Geometry and Algebra: Powerful When Together

Mix It Up Geometry and Algebra:Powerful When Together

The importance of the pictorial step • Remember • Introduce new concepts with manipulatives • Then go to a pictorial representation • Then teach the algorithm by relating the concrete and pictorial to the algorithm Mix It Up March 19, 2016

Models of Fractions • Area or region model • Usually part of whole • Pattern blocks, etc. (chapter activity) • Linear or measurement model • Part of whole, comparison, ratio • Fraction bars, Cuisenaire rods • Set model, only after have conservation of number • Ratio, comparison • Counters Mix It Up March 19, 2016

Multiplication of Common Fractions • Need conceptual understanding of meaning • More than 1-less than 1 is an important concept • Example: Will the answer be more than 1 or less than 1? • Will the answer be more than or less than ? • Will the answer be more than the first number or less than the first number? Mix It Up March 19, 2016

Division of Common Fractions • Need conceptual understanding of meaning (sometimes goes-in-to, sometimes what-part-of) • Example: • 6 ÷1/2 How many ½’s are in 6? More than 6, less than 6? • 6= 12/2; How many ½’s in 12/2? • 1/2 ÷3/2 “How many 3/2’s are in 1/2? More than 1, less than 1?” • With partner, use pattern blocks to explain answer ÷ Mix It Up March 19, 2016

Division of Common Fractions • More than 1, less than 1 is an important concept • Example: How many times goes into, more than 1 or less than 1? • How many times goes into, more than 1 or less than 1? • Will the answer be more than the first number or less than the first number? Mix It Up March 19, 2016

Using Number Lines Mix It Up March 19, 2016

Modeling division of fractions • Model with pattern blocks use one hexagon as one whole use two hexagons as one whole • Model with number lines • Model each with Cuisenaire rods l Mix It Up March 19, 2016

Homecoming Mums • If each streamer uses yards of ribbon, how many streamers can be made with 8 yards? • Hint: Use as 1 whole yard • How many streamers can be made from yards of white ribbon if each streamer needs yards of white ribbon. • Use Cuisenaire rods • Use a number line Mix It Up March 19, 2016

Habits of Mind Mathematical Habits of Mind are productive ways of thinking that support the learning and application of formal mathematics. The learning of mathematics is as much about developing these habits of mind as it is about understanding established results of mathematics. Patti talked about the Science Habits of Mind Mix It Up March 19, 2016

Mathematical Habits of Mind Geometric Habits of Mind • Reasoning with Relationships • Investigating Invariants • Generalizing Geometric Ideas • Balancing Exploration and Reflection Algebraic Habits of Mind • Doing-undoing • Building rules to represent functions • Abstracting from computation Mix It Up March 19, 2016

Square Table • A restaurant has square tables which seat four people. Stage 1 Mix It Up March 19, 2016

Square Table • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the square pattern blocks, create a concrete representation of this problem Stage 2 Stage 3 Mix It Up March 19, 2016

Triangle Table • A restaurant has trapezoidal tables which seat three people. Stage 1 Mix It Up March 19, 2016

Triangle Table • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the triangle pattern blocks, create a concrete representation of this problem Stage 3 Stage 2 Mix It Up March 19, 2016

Parallelogram Table • A restaurant has parallelogram tables which seat four people. Stage 1 Mix It Up March 19, 2016

Parallelogram Table • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the parallelogram pattern blocks, create a concrete representation of this problem Stage 3 Stage 2 Mix It Up March 19, 2016

Trapezoid Table • A restaurant has trapezoidal tables which seat five people. Stage 1 Mix It Up March 19, 2016

Trapezoid Table • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the trapezoidal pattern blocks, create a concrete representation of this problem Stage 3 Stage 2 Mix It Up March 19, 2016

Hexagonal Table • A restaurant has hexagonal tables which seat six people. Stage 1 Mix It Up March 19, 2016

Hexagonal Table • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the hexagonal pattern blocks, create a concrete representation of this problem Stage 3 Stage 2 Mix It Up March 19, 2016

Tables and Seats • On chart paper, illustrate your representation of the stages 1 through 5 • With a drawing of the tables and people • With a table • A graph • With a function (rule) • With a verbal description-be ableach number or term in your function is represented in the concrete model Mix It Up March 19, 2016

Thinking Back… • What did we do during this lesson? • If an outsider had walked into the room during this lesson, what observations could he/she have made? • What did he see? What did he hear? Where were the learners located and what were they doing? Where was the teacher located and what was he/she doing? • As a learner, were you engaged during this activity? How do you know? What evidence do you have that others were engaged? Mix It Up March 19, 2016

Mathematical Habits of Mind How did this activity address the habits of mind? Geometric Habits of Mind • Reasoning with Relationships • Investigating Invariants • Generalizing Geometric Ideas • Balancing Exploration and Reflection Algebraic Habits of Mind • Doing-undoing • Building rules to represent functions • Abstracting from computation Mix It Up March 19, 2016

Tables and Seats • Think about your classroom practice. • What is your role as teacher when a task such as the Table and Seats is being solved? • What is the role of your students? • Who is responsible for representing the ideas? • Are students given time and opportunity to represent their own thinking? • Are multiple representations shared and discussed publicly? Mix It Up March 19, 2016

Student Displays • Public record helps student remember the generalizations made, they can more easily build on these ideas. That is, they can use generalizations already established to build justifications for other conjectures. Mix It Up March 19, 2016

Representation-the process of representing an idea and the product, or result, of that process • Present information at three levels • Concrete – manipulatives • Pictorial – drawings, charts, graph • Abstract – number sentences • Representations can take many forms • Oral or written words, • symbols, • pictures, • diagrams, • tables, • graphs • or tactile forms with manipulatives Mix It Up March 19, 2016

As teach, consider … • What important ideas do I want to address in an activity? • What long-term learning goals does this activity relate to? • Which features of the habits of mind do I want this activity to elicit from my students? Mix It Up March 19, 2016

Questions For the Table Problem- List 2 or 3 questions that a teacher should ask that would encourage students to develop… • The three habits of algebraic thinking • Doing-undoing • Abstracting from computation • Building rules to represent functions • The four habits of geometric thinking • Reasoning with Relationships • Investigating Invariants • Generalizing Geometric Ideas • Balancing Exploration and Reflection Mix It Up March 19, 2016

Functions • Functions-the main concept in Algebra I in Texas • Describes how two (or more) quantities are related • Many times represented by tables and graphs • Example: number of eyes that 7 dogs have Mix It Up March 19, 2016

Patterns • Recognizing and using is critical to human thinking ability • Relate to expectations and predictions • Useful for predicting • Fundamental skill for algebraic thinking • Analyze what changes, what stays constant • Rate of change Mix It Up March 19, 2016

Recursive rule - each entry is found by adding subtracting, multiplying, dividing, etc. to previous entry (looks up or down a table) Explicit rule – uses a rule or equation based in the input term to determine the output (looks across the table) Mix It Up March 19, 2016

The Importance of Tables • As children record numbers in tables, they begin to develop idea of correspondence between the numbers • They attend to where the numbers go in the table and the meanings embodied in the position of the numbers • Learn certain #’s go in 1st column and certain # go in 2nd column • Helps them visually and cognitively look across columns and keep track of two numbers simultaneously-an important step in functional thinking • K-1 teacher may record • G2-3 students begin to record • G4-5 students record on t-table Mix It Up March 19, 2016

The importance of the pictorial step • Remember • Introduce new concepts with manipulatives • Then go to a pictorial representation • Then teach the algorithm by relating the concrete and pictorial to the algorithm • Clip-Rachel Mix It Up March 19, 2016

Geometry and Algebra: Powerful When Together