math curriculum guide 2011-12

Math Curriculum Guide

3rd Grade

2011 – 2012 School Year

Dear Colleague, Please use this curriculum guide to assist in your unit, lesson and assessment plans as you progress through the school year. As additional guidance, included below is further clarification regarding the “Content Key” symbols used throughout this document:

Content Key Description

√ Grade Level Content Content that is part of the current grade level set of standards. Students should be assessed fully on their ability to demonstrate an understanding of these concepts.

Previous Content

Content that was part of a previous grade level set of standards. Students should be assessed fully on their ability to demonstrate an understanding of these concepts. You may be able to condense material here based upon your students maintained knowledge of this information.

Exposure Content for Future Study

Content that is part of a future grade level set of standards. Students should be assessed, at an exposure level, on their ability to demonstrate an understanding of these concepts. It is not expected that all students develop a mastery level of understanding of these concepts at this time.

Additionally, in preparation for the upcoming implementation and transistion into the Common Core State Standards (CCSS), it is imperative that we begin fostering the mathematical mindsets and practices of our students that are embodied within these new standards. Also within these standards, there will be a stronger emphasis placed on specific areas of study. In following the guidance of the New York State Education Department, as a district, we have decided to plan for an intensive focus of study to occur within the topics of the meaning of multiplication and division as well as working with fractions. Time will be made available to develop these units of study (topics 5, 7, 12 and 13) together with district colleagues.

3rd Grade HHH Math Curriculum Guide Topic 1: Numeration Weeks of Sept. 6 & Sept. 12, 9 Days

Essential Understandings Lesson Number and Title

Content Key

Essential understandings or items that need supplementation or adjustment

1-1 Hundreds

1-2 Thousands Keep lessons as they exist. Expose the students to the content.

Greater Numbers

1-3

&

1-4 Ways to Name Numbers

Consider teaching 1-3 & 1-4 in one lesson. Expose the students to the content. Mastery is not essential.

1-5 Comparing Numbers Expose the students to the content. Mastery is not essential.

1-6 Ordering Numbers Expose students to the content. Mastery is not essential.

1-7 Counting Money

Incorporate an extra lesson in coin recognition and value before this lesson. Students are exposed to several money lessons in second grade. Possible money assessment before teaching this lesson to see where students are.

1-8 Making Change

Students should have been exposed to material in second grade. Expose students to material and move on. Use the bar diagram in addition to counting up. Expose students to real life experiences with coins.

• Our number system is based on groups of ten. Whenever we get 10 in one place value, we move to the next greater place value.

• The place-value periods ones, thousands, millions, and so forth are used to read and write large numbers.

• Two uses of numbers are to tell how many and to show order. A given number can be represented in different ways and not change its value.

• Place value can be used to compare and order numbers.

• When counting money, it is often easiest to start with the bills or coins that have the greatest value. You can use skip counting to count money. Some problems can be solved by generating a list of outcomes and organizing that list in a systematic way so all outcomes are accounted for.

• Counting up is usually an efficient way to make change.

1-9 Problem Solving Make an Organized List

Expose students to strategy (organized list) and continue to use in subsequent lessons.

√ Grade Level Content Previous Content Exposure Content for Future Study

3rd Grade HHH Math Curriculum Guide Topic 2: Adding Whole Numbers Weeks of Sept. 19 & Sept. 26 & Oct. 3, 11 Days


Content Key


2-1 Addition Meaning and Properties

Although difficult, students will revisit this in topic 5

2-2 Adding on a Hundred Chart

2-3 Using Mental Math to Add

Although challenging, it's necessary

2-4 Rounding Students may need additional time to master concept.

2-5 Estimating Sums Mixed Problem Solving

"Compatible" numbers not taught in 2nd grade. Emphasize ... numbers that make adding mentally easier

2-6 Adding 2-Digit Numbers

2-7 Models for Adding 3- Digit Numbers

2-8 Adding 3-Digit Numbers

2-9 Adding 3 or More Numbers

• Some real-world problems involving joining, separating, part-part-whole or comparison can be solved using addition. Two numbers can be added in any order; the sum of any number and 0 is that number; and three or more numbers can be grouped and added in any order.

• Patterns on a hundred chart can be used to add numbers and to develop mental math strategies and number sense.

• There is more than one way to do a mental calculation. Techniques for doing addition calculations mentally involve changing the numbers or the expression so the calculation is easy to do mentally.

• Rounding is a process for finding the multiple of 10,100, etc. closest to a given number. Different numerical expressions can have the sale value. Or, the value of one expression can be less than (or greater than) the value of the other expression.

• There is more than one way to estimate a sum. Rounding gives one way to estimate sums.

• The standard addition algorithm for 2-digit numbers breaks the calculation into simpler calculations using place value starting with the ones and then the tens.

• Models and the standard algorithm for adding 3-digit numbers are just an extension to the hundreds place of the models and standard algorithm for adding 2-digit numbers.

• Three or more whole numbers can be grouped and added in any order.

• Information in a problem can often be shown using a picture or diagram and used to understand and solve the problem. Some problems can be solved by writing and completing a number sentence or equation.

2-10 Problem Solving Draw a Picture


3rd Grade HHH Math Curriculum Guide Topic 3: Subtraction Number Sense Weeks of Oct. 3 & Oct. 11, 6 Days


Content Key


3-1 Subtraction Meanings

3-2 Subtracting on a Hundred Chart

3-3 Using Mental Math to Subtract

3-4 Estimating Differences Mixed Problem Solving

• Some real-world problems involving joining, separating, part-part-whole or comparison can be solved using subtraction. Fact families show addition and subtraction relationships.

• Patterns on a hundred chart can be used to subtract numbers and to develop mental math strategies and number sense.

• There is more than one way to do a mental calculation. Techniques for doing subtraction calculations mentally involve changing the numbers or the expression so the calculation is easy to do mentally.

• There is more than one way to estimate a difference. Rounding and substituting compatible numbers are two ways to estimate differences.

• Answers to problems should always be checked for reasonableness, and this can be done in different ways. Two ways are to use estimation when appropriate and to check the answer against the question and conditions in the problem.

3-5 Problem Solving Reasonableness


3rd Grade HHH Math Curriculum Guide Topic 4:Subtracting Whole Numbers to Solve Problems Weeks of Oct. 17 & Oct. 24, 8 Days


Content Key


4-1 Models for Subtracting 2-Digit Numbers

4-2 Subtracting 2-Digit Numbers

4-3 Models for Subtracting 3-Digit Numbers

4-4 Subtracting 3-Digit Numbers

4-5 Subtracting Across Zero

Need to spend extra time

• The standard subtraction algorithm for 2-digit numbers breaks the calculation into simpler calculations using place value starting with the ones and then the tens.

• Models and the standard algorithm for subtracting 3-digit numbers are just an extension to the hundreds place of the models and standard algorithm for subtracting 2-digit numbers. In the traditional subtraction algorithm, sometimes it is necessary to rename 1 ten as 10 ones, 1 hundred as 10 tends, 1 thousand as 10 hundreds, and so on.

• Place-value relationships can help simplify subtracting across zero.

• Information in a problem can often be shown using a picture or diagram and used to understand and solve the problem. Some problems can be solved by writing and completing a number sentence. 4-6

Problem Solving Draw a Picture and Write a Number Sentence

Be sure to use Bar Diagrams


3rd Grade HHH Math Curriculum Guide Topics 5 & 6:Multiplication Meanings and Fact Strategies: Use Known Facts Weeks of Oct. 24, Oct. 31, Nov. 7, Nov. 14 & Nov. 21, 18 Days


Content Key


5-1 Multiplication as Repeated Addition Supplement with multiplication tables

5-2 Arrays and Multiplication Mixed Problem Solving

5-3 Using Multiplication to Compare

5-4 Writing Multiplication Stories

5-5 Problem Solving Writing to Explain

5-6 2 and 5 as Factors

5-7 10 as a Factor

5-8 9 as a Factor

5-9 Multiplying with 0 and 1

5-10 Problem Solving Two-Question Problems

6-1 3 as a Factor Supplement with multiplication tables

6-2 4 as a Factor


6-4 8 as a Factor


6-6 Multiplying with 3 Factors

• Some real-world problems involving joining or separating equal groups or comparison can be solved using multiplication.

• Repeated addition involves joining equal groups and is one way to think about multiplication.

• An array involves joining equal groups and is one way to think about multiplication. Two numbers can be multiplied in any order.

• A "times as many" comparison is one way to think about multiplication.

• Mathematical explanations can be given using words, pictures, numbers or symbols. A good explanation should be correct, simple, complete, and easy to understand.

• There are patterns in the products for multiplication facts with factors of 2 and 5.

• Patterns can be used to find products involving factors of 10.• There are patterns in the products for multiplication facts

with a factor of 9. • The product of 0 and any number is 0. The product of 1 and

any number is that number. There are patterns in the products for multiplication facts with factors of 0 and 1.

• Sometimes the answer to one problem/question is needed to find the answer to another problem/question.

• Basic multiplication facts with 3, 4, 6, 7, 8, 11 or 12 as a factor can be found by breaking apart the unknown fact into known facts. The answers to the known facts are added to get the final product.

• Three or more numbers can be grouped and multiplied in any order.

• Some problems can be solved by first finding and solving a sub-problem(s) and then using that answer(s) to solve the original problem. 6-7 Problem Solving

Multiple-Step Problems

Students should understand the “break apart” strategy at a mastery level by the conclusion of topic 6.


3rd Grade HHH Math Curriculum Guide Topic 7: Division Meanings Weeks of Nov. 28 & Dec. 5, 7 Days


Content Key


7-1 Division as Sharing

7-2 Understanding Remainders Mixed Problem Solving

7-3 Division as Repeated Subtraction

7-4 Writing Division Stories

• Sharing involves separating equal groups and is one way to think about division. Some real-world problems involving joining or separating equal groups and is one way to think about division.

• The remainder when dividing must be less than the divisor. When solving a real-world problem that involves division with a remainder, the nature of the question asked determines how to interpret and use the remainder.

• Repeated subtraction involves separating equal groups and is one way to think about division. Some real-world problems involving joining or separating equal groups and is one way to think about division.

• Sharing and repeated subtraction both involve separating equal groups and are two ways to think about division. Some real-world problems involving joining or separating equal groups and is one way to think about division.

• Some problems can be solved by using objects to act out the problem or by drawing a picture to show the actions in the problem.

7-5 Problem Solving Using Objects and Draw a Picture


3rd Grade HHH Math Curriculum Guide Topic 8: Division Facts Weeks of Dec. 5 & Dec. 12, 8 Days


Content Key


8-1 Relating Multiplication and Division

8-2 Fact Families with 2,3,4, and 5

8-3 Fact Families with 6 and 7

8-4 Fact Families with 8 and 9

8-5 Dividing with 0 and 1

• Multiplication and division have an inverse relationship.

• Different numerical expressions can have the same value. Or, the value of one expression can be less than (or greater than) the value of the other expression. The inverse relationship between multiplication and division can be used to find division facts; every division fact has a related multiplication fact.

• The inverse relationship between multiplication and division can be used to find division facts; every division fact has a related multiplication fact.

• Any number (except 0) divided by itself is equal to 1. Any number divided by 1 is that number. Zero divided by any number (except 0) is zero. Zero cannot be a divisor.

• Information in a problem can often be shown using a picture or diagram and used to understand and solve the problem. Some problems can b e solved by writing and completing a number sentence or equation.

8-6 Problem Solving Draw a Picture and Write a Number Sentence


3rd Grade HHH Math Curriculum Guide Topic 9: Patterns and Relationships Weeks of Dec. 19 & Jan. 3, 9 Days


Content Key


Repeating Patterns

9-1

&

9-2 Number Sequences

Consider teaching 9-1 & 9-2 in one lesson

9-3 Extending Tables 2 Days of instruction might be needed for students to master these

concepts.

9-6 Geometric Patterns

9-4 Writing Rules for Situations Mixed Problem Solving

9-7 Equal or Unequal

• Some patterns consist of shapes or numbers arranged in a unit that repeats.

• Some numerical sequences have rules that tell how to generate more numbers in the sequence.

• Some real-world quantities have a mathematical relationship; the value of one quantity can be found if you know the value of the other quantity. Patterns can be used to identify some relationships.

• Some sequences of geometric objects change in predictable ways that can be described using mathematical rules.

• A numerical expression is equal to, greater than, or less than another numerical expression.

• Some problems can be solved by using objects to act out the information in the problem. Some problems can be solved by reasoning about the conditions in the problem.

9-8 Problem Solving Act It Out and Use Reasoning


3rd Grade HHH Math Curriculum Guide Topics 10 & 11: Solids, Shapes, Congruence and Symmetry Weeks of Jan. 9, Jan. 17 & Jan. 23, 13 Days


Content Key


Solid Figures 10-1

&

10-2 Relating Solids and Shapes


10-3 Lines and Line Segments

10-4 Angles

10-5 Polygons

10-6 Triangles

10-7 Quadrilaterals

10-8 Problem Solving Make and Test Generalizations

11-1 Congruent Figures and Motion 11-1 and 11-2 were combined in 2011

11-2 Line Symmetry 11-1 and 11-2 were combined in 2011

11-3 Drawing shapes with Lines of Symmetry 11-3 and 11-4 were combined in 2011

• Three-dimensional or solid figures have length, width, and height. Many can be described, classified, and analyzed by their faces, edges, and vertices. Many everyday objects closely approximate standard geometric solids.

• We can use shapes to describe some attributes of some solids.

• Lines and line segments are sets of points in space that can be used to describe parts of other geometric lines, shapes, and solids.

• An angle is formed by two rays with a common endpoint. Angles can be classified by their size.

• Two-dimensional or plane shapes have many properties that make them different from one another. Polygons can be described and classified by their sides and angles.

• Commonalities in attributes of objects or situations can be found and used to make and test generalizations about relationships.

• Congruent figures are the same shape and same size.

• Some shapes can be reflected across one or more lines passing through the shape so the shape folds onto itself exactly.

• Some problems can be solved by using objects to act out the actions in the problem. An angle is formed by two rays with a common endpoint. Angles can be classified by their size.

11-4 Problem Solving Use Objects

11-3 and 11-4 were combined in 2011

Consider assessing topics 10 and 11 together.


3rd Grade HHH Math Curriculum Guide Topics 12 & 13: Underatanding Fractions, Decimals and Money Weeks of Jan. 30, Feb. 6 & Feb. 13, 14 Days


Content Key


12-1 Dividing Regions into Equal Parts

12-2 Fractions and Regions 12-2 and 12-3 were combined in 2011

12-3 Fractions and Sets 12-2 and 12-3 were combined in 2011

12-4 Benchmark Fractions

12-5 Finding Equivalent Fractions Use manipulatives and models to represent equivalent fractions

12-6 Using Models to Compare fractions

12-7 Fractions on the Number Line Coach resources may be helpful here

Using Models to Add Fractions

12-8

&

12-9 Using Models to Subtract Fractions


12-10 Problem Solving Make a Table and Look for a Pattern

13-3 Adding and Subtracting Money


• A region can be divided into equal-sized parts in different ways. Equal-sized parts of a region have the same area but not necessarily the same shape.

• A fraction describes the division of a whole (region, set, segment) into equal parts. A fraction is relative to the size of the whole.

• Fractions can be approximated by other fractions that are close.

• The same fractional amount can be represented by an infinite set of different but equivalent fractions.

• To compare two fractions, the whole must be the same size. The fraction which represents a greater part of the whole is the greater fraction.

• Each fraction can be associated with a unique point on the number line.

• To add fractions with like denominators, add the numerators and write the sum over the same denominator.

• To subtract fractions with like denominators, subtract the numerators and write the difference over the same denominator.

• Some problems can be solved by recording and organizing data in a table and by finding and using numerical patterns in the table.

• Money can be used to model addition and subtraction of decimals.

• Information given in a problem can often be shown in a picture or diagram that can be used to solve the problem. Some problems can be solved by writing and completing a number sentence or equation.

• Some problems have data missing needed to find the answer, and some problems have extra data not needed to solve the problem.

13-5 Problem Solving Missing or Extra Information



3rd Grade HHH Math Curriculum Guide Topics 14 & 17:Customary Measurement, Time and Temperature Weeks of Feb. 27 & Mar. 5, 10 Days


Content Key


14-1 Understanding Measurement


14-2 Fractions of an Inch


14-3 Using Inches, Feet, Yards, and Miles

14-4 Customary Units of Capacity

14-5 Units of Weight

14-6 Problem Solving Act It Out and Use Reasoning

17-1 Time to the Half Hour and Quarter Hour


• Measurement compares a unit to the object being measured. The length of any object can be a measurement unit for length, but a standard unit, such as an inch, is always the same length. The smaller the unit used, the more units are needed to equal a given length. You do not need to start at 0 to use a ruler to measure length.

• Fractions of an inch give measurements that are closer to the actual lengths of objects than whole inches.

• Inches, feet, yards, and miles are standard units for measuring length and they are related to each other.

• Capacity is a measure of the amount of liquid a container can hold.

• The weight of an object is a measure of how heavy an object is.

• Some problems can be solved by using objects to act out the actions in the problem. Some problems can be solved by reasoning about the conditions in the problem.

• Time can be expressed using different units that are related to each other.

• The minute hand takes 5 minutes to move from one number to the next on a typical clock face. The minute hand takes 1 minute to move from one mark to the next on a typical clock face.

17-2 Time to the Minute




3rd Grade HHH Math Curriculum Guide Topic 20: Data, Graphs and Probability Weeks of Mar. 12 & Mar. 19, 9 Days


Content Key


20-1 Organizing Data 2 Days of instruction might be needed for students to master these

concepts.

20-2 Reading Pictographs and Bar Graphs Mixed Problem Solving

2 Days of instruction might be needed for students to master these concepts.

20-3 Making Pictographs

• Tally charts are useful in recording and organizing some kinds of data.

• Each type of graph is most appropriate for certain kinds of data.

• The key for a pictograph determines the number of pictures needed to represent each number in a set of data.

• In a bar graph, the scale determines how long the bar needs to be to represent each number in a set of data.

20-4 Making Bar Graphs


3rd Grade HHH Math Curriculum Guide Post NYS Assessment Topics: 9 & 20 Weeks of May 14 & May 21, 6 Days


Content Key


9-5 Translating Words to Expressions

Information to be discussed after the assessment

20-5 Ordered Pairs and Line Graphs


20-6 How Likely? Information to be discussed after the assessment

20-7 Outcomes and Experiments


20-8 Line Plots and Probability


• Some mathematical phrases can be represented using a variable in an algebraic expression.

• The Cartesian coordinate system is a scheme that uses perpendicular number lines intersecting at zero on each to name the location of points on the plane. A coordinate grid is a grid used to locate points. An ordered pair of numbers names a point on a grid. A line graph shows how data change over a period of time.

• The likelihood of an event occurring can be described using the words likely, unlikely, impossible (highly improbable), certain, more likely, less likely, equally likely, most likely, and least likely.

• There are techniques for representing and counting the number of outcomes from an experiment. The results of a probability experiment get closer to the expected results as the number of trials increases.

• The likelihood of an event occurring can be found using data on a line plot.

• Some problems can be solved by making, reading, and analyzing a graph.

20-9 Problem Solving Use Tables and Graphs to Draw Conclusion


3rd Grade HHH Math Curriculum Guide Post NYS Assessment Topics: 13, 17 & 15 Weeks of May 21, May 28 & June 4, 11 Days


Content Key


13-1 Fractions and Decimals Information to be discussed after the assessment

13-2 Using Money to Understand Decimals


17-3 Units of Time Information to be discussed after the assessment

17-4 Elapsed Time Information to be discussed after the assessment

17-5 Temperature Information to be discussed after the assessment

17-6 Problem Solving Work Backward


15-1 Using Centimeters and Decimeters


15-2 Using Meters and Kilometers


15-3 Metric Units of Capacity Information to be discussed after the assessment

15-4 Units of Mass Information to be discussed after the assessment

• A decimal is another name for a fraction. • Decimal place value is an extension of whole

number place value. Dollars, dimes, and pennies can be used to represent decimals through hundredths.

• There are different units for measuring time. Many clock times can be expressed in more than one way.

• The duration of an event can be measured if one knows the start and end times for the event.

• Temperature can be expressed using degrees Fahrenheit or degrees Celsius.

• Some problems with the initial data point unknown can be solved by starting with the end result, reversing the steps and processes, and working backward to the initial data point.

• Centimeters and decimeters are standard units for measuring length in the Metric System and they are related to each other.

• Meters and kilometers are standard units for measuring length in the Metric System and they are related to each other.

• Capacity is a measure of the amount of liquid a container can hold.

• Mass is a measure of the quantity of matter in an object. Weight and mass are different.

• Some problems can be solved by recording and organizing data in a table and by finding and using numerical patterns in the table.

15-5 Problem Solving Make a Table and Look for a Pattern



3rd Grade HHH Math Curriculum Guide Post NYS Assessment Topics: 18 & 19

Weeks of June 4 & June 11, 10 Days

Essential Understandings Lesson Number and Title Content Key Essential understandings or items that need

supplementation or adjustment

18-1 Using Metal Math to Multiply


18-2 Estimating Products Information to be discussed after the assessment

18-3 Multiplication and Arrays


18-4 Breaking Apart to Multiply


18-5 Using an Expanded Algorithm


18-6 Multiplying 2- and 3-Digit by 1-Digit Numbers




19-1 Mental Math Information to be discussed after the assessment

19-2 Estimating Quotients Information to be discussed after the assessment

19-3 Connecting Models and Symbols


19-4 Dividing 2-Digit Numbers


19-5 Dividing with Remainders


• Basic facts and place-value patterns can be used to find products when one factor is a multiple of 10, 100, or 1,000.

• Rounding gives one way to estimate products. The nature of the numbers used to make an estimate determines in an estimate is over or under the exact answer.

• Making an array with place-value blocks provides a way to visualize and find products.

• A 2-digit by 1-digit multiplication calculation can be broken into simpler problems: a basic fact and a 1-digit number times a multiple of 10. Answers to the simpler problems are added to give the product.

• There is an expanded algorithm for multiplying where the numbers are broken apart using place value and the parts are used to find partial products. The partial products are then added together to find the product.

• The standard multiplication algorithm is a shortcut for the expanded algorithm. Regrouping is used rather than showing all partial products. Properties such as the commutative property can be used to solve some equations.

• Information in a problem can often be shown using a picture or diagram and used to understand and solve the problem. Some problems can be solved by writing and completing a number sentence or equation.

• Basic facts and place-value patterns can help you mentally divide multiples of 10, 100, and 1,000 by 1-digit numbers.

• There are different ways to estimate quotients. Most involve replacing numbers with other numbers that are close and that make it easy to compute mentally.

• The sharing interpretation for division can be used to model the standard division algorithm.

• The standard division algorithm involves breaking the calculation into smaller, simpler calculations, using basic facts, place value, the relationship between multiplication and division, and estimation.

• When you divide whole numbers, sometimes there is a remainder. The remainder must be less than the divisor.

• Some problems can be solved by first finding and solving a sub-problem(s) and the using that answer(s) to solve the original problem.

19-6 Problem Solving Multiple-Step Problems


√ Grade Level Content Previous Content

Exposure Content for Future Study

3rd Grade HHH Math Curriculum Guide Post NYS Assessment Topic: 16 Week of June 18, 4 Days


Content Key


16-1 Understanding Perimeter Information to be discussed after the assessment

16-2 Perimeter of Common Shapes


16-3 Different Shapes with the Same Perimeter


16-4 Problem Solving Try, Check, and Revise


16-5 Understanding Area Information to be discussed after the assessment

16-6 Estimating and Measuring Area


16-7 Volume Information to be discussed after the assessment

• The distance around a figure its perimeter. • Different shapes can have the same perimeter. • Some problems can be solved by making a

reasoned first try for what the answer might be and then, through additional reasoning, arrive at the correct answer.

• The measure of the region inside a shape is its area, and area can be measured using square units. Area can be found by counting square units or sometimes by multiplying. Estimating the number of square units inside a figure gives an estimate of the area.

• Volume is a measure of the amount of space inside a solid figure. Volume can be measured by counting the number of cubic units needed to fill a three-dimensional object.

• Some problems can be solved by breaking apart or changing the problem into simpler ones, solving the simpler ones, and using those solutions to solve the original problem.

16-8 Problem Solving Solve a Simpler Problem



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