Using manipulatives to develop

numbers and operations/ Fractions don’t

have to be frustrating

Kevin Dykema

Mattawan (MI) Middle School

kdykema@mattawanschools.org

Math for Real

Show how math is

used to solve a real-

world 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 Start with a flat and subtract

l Make the flat represent 100, the long 10, and the unit 1- you could also use the cube as 1000

l Have the students roll the cubes 10 times each-every 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!

l 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

• …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.

NCSM Position Statement

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)

Research Summary

Inside cover of booklet

A

C

R

Concrete

Abstract

Representational

Hands-On Learning Instructional Cycle

When students are

exposed to hands-

on learning on a

weekly rather than a

monthly basis, they

prove to be 72% of

a grade level ahead

in mathematics (Page 27)

Impact on Student Performance

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