Using manipulatives to develop
numbers and operations/ Fractions don’t
have to be frustrating
Kevin Dykema
Mattawan (MI) Middle School
[email protected]
Math for Real
Show how math is
used to solve a realworld problem
Preferably from a
profession
300 words or less,
including several
problems
An ancient saying
I hear and I forget
I see and I remember
I do and I understand
I can’t Remember the formula
The definition of insanity
Doing the same thing over and over again and
expecting different results
implications for instruction
Concrete
Pictorial
Abstract
Representing Numbers
•
Use both color tiles and Base Ten Blocks to
represent numbers
•
Maybe also have the numbers on index card in
numeral form, in words, and with tally marks
•
To compare numbers, you can model each
number with color tiles (or other counters) and
line them up side by side to see which is taller
Linking Addition and
Subtraction
•
Make 2 trains of color tiles- 5 of one color and 3 of
another
•
Put the two trains together and write an addition
sentence that shows how you added the tiles together.
•
•
Now take away the 3 of the 2nd color and write the
subtraction sentence that shows your modeling.
The two different colors will benefit the visual kids.
Two Color Counter
Subtraction
Number lines
Make number lines with a Color Tile/ Unifix Cube/
Snap Cube/ Base 10 blocks as the unit
Base ten Number lines
10 More, 10 Less (1.NBT.5)
Subtracting a multiple of 10 from multiples of 10
(1.NBT.6)
l
On each turn
Roll 1or 2 number cubes and add the values
together
− Record the value of the roll
− Place the longs and units on the flat
− Add value of roll to previous score
−
l
If the roll takes you over 100, you lose
your turn
some variations
l
l
l
l
Start with a flat and subtract
Make the flat represent 100, the long 10, and the
unit 1- you could also use the cube as 1000
Have the students roll the cubes 10 times eachevery time they can decide if they are going to
add or subtract from their previous total- whoever
is closet to 100 at the end wins!
Use pennies, dimes, and dollars when working
with money
Base 10 Exchange
Roll a number cube. First person to 100 is the
winner
If you roll a 1, take 1 long
If you roll a 2, take 2 longs
If you roll a 3-6, take that number of units
Four Hundred



Each player starts with 400 points. The object is to
have the fewest points left after 10 rounds.
On each turn, the player tosses 2 number cubes,
places the digits in any order to form a number and
then subtracts the number from 400.
If you are unable to subtract with the number you
tossed, wait for your next turn.
4,321


Each player starts with 4,321 points. The object of
the game is to subtract a four-digit number from
4,321 and have the smaller difference.
On each turn, the player tosses four number cubes
and forms a number by combining the cubes in
any order. Then he or she subtracts the number
from 4,321


Each player compares the differences after each
round and the person with the smaller difference
earns a point.
Repeat game and the player with the more points
after 10 rounds win the game.
101 and Out


The goal is to arrive at a sum that is as close to
100 as possible without going over.
You will roll the number cube 6 times and each
time after you roll, you decide if you want it to go in
the 10s column or in the 1s column.
Target 300


The object is to be the player whose total is closest
to 300 after six rolls of a number cube- you may
be over or under.
After each of the 6 rolls, you decide if you want to
multiply the number by 10, 20, 30, 40 or 50.
Why manipulatives matter
NCSM Position Statement
• …in order to develop every student’s
mathematical proficiency, leaders and
teachers must systematically integrate the
use of concrete and virtual manipulatives
into classroom instruction at all grade
levels.
Research Summary
Research shows that the systematic use of visual
representations and manipulatives may lead to statistically
significant or substantively important positive gains in math
achievement. (Pages 30-31)
The evidence indicates, in short, that manipulatives can provide
valuable support for student learning when teachers interact
over time with the students to help them build links between the
object, the symbol, and the mathematical idea both represent.
(Page 354)
Inside cover of booklet
Hands-On Learning Instructional Cycle
C
Concrete
R
A
Representational
Abstract
Impact on Student Performance
When students are
exposed to handson learning on a
weekly rather than a
monthly basis, they
prove to be 72% of
a grade level ahead
in mathematics
(Page 27)
Inside cover of booklet
Virtual Manipulatives
• Virtual manipulatives are important tools for
teacher modeling and demonstration….
• …virtual manipulatives do not replace the
power of physical objects in the hands of
learners.
NCSM Position Statement
Fraction Language
Use partitioning (rather than dividing)
Use “ths” (rather than “over”)
Use simplify (rather than reduce)
Call them fractions (not improper)
Mathematics Teaching in the Middle School,
September 2013 by Jennifer- Bay Williams
FRACTIONS
K-2: Foundations laid with geometry and
with sharing
3-5: Meat of Fractions
6-8: Applied in proportional reasoning
and with slope
Color Tiles
Multiplication as Rectangular Array
Commutative Property
Distributive Property
Factors and Multiples
Primes and Composites
Benefits of Manipulative-Based Teaching
1.Students learn and achieve at a higher level
–Efficacy documented by decades of research studies
–Endorsed by leading organizations such as NCSM and NCTM, as well as
many teaching experts, such as Marilyn Burns
2.Mathematical process standards are fully
integrated into instruction
–Students are analyzing mathematical relationships as they communicate
mathematical ideas.
–Students are selecting tools and techniques to solve problems.
–Students are creating and using a variety of representations of
mathematical ideas
Benefits of Manipulative-Based Teaching
3.Mathematical concepts are taught consistently
with a common meaning across multiple
aspects of mathematics
–These ideas extend beyond whole numbers to fractions & decimals and
to algebraic expressions.
4.Students develop critical problem solving and
strategic thinking skills
–Variety of tools & strategies for different problems and settings
–More versatile and resilient than memorizing algorithms
two-digit multiplication with
base ten blocks
Loose Links- the Introduction
• Make a pile of 19 color tiles. We are going to
use these to make “chains” of equal length.
• If I were to roll a 5 on a number cube, this means
we need to make 5 chains and then we'll set
aside the leftover tiles or “loose links”.
• Now we'll do the same thing with the remaining
15 color tiles.
Tips for Getting Started
• Talk with your students about why
manipulatives are important tools for learning
mathematics.
• Set ground rules for using materials.
• Set up a system for storing materials.
• Provide time for free exploration of materials.
• Post class charts for reference.
• Make sure students handle the materials.
• Let parents get their hands on
manipulatives.
Booklet pages 29-33
“Teacher demonstrations alone
are as shallow as eating a
papaya in front of the class and
expecting them to know how it
tastes.”
Marilyn Burns
Manipulatives and Questioning
• Can you make a model to show that?
• What did you notice when…?
• Does anyone have the same answer but a
different way to explain it?
• What ideas have we learned before that
could be useful in solving this problem?
• Can you describe your method to us?
Can you explain why it works?
PBS TeacherLine Math Questioning Cards
http://www-tc.pbs.org/teachers/_files/pdf/TL_MathCard.pdf
Results from a Study on
Learning styles
In 1996
35-50% were auditory
35% were visual
15-30% were kinesthetic
In 2005
5-20% were auditory
Resources
Hands-on Standards series of books by
ETAhand2mind
Facebook- ETAhand2mind
Edweb.net- Implementing Common Core
Standards in Math