# Numbers & Operations

##### Pre-Kindergarten
Jimmy was doing math years before he started school. In fact, from infancy, Jimmy could recognize small quantities. The process is called subitizing, instantly seeing quantity without counting. With practice, this innate ability leads to a conceptual understanding of addition and subtraction before Jimmy can even count to 12! It is important in this time to nourish a positive relationship with numbers to prevent intimidation later on. Before Kindergarten, Jimmy should be able to recognize the association between numbers and quantity- that one number correlates to only one quantity. He should also be able to recognize the patterns on dice and move towards knowing those numbers without counting the dots. This is the time kids begin to associate numerals with their quantities. Games that require Jimmy to roll a dice or spin a wheel and count out that amount- like Candyland, Chutes and Ladders, and Hi-Ho Cherrio- reinforce these skills.
##### Kindergarten - 2nd Grade
In early elementary school, Jimmy will reaffirm the connection between numerals and quantity, and should fluently be able to describe amounts using numbers. He will be establishing a foundation in basic operations (addition and subtraction), then building on it with simple fractions of ½ and ¼, an understanding of place value up to 1000, and the base 10 system. It is crucial in this step that Jimmy understands not only that 9 + 3 = 12 but also why 9 + 3 = 12. He should be able to break apart the number to explain his reasoning, to explain that 9 + 1 = 10 and 2 more equals 12. This knowledge is called Associative Property of Addition and is an important concept for Jimmy to grasp to deepen his understanding of numbers. Developmentally, Jimmy is at a great stage for cooperation. He is starting to be able to learn from peers and thrive off of fellow students’ excitement to learn. Studies have shown that students become much more engaged and motivated when they are working in cooperative Groups and more involved in the learning process (Steele, D. 1993). In addition to encouraging involvement and building confidence, this cooperative learning encourages Jimmy to talk about math with his classmates even outside the classroom, facilitating meaningful discussions that reinforce critical ideas about numbers (Fosnot and Dolk 2001; Gravemeijer and van Galen 2003; Van de Walle 2004). Jimmy is probably really excited at this age to show his knowledge, and it will be a point of pride to be able to add objects. It’s a great time to introduce the concept of money- that a single object can be worth more than 1; a dime is worth 10 while a penny is worth 1. At the end of 2nd grade, Jimmy should be able to solve this number sense problem: 653 – 243 = ? He should be able to walk through this using his knowledge of place value.
##### 3rd Grade - 5th Grade
Using his strong foundation of number relationships and simple operations, Jimmy will have an easy time building upon that foundation with multiplication and division, fractions and decimal places in 3rd – 5th grade. It is beneficial to encourage Jimmy to think of multiplication as quantity in Groups; division as the separation of the whole into Groups; and that a fraction is another way to express a division problem. That concept will naturally allow him to find patterns in multiplication and division facts and provide strategies to find answers, instead of learning them only by rote memorization. Fractions and rational numbers also come more naturally with this conceptual foundation of factors and multiples. At 11 years old, Jimmy’s brain has twice as many synapses as an adult brain, which makes it the ideal time to ensure strong mathematical hardwiring (SECPTAN). In addition to brain development, this is a crucial social development period for Jimmy, as peer pressures and learned attitudes can easily affect his affinity for learning. He might decide or be told that he is “bad” at math and lose the enthusiasm he had in early elementary school. Providing opportunities to succeed is crucial, and supporting the idea that math isn’t about speed- it’s about a thought process- will help Jimmy maintain his confidence to learn. Reinforcing the importance of math in Jimmy’s daily life will help keep his learning active. He will be more excited about it if he makes the connection that we use multiplication, division, fractions and decimals every day when we figure out things like how to divide up candy or how many Groups of kids will ride the Ferris Wheel before it’s our turn. Jimmy’s attention span is increasing rapidly, allowing for more complex thought and multi-step equations and word problems. For Jimmy’s classmates who have difficulty with attention span, it helps if they can visualize and express each step in a systematic way to help with comprehension. Mnemonic devices can be useful, like using the phrase “Dirty Monkeys Smell Bad” to help students understand the steps of a division algorithm (divide, multiply, subtract, bring down). It often helps for them to say each step as they are calculating. At the end of 5th grade, Jimmy should be able to solve these number sense problems: 1. Half of a school auditorium is needed to seat 3 equal-sized fifth grade classes. Make a visual fraction model to represent the whole auditorium when each class is seated in separate sections. Write an expression and solve to determine what fractional part of the auditorium a single fifth grade class will need. 2. How many times greater is the value of the digit 5 in 583,607 than the value of the digit 5 in 362,501?
##### 6th Grade - 8th Grade
By now, Jimmy’s brain is fully developed and he is starting to define who he is in relation to others. He is capable of more abstract thinking and has the attention span to manipulate multiple-step processes in his head. At this stage, he should have a firm understanding of the concept of fractions, decimals, and ratios; so it is a logical step to be able to multiply and divide them. He is learning about exponents, variables and formulas and how to use the order of operations to solve any math problem he faces. It should be natural for Jimmy to think through multiple step problems, using his understanding of base 10 to quickly reach solutions in his head. Jimmy should have a firm grasp of whole numbers, integers (positive and negative numbers), rational numbers (integers, fractions, terminating and repeating decimals) and irrational numbers (non-terminating and non-repeating decimals). He should have a visual understanding of fractions and decimals and when to add, subtract, multiply or divide them and what happens to the number when this is done. At the end of 8th grade, Jimmy should be able to solve this number sense problem: You earn \$8 an hour working at a café. For the next two years, you get a 10 percent raise per year. How much are you earning after your second raise?
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