Building Addition and Subtraction Facts Through Number Sense

1
Building Addition and Subtraction Facts Through
Number Sense
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Oregon Kindergarten Grade-Level Foundations for
Calculations and Estimations related to Basic
Facts

• Read, write, order and identify whole numbers
  less than 10
• Use objects or pictures to decompose whole
  numbers
• Add and subtract pairs of numbers using less than
  10 concrete objects
• Mentally find one more or one less than a
  single-digit number
• Judge whether sets of objects have less than,
  more than or the same number as a reference set.

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Oregon 1st Grade Level Foundations for
Calculations and Estimations Related to Basic
Facts

• Use objects or pictures to decompose whole
  numbers to 10 (e.g., 541)
• Add and subtract with concrete objects
• Apply with fluency sums to nine and related
  subtraction facts
• Mentally add 10 to a single-digit number
• Represent situations using models of addition and
  subtraction

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Oregon 2nd Grade Level Standards for
Calculations and Estimations Related to Basic
Facts

• Develop and evaluate strategies for adding and
  subtracting whole numbers
• Apply with fluency sums to 18 and related
  subtraction facts
• Understand the relationship between addition and
  subtraction
• Use the commutative (42)(24) and associative
  (43)7 (37)4 properties of addition to
  simplify calculations
• Demonstrate the zero property for addition and
  subtraction

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Number Sense and Basic Facts

• The learning of basic facts, place value, and
  computation is directly affected by how well
  early number concepts have been developed.
• Elementary and Middle School Mathematics,
  Teaching Developmentally
• John A. Van De Walle

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Procedural Skills versus Number Concepts

• Procedural skills refer to things that are
  done by rote and simply memorized. Number
  concepts refer to understanding the concept not
  simply memorizing.

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Procedural Skills

• Rote Counting Naming the counting sequence in
  order. Children can often do this before they
  understand what it means to count.
• Numeral Recognition Recognizing the symbol for a
  given numeral (1-9) and saying its corresponding
  name.
• Numeral Writing Being able to write the symbol
  for a given numeral.

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Number Concepts

• One-to-one correspondence Understanding that
  each item in a group must be counted one and only
  one time. In order to do this, children need to
  develop a way of organizing and keeping track of
  the objects they are counting.
• Counting with meaning In order to count with
  meaning, a child must be able to say the counting
  number, have one-to-one correspondence AND
  understand that the last number they say names
  the quantity of the set. Example When asked how
  many objects are in a group, the child counts,
  one, two, three, four and then says there are
  four. This confirmation that there are four is
  an indicator that the child is counting with
  meaning.

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Number Concepts

• More, Less, Same Children develop these
  fundamental concepts by comparing sets of
  objects, NOT by looking at numerals.
• Recognizing Patterned Sets This fundamental
  skill refers to the ability to tell how many
  objects are in a set when they are arranged in an
  organized pattern. Common examples are dots on
  dice or dominos and dots in ten-frames.

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Number Concepts

• Number Relationships One More/Less Students
  need to actually understand that there is a
  relationship that never changes seven is one
  more than six and six is one less than seven.
  This can be practiced with ten frame cards and
  then numeral cards.
• Number Relationships Anchoring to 5 10
  Students need to build an understanding of the
  relationship of a given number to the numbers 5
  and 10. For example, students need to recognize
  that 7 is 2 more than 5 and 3 less than 10. A
  ten-frame is very important tool for building
  understanding of these relationships.

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Number Concepts

• Number Relationships Part-part-whole It is
  critical for children to develop the
  understanding that a number can be made up of two
  or more parts. For example, 6 can be thought of
  as a set of five and a set of one, or a set of
  four and a set of two, or a set of three and a
  set of three. You can give students 6 counters
  and ask them to arrange them in different ways on
  a ten frame. See the example below

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Activities to Build Number Sense

• Find the Same Amount A game to play to develop
  one-to-one correspondence, counting with meaning
  and the concept of more, less and the same.
• Quick Images Use dot cards or ten-frame cards
  for this game to help develop the ability to
  recognize patterned sets.
• Build it in Parts Children find how many
  different combinations for a particular number
  they can make using two parts. This focuses on
  the concept of part-part-whole.

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Stages for Learning Basic Facts

• Modeling the problem with objects or pictures
  Children commonly use this strategy while they
  are developing understanding of the concept.
• Counting Strategies A developmental stage that
  children go through after developing
  understanding of the concept. Eventually we want
  them to use thinking strategies to know or figure
  out their basic facts.
• Thinking Strategies Strategies students can
  quickly do mentally in order to solve basic facts.

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Counting Strategies for Addition and Subtraction

• Counting All The child counts the first set of
  objects, counts the second set of objects and
  then puts them together and counts all the
  objects.
• Counting On The child starts with one of the two
  addends then continues counting to add the other
  number.
• Order Property This is also called the
  Commutative property. As children develop an
  understanding of addition they realize that it
  doesnt matter in which order the numbers are
  added. They can switch the numbers around for
  ease in adding.

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Counting Strategies for Addition and Subtraction

• Counting Back The child starts with the original
  number and counts backward the number they are
  subtracting, usually using fingers or counters to
  keep track. This subtraction strategy works well
  when the structure of the problem is take away
  and the number being subtracted is small.
• Counting Up The child starts with the number
  being subtracted and counts up to the larger
  number. This subtraction strategy works well when
  the structure of the problem is comparing or
  finding the difference.

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Thinking Strategies that are Explicitly taught in
Bridges

• Zero Property The zero property is the identity
  property for addition and subtraction. When zero
  is added or subtracted from any number the answer
  is the original number. It retains its identity.
• Think One More/Less The strategy used when
  adding or subtracting 1. Think (not count) what
  number is one more/less.
• Think Two More/Less The strategy used when
  adding or subtracting 2. Think (not count) what
  number is one more/less.
• Combinations of Ten Knowing the pairs of numbers
  that go together to make ten. Students can get to
  ten from any one digit number. If I say 8 they
  say how many they need to get to ten.
• Doubles Adding two of the same number. (448)
• Neighbors An addition problem in which one
  addend is one more than the other (45). It can
  be solved by using a known doubles fact and
  mentally adding one more (448 so 459).

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Other Thinking Strategies

• Sharing Breaking numbers apart in order to add
  numbers that are friendly or easy to add.
  Example In solving 68, the child thinks 8 can
  be 6 and 2. So I have 662, I know 6612 and 2
  more is 14.
• Make a Ten This is a form of sharing in which a
  child works with numbers in chunks of 10. It is
  commonly used when adding 8 or 9. Example In
  solving 59, you can take one from the 5 to make
  the 9 a ten. Then you have 410.
• Think Addition Think addition to solve
  subtraction facts. For 10-6 you think 6 plus how
  many equal 10.
• Building up through Ten Used in subtraction,
  commonly when subtracting 8 or 9. Example In
  solving 14-8, think of adding on to 8 to get to
  14. If I start at 8, I need 2 more to get to 10.
  Then 4 more to get to 14. 246 so 14-86.
• Backing down through Ten This is a sharing
  strategy for subtraction similar to Make a Ten.
  Example For 15-7 a child thinks, 15-510 and I
  need to subtract 2 more because 752 so 10-2
  equals 8.

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Zeros The Identity Property of Addition

• Zero Property Adding any number to zero results
  in the number. In other words it retains its
  identity.
• Most first graders are ready for this strategy.

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Think One More/Less

• Students can think one more instead of having to
  count on. Understanding the concept of more and
  less is essential for this strategy.
• The subtraction version of this strategy is Think
  one less.
• Most first graders are ready for this strategy.

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Think Two More/Less

• This strategy is the same as think one more/less,
  but they are thinking two more/less. It requires
  more number sense and understanding of number
  relationships.
• The subtraction version of this strategy is Think
  two less.
• Most second graders are ready for this strategy.

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Adding Ten

• Understanding that when adding ten to any number
  that that number remains in the ones place and
  the tens place increases by 10.
• This strategy is explicitly taught in Bridges,
  Grade 1.

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Make a Ten (for adding nine)

• Once students can add ten to a number easily they
  are ready to use that to help them add nine.
• Students can think ten and then mentally
  subtract one from the answer. Ex. 9 6_ Think
  10616 so 9615.
• Students can also move one from the addend to the
  nine to make a ten. Ex. 96___ So take one from
  the 6 and give it to the 9 to make 10. Now you
  have 10515.
• This is an example of a sharing strategy,
  specifically, Make a Ten.

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Doubles

• These 11 facts are powerful for students to learn
  so they can use them to solve other facts
  (neighbors).
• First grade students can learn the doubles for
  0-5 and 10. That just leaves four doubles for
  second graders to learn.

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Neighbors (Doubles Plus One)

• This strategy uses the doubles. In order to use
  this strategy students must know their doubles
  facts.
• The strategy is to double the smaller number and
  add one.
• Some students will double the larger number and
  subtract one. This also works. Students can use
  which ever makes sense to them.
• This strategy is explicitly taught in Bridges,
  Grade 2.

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Commutative Property and Combinations of 10

• There are now 20 facts left to learn.
• Understanding the Order or Commutative Property
  cuts the facts down to 10.
• There are 2 facts in there that are combinations
  of 10 64 and 73. Students work on combinations
  to ten in Grade 1 Bridges
• That leaves 8 facts to learn.

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Sharing and Make a Ten

• Students can think three more or use a counting
  on strategy for 53 and 63.
• For the remaining facts students can use the
  strategies of Sharing or Make a Ten. Using a
  ten-frame to actually share the counters is a
  great way to show this strategy.

• Facts that are Left to Learn
• 53 63 83
• 74 84
• 85
• 86
• Example of using the Sharing strategy of Make a
  Ten to solve 7 4 I can break 4 into 3 and 1, I
  know that 7310 then I add the 1 to the 10 to
  get 11. So 7411.

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Time for Some Practice

• Play Class Games
• Zero Property Around the Class A child says a
  number and everyone adds or subtracts 0. You may
  have them say the whole equation.
• Add Ten A child says a number and everyone
  mentally adds 10 to the number. You may have them
  say the whole equation.
• Say the Ten Hold up a ten-frame or numeral card
  and have the children say the ten fact. For a
  card with 7 the response is 7310.
• Doubles Around the Class A child says a number,
  teacher says double it and everyone doubles it. A
  variation is after the child says a number (n)
  the teacher says, Class nn? and the class says
  the answer. You may have them say the whole
  equation.
• Double Dice Plus One Roll a ten sided die,
  double the number rolled and add one. Say the
  complete equation. For example, if you roll a 7
  you say 7815.
• Interval Training Students listen to a tape of
  numbers or the teacher saying numbers at specific
  intervals. They record the number said and then
  the apply the strategy you are working on to the
  number and write the answer.

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Games that Build Number Sense and Practice
Addition Basic Facts

• Ten-Frame Games
• The following three games are played like War
• Who Has More? Practices the concept of more and
  less.
• Terrific Tens Practices adding 10 to any single
  digit number
• Nifty Nines/Excellent Eights/Fantastic Fives The
  same as Terrific Tens, but practicing strategies
  for adding 9, 8 or 5. This is good game for
  practicing Sharing and Make a Ten strategies.
• The next game is played like Go Fish
• Make a Ten Students are practicing finding
  combinations with sums to ten.
• All of these games can be played with 2 to 4
  players. They can be adapted to practice other
  strategies.

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Strategy Building Flash Cards

• 6
• 6

• A dozen eggs 12
• They come in rows of 6
• 6 6

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Strategy Building Flash Cards

8 8
2 6 4
8
6
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What about Subtraction?

• Begin by focusing on the concept of subtraction.
• Remember the stages modeling, counting
  strategies, thinking strategies.
• Not all subtraction is take away, remember to
  model the difference model.
• Relate subtraction to addition.

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Some Subtraction Strategies

• Think Addition Think addition to solve
  subtraction facts. For 10-6 you think 6 plus how
  many equal 10.
• Building up through Ten Used in subtraction,
  commonly when subtracting 8 or 9. Example In
  solving 14-8, think of adding on to 8 to get to
  14. If I start at 8, I need 2 more to get to 10.
  Then 4 more to get to 14. 246, so 14-86.
• Backing down through Ten This a sharing strategy
  for subtraction similar to Make a Ten. Example
  For 15-7 a child thinks 15-510 and I need to
  subtract 2 more because 752 so 10-2 equals 8.

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More on Subtraction

• Focus on understanding subtraction in first
  grade. Some subtraction strategies are taught in
  Grade 2 Bridges
• Play the same games you did to practice addition
  strategies, but substitute a subtraction
  strategy. Example Play Think Half, or Subtract 1
• Make Strategy Building Flash Cards
• Use Interval Training

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What Are Reasonable Goals?

• Kindergarten should focus on procedural skills
  and number concepts, NOT basic facts.
• First grade should focus on number concepts,
  understanding the concepts of addition and
  subtraction, counting strategies and the thinking
  strategies of /- 0, /- 1, 10, doubles to 10
  and making combinations of ten.

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What Are Reasonable Goals?

• Second grade should focus on thinking strategies
  for addition and subtraction. So that by the end
  of second grade students have efficient
  strategies for sums to 18 and their related
  subtraction facts.

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How Do We Fit it In?

• Basic Fact practice should take place in very
  short chunks of time Number Corner, the first
  few minutes of math, a few extra minutes before
  lunch or going home are good places for this.
• Different grade levels have different needs and
  should have different goals.
• Some of the practice fits logically into
  curriculum units.

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It is Attainable!!!

• When learning basic facts focuses on using
  strategies, it is possible for ALL students to
  know their facts by the time they leave
  elementary school.