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Geometry, the Common Core, and Proof

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Geometry, the Common Core, and Proof Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Geometry, the Common Core, and Proof Overview From Geometry to Numbers John T. Baldwin, Andreas Mueller Proving the field axioms Interlude on Circles December 14, 2012 An Area function Side-splitter Pythagorean Theorem Irrational Numbers Outline Geometry, the Common Core, and Proof 1 Overview John T. Baldwin, 2 From Geometry to Numbers Andreas Mueller 3 Proving the field axioms Overview From Geometry to 4 Interlude on Circles Numbers Proving the 5 An Area function field axioms Interlude on Circles 6 Side-splitter An Area function 7 Pythagorean Theorem Side-splitter Pythagorean 8 Irrational Numbers Theorem Irrational Numbers Agenda Geometry, the Common Core, and Proof John T. Baldwin, Andreas 1 G-SRT4 { Context. Proving theorems about similarity Mueller 2 Proving that there is a field Overview 3 Areas of parallelograms and triangles From Geometry to Numbers 4 lunch/Discussion: Is it rational to fixate on the irrational? Proving the 5 Pythagoras, similarity and area field axioms Interlude on 6 reprise irrational numbers and Golden ratio Circles 7 resolving the worries about irrationals An Area function Side-splitter Pythagorean Theorem Irrational Numbers Logistics Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Common Core Geometry, the Common Core, and Proof John T. Baldwin, G-SRT: Prove theorems involving similarity Andreas Mueller 4. Prove theorems about triangles. Theorems include: a line Overview parallel to one side of a triangle divides the other two From proportionally, and conversely; the Pythagorean Theorem Geometry to Numbers proved using triangle similarity. Proving the field axioms This is a particularly hard standard to understand. What are Interlude on Circles we supposed to assume when we give these proofs. An Area Should the result be proved when the ratio of the side length is function irrational? Side-splitter Pythagorean Theorem Irrational Numbers Activity: Dividing a line into n-parts: Diagram Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Side-splitter Theorem Geometry, the Common Core, and Proof Theorem John T. Euclid VI.2 CCSS G-SRT.4 If a line is drawn parallel to the Baldwin, Andreas base of triangle the corresponding sides of the two resulting Mueller triangles are proportional and conversely. Overview From CME geometry calls this the `side-splitter theorem' on pages Geometry to Numbers 313 and 315 of CME geometry. Proving the Two steps: field axioms Interlude on 1 Understand the area of triangle geometrically. Circles bh 2 An Area Transfer this to the formula A = 2 . function Side-splitter The proof uses elementary algebra and numbers; it is Pythagorean formulated in a very different way from Euclid. To prove the Theorem side-splitter theorem, we have to introduce numbers. Irrational Numbers Towards proving Side-splitter Theorem Geometry, the Common Core, and Proof John T. Baldwin, CME geometry calls this the `side-splitter theorem' on pages Andreas 313 and 315 of CME geometry. Mueller Two steps: Overview From 1 Understand the area of triangle geometrically. Geometry to Numbers bh 2 Transfer this to the formula A = 2 . Proving the field axioms The proof uses elementary algebra and numbers; it is Interlude on formulated in a very different way from Euclid. To prove the Circles side-splitter theorem, we have to introduce numbers. An Area function But we won't do it via limits. Side-splitter Pythagorean Theorem Irrational Numbers Summary -transition Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller We were able to prove division of lines into n-equal segments Overview using congruence and parallelism. From Geometry to We could apply the side-splitter theorem. Numbers Proving the But we haven't proved the full-strength of side-splitter and we field axioms don't need it for division into equal segments. Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Geometry, the Section 1: From Geometry to Numbers Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers From geometry to numbers Geometry, the Common Core, and Proof We want to define the addition and multiplication of numbers. John T. Baldwin, We make three separate steps. Andreas Mueller 1 identify the collection of all congruent line segments as Overview having a common `length'. Choose a representative From segment OA for this class . Geometry to Numbers 2 define the operation on such representatives. Proving the field axioms 3 Identify the length of the segment with the end point A. Interlude on Now the multiplication is on points. And we define the Circles addition and multiplication a little differently. An Area function We focus on step 2; it remains to show the multiplication and Side-splitter addition satisfy the axioms. Pythagorean Theorem Irrational Numbers No. For this we need `addition on points'. We will worry about this later. For the moment think of the algebraic properties of fa : a 2 <; a ≥ 0g with ordinary + and ×. Properties of segment addition Geometry, the Common Adding line segments Core, and Proof The sum of the line segments AB and AC is the segment AD John T. Baldwin, obtained by extending AB to a straight line and then choose D Andreas Mueller on AB extended (on the other side of B from A) so that BD =∼ AC. Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Is segment addition associative? Theorem Does it have an additive identity? Irrational Numbers Does it have additive inverses? Properties of segment addition Geometry, the Common Adding line segments Core, and Proof The sum of the line segments AB and AC is the segment AD John T. Baldwin, obtained by extending AB to a straight line and then choose D Andreas Mueller on AB extended (on the other side of B from A) so that BD =∼ AC. Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Is segment addition associative? Theorem Does it have an additive identity? Irrational Numbers Does it have additive inverses? No. For this we need `addition on points'. We will worry about this later. For the moment think of the algebraic properties of fa : a 2 <; a ≥ 0g with ordinary + and ×. Defining Multiplication Geometry, the Common Core, and Proof John T. Baldwin, Andreas Consider two segment classes a and b. To define their product, Mueller define a right triangle1 with legs of length a and b. Denote the Overview angle between the hypoteneuse and the side of length a by α. From Now construct another right triangle with base of length b with Geometry to Numbers the angle between the hypoteneuse and the side of length b Proving the congruent to α. The length of the vertical leg of the triangle is field axioms Interlude on ab. Circles An Area function Side-splitter Pythagorean Theorem 1 Irrational The right triangle is just for simplicity; we really just need to make the Numbers two triangles similar. Defining segment Multiplication diagram Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers On the one hand, we can think of laying 3 segments of length a end to end. On the other, we can perform the segment multiplication of a segment of length 3 (i.e. 3 segments of length 1 laid end to end) by the segment of length a. Prove these are the same. Discuss: Is multiplication just repeated addition? Is multiplication just repeated addition? Geometry, the Common Core, and Proof Activity John T. Baldwin, We now have two ways in which we can think of the product Andreas Mueller 3a. What are they? Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Is multiplication just repeated addition? Geometry, the Common Core, and Proof Activity John T. Baldwin, We now have two ways in which we can think of the product Andreas Mueller 3a. What are they? Overview From On the one hand, we can think of laying 3 segments of length a Geometry to Numbers end to end. Proving the field axioms On the other, we can perform the segment multiplication of a Interlude on segment of length 3 (i.e. 3 segments of length 1 laid end to Circles end) by the segment of length a. An Area function Side-splitter Prove these are the same. Pythagorean Theorem Discuss: Is multiplication just repeated addition? Irrational Numbers Geometry, the Section 2: Proving the field Axioms Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers Addition and multiplication are associative and commutative. There are additive and multiplicative units and inverses. Multiplication distributes over addition. What are the axioms for fields? Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview From Geometry to Numbers Proving the field axioms Interlude on Circles An Area function Side-splitter Pythagorean Theorem Irrational Numbers What are the axioms for fields? Geometry, the Common Core, and Proof John T. Baldwin, Andreas Mueller Overview Addition and multiplication are associative and commutative.
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