MATH STRATEGIES
FOR GRADES 3-5
IDEAS 2015
June 3, 2015
By Dr. Jenny Williams and
Dr. Nora Swenson
INTRODUCTIONS
Dr. Jenny Williams, Ed. D. SLP
Jenny Williams Educational Consulting
Adjunct Professor -
Coastal College of Georgia
Dr. Nora Swenson, Ed. D., CCC-SLP
Swenson Educational Consulting
MC
SC
BC
MC
Wt = 15 lbs
Wt = 8 lbs
SC
SC
MC
BC
Wt = 11 lbs
BC
Wt = ?
Visual Strategies
• Multiple representations for Concepts
• Mental math
• Building mathematical language
• Early emphasis on building students’
understanding of “ten”
• Number bonds, ten frames, place value
charts
• Concrete to Pictorial to Abstract (CPA)
• Model Drawing
Where are strategies used?
 In Co-taught classes
 Specialized Instruction used to focus and teach
concepts to students who are lacking skills
(IEPed and students who missed concept)
 Format- Station, parallel or alternative groups
 Accommodations or Universal Design for Learning -
for all members in class
 Give multiple representations for concept to enhance
learning for all
 Format-Whole group, station or alternative
Number Sense and Place Value Development
• Develop Cardinality (The final number
stated while counting a group is the total
number of the set.)
• Conservation of number (One-to-One
Correspondence)
• Subitization (Instantly recognizing units
of a number – usually 1-6)
Exemplars
Why Use Manipulatives?
• Makes abstract ideas concrete
• Gives students a way to get their hands on ideas
• Builds mathematical confidence
• Useful tools for problem -solving
• Makes learning math interesting and more
enjoyable
Turn to your partner:
• Identify three spacial words
• Briefly describe the previous mat
using those spacial words
• Identify three numeral words
and have the other partner
describe the mat with those
words
Use Literature for Context
• Read a story with numerical concepts
• Use blocks to count
• Divide blocks into “tens’ towers with students
counting up to 10 to help break into towers
• Students count “10s “ towers
• Count extra by 1’s
• Use literature as context for presenting object—
• Match blocks to numbers presented
• Put blocks into long string
• Break string into 10s towers
• Count by 10s and extras by 1s
Review Articles
• As a group we will define:
• Number Sense
• Subitizing
Sense of Number
(Exemplars)
• Basic Counting
• Understanding Size
• Number Relationships
• Patterns
• Operations
• Place Value
Subitizing
• See article on Subitizing
• Read and discuss with your table.
• What are some new ideas you gained from this article?
Subitizing
• Math Dictionary
• Subitizing is instantly recognizing a number when
group is presented
a
SUBITIZING
This exercise lets you know if the student
has a concept of numbers and what they
represent.
If they have to “peek” to see
how many fingers they are holding up,
they don’t understand numeracy.
What strategies can you use to help
them grasp this concept?
Hands for Counting-• If ten frames are difficult- relate back to Hands for
counting
Ten Frames
• Establish recognition of
numbers of objects –
• Counting, grouping objects
and leading to identifying
groups without counting
• Focus on multiples of 10
• Leads to number bonds or
groups of numbers that equal
10- and
• Do your students have
these?
• Where do you need to start?
Ten Frames - Game
• Do your students need practice at this
level?
• Do they all have this concept or do you
need to go back to the
Ten Frames originated with – “Hands”
Number Bracelets is
another subitizing format
• This is another way to orient students to
grouping numbers with “rekenrek” or
“arithmetic rack”
• Number lines made of beads are color
coded to help students group numbers
and increase “mental math” skills
Number Bracelets with Rekenrek Pattern
NUMBER BRACELETS
You will need the following:
10 chenille stems (pipe cleaners), all one color
55 beads, 40 red and 15 white
Choose one color of pipe cleaner and add red
beads for first 5 beads on
each bracelet
Numbers 6-10 will be white
Make a number bracelet for each number 1-10
Number Bracelets
Add beads to pipe cleaners, in increasing
numbers.
• Beads 1-5 will all be red.
• Bracelets 6-10 will have the first five beads red
and the remainder are white.
• After adding beads, bend pipe cleaners into
circles and twist ends together to make circle
large enough to go over your hand for a bracelet.
Number Bracelets Activity
• Each student will have 10 bracelets, one for each
number to use like an abacus counting frame.
• Sequence bracelets on arm from smallest number
to greatest numbers of beads.
• Have one students “draw” bracelets from a bag to
see who can call out the correct number of beads
first.
• http://www.thinkingmathematically.com
Rekenrek Rods
• Another way to stress
Focus on Base 10
• An Arithmetic Rack
To make a Rekenrek Rod
• One tongue blade
• 6 inch elastic string
• 5 red beads (right)-5 white beads (left)
• Put string through hole at one end of
tongue blade
• Add beads and add other end of string
through hole at opposite end of tongue
blade
Learning to Think Mathematically
With the Rekenrek
A Resource for Teachers
A Tool for Young Children
Jeff Frykholm
From Website:
Let’s try another
subitizing tool
• Make and Take:
• Make DOT CARDS
• You need:
• 5 x 7 cards
• Dot Markers
• Dot patterns
Domino Patterns to 6
Subitizing with dots
• Most adults see a dice and never count –
they know the number by the pattern
• Dot “patterns” help students identify the
number
• Common dot patterns are used with “number
cubes” like domino or dice patterns
• Use these patterns to make sets of cards
Dot Pattern Activity
• Use cards you made as flashcards
• Have group identify numbers when
patterns are flashed
PROBLEM SOLVING
WITH MODEL DRAWING
• The model drawing approach takes students
•
•
•
•
from the concrete to the pictorial to the abstract
stage.
Students create bars and break them down into
“units.”
The units create a bridge to the concept of an
"unknown” quantity that must be found.
Students can learn to use this strategy in the
primary grades and continue with it through the
middle grades.
There are two types of model drawings: discrete
and continuous
Discrete Model
• “crete” like concrete blocks
• Used with smaller numbers
• One-to-one correspondence
• Use with fraction problems
• Use with percentage problems
Concrete: Unifix cubes
Jan has 5 yellow cubes. Bill has 4
red cubes. How many cubes
altogether?
Jan’s
cubes
Bill’s
cubes
}
5
4
9
Concrete to pictorial
Jan has 5 dogs. Bill has 4 dogs.
How many dogs altogether?
Jan’s
dogs
Bill’s
dogs
}
5
9
4
Pictures to Dots
Jan has 5 dogs. Bill has 4 dogs.
How many dogs altogether?
Jan’s
dogs
• • • • •
Bill’s
dogs
• • • •
5
4
}
9
Numbers on the inside
Jan has 5 dogs. Bill has 4 dogs.
How many dogs altogether?
You may want to use large grid paper at first
Jan’s
dogs
Bill’s
dogs
1
1
1
1
1
1
1
1
1
4
5
9
Numbers on the outside
Jan has 5 dogs. Bill has 4 dogs.
How many dogs altogether?
First Grade
Jan’s
dogs
Bill’s
dogs
}
5
4
9
8 Steps of Model Drawing
1. Read the entire problem.
Jan has 5 yellow cubes. Bill has 4 red
cubes. How many cubes altogether?
2. Decide who is involved in the
problem.
Jan
Bill
Jan has 5 yellow cubes. Bill has 4 red cubes.
How many cubes altogether?
3. Decide what is involved in
the problem.
Jan’s cubes
Bill’s cubes
Jan has 5 yellow cubes. Bill has 4
red cubes. How many cubes
altogether?
4. Draw units of equal length.
Jan’s cubes
Bill’s cubes
STOP
Jan has 5 yellow cubes. Bill has
4
STOP
red cubes. How many cubes
altogether?
5. Reread the problem, one sentence at
a time, saying the word stop at each
comma or period and draw the
information on the unit bars.
5
Jan’s cubes
Bill’s cubes
4
Jan has 5 yellow cubes. Bill has 4
red cubes. How many cubes
altogether?
6. Determine the question and place the
question mark in the appropriate place in
the drawing.
Jan’s cubes
Bill’s cubes
5
4
?
Jan has 5 yellow cubes. Bill has 4
red cubes. How many cubes
altogether?
7. Work all the computation to the side or
underneath the drawing.
5
Jan’s cubes
Bill’s cubes
5+4=9
4
9
?
Jan has 5 yellow cubes. Bill has 4
red cubes. How many cubes
altogether?
8. Answer the question in a complete
sentence.
5
1
1
1
1
Jan’s cubes 1
Bill’s cubes
1
1
1
1
4
Jan and Bill have 9 cubes altogether.
9
Model Drawing Problems
• Jan has 2 more kittens than Sally. Sally has 6
kittens. How many kittens are there in all?
Continuous Model
• Use with larger numbers
• Talk about what part-part-whole means
• Talk about what whole –part-part means
• Adam has 65 matchbox cars. He has 13
more than Peter. How many cars does
Peter have? How many cars are there in
all?
• A necklace costs $15. Meg had $3 left
after buying the necklace. How much
money did Meg have at first?
Together Kim and Chris have 35 cards. Kim has
5 more cards than Chris. How many cards did
Kim have?
One basket holds 10 apples. How many
apples will 6 baskets hold?
Grandma baked 25 cookies. There were
6 children. She gave each child 4
cookies. How many cookies were left?
BRANCHING
Mental math activity
Number bonds come
before branching
Frames of ten come
before branching
Helps with understanding
the abstract of addition
and subtraction.
Branching with single-digit
addition: “Watch me”
What goes
with 7 to
make 10?
7 + 9
3 6
What goes
with 3 to
make 9?
10 + 6 = 16
Look at
handout
and change
numbers to
5 +8
5 + 8
5
Always keep
the tens on
the outside!
10 +3 = 13
3
14 + 5
10 + 4
5
10 + 9 = 19
21
20
Always keep
the tens on
the outside.
+
1
Always
think of
making
the
number
10.
18
8
20 + 10 + 9 = 39
10
Mental Math
• Mental math starts with number bonds in
kindergarten
• Number bonds are a precursor to mental
math
• Number bonds are used after students
have numeral recognition
• Number bonds are used after the
concrete stage
ESSENTIAL QUESTION
FOR THIS WORKSHOP
What math strategies can be used to
differentiate math lessons and improve
math performance?
Websites
• http://pages.cms.k12.nc.us/michaelwhite/
• http://www.nzmaths.co.nz/
• http://www.bbc.co.uk/education/mathsfile/ind
•
•
•
•
•
•
ex.shtml
http://www.quickmath.com/
http://mathforum.org/dr.math/
Mathwire (mathwire.com)
J. Meacham (jmeacham.com)
http://www.thinkingmathematically.com
Materials: www. Crystalspringsbooks.com
Key Concepts for K-2
• Provided multiple ways to develop
• “Number Sense” and
• Subitizing
with your students