Author: Capital Region BOCES

Four Operations with Decimals

Students will deepen their understanding of decimals as they work with decimals to the thousandths place. Students will extend their strategies for computing with whole numbers to add, subtract, multiply, and divide decimals.

Note: Lessons will vary in length, depending on the amount of time you have with students, the resources that you choose to accompany the unit, the level of rigor within each learning target, and any other factors that may contribute to the pacing of your learning progressions. It is recommended that you adjust the pace and length of each learning progression(s) accordingly in response to these factors.

These learning progressions were developed using Next Generation Learning Standards and were crosswalked with the Common Core Standards.

Download the complete Grade 5 Math Unit 2 framework to customize for your own planning.

Essential Outcomes

Numbers in Base Ten

• NY-5.NBT.7 – Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between operations:
  • add and subtract decimals to hundredths;
  • multiply and divide decimals to hundredths.
  • Relate the strategy to a written method and explain the reasoning used.
• NY-5.NBT.3 Read, write, and compare decimals to thousandths.
• NY-5.NBT.3b Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
• NY-5.NBT.4 – Use place value understanding to round decimals to any place.

Other Standards Addressed in this Unit

Numbers in Base Ten

• NY-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.
• NY-5.NBT.2 – Use whole-number exponents to denote powers of 10. 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.

Essential Questions and Big Ideas

• How do numbers with decimals relate to the base ten number system?
  • Decimals maintain the relationships between digits that are based on powers of ten.
  • A digit in one place is ten times the value of the same digit in the place to the right.
  • A digit in one place is 1/10 the value of the same digit in the place to the left.
  • The patterns in place values continue in the decimal places (tenths, hundredths, thousandths).
  • Numbers with decimals can be written in expanded form and word form.
• How do I add and subtract with decimals?
  • Addition and subtraction is based on the base ten number system.
  • Like place values are added together.
  • If there is not a digit in a certain place value, a zero can be used as a placeholder.
• How do I multiply with decimals?
  • Numbers with decimals can be multiplied in a similar way to whole numbers.
  • When multiplying tenths by tenths, hundredths are created.
  • Multiplying decimal places create smaller decimal places.
• How do I divide with decimals?
  • Numbers with decimals can be divided in a similar way to whole numbers.
  • When dividing by an amount with a decimal, multiply by a power of ten to create a whole number. Multiply the dividend by the same power of ten to keep the relationship between amounts the same.
  • When dividing a decimal by a whole number, maintain the location of the decimal place.
• How do I solve multi-step problems?
  • When solving multi-step problems, identify the unknowns.
  • After identifying the unknowns, identify the actions and relationships within the problem.
  • Write number sentences to represent the actions and relationships within a problem.

Download the complete Grade 5 Math Unit 2 framework to customize for your own planning.

Intro to Place Value through Addition and Subtraction within 20

Unit description: This unit serves as a bridge from problem solving within 10 to work within 100 as students begin to solve addition and subtraction problems involving teen numbers. In unit 1, students were encouraged to move beyond the beginning strategy of counting all to the more efficient counting on. Now, they go beyond that level to decomposition and composition strategies, informally called make ten or take from ten. Students will work on the concept of addition and subtraction within 20 and focus on building fluency within 10. Mastery of facts is not expected at this point in the year. They will develop an understanding of ten as a unit to analyze teens as ten and some ones and use modalities to build individual numbers with tens/ones while counting. Students will compare two two-digit numbers using symbols (<,>,=).

Essential Outcomes of the Unit

Numbers in Base Ten

Understand place value

• 1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones.
• 1.NBT.2a Understand 10 can be thought of as a bundle of ten ones, called a “ten”.
• 1.NBT.2b Understand the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
• 1.NBT.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Other Standards Addressed in this Unit

Operations in Algebraic Thinking

Represent and solve problems involving addition and subtraction.

• 1.OA.1 Use addition and subtraction within 20 to solve one-step word problems involving situations of adding to, taking from, putting together, taking apart, and/or comparing, with unknowns in all positions.
  • Note: Problems should be represented using objects, drawings, and equations with a symbol for the unknown number. Problems should be solved using objects or drawings, and equations.
• 1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.

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

• 1.OA.3 Apply properties of operations as strategies to add and subtract.
  • (Note: Students need not use formal terms for these properties. When students use the making ten strategy (NY-1.OA.6), they are applying the Associative property of addition.
• 1.OA.4 Understand subtraction as an unknown-addend problem within 20.

All work with properties (NY-1.OA.3) and place value (e.g., NY-1.NBT.2 & 4) should be seen as an investigation and use of the structure of the number system and of arithmetic.

Add and subtract within 20.

• 1.OA.5 Relate counting to addition and subtraction.
• 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. e.g., Which of the following equations are true and which are false?
  6 = 6   7 = 8 – 1   5 + 2 = 2 + 5   4 + 1 = 5 + 2
• 1.OA.8 Determine the unknown whole number in an addition or subtraction equation with the unknown in all positions. e.g., Determine the unknown number that makes the equation true in each of the equations
  8 + ? = 11   – 3 = 5   6 + 6 = □

Essential Questions and Big Ideas

1. Why is it important to know multiple strategies in solving addition/subtraction problems?

• • Strategies can be used to decompose complex problems to make an easier problem.
  • Strategies help to solve addition and subtraction problems within 20 quicker.

2. How are problem solving strategies connected to number relationships?

• • Problem solving structures reinforce part/part/whole and number combinations within 20.
  • Each type of word problem situation (adding to, taking from, putting together, taking apart, comparing) reflects number relationships.

3. What is significant about the teen numbers (related to 10)?

• • Explain the value of each digit in a two digit number.
  • Represent a 2 digit numeral using “tens” and “ones.”
  • Build and decompose numbers into tens and ones.
  • Identify a bundle of 10 ones as a “ten”.

4. How is counting connected to quantity in a number?

• • Use comparison words greater than, less than, and equal to communicate understanding of the relationship between the numbers.

5. How does using objects and drawings help me represent problems in multiple ways?

• • Use models to represent 2 sets of numbers.
  • When given a set of objects (ranging from 0-120), represent the quantity with a written numeral.
  • Represent a problem situation involving 2-digit numbers using any combination of words, numbers, physical objects, or symbols.

Download the complete Grade 1 Math Unit 2 framework to customize for your own planning.

Addition and Subtraction within 1,000

Students will develop deeper understandings of the base ten number system, which will strengthen their abilities to round as well as add and subtract numbers.

Note: Lessons will vary in length, depending on the amount of time you have with students, the resources that you choose to accompany the unit, the level of rigor within each learning target, and any other factors that may contribute to the pacing of your learning progressions. It is recommended that you adjust the pace and length of each learning progression(s) accordingly in response to these factors.

These learning progressions were developed using Next Generation Learning Standards and were cross-walked with the Common Core Standards.

Download the complete Grade 3  Unit 2 framework to customize for your own planning.

Essential Outcomes

Numbers in Base Ten

• 3-NY-NBT.1: Use place value understanding to round whole numbers to the nearest 10 or 100.
• 3-NY-NBT.2: Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Operations and Algebraic Thinking

• NY-3.OA.8: Solve two-step word problems posed with whole numbers and having whole-number answers using the four operations.

Other Standards Addressed in this Unit

Numbers in Base Ten

• NY-3.NBT.4a: Understand that the digits of a four-digit number represent amounts of thousands, hundreds, tens, and ones. e.g., 3,245 equals 3 thousands, 2 hundreds, 4 tens, and 5 ones.
• NY-3.NBT.4b: Read and write four-digit numbers using base-ten numerals, number names, and expanded form. e.g., The number 3,245 in expanded form can be written as 3,245= 3,000 + 200 + 40 + 5.

Essential Questions and Big Ideas

• How does understanding place value help you to better understand numbers?
  • The base ten number system determines the value of the digits within numbers.
  • Numbers are broken into places which are based on powers of 10.
  • Rounding numbers shows about how much of a certain place value they hold.
  • Expanded form is a way to write numbers broken into the values of each place value.
• What are efficient strategies to add multi digit whole numbers?
  • Digits in a number can be added based on place value.
  • Add like place values.
  • Only 9 of a place can fit in each place.
  • Ten of one place makes one of the next largest place value.
• What are efficient strategies to subtract multi digit whole numbers?
  • Digits in a number can be subtracted based on place value.
  • Subtract like place values.
  • If you do not have enough of a place value, regroup one of the next largest place value to create ten more of that place value.
  • Subtraction is the inverse to addition.
  • Adding on can be used to find a difference.
• How can I solve two-step word problems with addition and subtraction?
  • When a word problem has more than one unknown, there is more than one step.
  • Finding a difference or comparing requires subtraction.
  • Finding a total or combining amounts requires addition.
  • Finding a missing part requires subtraction.

Download the complete Grade 3  Unit 2 framework to customize for your own planning.