Maria has not started school full time yet; she spends her days observing the world, making connections, and practicing hard skills. Algebra is not in Maria’s vocabulary, but she is still gaining the foundational knowledge to understand algebraic concepts. Number sense provides that foundation – counting, number association, and recognizing and associating patterns. Studies have found that a child’s perceptual recognition and their ability to distinguish small Groups of objects play “an indispensable role in the development of arithmetic operations” (Von Glasersfeld, Subitizing: The Role of Figural Patterns in the Development of Numerical Concepts). Show Maria, at 12 months, three cookies, put them behind your back, and only give her two of them. She will know the difference! Having her follow simple patterns like red, yellow, green, red, yellow, green will help her to recognize and expand those patterns on her own. The focus of pre-K education should be to reinforce this abstract recognition, as well as more concrete numeral association.
Like in Numbers and Operations, the goal at this age is to establish a strong foundation in positive mathematical experiences and number relationships. Maria will start to understand number families and simple inverse operations. She will understand subtraction more fluently in conjunction with addition, instead of as a separate, intimidating operation. If she learns to “put together” and “take away” at the same time, she will have a firmer grasp of the underlying concepts of operation (Van de Walle & Lovin, Teaching Student-Centered Mathematics). As she learns to express these concepts, she will find applications in her everyday life- when building with blocks, playing games, and eating snacks- numbers and counting are everywhere! Her ability to notice patterns in objects like colored Blocks and small toys is a great first step to understanding Algebra. Start simple with AB patterns, and move up to ABBC patterns as her understanding progresses. Lining up things like bear counters so that the bears are different sizes and colors and positions gets Maria thinking about the number of different attributes and patterns that one simple object may possess. It is also helpful when presenting her with equations that she understand that the equal sign means both sides are the same or balanced and not the misconception that the equal sign means one side “produces” another side. You can use an empty box, a letter, or a symbol to represent the unknown. Be sure to put the variable both before and after the equal sign so Maria starts to understand balanced equations. For instance:
__ + 4 = 7 7 – __ = 4 4 + 3 = __
This will illustrate the relationships of the operations and allow for an easy transition to using variables in equations. “Skip” counting, or counting by a multiple of a number, as well as lining up plastic animals in rows with the same number and counting them out by the total number in each row sets her up for understanding multiplication.
At the end of 2nd grade, Maria should be able to mentally solve these algebra problems:
1. Write the following sentence as a mathematical expression: Subtract 15 from 45, and then divide by 3.
2. 37 + ___ = 100
Maria has been doing math since she was in diapers; she just didn’t know it. Now, though, she is starting to be able to name what she is doing, which, at this age, is lots of multiplication and division! Maria should learn these operations using patterns and conceptual understanding, not just rote memorization. If she can see reversibility, like: 32 ÷ 8=? is the same as 8 x ? = 32, she will have a firmer understanding of how multiplication and division are related. Similarly, Maria should understand the properties of multiplication: commutative property (6 x 4 = 24 and 4 x 6 = 24), associative property ([3 x 5] x 2 = [3 x 5 = 15] x 2= 30, or [5 x 2 = 10] x 3 = 30), and distributive property (8 x 5 = 40 and 8 x 2 = 16, so 8 x 7 = 8 x (5+2) = 56).
Using real world math problems encourages students to practice math all the time. Introducing distance, time, and money word problems that involve solving for the unknown may inspire Maria to figure out on her own how long it will take her mom to get to school, or how much change her mom should get at the grocery store.
At the end of 5th grade, Maria should be able to solve this algebra problem:
25, 36, ?, 64, 81, 100 What is the missing number in this pattern?
Maria should be getting a good grasp on the basics of algebra. She should know and be able to explain words like variable, exponent, and integer. She should have a clear understanding of fractions with integers, including negatives. Many students begin using calculators at this time; it is crucial that Maria truly learns the facts and concepts behind them, and uses the calculator only as a supporting tool. There are plenty of real world applications for what Maria is learning in algebra: figuring out the new price of a shirt that is marked 15% off, calculating the tip when her family eats out, knowing how long it will take to travel distances in the car, on the bike, or on foot.
Maria should be able to understand, via reading and writing, expressions and be able to solve problems using known vocabulary like quotient, product, double, triple, sum & difference. She should know how to work with radicals and integer exponents. She will be able to make connections between proportional relationships and line and linear equations. For examples, comparing a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
At the end of 8th grade, Maria should be able to solve this algebra problem:
Two hundred tickets for the school play were sold. Tickets cost $2.00 for students and $3.00 for adults. The total amount collected was $490.00. How many student tickets were sold?