## Numbers & Operations

##### Pre-kindergarten

##### Kindergarten through 2nd grade

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 through 5th grade

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 through 8th grade

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?