Building a Solid Foundation in Number Sense

Lynn Rule lynn@mathrack.com

You can download this presentation at www.mathrack.com

“If you have built your castles in the air, your work need not be lost: that is where they should be. Now put the foundations under them.”

--David Thoreau

In this session:1. What is Number Sense?2. What are Number Relationships?3. What are the Four Fact Strategies?4. What is Fluency?

Learning is a messy business; constructing understanding is hard work …

We can’t simply tell children about numbers and think that they

will ‘know’ them. Children will not develop numeracy development

by merely circling answers and writing in workbooks. They have

to construct these understandings and build these relationships

in their minds, through experiences over time and through

discussing with others the relationships they encounter.

What Does Number Sense Mean to You?

• Making sense of numerical situations, and use what is known to figure out what is unknown

• Understanding number meanings, knowing relationships between numbers, knowing the size of numbers, and knowing the effects of operations on numbers

• Construct an understanding of number and build relationships, through experiences over time and through discussion

• Develop solid relationships with smaller numbers and use them as tools for understanding larger numbers

• Number sense is never complete—It is a lifelong process that is promoted through many and varied experiences with using and applying numbers

Do You Have A Good Understanding of What It Means for a Child to Have Number Sense?

To have good number sense…

Children must understand the following basic concepts:• Classification

• Patterning

• Subitizing

• Counting and Cardinality

• Number Relationships

• Decomposing and Composing Numbers

• Landmark Numbers

• Strategies for Computation

The Teacher’s Role in Developing Number Sense

• Encourage children to make sense of situations

• Invest time in the early years to allow children to develop solid number understandings

• Meaningful Discussions/ Math Number Talk• How did you get your answer? Can you prove it?• Can you explain it another way?• Did anyone think about it differently?• What is the most efficient strategy?• I agree or disagree because…• So you’re saying _________

So Where Do We Begin?

Daily Numeracy Routine

Math Routine• Read Aloud Connected to Math Standard• Numeral Identification (100’s Chart)• Build the number using various math models• One/ Two more and One/ Two less• Ten more and Ten less• How many more to get to the next friendly number• Counting Circle• Telling Time

Book Review

Context for Learning

Strong Arithmetic Knowledge

• Mathematics is not memorizing the basic facts and mastering the algorithms for the four operations

• By contrast children need to develop strong integrated mathematical knowledge

• Students need to know a lot about numbers

• Students need to solve many kinds of number problems

• Learning about numbers happens together with calculating and organizing numbers

Let’s think about 48?• What is 1 less than 48?

• What is 10 more than 48?

• How far back to 40?

• How many more to get to 50?

• Where is 48 positioned on a number line?

• What is half of 48?

Focus on Relationships• When we focus on relationships, it

helps give children flexibility when dealing with their basic facts and extending their knowledge to a new task. When we build a child’s number sense it promotes thinking instead of just computing.

Number Relationships• Spatial Patterns: Recognizing how many without

counting by seeing a visual pattern

• One/Two More or Less: Knowing which numbers are

one/two more and which are one/two less

• Landmarks of 5 and 10: How any number relates to 5

and 10

• Part Part Whole: Ability to conceptualize a number

as being made up of 2 or more parts

Activities to Develop Early Number Concepts and Number Sense

Spatial Relationships Subitizing • The ability to identify the number in a collection or group

without counting

• Collections that number between 1 and 5

• Saves time in counting and is often more accurate

• De-emphasizes a counting by ones strategy

• Assists in simple addition and subtraction

• Emphasis on automatic recognition of the number of items in a

collection

Spatial Relationships—Counting Recognizing how many without counting by seeing a visual pattern

• Activities that utilize concrete math tools (MathRack, ten frames, dice patterns)– Roll the Dice– Three Dice Roll– Write an Equation– Dot Plate Flash– Five Frame- also parts of 5 (Counting Spots)

– Spinner - also fact strategies (Counting Spots)

– Playing Cards (Counting Spots)

– Sponge Ten/Five Frames (Counting Spots)

– Dominoes

• “In the United States, the manipulatives most commonly used with young children are single objects that can be counted-Unifix cubes, bottle caps, chips, or buttons, while these manipulatives have great benefits in the very early stages of counting and modeling problems, they do little to support the development of the important strategies needed for automaticity.” (Fosnot)

How many dots are there?

How many dots are there?

How to use a MathRack?

• The MathRack has a built-in structure the

encourages children to use their knowledge about

numbers instead of counting one to one.

Mastering the MathRackTo Build Mathematical Minds

How to use the MathRack

Start position-all beads to the right.

How to use the MathRack

Read this side

One/Two More and Less• Allows students to be flexible thinkers and aides in

mental computation

• 9 + 5 = 10 + 5 - 1• 59 + 25 = 60 + 25 - 1

27

How many beads? How do you know?

One/Two More or Less: Counting On/Counting Back

Knowing which numbers are one/two more and which are one/two less

Activities that enhance the forward and backward number sequence –Hop the Line–Before and After–Counting On–Plus One–One More One Less–More or Less –Five frame spinner + - 1,2,3

Landmark Numbers

• Numbers important to assist mental computation, addition and subtraction

• Facilitate more effective counting and operation strategies

• Finger Patterns and Spatial Patterns

• Using counters with five frames, ten frames, and the MathRack to facilitate spatial relationships

How many beads? How do you know?

Benchmarks of 5 and 10

• Help children see how numbers relate to 5 and 10 becomes useful as they start to compute with numbers. ex. If you know that 7 is 5 + 2 or it is three less than 10 you could solve:

• 7 + 8 13 - 7 47 + 6• 5 2 3 4

• 5 + (2+8) (13-3) - 4

•

32

Turn and Talk What are all the possible ways

children will figure out how many?

Thinking Flexibly• 7 + 7 + 1

• 8 + 8 - 1

• 5 + 5 + 5

• 10 + 5

• 8 + (2 + 5)

• 20 - 5

Using The Number Path• Show 8 + 7 on your MathRack

• Model what you did on your Number Path

• Write down a number sentence that represents how you determine the total number of beads shown

• Show 9 + 8 on your MathRack

• Model what you did on your Number Path

• Write down a number sentence that represents how you determine the total number of beads shown

© 2013, Mathematically Minded

Using the MathRack to Subtract

• Show 15 - 9 on your MathRack

• Write down a number sentence that represents how you determine the total number of beads left

• What relationship did you use?

• Model what you did on your Number Path?

How many beads? How do you know?

Landmarks of 5 and 10 How any number relates to 5 and 10

Activities to build 5 plus and 10 plus facts– 5 plus Match–10 Plus Match– 5 plus Bingo– 10 plus Bingo– Rolling Cube (Counting Spots)

– Greg Tang gregtangmath.com/

Part-Part-Whole• The three prior relationships help build the

part-part-whole concept:

• Children who can decompose numbers, understand a number’s relationship to 5 and 10, and know one/two more and less will see both 8 + 7 and 38 + 7 in the same way.

• When we build a child’s number sense it promotes thinking instead of just computing.

Activities to build part part whole relationships

• Three sectioned plates• 2 Color Count• Turn Arounds• Seven Sentences• Whole--What are my parts?• How many more to make 5, 9, 10• Part- Part Whole Cards

Part - Part - WholeAbility to conceptualize a number as being made up of 2 or more parts

Number Sense Assessment Numbers 0-20

by Christina Tondevold www.therecoveringtraditionalist.com

• Spatial

• One/Two More and Less

• Benchmarks 5 and 10

• Part-Part-Whole

✴ Rationale, Assessment Checklist, Administering the Assessment for each relationship!

How We Learn Best! Memorize this eleven digit number:

25811141720

Now look for a connection (relationship) within numbers

2 5 8 11 14 17 20

Four Fact Strategies

• Plus Zero

• Doubles

• Make 10

• 10 plus something

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

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

1 1 2 3 4 5 6 7 8 9 10 11

2 2 3 4 5 6 7 8 9 10 11 12

3 3 4 5 6 7 8 9 10 11 12 13

4 4 5 6 7 8 9 10 11 12 13 14

5 5 6 7 8 9 10 11 12 13 14 15

6 6 7 8 9 10 11 12 13 14 15 16

7 7 8 9 10 11 12 13 14 15 16 17

8 8 9 10 11 12 13 14 15 16 17 18

9 9 10 11 12 13 14 15 16 17 18 19

10 10 11 12 13 14 15 16 17 18 19 20

Teach Relationships NOT Isolated Facts!121 facts--Purple = plus 0, Orange = doubles, Green = Make 10, Blue = 10 plus something, Lighter shades +1 - 1 or 2

What is Fluency?Fluent-Mathematically Proficient

• Accuracy-ability to produce an accurate answer

• Efficiency-ability to choose an appropriate, expedient strategy

• Flexibility-ability to use number relationships with ease in computation-compose and decompose numbers

Fluency and Flexibility

• Fluency - efficient and accurate

• Flexibility - multiple solution strategies determined by the problem

• Fluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective, inefficient, and damaging to student learning. --Henry and Brown

Kindergarten TEKSCount with and without objects forward and backward to at least 20Read, write, and represent whole numbers from 0 to at least 20, with and without objects or picturesCount a set of objects, up to at least 20, and demonstrate that the last number said tells the number of objects in the set regardless of their arrangementRecognize instantly the quantity of a small group of objects in organized and random arrangementsGenerate a set using concrete and pictorial models that represents a number that is more than, less than, and equal to a given number, up to 20Generate a number that is one more than or one less than another number up to at least 20Compare sets of objects up to at least 20 in each set using comparative languageCompose and decompose numbers up to 10 with objects and picturesModel the action of joining to represent addition and the action of separating to represent subtractionSolve word problems using objects and drawings to find sums up to 10 and differences within 10.Explain the strategies used to solve problems involving adding and subtracting within 10 using spoken words, concrete and pictorial models, and number sentencesRepresent addition and subtraction with objects, drawings, acting out situations, verbal explanations, or number sentences

1st Grade TEKSRecognize instantly the quantity of structured arrangements such as seen on a die or a ten-frameUse concrete and pictorial models to compose and decompose numbers up to 120 as so many hundreds, so many tens, and so many ones in more than one wayUse objects, pictures, and expanded and standard form to represent numbers up to 120Use concrete and pictorial models to determine the sum of a multiple of ten and a one-digit number in problems up to 99Use objects and pictorial models to solve word problems involving joining, separating, and comparing sets within 20 and unknowns as any one of the terms in the problemCompose 10 with two or more addends with and without concrete objectsApply basic fact strategies to add and subtract within 20 using strategies, including making a 10 and decomposing a number leading to a 10Explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentencesSkip count by twos, fives, and tens to 100Skip count by twos, fives, and tens to determine the total number of objects up to 120 in a setUse relationships to determine the number that is 10 more and 10 less than a given number up to 120Represent word problems involving addition and subtraction of whole numbers to 20 using concrete and pictorial models and number sentencesUnderstand that the equal sign represents a relationship where statements on each side of the equal sign are trueDetermine the unknown whole number in an addition or subtraction equation when the unknown may be any one of the three or four terms in the equationIdentify relationships between addition facts and related subtraction sentences such as 3+2=5 and 5-2=3Apply properties of operations as strategies to add and subtract such as if 2+3=5 is known, then 3+2=5

2nd Grade TEKSLocate the position of a given whole number on an open number lineName the whole number that corresponds to a specific point on a number lineRecall basic facts to add and subtract within 20 with automaticityUse mental strategies, flexible methods, and algorithms based on knowledge of place value and equality to add and subtract two-digit numbersSolve one-step and multistep word problems involving addition and subtraction of two-digit numbers using a variety of strategies based on place value, including algorithmsGenerate and solve problems situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 100Model, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joinedUse relationships and objects to determine whether a number up to 40 is even or oddUse relationships to determine the number that is 10 or 100 more or less than a given number up to 1,200Represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem

In this session...• We learned that Number Sense is never complete—It is a

lifelong process that is promoted through many and varied experiences with using and applying numbers.

• We learned the importance of Relationships--it helps give children flexibility when dealing with their basic facts and extending their knowledge to a new task. When we build a child’s number sense it promotes thinking instead of just computing.

• We learned the importance of using Fact Strategies in combination with relationships leads to fact fluency through understanding...not memorization

• We learned that to be Fluent with our facts we need to be efficient, accurate, and think flexibly.

“If you have built your castles in the air, your work need not be lost: that is where they should be. Now put the foundations under them.”

--David Thoreau

Resources

• www.mathrack.com

• www.therecoveringtradionalist.com/

• www.mathematicallyminded.com

• www.countingspots.com

• Contexts for Learning Mathematics by Catherine Fosnot

• How the Brain Learns Mathematics by David Sousa

• Last to Finish by Susan Allen and Jane Lindaman

• Mastering the MathRack by Christina Tondevold

• Number Talks by Sherry Parrish

• Teaching Student Centered Mathematics K-3 by John Van de Walle