Meaningful Use of Symbols Two types of symbols most important to algebra, but unfortunately not well understood by many students: Equal Sign (=) and Inequality Signs (˂, ≤, ˃,≥) Variables Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Meaningful Use of Symbols Border Tiles Equal Sign and Inequality Signs Conceptualizing the Equal and Inequality Signs with a Balance True/False Sentences Relational Thinking After Border Tiles appears, have participants build an 8 x 8 square and “border” it. Generate expressions to determine the # of border tiles needed without counting. Discussion. 2. Equal Sign and Inequality Signs--#1, #2 Equal/Inequality with a Balance--#3, Tilt or Balance 4. True or False? #4 5. Relational Thinking—next 2 slides, then back to #4, bottom of page Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Relational Thinking Takes place when a student observes and uses numeric relationships between two sides of the equal sign rather than actually computing the amounts This type of thinking is a first step toward generalizing the relationships found in arithmetic to the relationships used when variables are involved Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Consider two different explanations for placing a 3 in the box for this open sentence: 2.4 ÷ = 4.8 ÷ 6 “Since 4.8 ÷ 6 is 0.8, then 2.4 ÷ something is also 0.8, so that must be 3.” “I noticed that 2.4 is half of 4.8, so I need to divide by a number half the size of 6 in order to maintain equivalence, so the number is 3.” Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Meaningful Use of Symbols Border Tiles Equal Sign Inequality Signs Equality and Inequality Signs with a Balance True/False Sentences Relational Thinking Open Sentences #5 then review relational thinking and do bottom of page. Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Solving Equations Using a Balance Scale #6 Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

The Meaning of Variables Variables Used as Unknown Values: One-Variable Situations Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Gary ate 14 strawberries, and Jeremy at some, too Gary ate 14 strawberries, and Jeremy at some, too. The container of 25 strawberries was empty! How many strawberries did Jeremy eat? How could we use what we learned about open sentences to write a statement representing this story problem? Can you use variables instead of ? Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Two or More Variable Situations: Systems of Equations Five Problems With Multiple Variables What’s My Weight? After first 4 problems, #7 Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

The Meaning of Variables Variables Used as Unknown Values: One-Variable Situations Two-or-More-Variables Situations Systems of Equations and Reflect How could you help students bridge the connection from these informal ways of solving to a more formal understanding of systems of equations? Stop and reflect is at bottom of #7 Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Use relational reasoning to determine which ones of the following systems of equations can be solved for or without using algebra. Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Simplifying Expressions and Solving Equations My Favorite No https://www.youtube.com/watch?v=Rulmok_9HVs Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

Variables Used as Quantities That Vary a) If you have $10 to spend on $2 granola bars and $1 fruit bars, how many ways can you spend all your money without receiving change? Use the table below to explore ways to spend your money. Number of $2 granola bars Number of $1 fruit bars Total amount spent ($10) Equation? #9 (a) on problem sheet Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp

b) You bought $1. 75 pencils and $1 b) You bought $1.75 pencils and $1.25 erasers from the school store, and you spent exactly $35.00. What might you have purchased? What equation represents this situation? Use the table below to explore values that work and to look for patterns. Number of $1.75 pencils Number of $1.25 erasers Total amount spent ($35) Equation? #9 (b) Teaching Student-Centered Mathematics by Van de Walle, Bay-Williams, Lovin and Karp