Connecting Models and Concepts in Eureka Math

Getting students to understand math concepts is to be able to SHOW them what is happening in the computations. 

When students use manipulatives as concrete examples and relate to picture models, they make strong connections and deep understanding. 

This type of learning is connected to long term memory and will be a platform for students to build on as they continue up in the program.
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Each type of concept has a model that is most appropriate for teachers to use at that students developmental level.  Models are not segregated into just specific age groups though, they can be repeated and seen again at various levels.

Number Towers
In the lower grades, number towers are both fun and helpful.  Students use unifix cubes of the same color to make a group of ten.  Then when they build on they start a different color. This is a way to build the concept of ten as a benchmark.  Students can easily count on when they see a ten first and add to it.

Number Path
A number path is a picture of cubes in a line showing the numerals on them.  This works for students like a number line.  The students can visually make jumps from one number to another to either find the sum or find the difference.

Number Bond
A number bond is a visual representation of part- part- total.  Students will learn the number bond visual in kindergarten and it will be used as a pictorial model year after year.  Students will learn to relate the number bond to addition and subtraction depending on whether the bond is missing one part or the total.

​Addition and Subtraction
A ten-frame strategy is introduced to students at the foundation.  Students see a ten frame as ten open squares.  Some of the squares are filled with dots.  The open squares are what the other part of ten is.  Students build fact fluency to ten using this visual.

Bundles of ten
These are pictorial model to show students how ten ones make a ten, then ten tens make a hundred, and ten hundreds make a thousand. 

​By showing this visually with bundles of straws, students can see the pattern and make the connection that base ten place value will also have ten to make the next place value.

Place Value Chart
A hundreds charts or place value chart is a mat or chart that students will use to visually see the different place values of a number. ​

Base-Ten Blocks
Base ten blocks are a similar concept to the bundles.  Instead of straws however, it is squares.  One single square is a one.  Ten squares in a stick are a ten.  Then one hundred squares or ten ten sticks are one hundred.  As students progress, they will begin to see the visual of a thousand.

Number Disks
Number disks are coin shaped pieces with place value numbers on them.  This can make it easy for students count by 1, 10, 100, or 1,000. 

​Students use these disks in place of bundles or base-ten blocks. ​

Money
Students learn the value and names of coins to count money.  When students are learning money it is helpful to provide coin manipulatives first so they can see the front and back of each coin.  After that, coins are presented to students in picture form.  They may either see the front or back of a coin at any time.

Multiplication Model
An array is a picture of equal groups. 

It is an arrangement of rows and columns that shows students how to represent repeated addition.

​It is a visual representation of a  a multiplication problem.

Fractions
In fractions, students need to rely on visuals to help them understand the concept of part of a whole.  First, they are introduced using area models showing a part shaded and the total amount of equal parts.  When students progress, they are shown fractions on a number line to use as a visual representation.

Tape Diagrams
Another term for tape diagrams can also be bar models.  Students are given this visual representation when solving story problems.  They see the bar model as a rectangle.  One represents the whole another in comparison to it means the part and then there is focus on the missing part to be found.  It can also be used reverse to show the two parts and the missing part represented as the total.  Students draw bar models throughout the Eureka Math program to solve.

 Students learn concepts and build on them using these pictorial models.  They are a firm foundation for the students to expand on as they progress through the Eureka Math program.