9)

Division with decimal quotients (Fifth grade – J.3)

Division with decimal quotients and rounding (Fifth grade – J.4)

Division with decimal quotients: word problems (Fifth grade – J.5)

Add, subtract, multiply, and divide decimals: word problems (Fifth grade – O.6)

Price lists (Fifth grade – R.1)

Unit prices (Fifth grade – R.2)

### 5.5.NF Number and Operations-Fractions

#### 5.5.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.

Equivalent fractions (Fifth grade – K.3)

Reduce fractions to lowest terms (Fifth grade – K.4)

Convert between improper fractions and mixed numbers (Fifth grade – K.5)

Add up to 4 fractions with denominators of 10 and 100 (Fifth grade – L.7)

Subtract fractions with unlike denominators using models (Fifth grade – L.9)

Subtract fractions with unlike denominators (Fifth grade – L.10)

Add 3 or more fractions with unlike denominators (Fifth grade – L.12)

Subtract mixed numbers with unlike denominators (Fifth grade – L.19)

Complete addition and subtraction sentences with mixed numbers (Fifth grade – L.22)

Inequalities with addition and subtraction of mixed numbers (Fifth grade – L.23)

#### 5.5.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 assess the reasonableness of answers.

Add and subtract fractions with like denominators: word problems (Fifth grade – L.4)

Add and subtract fractions with unlike denominators: word problems (Fifth grade – L.11)

Compare sums and differences of unit fractions (Fifth grade – L.14)

#### 5.5.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.

Fractions review (Fifth grade – K.1)

Divide fractions by whole numbers (Fifth grade – N.4)

#### 5.5.NF.4.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.

Multiply fractions by whole numbers I (Fifth grade – M.5)

Multiply fractions by whole numbers II (Fifth grade – M.8)

Multiply fractions by whole numbers: input/output tables (Fifth grade – M.11)

#### 5.5.NF.4.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.

Multiply two unit fractions using models (Fifth grade – M.12)

Multiply two fractions using models: fill in the missing factor (Fifth grade – M.13)

Multiply two fractions using models (Fifth grade – M.14)

Area of squares and rectangles (Fifth grade – Z.16)

Area and perimeter: word problems (Fifth grade – Z.22)

#### 5.5.NF.5.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.

Scaling whole numbers by fractions (Fifth grade – M.17)

Scaling fractions by fractions (Fifth grade – M.18)

Scaling mixed numbers by fractions (Fifth grade – M.19)

#### 5.5.NF.5.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.

Multiply two fractions using models (Fifth grade – M.14)

#### 5.5.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.

Multiply fractions by whole numbers: word problems (Fifth grade – M.10)

Multiply two fractions (Fifth grade – M.15)

Multiply two fractions: word problems (Fifth grade – M.16)

Multiply a mixed number by a whole number (Fifth grade – M.23)

Multiply a mixed number by a fraction (Fifth grade – M.24)

Multiply two mixed numbers (Fifth grade – M.25)

Multiplication with mixed numbers: word problems (Fifth grade – M.27)

Multiply fractions and mixed numbers in recipes (Fifth grade – M.28)

Add, subtract, multiply, and divide fractions and mixed numbers (Fifth grade – O.7)

Add, subtract, multiply, and divide fractions and mixed numbers: word problems (Fifth grade – O.8)

#### 5.5.NF.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

Divide unit fractions by whole numbers (Fifth grade – N.1)

Divide fractions by whole numbers (Fifth grade – N.4)

#### 5.5.NF.7.b Interpret division of a whole number by a unit fraction, and compute such quotients.

Divide whole numbers by unit fractions (Fifth grade – N.2)

Divide whole numbers by unit fractions using models (Fifth grade – N.7)

Divide whole numbers by fractions (Fifth grade – N.8)

#### 5.5.NF.7.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.

Divide whole numbers and unit fractions (Fifth grade – N.3)

Divide unit fractions by whole numbers: word problems (Fifth grade – N.6)

### 5.5.MD Measurement and Data

#### 5.5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Compare and convert customary units of length (Fifth grade – Y.2)

Compare and convert customary units of weight (Fifth grade – Y.3)

Compare and convert customary units of volume (Fifth grade – Y.4)

Compare and convert customary units (Fifth grade – Y.5)

Conversion tables – customary units (Fifth grade – Y.6)

Compare and convert metric units of length (Fifth grade – Y.8)

Compare and convert metric units of weight (Fifth grade – Y.9)

Compare and convert metric units of volume (Fifth grade – Y.10)

Compare and convert metric units (Fifth grade – Y.11)

Conversion tables – metric units (Fifth grade – Y.12)

Compare customary units by multiplying (Fifth grade – Y.13)

Convert customary units involving fractions (Fifth grade – Y.14)

Convert mixed customary units (Fifth grade – Y.15)

#### 5.5.MD.2 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 for this grade to solve problems involving information presented in line plots.

Interpret line plots (Fifth grade – V.10)

Create line plots (Fifth grade – V.11)

Create and interpret line plots with fractions (Fifth grade – V.12)

#### 5.5.MD.3.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.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

#### 5.5.MD.3.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.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

#### 5.5.MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

#### 5.5.MD.5.a 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. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

Volume of rectangular prisms made of unit cubes (Fifth grade – Z.23)

Volume of cubes and rectangular prisms (Fifth grade – Z.25)

#### 5.5.MD.5.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.

Volume of cubes and rectangular prisms (Fifth grade – Z.25)

#### 5.5.MD.5.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.

Volume of irregular figures made of unit cubes (Fifth grade – Z.24)

### 5.5.G Geometry

#### 5.5.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).

Coordinate graphs review – whole numbers only (Fifth grade – T.1)

Coordinate graphs with decimals and negative numbers (Fifth grade – T.2)

Graph points on a coordinate plane (Fifth grade – T.3)

#### 5.5.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.

Graph points on a coordinate plane (Fifth grade – T.3)

Coordinate graphs as maps (Fifth grade – T.4)

#### 5.5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

Identify 2-dimensional and 3-dimensional shapes (Fifth grade – Z.1)

Types of triangles (Fifth grade – Z.2)

Open and closed shapes and qualities of polygons (Fifth grade – Z.3)

Regular and irregular polygons (Fifth grade – Z.4)

Number of sides in polygons (Fifth grade – Z.5)

Which figure is being described? (Fifth grade – Z.6)

## THIRD GRADE 3.NF Number and Operation: Fractions

### FOURTH GRADE 4.NF Number and Operations | Fractions

#### Understand decimal notation for fractions, and compare decimal fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator
100, and use this technique to add two fractions with respective denominators 10 and
100.4 For example, express 3=10 as 30=100, and add 3=10 + 4=100 = 34=100.
6. Use decimal notation for fractions with denominators 10 or 100. For example,
rewrite 0:62 as 62=100; describe a length as 0:62 meters; locate 0:62 on a number line
diagram.
7. Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the
results of comparisons with the symbols >;=, or <, and justify the conclusions, e.g., by using a visual model
4Students who can generate equivalent fractions can develop strategies for adding fractions with
unlike denominators in general. But addition and subtraction with un-like denominators in general
is not a requirement at this grade.

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

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 equiv-
alent sum or di erence 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.)
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 assess the reasonableness of answers. For example,
recognize an incorrect result 2=5 + 1=2 = 3=7, by observing that 3=7 < 1=2.

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

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 re-
lationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used.
Apply and extend previous understandings of multiplication and division to
multiply and divide fractions.
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
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
rectangles6 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 nd areas of rectangles, and represent fraction products as rectangular areas.
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 num-
bers 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 e ect of multiplying a=b by 1.
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.
7. Apply and extend previous understandings of division to divide unit fractions by
whole numbers and whole numbers by unit fractions.7
a. Interpret division of a unit fraction by a non-zero whole number, and compute
6In the original, it is incorrectly stated as \squares”.
7Students able to multiply fractions in general can develop strategies to divide fractions in general,
by reasoning about the relationship between multiplication and division. But division of a fraction
by a fraction is not a requirement at this grade.

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?

##### Understand ratio concepts and use ratio reasoning to solve problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio rela-
tionship between two quantities. For example, \The ratio of wings to beaks in the bird
house at the zoo was 2 : 1, because for every 2 wings there was 1 beak. \For every vote
2. Understand the concept of a unit rate a=b associated with a ratio a : b with b 6= 0,
and use rate language in the context of a ratio relationship. For example, \This recipe
has a ratio of 3 cups of
our to 4 cups of sugar, so there is 3=4 cup of
our for each cup
of sugar. \We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.”11
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams,
or equations.
b. Solve unit rate problems including those involving unit pricing and constant
speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns
could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means
30/100 times the quantity); solve problems involving nding the whole, given a part and
the percent.

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

11Expectations for unit rates in this grade are limited to non-complex fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving
division of fractions by fractions, e.g., by using visual fraction models and equations to
represent the problem. For example, create a story context for (2=3)  (3=4) and use
a visual fraction model to show the quotient; use the relationship between multiplica-
tion and division to explain that (2=3)  (3=4) = 8=9 because 3=4 of 8=9 is 2=3. (In
general, (a=b)  (c=d) = ad=bc:) How much chocolate will each person get if 3 people
share 1=2 lb of chocolate equally? How many 3=4-cup servings are in 2=3 of a cup of
yogurt? How wide is a rectangular strip of land with length 3=4 mi and area 1=2 square mi?
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.

##### 7.RP Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or di fferent units.
For example, if a person walks 1/2 mile in each 1/4-hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.

## Progression – Fractions – Grade 4

 Progression: 3-5 Number and Operations – Fractions Grade 4 1   Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction. 2.  This property forms the basis for much of their other work in Grade 4, including the comparison, addition, and subtraction of fractions and the introduction of finite decimals. 3   Equivalent fractions 4   Students can use area models and number line diagrams to reason about equivalence. (4.NF.1) 5   4.NF.1   Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b)  by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 6   They see that the numerical process of multiplying the numerator and denominator of a fraction by the same number, n. corresponds physically to partitioning each unit fraction piece into n smaller equal pieces. 7   The whole is then partitioned into ￿ times as many pieces, and there are ￿ times as many smaller unit fraction pieces as in the original fraction. 8   This argument, once understood for a range of examples, can be seen as a general argument, working directly from the Grade 3 understanding of a fraction as a point on the number line. 9   Using an area model to show that 2/3 = (4 x 2)/(4 x 3). FIGURE The whole is the square, measured by area. On the left it is divided horizontally into 3 rectangles of equal area, and the shaded region is 2 of these and so represents 2/3. On the right it is divided into 4 x 3 small rectangles of equal area, and the shaded area comprises 4 x 2 of these, and so it represents (4 x 2)/(4 x 3) . 10   Using the number line to show that 4/3 = (5 x 4)/(5 x 3)   PICTURE 4/3  is 4 parts when each part is 1/3 , and we want to see that this is also 5 x 4 parts when each part is 1/(5 x 3). Divide each of the intervals of length 1/3 into 5 parts of equal length. There are 5 x 3 parts of equal length in the unit interval, and 4/3 is 5 x 4 of these. Therefore 4/3 = (5 x 4)/(5 x 3) = 20/15 . 11   The fundamental property can be presented in terms of division, as in, e.g.     28/36 = (28   7/9 12   Because the equations 28 4 = 7 and 36 4 = 9 tell us that 28 = 4 x 7 and 36 = 4 x 9, this is the fundamental fact in disguise: (4 x 7)/(4 x 9) = 7/9 13   It is possible to over-emphasize the importance of simplifying fractions in this way. 14   There is no mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases. 15   Grade 4 students use their understanding of equivalent fractions to compare fractions with different numerators and different denominators.  (4.NF.2) 16   4.NF.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , < and justify the conclusions, e.g., by using a visual fraction model. 17  For example, to compare 5/8 and 7/12 they rewrite both fractions [with same denominator] as 60/96 (= (12 x 5)/(12 x 8) and 56/96 ( = (7 x 8)/(12 x 8) 18   Because 60/96 and 56/96 have the same denominator, students can compare them using Grade 3 methods and see that 56/96 is smaller, so        7/12 <  5/8 19   Students also reason using benchmarks such as ½ and 1. 20   For example, they see that 7/8 < 13/12 because 7/8 is less than 1 (and is therefore to the left of 1) but 13/12 is greater than 1 (and is therefore to the right of 1). 21   Grade 5 students who have learned about fraction multiplication can see equivalence as “multiplying by 1″: 7/9 = 7/9 x 1 = 7/9 x 4/4 = 28/36 22   However, although a useful mnemonic device, this does not constitute a valid argument at this grade, since students have not yet learned fraction multiplication. 23   Adding and subtracting fractions 24   The meaning of addition is the same for both fractions and whole numbers, even though algorithms for calculating their sums can be different. 25   Just as the sum of 4 and 7 can be seen as the length of the segment obtained by joining together two segments of lengths 4 and 7, so the sum of 2/3 and 8/5 can be seen as the length of the segment obtained joining together two segments of length 2/3 and 8/5. 26   It is not necessary to know how Representation of 23 85 as a length 2 3 8 5 Using the number line to see that 53 13 13 13 13 13 Segment of length 13 0 1 2 3 4 5 segments put end to end 53 13 13 13 13 13 much 23 85 is exactly in order to know what the sum means. This is analogous to understanding 51 78 as 51 groups of 78, without necessarily knowing its exact value.