* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Algebraic Thinking - Math Methods 5360 ePortfolio
Survey
Document related concepts
Foundations of mathematics wikipedia , lookup
Mathematical model wikipedia , lookup
Ethnomathematics wikipedia , lookup
History of mathematics wikipedia , lookup
Mathematics and architecture wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
History of mathematical notation wikipedia , lookup
Patterns in nature wikipedia , lookup
Elementary algebra wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
System of polynomial equations wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Elementary mathematics wikipedia , lookup
Transcript
Algebraic Thinking Algebraic Reasoning What is Algebraic Thinking? What is Algebraic Thinking? Algebraic thinking can be broken into two categories: 1. Algebraic ideas 2. Mathematical thinking tools Algebraic Ideas Algebraic ideas are the building blocks of algebraic thinking: 1. Patterns 2. Functions 3. Variables Mathematical Thinking Tools Mathematical thinking tools are the analytical habits of mind, such as: 1. Problem solving 2. Representation 3. Reasoning When do students begin to think algebraically? Algebraic Thinking through the Grades Kindergarten Represent addition and subtraction in many ways Solve addition and subtraction problems within 10 Break apart numbers into pairs in more than one way (e.g. 5 = 4+1 5= 2 + 3 5 = 5 + 0) Find the number that makes 10 when added to given number: (e.g. If you have 6 what number is needed to make 10 Fluently add and subtract with 5. Algebraic Thinking Through the Grades First Grade Add and subtract within 20, using 2 or 3 whole numbers, to solve word problems Use commutative and associate properties Understand the meaning of the equal sign and determine if equations are true. Find the missing number in an addition or subtraction problem. Algebraic Thinking Through the Grades Third Grade Represent and solve problems involving multiplication and division Understand properties of multiplication and the relationship between multiplication and division Multiply and divide within 100 Solve problems involving the four operations Identify and explain patterns in arithmetic Algebraic Thinking Through the Grades Second Grade Add and subtract within 100 to solve one- and twostep problems. Mentally add and subtract within 20 Determine odd or even numbers and write and equation to express an even number. Skip count Use addition to find the total number of objects in rows and columns. Algebraic Thinking Through the Grades Fourth Grade Use the four operations with whole numbers to solve problems Gain familiarity with factors and multiples Generate and analyze patterns Algebraic Thinking Through the Grades Fifth Grade Write and interpret numerical expressions Analyze patterns and relationships Algebraic Thinking Through the Grades Sixth Grade Apply and extend previous understandings of arithmetic to algebraic expressions. Apply and extend previous understandings of numbers to the system of rational numbers. Reason about and solve one-variable equations and inequalities. Represent and analyze quantitative relationships between dependent and independent variables. Understand ratio concepts and use ratio reasoning to solve problems. Patterns, Functions, Variables Patterns are key factors in understanding mathematical concepts. The relationship between two sets of numbers is called a function. Looking for patterns in the relationships between two sets of numbers is a key way to develop students’ understanding of functions. Variables 1. 2. 3. Place holders for a specific number (2 + 3 = n) Unknown values (A=lw or n + n = 7) Represent quantities that vary (y = 2x) An important convention to remember is that the same symbol in an equation stands for the same number every time it occurs in the equation. Expressions and Equations Equations include an equal sign and state the equality of two expressions. Expressions are symbolic statements. They can either be arithmetic expressions such as 3 + 4, or algebraic expressions such as 2x – 3. Coefficients, constants, terms Ratios Ratios compare values A ratio says how much of one thing there is compared to another thing. 3 : 1 There are 3 red squares to 1 yellow square. Ratios Ratios can be shown in different ways: Using “ : “ to separate the values 3:1 Instead of using the “ : “ we can use the word “to” 3 to 1 Or write it like a fraction 3 ⁄ 1 Rates A rate is a little bit different than a ratio. It is a special ratio. It is a comparison of measurements that have two different units. 2 cases of soda for $10.00 Sara types 10 words in 5 seconds 4 lbs of hamburger costs $20.00 I drove 250 miles in 4 hours. Unit Rates Unit rates describe how many units of the first type of quantity corresponds to ONE unit of the second type of quantity. Miles per hour Earnings per week Photos per page People per vehicle Cost per pound $.75 for 3 apples = .75 ÷3÷ = 25¢ per apple Proportions Proportion says that two ratios (or fractions) are equal.