# Mathematics

## Kindergarten

### Standards and Tasks

Council of the Great City Schools provides guidance to parents about what their children will be learning and how they can support that learning.

- Counting and Cardinality
- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Measurement and Data
- Geometry
- Fluency

## Counting and Cardinality

## Know Number Names and the Count Sequence

## K.CC.A.1

Count to 100 by ones and by tens.

#### Tasks by Illustrative Mathematics

Assessing Counting Sequences Part 1

## K.CC.A.2

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

#### Tasks by Illustrative Mathematics

Assessing Counting Sequences Part I

Assessing Counting Sequences Part II

## K.CC.A.3

## Count to Tell the Number of Objects

## K.CC.B.4

Understand the relationship between numbers and quantities; connect counting to cardinality.

- When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.
- Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.
- Understand that each successive number name refers to a quantity that is one larger.

#### Tasks by Illustrative Mathematics

## K.CC.B.5

## Compare Numbers

## K.CC.C.6

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

#### Tasks by Illustrative Mathematics

Which number is greater? Which number is less? How do you know?

## K.CC.C.7

## Operations and Algebraic Thinking

## Understanding Addition and Subtraction

## K.OA.A.1

## K.OA.A.2

## K.OA.A.3

## K.OA.4

## K.OA.5

## Number and Operations in Base Ten

## Work with numbers 11-10 to gain foundations for place value

## K.NBT.A.1

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

#### Tasks by Illustrative Mathematics

## Measurement and Data

- Describe and compare measureable attributes
- Classify objects and count the number of objects in each category

## Describe and compare measureable attributes

## K.MD.A.1

## K.MD.A.2

Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

#### Tasks by Illustrative Mathematics

Which weighs more? Which weighs less?

## Classify objects and count the number of objects in each category

## K.MD.B.3

## Geometry

## Identify and describe shapes

## Analyze, compare, create, and compose shapes

## K.G.B.4

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).

#### Tasks by Illustrative Mathematics

## K.G.B.5

## K.G.B.6

## Fluency

Students will fluently add and subtract with 5, by the end of their Kindergarten year.

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

Add within 5 | ||
---|---|---|

0 + 0 | ||

1 + 0 | 1 + 1 | |

2 + 0 | 2 + 1 | 2 + 2 |

3 + 0 | 3 + 1 | 3 + 2 |

4 + 0 | 4 + 1 | |

5+0 |

Subtract within 5 | |||||
---|---|---|---|---|---|

0 - 0 | |||||

1 - 0 | 1 - 1 | ||||

2 - 0 | 2 - 1 | 2 - 2 | |||

3 - 0 | 3 - 2 | 3 - 3 | |||

4 - 0 | 4 - 1 | 4 - 2 | 4 - 3 | 4 - 4 | |

5 - 0 | 5 - 1 | 5 - 2 | 5 - 3 | 5 - 4 | 5 - 5 |

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

A new study led by Johns Hopkins University psychologists shows Bedtime Math’s Crazy 8s club significantly reduces children’s feelings of math anxiety after eight weeks of participation in the club.

### National Library

of Virtual Manipulatives

A library of uniquely interactive, web-based virtual manipulatives and concept tutorials.

This site requires a Java applet to run appropriately.

### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...

## First Grade

### Standards and Tasks

Council of the Great City Schools provides guidance to parents about what their children will be learning and how they can support that learning.

- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Measurement and Data
- Geometry
- Fluency

## Operations and Algebraic Thinking

- Represent & solve problems involving addition and subtraction
- Understand and apply properties of operations and the relationship between addition and subtraction
- Add and subtract within 20
- Work with addition and subtraction equations

## Represent & solve problems involving addition and subtraction

## 1.OA.A.1

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

#### Tasks by Illustrative Mathematics

## 1.OA.A.2

## Understand and apply properties of operations and the relationship between addition and subtraction

## 1.OA.B.3

Apply properties of operations as strategies to add and subtract; Examples: If 8+3=11 is known, then 3+8=11 is also known (Commutative property of addition); To add 2+6+4, the second two numbers can be added to make a ten, so 2+6+4=2+10=12 (Associative property of addition).

#### Tasks by Illustrative Mathematics

## 1.OA.B.4

## Add and subtract within 20

## 1.OA.C.5

## 1.OA.C.6

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10; Use strategies such as counting on; making ten (e.g., 8+6=8+2+4=10+4=14); decomposing a number leading to a ten (e.g., 13−4=13−3−1=10−1=9); using the relationship between addition and subtraction (e.g., knowing that 8+4=12, one knows 12−8=4); and creating equivalent but easier or known sums (e.g., adding 6+7 by creating the known equivalent 6+6+1=12+1=13).

#### Tasks by Illustrative Mathematics

## Work with addition and subtraction equations

## 1.OA.D.7

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6=6, 7=8−1, 5+2=2+5, 4+1=5+2.

#### Tasks by Illustrative Mathematics

Using lengths to represent equality

## 1.OA.D.8

## Number and Operations in Base Ten

## Extend the Counting Sequence

## 1.NBT.A.1

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

#### Tasks by Illustrative Mathematics

"Crossing the Decade" Concentration

## Understand Place Value

## 1.NBT.B.2

Understand that the two digits of a two-digit number represent amounts of tens and ones; Understand the following as special cases:

- 10 can be thought of as a bundle of ten ones – called a “ten.”
- The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
- The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

#### Tasks by Illustrative Mathematics

## 1.NBT.B.3

## Using Place Value

## 1.NBT.C.4

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, 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. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

#### Tasks by Illustrative Mathematics

## 1.NBT.C.5

## 1.NBT.C.6

Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), 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.

#### Tasks by Illustrative Mathematics

N/A

## Measurement and Data

- Measure lengths indirectlly and by iterating length units
- Tell and write time
- Represent and interpret data

## Measure lengths indirectlly and by iterating length units

## 1.MD.A.1

## 1.MD.A.2

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps; Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

#### Tasks by Illustrative Mathematics

## Tell and write time

## Represent and interpret data

## 1.MD.C.4

## Geometry

## Reason with shapes and their attributes

## 1.G.A.1

## 1.G.A.2

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

#### Tasks by Illustrative Mathematics

## 1.G.A.3

Partition circles and rectangles into two and four equal shares, describe the shares using the words *halves*, *fourths*, and *quarters*, and use the phrases *half of*, *fourth of*, and *quarter of*; Describe the whole as two of, or four of the shares; Understand for these examples that decomposing into more equal shares creates smaller shares.

#### Tasks by Illustrative Mathematics

## Fluency

Students will demonstrate fluency with adding and subtraction within 10, by the end of their 1st grade year.

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

Addition within 10 | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 + 0 | 1 + 1 | 1 + 2 | 1 + 3 | 1 + 4 | 1 + 5 | 1 + 6 | 1 + 7 | 1 + 8 | 1 + 9 |

2 + 0 | 2 + 1 | 2 + 2 | 2 + 3 | 2 + 4 | 2 + 5 | 2 + 6 | 2 + 7 | 2 + 8 | |

3 + 0 | 3 + 1 | 3 + 2 | 3 + 3 | 3 + 4 | 3 + 5 | 3 + 6 | 3 + 7 | ||

4 + 0 | 4 + 1 | 4 + 2 | 4 + 3 | 4 + 4 | 4 + 5 | 4 + 6 | |||

5 + 0 | 5 + 1 | 5 + 2 | 5 + 3 | 5 + 4 | 5 + 5 | ||||

6 + 0 | 6 + 1 | 6 + 2 | 6 + 3 | 6 + 4 | |||||

7 + 0 | 7 + 1 | 7 + 2 | 7 + 3 | ||||||

8 + 0 | 8 + 1 | 8 + 2 | |||||||

9 + 0 | 9 + 1 | ||||||||

10 + 0 |

Subtraction within 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 - 0 | ||||||||||

1 - 0 | 1 - 1 | |||||||||

2 - 0 | 2 - 1 | 2 - 2 | ||||||||

3 - 0 | 3 - 1 | 3 - 2 | 3 - 3 | |||||||

4 - 0 | 4 - 1 | 4 - 2 | 4 - 3 | 4 - 4 | ||||||

5 - 0 | 5 - 1 | 5 - 2 | 5 - 3 | 5 - 4 | 5 - 5 | |||||

6 - 0 | 6 - 1 | 6 - 2 | 6 - 3 | 6 - 4 | 6 - 5 | 6 - 6 | ||||

7 - 0 | 7 - 1 | 7 - 2 | 7 - 3 | 7 - 4 | 7 - 5 | 7 - 6 | 7 - 7 | |||

8 - 0 | 8 - 1 | 8 - 2 | 8 - 3 | 8 - 4 | 8 - 5 | 8 - 6 | 8 - 7 | 8 - 8 | ||

9 - 0 | 9 - 1 | 9 - 2 | 9 - 3 | 9 - 4 | 9 - 5 | 9 - 6 | 9 - 7 | 9 - 8 | 9 - 9 | |

10 - 0 | 10 -1 | 10 - 2 | 10 - 3 | 10 - 4 | 10 - 5 | 10 - 6 | 10 - 7 | 10 - 8 | 10 - 9 | 10 - 10 |

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

A new study led by Johns Hopkins University psychologists shows Bedtime Math’s Crazy 8s club significantly reduces children’s feelings of math anxiety after eight weeks of participation in the club.

### National Library

of Virtual Manipulatives

A library of uniquely interactive, web-based virtual manipulatives and concept tutorials.

This site requires a Java applet to run appropriately.

### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...

## Second Grade

### Standards and Tasks

Council of the Great City Schools provides guidance to parents about what their children will be learning and how they can support that learning.

- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Measurement and Data
- Geometry
- Fluency

## Operations and Algebraic Thinking

- Represent & solve problems involving addition and subtraction
- Work with equal groups of objects to gain foundations for multiplication

## Represent & solve problems involving addition and subtraction

## 2.OA.A.1

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

#### Tasks by Illustrative Mathematics

## 2.OA.B.2

## Work with equal groups of objects to gain foundations for multiplication

## 2.OA.C.3

## 2.OA.C.4

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

#### Tasks by Illustrative Mathematics

Partitioning a Rectangle into Unit Squares

## Number and Operations in Base Ten

## Understand Place Value

## 2.NBT.A.1

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

- 100 can be thought of as a bundle of ten tens – called a “hundred.”
- The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

#### Tasks by Illustrative Mathematics

Looking at Numbers Every Which Way

One, Ten, and One Hundred More and Less

Three Composing/Decomposing Problems

## 2.NBT.A.2

## 2.NBT.A.3

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

#### Tasks by Illustrative Mathematics

Looking at Numbers Every Which Way

## 2.NBT.A.4

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

#### Tasks by Illustrative Mathematics

Using Pictures to Explain Number Comparisons

## Using Place Value

## 2.NBT.B.5

## 2.NBT.B.6

## 2.NBT.B.7

Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

#### Tasks by Illustrative Mathematics

How Many Days Until Summer Vacation?

Peyton and Presley Discuss Addition

## 2.NBT.B.8

## 2.NBT.B.9

Explain why addition and subtraction strategies work, using place value and the properties of operations.

#### Tasks by Illustrative Mathematics

Peyton and Presley Discuss Addition

## Measurement and Data

- Measure and estimate lengths in standard units
- Relate addition and substraction to length
- Work with time and money
- Represent and interpret data

## Measure and estimate lengths in standard units

## 2.MD.A.1

## 2.MD.A.2

## 2.MD.A.3

## 2.MD.A.4

## Relate addition and substraction to length

## 2.MD.B.5

## 2.MD.B.6

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, …, and represent whole-number sums and differences within 100 on a number line diagram.

#### Tasks by Illustrative Mathematics

Frog and Toad on the Number Line

## Work with time and money

## 2.MD.C.7

## 2.MD.C.8

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

#### Tasks by Illustrative Mathematics

Alexander, Who Used to be Rich Last Sunday

## Represent and interpret data

## 2.MD.D.9

## 2.MD.D.10

## Geometry

## Reason with shapes and their attributes

## 2.G.A.1

## 2.G.A.2

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

#### Tasks by Illustrative Mathematics

Partitioning a Rectangle into Unit Squares

## 2.G.A.3

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words *halves*, *thirds*, *half of*, *a third of*, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

#### Tasks by Illustrative Mathematics

Representing Half of a Rectangle

Which Pictures Represent One Half?

## Fluency

Students can:

- Fluently add and subtract within 20 using mental strategies. By end of grade 2, know from memory all sums of two one-digit numbers.
- Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

Addition Facts within 20 | |||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 + 0 | 1 + 1 | 1 + 2 | 1 + 3 | 1 + 4 | 1 + 5 | 1 + 6 | 1 + 7 | 1 + 8 | 1 + 9 | 1 + 10 | 1 + 11 | 1 + 12 | 1 + 13 | 1 + 14 | 1 + 15 | 1 + 16 | 1 + 17 | 1 + 18 | 1 + 19 |

2 + 0 | 2 + 1 | 2 + 2 | 2 + 3 | 2 + 4 | 2 + 5 | 2 + 6 | 2 + 7 | 2 + 8 | 2 + 9 | 2 + 10 | 2 + 11 | 2 + 12 | 2 + 13 | 2 + 14 | 2 + 15 | 2 + 16 | 2 + 17 | 2 + 18 | |

3 + 0 | 3 + 1 | 3 + 2 | 3 + 3 | 3 + 4 | 3 + 5 | 3 + 6 | 3 + 7 | 3 + 8 | 3 + 9 | 3 + 10 | 3 + 11 | 3 + 12 | 3 + 13 | 3 + 14 | 3 + 15 | 3 + 16 | 3 + 17 | ||

4 + 0 | 4 + 1 | 4 + 2 | 4 + 3 | 4 + 4 | 4 + 5 | 4 + 6 | 4 + 7 | 4 + 8 | 4 + 9 | 4 + 10 | 4 + 11 | 4 + 12 | 4 + 13 | 4 + 14 | 4 + 15 | 4 + 16 | |||

5 + 0 | 5 + 1 | 5 + 2 | 5 + 3 | 5 + 4 | 5 + 5 | 5 + 6 | 5 + 7 | 5 + 8 | 5 + 9 | 5 + 10 | 5 + 11 | 5 + 12 | 5 + 13 | 5 + 14 | 5 + 15 | ||||

6 + 0 | 6 + 1 | 6 + 2 | 6 + 3 | 6 + 4 | 6 + 5 | 6 + 6 | 6 + 7 | 6 + 8 | 6 + 9 | 6 + 10 | 6 + 11 | 6 + 12 | 6 + 13 | 6 + 14 | |||||

7 + 0 | 7 + 1 | 7 + 2 | 7 + 3 | 7 + 4 | 7 + 5 | 7 + 6 | 7 + 7 | 7 + 8 | 7 + 9 | 7 + 10 | 7 + 11 | 7 + 12 | 7 + 13 | ||||||

8 + 0 | 8 + 1 | 8 + 2 | 8 + 3 | 8 + 4 | 8 + 5 | 8 + 6 | 8 + 7 | 8 + 8 | 8 + 9 | 8 + 10 | 8 + 11 | 8 + 12 | |||||||

9 + 0 | 9 + 1 | 9 + 2 | 9 + 3 | 9 + 4 | 9 + 5 | 9 + 6 | 9 +7 | 9 + 8 | 9 + 9 | 9 + 10 | 9 + 11 | ||||||||

10 + 0 | 10 + 1 | 10 + 2 | 10 + 3 | 10 + 4 | 10 + 5 | 10 + 6 | 10 + 7 | 10 + 8 | 10 + 9 | 10 + 10 | |||||||||

11 + 0 | 11 + 1 | 11 +2 | 11 + 3 | 11 + 4 | 11 + 5 | 11 + 6 | 11 + 7 | 11 +8 | 11 + 9 | ||||||||||

12 + 0 | 12 + 1 | 12 + 2 | 12 + 3 | 12 + 4 | 12 + 5 | 12 + 6 | 12 + 7 | 12 + 8 | |||||||||||

13 + 0 | 13 + 1 | 13 + 2 | 13 + 3 | 13 + 4 | 13 + 5 | 13 + 6 | 13 +7 | ||||||||||||

14 + 0 | 14 + 1 | 14 + 2 | 14 + 3 | 14 + 4 | 14 + 5 | 14 + 6 | |||||||||||||

15 + 0 | 15 + 1 | 15 +2 | 15 + 3 | 15 + 4 | 15 + 5 | ||||||||||||||

16 + 0 | 16 +1 | 16 + 2 | 16 + 3 | 16 + 4 | |||||||||||||||

17 + 0 | 17 + 1 | 17 + 2 | 17 + 3 | ||||||||||||||||

18 + 0 | 18 + 1 | 18 + 2 | |||||||||||||||||

19 + 0 | 19 + 1 | ||||||||||||||||||

20 + 0 |

Subtraction within 20 | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 - 0 | ||||||||||||||||||||

1 - 0 | 1 - 1 | |||||||||||||||||||

2 - 0 | 2 - 1 | 2 - 2 | ||||||||||||||||||

3 - 0 | 3 - 1 | 3 - 2 | 3 - 3 | |||||||||||||||||

4 - 0 | 4 - 1 | 4 - 2 | 4 - 3 | 4 - 4 | ||||||||||||||||

5 - 0 | 5 - 1 | 5 - 2 | 5 - 3 | 5 - 4 | 5 - 5 | |||||||||||||||

6 - 0 | 6 - 1 | 6 - 2 | 6 - 3 | 6 - 4 | 6 - 5 | 6 - 6 | ||||||||||||||

7 - 0 | 7 - 1 | 7 - 2 | 7 - 3 | 7 - 4 | 7 - 5 | 7 - 6 | 7 - 7 | |||||||||||||

8 - 0 | 8 - 1 | 8 - 2 | 8 - 3 | 8 - 4 | 8 - 5 | 8 - 6 | 8 - 7 | 8 - 8 | ||||||||||||

9 - 0 | 9 - 1 | 9 - 2 | 9 - 3 | 9 - 4 | 9 - 5 | 9 - 6 | 9 - 7 | 9 - 8 | 9 - 9 | |||||||||||

10 - 0 | 10 -1 | 10 - 2 | 10 - 3 | 10 - 4 | 10 - 5 | 10 - 6 | 10 - 7 | 10 - 8 | 10 - 9 | 10 - 10 | ||||||||||

11 - 0 | 11 - 1 | 11 - 2 | 11 - 3 | 11 - 4 | 11 - 5 | 11 - 6 | 11 - 7 | 11 - 8 | 11 - 9 | 11- 1 0 | 11 - 11 | |||||||||

12 - 0 | 12 - 1 | 12 - 2 | 13 - 3 | 12 - 4 | 12 - 5 | 12 - 6 | 12 - 7 | 12 - 8 | 12 - 9 | 12 - 10 | 12 - 11 | 12 - 12 | ||||||||

13 - 0 | 13 - 1 | 13 - 2 | 13 - 3 | 13 - 4 | 13 - 5 | 13 - 6 | 13 - 7 | 13 - 8 | 13 - 9 | 13 - 10 | 13 -11 | 13 - 12 | 13 - 13 | |||||||

14 - 0 | 14 - 1 | 14 - 2 | 14 - 3 | 14 - 4 | 14 - 5 | 14 - 6 | 14 - 7 | 14 - 8 | 14 - 9 | 14 - 10 | 14 - 11 | 14 - 12 | 14 - 13 | 14 - 14 | ||||||

15 - 0 | 15 - 1 | 15 - 2 | 15 - 3 | 15 - 4 | 15 - 5 | 15 - 6 | 15 - 7 | 15 - 8 | 15 - 9 | 15 - 10 | 15 - 11 | 15 - 12 | 15 - 13 | 15 - 14 | 15 - 15 | |||||

16 - 0 | 16 - 1 | 16 - 2 | 16 - 3 | 16 - 4 | 16 - 5 | 16 - 6 | 16 - 7 | 16 - 8 | 16 - 9 | 16 - 10 | 16 - 11 | 16 - 12 | 16 - 13 | 16 - 14 | 16 - 15 | 16 - 16 | ||||

17 - 0 | 17 - 1 | 17 - 2 | 17 - 3 | 17 - 4 | 17 - 5 | 17 - 6 | 17 - 7 | 17 - 8 | 17 - 9 | 17 - 10 | 17 - 11 | 17 - 12 | 17 - 13 | 17 - 14 | 17 - 15 | 17 - 16 | 17 - 17 | |||

18 - 0 | 18 - 1 | 18 - 2 | 18 - 3 | 18 - 4 | 18 - 5 | 18 - 6 | 18 - 7 | 18 - 8 | 18 - 9 | 18 - 10 | 18 - 11 | 18 - 12 | 18 - 13 | 18 - 14 | 18 - 15 | 18 - 16 | 18 - 17 | 18 - 18 | ||

19 - 0 | 19 - 1 | 19 - 2 | 19 - 3 | 19 - 4 | 19 - 5 | 19 - 6 | 19 - 7 | 19 - 8 | 19 - 9 | 19 - 10 | 19 - 11 | 19 - 12 | 19 - 13 | 19 - 14 | 19 - 15 | 19 - 16 | 19 - 17 | 19 - 18 | 19 - 19 | |

20 - 0 | 20 - 1 | 20 - 2 | 20 - 3 | 20 - 4 | 20 - 5 | 20 - 6 | 20 - 7 | 20 - 8 | 20 - 9 | 20 -10 | 20 - 11 | 20 - 12 | 20 - 13 | 20 - 14 | 20 - 15 | 20 - 16 | 20 - 17 | 20 - 18 | 20 - 19 | 20 - 20 |

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

A new study led by Johns Hopkins University psychologists shows Bedtime Math’s Crazy 8s club significantly reduces children’s feelings of math anxiety after eight weeks of participation in the club.

### National Library

of Virtual Manipulatives

A library of uniquely interactive, web-based virtual manipulatives and concept tutorials.

This site requires a Java applet to run appropriately.

### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...

## Third Grade

### Standards and Tasks

- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Number and Operations – Fractions
- Measurement and Data
- Geometry
- Fluency

## Operations and Algebraic Thinking

- Represent & solve problems involving multiplication and division
- Multiply and divide within 100
- Work with equal groups of objects to gain foundations for multiplication
- Solve problems involving the four operations...

## Represent & solve problems involving multiplication and division

## 3.OA.A.1

## 3.OA.A.2

Interpret whole-number quotients of whole numbers, e.g., interpret 56÷8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56÷8.

#### Tasks by Illustrative Mathematics

## 3.OA.A.3

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

#### Tasks by Illustrative Mathematics

Analyzing Word Problems Involving Multiplication

Gifts from Grandma, Variation 1

Two Interpretations of Division

## 3.OA.A.4

Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8×?=48, 5=?÷3, 6×6=?

#### Tasks by Illustrative Mathematics

Finding the unknown in a division equation

## Multiply and divide within 100

## 3.OA.B.5

Apply properties of operations as strategies to multiply and divide. Examples: If 6×4=24 is known, then 4×6=24 is also known. (Commutative property of multiplication.) 3×5×2 can be found by 3×5=15, then 15×2=30, or by 5×2=10, then 3×10=30. (Associative property of multiplication.) Knowing that 8×5=40 and 8×2=16, one can find 8×7 as 8×(5+2)=(8×5)+(8×2)=40+16=56. (Distributive property.)

#### Tasks by Illustrative Mathematics

## 3.OA.B.6

## Work with equal groups of objects to gain foundations for multiplication

## 3.OA.C.7

Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8×5=40, one knows 40÷5=8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

#### Tasks by Illustrative Mathematics

Kiri's Multiplication Matching Game

## Solve problems involving the four operations...

## 3.OA.D.8

## 3.OA.D.9

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

#### Tasks by Illustrative Mathematics

Patterns in the Multiplication Table

Symmetry of the Addition Table

## Number and Operations in Base Ten

## Use place value understanding and properties of operations to perform multi-digit arithmetic

## 3.NBT.A.1

Use place value understanding to round whole numbers to the nearest 10 or 100.

#### Tasks by Illustrative Mathematics

Rounding to the Nearest Ten and Hundred

Rounding to the Nearest 100 and 1000

## 3.NBT.A.2

## 3.NBT.A.3

## Number and Operations – Fractions

## Develop understanding of fractions as numbers

## 3.NF.A.1

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

#### Tasks by Illustrative Mathematics

Naming the Whole for a Fraction

## 3.NF.A.2

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

- Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
- Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

#### Tasks by Illustrative Mathematics

Locating Fractions Less than One on the Number Line

Locating Fractions Greater than One on the Number Line

Finding 1 starting from 5/3, Assessment Variation

Find 7/4 starting from 1, Assessment Variation

## 3.NF.A.3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

- Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
- Recognize and generate simple equivalent fractions, e.g., 1/2=2/4, 4/6=2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3=3/1; recognize that 6/1=6; locate 4/4 and 1 at the same point of a number line diagram.
- Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

#### Tasks by Illustrative Mathematics

Comparing Fractions with a Different Whole

Comparing Fractions with the Same Denominator, Assessment Variation

Comparing Fractions with the Same Numerators, Assessment Variation

Fraction Comparisons with Pictures, Assessment Variation

## Measurement and Data

- Solve problems involving measurement and estimation of intervals of time, liquid volumes...
- Represent and interpret data
- Geometric measurement: Concepts of area
- Geometric measurement: Perimeter

## Solve problems involving measurement and estimation of intervals of time, liquid volumes...

## 3.MD.A.1

## 3.MD.A.2

Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

#### Tasks by Illustrative Mathematics

## Represent and interpret data

## 3.MD.B.3

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

#### Tasks by Illustrative Mathematics

## 3.MD.B.4

## Geometric measurement: Concepts of area

## 3.MD.C.5

Recognize area as an attribute of plane figures and understand concepts of area measurement.

- A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.
- A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

#### Tasks by Illustrative Mathematics

N/A

## 3.MD.C.6

## 3.MD.C.7

## Geometric measurement: Perimeter

## 3.MD.D.8

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

#### Tasks by Illustrative Mathematics

N/A

## Geometry

## Reason with shapes and their attributes

## 3.G.A.1

Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

#### Tasks by Illustrative Mathematics

N/A

## 3.G.A.2

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

#### Tasks by Illustrative Mathematics

Geometric pictures of one half

## Fluency

Students will demonstrate fluency with:

- Multiplying and dividing within 100, using strategies or properties of operations.
- Knowing from memory all products of tow-one-digit numbers, by the end of their 3rd grade year.
- Adding and subtraction within 1,000

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

Multiply Two, One-Digit Numbers | |||||||||
---|---|---|---|---|---|---|---|---|---|

0 x 0 | |||||||||

1 x 0 | 1 x 1 | ||||||||

2 x 0 | 2 x 1 | 2 x 2 | |||||||

3 x 0 | 3 x 1 | 3 x 2 | 3 x 3 | ||||||

4 x 0 | 4 x 1 | 4 x 2 | 4 x 3 | 4 x 4 | |||||

5 x 0 | 5 x 1 | 5 x 2 | 5 x 3 | 5 x 4 | 5 x 5 | ||||

6 x 0 | 6 x 1 | 6 x 2 | 6 x 3 | 6 x 4 | 6 x 5 | 6 x 6 | |||

7 x 0 | 7 x 1 | 7 x 2 | 7 x 3 | 7 x 4 | 7 x 5 | 7 x 6 | 7 x 7 | ||

8 x 0 | 8 x 1 | 8 x 2 | 8 x 3 | 8 x 4 | 8 x 5 | 8 x 6 | 8 x 7 | 8 x 8 | |

9 x 0 | 9 x 1 | 9 x 2 | 9 x 3 | 9 x 4 | 9 x 5 | 9 x 6 | 9 x 7 | 9 x 8 | 9 x 9 |

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

### National Library

of Virtual Manipulatives

This site requires a Java applet to run appropriately.

### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...

## Fourth Grade

### Standards and Tasks

- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Number and Operations – Fractions
- Measurement and Data
- Geometry
- Fluency

## Operations and Algebraic Thinking

- Use the four operations with whole numbers to solve problems
- Gain familiarity with factors and multiples
- Work with equal groups of objects to gain foundations for multiplication

## Use the four operations with whole numbers to solve problems

## 4.OA.A.1

Interpret products of whole numbers, e.g., interpret 5×7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5×7.

#### Tasks by Illustrative Mathematics

Thousands and Millions of Fourth Graders

## 4.OA.A.2

## 4.OA.A.3

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

#### Tasks by Illustrative Mathematics

## Gain familiarity with factors and multiples

## 4.OA.B.4

Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

#### Tasks by Illustrative Mathematics

Multiples of 3, 6, and 7 (cluster task)

Numbers in a Multiplication Table (cluster task)

## Work with equal groups of objects to gain foundations for multiplication

## 4.OA.C.5

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

#### Tasks by Illustrative Mathematics

## Number and Operations in Base Ten

- Generalize place value understanding for multi-digit whole numbers
- Use place value understanding and properties of ooperations to perform multi-digit arithmetic

## Generalize place value understanding for multi-digit whole numbers

## 4.NBT.A.1

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700÷70=10 by applying concepts of place value and division.

#### Tasks by Illustrative Mathematics

Thousands and Millions of Fourth Graders

## 4.NBT.A.2

## 4.NBT.A.3

Use place value understanding to round multi-digit whole numbers to any place.

#### Tasks by Illustrative Mathematics

Rounding to the Nearest 100 and 1000

## Use place value understanding and properties of ooperations to perform multi-digit arithmetic

## 4.NBT.B.4

## 4.NBT.B.5

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

#### Tasks by Illustrative Mathematics

Thousands and Millions of Fourth Graders

## 4.NBT.B.6

Find whole-number quotients and remainders with up to four-digit dividends and one-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.

#### Tasks by Illustrative Mathematics

## Number and Operations – Fractions

- Extend understanding of fraction equivalence and ordering
- Build fractions from unit fractions
- Understand decimal notation for fractions, and compare decimal fractions

## Extend understanding of fraction equivalence and ordering

## 4.NF.A.1

Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×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.

#### Tasks by Illustrative Mathematics

Explaining Fraction Equivalence with Pictures

## 4.NF.A.2

Compare 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.

#### Tasks by Illustrative Mathematics

Comparing Fractions Using Benchmarks Game

Doubling Numerators and Denominators

Listing Fractions in Increasing Size

Using Benchmarks to Compare Fractions

## Build fractions from unit fractions

## 4.NF.B.3

Understand a fraction a/b with a>1 as a sum of fractions 1/b.

- Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
- Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 38=18+18+18; 38=18+28; 218=1+1+18=88+88+18.
- Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
- Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

#### Tasks by Illustrative Mathematic

Comparing Sums of Unit Fractions

Making 22 Seventeenths in Different Ways

Writing a Mixed Number as an Equivalent Fraction

## 4.NF.B.4

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

- Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5×(1/4), recording the conclusion by the equation 5/4=5×(1/4).
- Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3×(2/5) as 6×(1/5), recognizing this product as 6/5. (In general, n×(a/b)=(n×a)/b.)
- Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

#### Tasks by Illustrative Mathematics

Extending Multiplication from Whole Numbers to Fractions

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

## 4.NF.C.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. For example, express 3/10 as 30/100, and add 3/10+4/100=34/100.

#### Tasks by Illustrative Mathematics

Expanded Fractions and Decimals

How Many Tenths and Hundredths?

## 4.NF.C.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.

#### Tasks by Illustrative Mathematics

Expanded Fractions and Decimals

How Many Tenths and Hundredths?

## 4.NF.C.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.

#### Tasks by Illustrative Mathematics

## Measurement and Data

- Solve problems involving measurement and conversion
- Represent and interpret data
- Geometric measurement: Concepts of angles

## Solve problems involving measurement and conversion

## 4.MD.A.1

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,12),(2,24), (3,36), …

#### Tasks by Illustrative Mathematics

## 4.MD.A.2

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

#### Tasks by Illustrative Mathematics

## 4.MD.A.3

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

#### Tasks by Illustrative Mathematics

## Represent and interpret data

## 4.MD.B.4

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 addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

#### Tasks by Illustrative Mathematics

## Geometric measurement: Concepts of angles

## 4.MD.C.5

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

- An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
- An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

#### Tasks by Illustrative Mathematics

N/A

## 4.MD.C.6

## 4.MD.C.7

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

#### Tasks by Illustrative Mathematics

## Geometry

## Draw and identify lines and angles, and classify shapes

## 4.G.A.1

## 4.G.A.2

Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

#### Tasks by Illustrative Mathematics

Defining Attributes of Rectangles and Parallelograms

## 4.G.A.3

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

#### Tasks by Illustrative Mathematics

Lines of Symmetry for Triangles

Lines of Symmetry for Quadrilaterals

## Fluency

Students can fluently add and subtract multi-digit whole numbers using the standard algorithm (1,000,000).

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

### National Library

of Virtual Manipulatives

This site requires a Java applet to run appropriately.

### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...

## Fifth Grade

### Standards and Tasks

- Operations and Algebraic Thinking
- Number and Operations in Base Ten
- Number and Operations – Fractions
- Measurement and Data
- Geometry
- Fluency

## Operations and Algebraic Thinking

## Write and interpret numerical expressions

## 5.OA.A.1

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

#### Tasks by Illustrative Mathematics

Using Operations and Parentheses

## 5.OA.A.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×(8+7). Recognize that 3×(18932+921) is three times as large as 18932+921, without having to calculate the indicated sum or product.

#### Tasks by Illustrative Mathematics

## Analyze patterns and relationships

## 5.OA.B.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.

#### Tasks by Illustrative Mathematics

## Number and Operations in Base Ten

- Understand the place value system
- Perform operations with multi-digit whole numbers and with decimals to hundredths

## Understand the place value system

## 5.NBT.A.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.

#### Tasks by Illustrative Mathematics

Millions and Billions of People

## 5.NBT.A.2

## 5.NBT.A.3

Read, write, and compare decimals to thousandths.

- 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) - Compare two decimals to thousandths based on meanings of the digits in each place, using > =, and < symbols to record the results of comparisons.

#### Tasks by Illustrative Mathematics

Drawing Pictures to Illustrate Decimal Comparisons

Comparing Decimals on the Number Line

Placing Thousandths on the Number Line

## 5.NBT.A.4

Use place value understanding to round decimals to any place.

#### Tasks by Illustrative Mathematics

Rounding to Tenths and Hundredths

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

## 5.NBT.B.5

## 5.NBT.B.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.

#### Tasks by Illustrative Mathematics

## 5.NBT.B.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.

#### Tasks by Illustrative Mathematics

## Number and Operations – Fractions

- Use equivalent fractions as a strategy to add and subtract fractions
- Apply and extend previous understandings of multiplication and division to...

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

## 5.NF.A.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.)

#### Tasks by Illustrative Mathematics

Mixed Numbers with Unlike Denominators

Finding Common Denominators to Add

Finding Common Denominators to Subtract

## 5.NF.A.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.

#### Tasks by Illustrative Mathematics

## Apply and extend previous understandings of multiplication and division to...

## 5.NF.B.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?

#### Tasks by Illustrative Mathematic

Converting Fractions of a Unit into a Smaller Unit

## 5.NF.B.4

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

- 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.)
- 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.

#### Tasks by Illustrative Mathematics

Connor and Makayla Discuss Multiplication

Connecting the Area Model to Context

Mrs. Gray's Homework Assignment

## 5.NF.B.5

Interpret multiplication as scaling (resizing), by:

- 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.
- 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.

#### Tasks by Illustrative Mathematics

Reasoning about Multiplication

Comparing a Number and a Product

Comparing Heights of Buildings

## 5.NF.B.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.

#### Tasks by Illustrative Mathematics

To Multiply or Not to Multiply?

Comparing Heights of Buildings

To Multiply or Not to Multiply, Variation 2

## 5.NF.B.7

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

- 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.
- 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.
- 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?

#### Tasks by Illustrative Mathematics

## Measurement and Data

- Convert like measurement units within a given measurement system
- Represent and interpret data
- Geometric measurement: Concepts of volume

## Convert like measurement units within a given measurement system

## 5.MD.A.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.

#### Tasks by Illustrative Mathematics

Converting Fractions of a Unit into a Smaller Unit

## Represent and interpret data

## 5.MD.B.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. 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.

#### Tasks by Illustrative Mathematics

## Geometric measurement: Concepts of volume

## 5.MD.C.3

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

- 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.
- 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.

#### Tasks by Illustrative Mathematics

N/A

## 5.MD.C.4

## 5.MD.C.5

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

- 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.
- 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.
- 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.

#### Tasks by Illustrative Mathematics

You Can Multiply Three Numbers in Any Order

Using Volume to Understand the Associative Property of Multiplication

Breaking Apart Composite Solids

## Geometry

- Graph points on the coordinate plane to solve real-world mathematical problems
- Classify two-dimensional figures into categories based on their properties

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

## 5.G.A.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).

#### Tasks by Illustrative Mathematics

## 5.G.A.2

## Classify two-dimensional figures into categories based on their properties

## 5.G.B.3

## 5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.

#### Tasks by Illustrative Mathematics

What Do These Shapes Have in Common?

## Fluency

Students can fluently multiply multi-digit whole numbers using the standard algorithm.

Fluency is:

**Accuracy:**Precision and correctness**Flexibility:**The ability to use other strategies**Efficiency:**Speed in knowing from memory or speed of using a strategy.

The NRICH Project aims to enrich the mathematical experiences of all learners.

### Mathematical Practices

- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.

### Resources

### Bedtime Math

### National Library

of Virtual Manipulatives

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### Would You Rather Math

Applying mathematics to authentic situations and justifying why you would rather...