**I am so very pleased to welcome Dr. Shari Beck of Teaching to Make a Difference to Minds in Bloom. Her in-depth post on Developing Algebraic Thinking shows you exactly how to apply the standards to each elementary grade level.**

content strands included in the Common Core State Standards for Mathematics,

and a vertical progression of algebraic concepts can be seen in the specified

standards from one elementary grade level to the next. Beginning in Kindergarten, students are asked

to find missing numbers that sum with another number to make 10. In Grades 1-2, students must find unknown

whole numbers in all positions of addition and subtraction problems. The objective of finding missing unknowns is

extended to include multiplication and division in Grade 3 where students also

begin searching for arithmetic patterns.

By Grade 4, students are required to solve multistep word problems where

they begin to represent unknown quantities in an equation with a letter. In Grade 5, a focus on patterns is included

by having students identify apparent relationships between the corresponding

terms of two numerical patterns.

Students are actually required to write ordered pairs of numbers and

graph these ordered pairs in a coordinate plane.

can be an intimidating thought for some, especially for teachers who may have

struggled with an Algebra class in high school or college. While some may think that algebraic thinking

requires higher order thinking skills and may not be feasible for all students,

the developers of the Common Core integrated this type of thought process into

the standards with the idea that students should be exposed to algebraic

thought beginning as young as Kindergarten.

Teachers can accomplish the goal of each standard by using various

representations in the classroom, and this type of approach can foster creative

and critical thought processes. When

teachers are able to provide problems that actually relate to the real world of

the student and have those students relate ideas among the various

representations, the development of algebraic thinking with learners at all

elementary levels can occur and build a foundation for the learning of algebra

in future grade levels.

taught through the use of five representations which include:

- the concrete representation
- the verbal

representation - a representation by a table
- representation by a graph
- representation by an algebraic equation

given algebraic thinking problem can be increased as students progress and as

the grade level increases. The teacher

can first model each of the five representations for the students and then

allow students to creatively and critically develop their own representations

with future problems. Once students are

efficient and taking a given problem and developing the representations for it,

the teacher can then simply give a pattern and ask the students to creatively

write a problem to fit it along with critically thinking about the development

of the five representations. To further

explain the five representations, the following example problem, which would be

appropriate for students as young as Kindergarten, will be used.

As a reward for good behavior in class, the

teacher hands out stickers. For each day

that Ben is good, he earns 2

stickers. If Ben currently has 4

stickers, how many more days must he

have good behavior to have a total of 10 stickers?

problem, a teacher can use basic centimeter cubes (or any other object which

can be laid out in groups). A student begins by laying out 2 objects to represent

the number of stickers that Ben has received after the first day that he had

good behavior. Then, the student lays

out 4 objects in a separate pile to represent the total number of stickers that

Ben had received after his second day of good behavior. Thus, the second pile represents two days of

good behavior. Next, the student lays

out 6 objects in a third pile to represent the total number of stickers that

Ben had received after his third day of good behavior. At this point, the teacher can remind the

students of the posed question of “…how many more days must he have good

behavior to have a total of 10 stickers?”

Students continue laying out the next pile in the pattern until a total

of 10 objects are in the pile. For the

Kindergarten standard which focuses on algebraic thinking, students are being

required to find the number that makes 10.

First and second graders are finding the unknown where 6 + ? = 10. Third graders are being required to identify

an arithmetic pattern where there is a common difference from one pile of

objects to the next. All of these tasks

relate to the algebraic thinking standards for the specified grade levels.

asked to provide a description of the pattern by discussing how the number of

stickers is changing from one day to the next.

For example, a student may say that “Ben has 2 stickers after his first

day of good behavior. Ben has 4 stickers

after his second day of good behavior.

Ben has 6 stickers after his third day of good behavior, and his total

number of stickers is increasing by 2 each time. In order for Ben to have 10 total stickers,

he has to earn 4 more stickers. This would

require that he has had good behavior for a total of 5 days.” In grade levels where students are able to

write out the description, the teacher should require that it be written. Younger students may simply orally give the

description of the pattern.

table, two columns are used with a label of total days of good behavior and

number of stickers. Students are asked

to complete the table based on either the concrete representation or their

verbal description. As students progress

with the concept of algebraic thought, the teacher can require that they

determine the labels of the columns in the table. Students in Grade K-5 should be able to reach

this third level of representation for the problem. The remaining two representations may only be

posed in Grades 4 and above, depending on the level of the students. Advance students in Grades K-3 can be

challenged with the remaining two representations in an effort to further

develop their critical thought processes.

the table to plot the points in order to represent the problem with a

graph. Be sure that the students are

labeling the axes as appropriate. In

this case, the horizontal axis would be labeled based on the first column of the

table (total days of good behavior), and the second column would be labeled

based on the second column of the table (total number of stickers).

then be required to find a pattern in the number of total stickers. When the student realizes that the number of

stickers is twice that of the total number of days of good behavior, the

student can represent this expression as 2*D (or whatever letter they choose to

select for total number of days. Then,

students can choose a letter to represent the total number of stickers, such as

S, so that an equation can be formed as S=2*D.

While the “letters” are not yet referred to as “variables”, students are

laying the foundation for solving algebraic problems in higher grade levels

that involve variables.

to develop algebraic thinking in Grades K-5 can be developed through the use of

a graphic organizer such as the one shown below.

explanation of how to use this type of organizer in class can actually be

downloaded for free from my TPT store.

You can download a copy of Developing

Algebraic Thinking Through Five Representations by simply clicking on the link.

professor at Navarro College where she has taught for the past 12 years. Her career in education spans the last 20

years where she has a B.S. and M.S. in Mathematics and a Ph.D. in Curriculum

and Instruction. She enjoys working with

preservice teachers focusing on EC-Grade 8 and providing professional training

for inservice teachers in the field of mathematics. To view other products created by Shari, you

can visit her TPT

Store by clicking on the link.

Alicia says

Great post! I actually love teaching algebra to young students. It's like a game, or puzzle with a missing piece.

Shari Beck says

Thanks for the comment Alicia!