Teaching CPA- Concrete, Pictorial, Abstract

When we teach a new strategy we always get out the manipulatives.  We collaborate and ask each other, "How can we teach this concretely so the students can gain a deeper understanding and show a visual model?"  Once they practice with concrete manipulatives,  they show it using a pictorial representation.  We might use other manipulatives to show multiple representations and to tap into the students different learning styles.  The second picture shows a student showing fractions using a bar model.  After she shows it with a bar model (pictorial), we have asked her to write a story problem to show how it can be used in the real world.  Last we teach the algorithm.  We have found when we take the time to teach CPA the students have a deeper understanding and need less time learning the algorithm.  We also try to tie in a game or activity to reinforce their learning.  They can work with partners or teams.  Many times we put a high student with a low student.  This gives one of them the teaching role and the other a role of learning from their peer. 

 Activities-Here is a fun concrete/pictorial activity to reinforce Place Value.  Each student has a place value mat on their desk.  These are free at  http://www.teacherspayteachers.com/Store/Barbara-Child/   

http://www.teacherspayteachers.com/Product/Place-Value-Mats-Ones-Thousands-Millions-Billions-Decimals-FREEBIE-769003  JANCY

The Place value strips are also available at http://www.teacherspayteachers.com/Product/Small-Place-Value-Card-Templates for only $2.00.  I keep them organized in folders (shown in Place Value).

 

PLACE VALUE

Place Value-This is in the front of my room so my students see multiple representations of Place Value. (There's a dime and a penny to compare to the base10 units.)

Place Value-
It is very important that students have a strong foundation of Place Value.  In upper grades we teach place value and review it the first three weeks of school and continue reviewing throughout the school year.  We use concrete and multiple representations to help the students visualize and make connections.  It is also an opportunity to tap into the variety of learning styles within a classroom.  We first start with base 10 blocks, then move to discs.  We use the discs so we can work with larger numbers.  Jana Hazekamp's book below shows other ways to use the discs.

Multiple Representations of Place Value

Organizing Base 10 Blocks

I got this idea from Bomb-Diggity Classroom.  Thank you!
My Base 10 Blocks are always disorganized in plastic bags!
I'm excited to finally have them in their own compartments!

Organizing Place Value Strips

http://www.teacherspayteachers.com/Product/Small-Place-Value-Card-Templates

I organize my place value strips in stronger folders I purchased at Wal-Mart.  I just stapled the sections with a long stapler.  These are smaller place value strips that are easier for the students to manipulate.  They are available using the link above.  I copied them on colored card stock then laminated.

Organizing Place Value Discs

Our Math coach, Barbara Child, bought these Utility Boxes through Maurice Sporting Goods.  We tried several others and these work well with bingo chips and the tiles.

Place Value Mats  -I love using these mats to teach Place Value!  You can use the periods you need and set the periods side by side.

FREE on TEACHERS PAY TEACHERS

SINGAPORE MODEL DRAWING

Model drawing has helped my students become very confident in solving word problems.  Instead of guessing, or giving up on word problems, they have strategies they can use to solve the problem and be successful.  Step 1 is to READ the entire problem.  Step 2 is to REWRITE the question in sentence form.  Step 3 is to determine WHO or WHAT is involved in the problem.  I tell my students they can do Steps 1, 2, and 3.  Once they do the first three steps it is easier for them to understand what the problem is asking.  When they get to Step 4- Draw the unit bars they are showing their work using a visual model.  Step 5.  They are re-reading or CHUNKING the problem.

 

This poster is on my wall for my students to refer to.  They also have a small copy in their own Singapore Model Drawing Books.

I love Model Drawing and I couldn't imagine teaching anything other than model drawing with my students!  I wish my own children would have learned problem solving this way.

My students struggle when they see a problem like the one below.  When they see the three times they want to show 4 unit bars.  I explain it first using pencils (below).  This makes it easier for them to understand.

  This book is a great resource if you are just starting out with model drawing.  Char Forsten is amazing and her ideas are very helpful.  When I started model drawing I was taught by Char Forsten and the SDE staff in Boston.  Before I went to Boston I used this book (actually her older version) with my students.

DECIMALS

 
http://www.teacherspayteachers.com/Product/Small-Decimal-Place-Value-Cards
Another great idea from Barbara Child, our math coach.
Here is a great way to organize place value discs and tiles.  We have one student show a number on the place value chart.  We use dice to make a number and another student will make the number using decimal place value strips.          

I help my students understand decimals by having them use base ten blocks (concrete).  This helps them break up 1 whole into tenths and show it on a number line.  It also gives them a deeper understanding of decimals.

Showing tenths on a number line.

      

Showing hundredths on a number line.

 

SPIRAL REVIEW of CONCEPTS --- ZOLTON DIENES

Beginning of the Year Calendar

Using a Math Calendar provides a great review of math concepts. This calendar reviews many concepts taught in 4th grade and reinforces knowledge of place value and bonding. It is a great way to tie math information into a calendar and teach math concepts at the same time.

Calendar also available on smart board to do with your students.

Teachers Pay Teachers

I review math concepts throughout the year.   I review with visuals in my room, mountain math (changed as to what I teach), in the lunch line,  on coat racks, and on my ceiling.  This is a coordinate grid I have on my ceiling.  After I read the book, The Fly on the Ceiling, it makes sense to the students.  I also have a yardstick with a big magnet on it where I can hang it at any coordinate.  There are also flies and spiders on certain coordinates.

Mtn. Math in corner              

My coat rack where numbers are colored according to Prime & Composite.

RECOMMENDED BOOKS

                                   Las Vegas Presentation
During our presentation we forgot to mention this valuable book by Jana Hazekamp.  This book gives ideas and steps on Why and How to teach basic math strategies.  Some of the teachers in our district did a book study using this book.  

Some of you might have seen Jana present at SDE.  She is another one of their amazing presenters!

SINGAPORE MATH

What is Singapore Math!  by Anni Stipek, Singapore Math Trainer

You may be wondering what Singapore math is all about, and with good reason.  What you may not know is that Singapore has led the world in math mastery for over a decade; its students become competent and proficient mathematicians at very early ages.  Even better, they grow at to be capable problem solvers who think mathematically with ease.  Wouldn't it be nice if your child could enjoy the same success with math?

First, you need to know that Singapore Math takes a slightly different mathematical approach than what you may be used to.  It revolved around several key number-sense strategies: 

(1) building number sense through part-whole thinking, (2) understanding place value, and (3) breaking numbers into decomposed parts or friendlier numbers, ones that are easier to work with in the four operations (addition, subtraction, multiplication and division).  Second, Singapore Math does something dramatically different when it comes to word problems.  It relies on model drawing, which uses units to visually represent a word problem.  Students learn to visualize what a word problem is saying so they can understand the meaning and thus how to solve the problem.
Third, we have mental math, which teaches students to calculate in their heads to memory, but mental math will teach him or her to do calculations without using paper and pencil.  Sure, your child will still need to commit facts strategies that don't require paper and pencil.
Fourth, the strategies taught in Singapore are layered upon one another.  One strategy is the foundation for another one.  For example, students need prior knowledge of bonding in order to be successful at strategies they will learn later on (like vertical addition) beginning with concrete (using manipulatives such as counters, number disks, dice, and so on), then moving to pictorial (solving problems where pictures are involved), and finally
Last Singapore Math teaches students to understand math in stages working in the abstract process, students learn numerous strategies to work with numbers and build (where numbers represent symbolic values) through the conceptual understanding. 

THINKING MAPS

This is the second year our school district has been using Thinking Maps.  Thinking Maps are effective not only in Language Arts, but it also helps the student mentally make connections of different math concepts.  Here are a couple we have made this year.  (The students draw their own maps.)

 MEASUREMENT

I have made this on the Smart Board.  It is on Teachers Pay Teachers for only  $1.00

Teachers Pay Teachers

http://www.teacherspayteachers.com/Product/Customary-Units-of-Measure-Thinking-Maps-790223