Developing Algebraic Thinking for 1st Graders


Common Core math expects that students will understand math concepts rather than learn processes to apply.  Because of this, often the practices of early grade math cause frustrations for parents as they ‘see’ the way to do the problems without going through the steps that are required by the practice assignments.  The problem lies when the steps that are asked of the students at these early grades are building a foundation for understanding, but seem to be more complication that necessary from the parents perspectives. 1st grade math standards in building the Algebraic foundations have several of these types of standards.  They are explained below:

Represent and solve problems involving addition and subtraction.

A-1.  Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
In this standard, students would be expected to use cubes or other manipulative objects or drawings to represent problems.  So, given the problem “If Sally has 7 cookies and gets 4 more from John”  the student would be able to explain the problem by drawing (or using objects – cubes, etc.) to show the 7 cookies and then adding 4 more and then, count the total to get the solution.  Additionally students should be able to represent the problem with an equation: 7 + 4 = 11.
A – 2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
This standards expects the same as the previous, but extends the expectation with the addition of a third number in the problem.

Understand and apply properties of operations and the relationship between addition and subtraction.

B-3  Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
This standard expects that students will begin to make some associations with general understandings about addition and subtraction.  By using those understandings, the students should begin to apply those understandings to mental math to become more fluent in their math skills.  So, if a student ‘knows’ that 5 + 7 = 12 then they will quickly recognize that 7 + 5 = 12.  Also, they should begin to see number relationships (for example, addends of 10). 
B-4  Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.

Add and subtract within 20.

C – 5  Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
This standard expects that students can use a number-line (for example) to ‘start on a number’ and count forward to represent addition and, for subtraction, count backwards. So, for example, “If Sally has 4 cookies and John gives her 5 more” the student could explain that starting ‘from 4 count 5 more’ to get to 9, or, “If Sally had 8 cookies and gave 3 to John” the student would start at 8 and count back 3 to get to 5.
C – 6  Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
This standards often gives parents difficulty because the practice problems often requires students to ‘break apart’ numbers in order to ‘make ten’, and then use that information to arrive at the sum.  Parents often feel that this step is unnecessary and causes them frustration.  It does provide practice in a skill that will later help with math fluency when the numbers get larger.  For example:  When students are asked to add 54 and 78, the students will be able to quickly recognize that ‘if they take 6 from the 78 to make 60 , they will have 60 +72, which become an easier mental math problem.  But, as 1st graders this seems like an unnecessary step.

Work with addition and subtraction equations.

D – 7  Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
This standard expects that students will understand an ‘equal sign’ and will recognize when ‘both sides are equal’.  This is a fundamental concept for algebra.
D – 8  Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. 
For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ – 3, 6 + 6 = _.  Again, another fundamental concept for algebra.