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 {a : a ∈ <, a ≥ 0} 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 {a : a ∈ <, a ≥ 0} 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. From Geometry to There are additive and multiplicative units and inverses. Numbers Multiplication distributes over addition. Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Geometry, the Section 3 Reprise: Some theorems on Circles 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 Circles in the common core
Geometry, the Common Core, and Proof
John T. Baldwin, Common Core Standards G-C2, G-C3 Andreas Mueller 2. Identify and describe relationships among inscribed angles,
Overview radii, and chords. Include the relationship between central,
From inscribed, and circumscribed angles; inscribed angles on a Geometry to Numbers diameter are right angles; the radius of a circle is perpendicular
Proving the to the tangent where the radius intersects the circle. field axioms
Interlude on 3. Construct the inscribed and circumscribed circles of a Circles triangle, and prove properties of angles for a quadrilateral An Area function inscribed in a circle. Side-splitter
Pythagorean Theorem
Irrational Numbers Central and Inscribed Angles
Geometry, the Common Theorem Core, and Proof [Euclid III.20] CCSS G-C.2 If a central angle and an inscribed John T. Baldwin, angle cut off the same arc, the inscribed angle is congruent to Andreas half the central angle. Mueller
Overview
From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Prove this theorem. Look at Euclid’s proof.
Diagram for proof
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 Are there any shortcuts hidden in the diagram? Pythagorean Theorem
Irrational Numbers Diagram for proof
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 Are there any shortcuts hidden in the diagram? Pythagorean Theorem Look at Euclid’s proof. Irrational Numbers Another Diagram for proof
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 Cyclic Quadrilateral Theorem
Geometry, the CCSS G-C.3 Common Core, and Proof Theorem John T. Let ABCD be a quadrilateral. The vertices of ABCD lie on a Baldwin, Andreas circle (the ordering of the name of the quadrilateral implies A Mueller and B are on the same side of CD) if and only if Overview ∼ ∠DAC = ∠DBC. From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Multiplication
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas The multiplication defined on points satisfies. Mueller 1 For any a, a · 1 = 1 Overview 2 For any a, b From Geometry to ab = ba. Numbers
Proving the 3 For any a, b, c field axioms (ab)c = a(bc). Interlude on Circles 4 For any a there is a b with ab = 1. An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Proving these properties
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview From Do activity multpropact.pdf. Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Commutativity of Multiplication
Geometry, the Common Given a, b, first make a right triangle 4ABC with legs 1 for Core, and Proof AB and a for BC. Let α denote ∠BAC. Extend BC to D so ∼ John T. that BD has length b. Construct DE so that ∠BDE = ∠BAC Baldwin, Andreas and E lies on AB extended on the other side of B from A. The Mueller segment BE has length ab by the definition of multiplication. Overview
From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Commutativity of Multiplication: finishing the proof
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas ∼ Mueller Since ∠CAB = ∠EDB by the cyclic quadrilateral theorem, ACED lie on a circle. Now apply the other direction of the yclic Overview quadrilateral theorem to conclude DAE =∼ DCA (as they From ∠ ∠ Geometry to both cut off arc AD. Now consider the multiplication beginning Numbers with triangle 4DAE with one leg of length 1 and the other of Proving the field axioms ∼ length b. Then since ∠DAE = ∠DCA and one leg opposite Interlude on Circles ∠DCA has length a, the length of BE is ba. Thus, ab = ba.
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Geometry, the Section 4: An Area function 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 Equal Content
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller Definition Overview [Equal content] Two figures P, Q have equal content if there From 0 0 0 0 Geometry to are figures P1 ... Pn, Q1 ... Qn such that none of the figures Numbers 0 0 overlap, each Pi and Qi are scissors congruent and Proving the 0 0 0 0 field axioms P ∪ P1 ... ∪ Pn is scissors congruent with Q ∪ Q1 ... ∪ Qn. Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Equal Content for parallelogram
Geometry, the We showed. Common Core, and Proof [
John T. Baldwin, Euclid I.35, I.38] Andreas Mueller Parallelograms on the same base and in the same parallels have the same area. (actually content) Overview Triangles on the same base and in the same parallels have the From Geometry to same area. Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational I-35.jpg Numbers Properties of Area
Geometry, the Common Core, and Proof Area Axioms John T. Baldwin, The following properties of area are used in Euclid I.35 and Andreas Mueller I.38. We take them from pages 198-199 of CME geometry.
Overview 1 Congruent figures have the same area. From Geometry to 2 The area of two ‘disjoint’ polygons (i.e. meet only in a Numbers point or along an edge) is the sum of the two areas of the Proving the field axioms polygons. Interlude on Circles 3 Two figures that have equal content have the same area. An Area 4 If one figure is properly contained in another then the area function
Side-splitter of the difference (which is also a figure) is positive.
Pythagorean Theorem
Irrational Numbers Area function to come.
Equal area
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview What are these axioms about? Is there a measure of area? From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Equal area
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview What are these axioms about? Is there a measure of area? From Geometry to Area function to come. Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Equal Content as equal area
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview Clearly equal content satisfies the first 3 conditions for equal
From area. Geometry to Numbers It doesn’t satisfy the last unless we know there are no figures of
Proving the zero area. field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers A crucial lemma Activity
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview From Do the activity– a crucial lemma - Massaging the picture. Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers A crucial lemma
Geometry, the Common Core, and Proof John T. Lemma Baldwin, Andreas Mueller If two rectangles ABGE and WXYZ have equal content there is a rectangle ACID, with the same content as WXYZ and Overview satisfying the following diagram. Further the diagonals of AF From Geometry to and FH are collinear. Numbers Proving the Proof. Suppose AB is less than WX and YZ is less than AE. field axioms
Interlude on Then make a congruent copy of WXYZ as ACID below. Let F Circles be the intersection of BG and DI . Construct H as the An Area function intersection of EG extended and IC extended. Now we prove F
Side-splitter lies on AH.
Pythagorean Theorem
Irrational Numbers Crucial Lemma: 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 Crucial Lemma: proof continued
Geometry, the Common Core, and Proof
John T. Baldwin, Suppose F does not lie on AH. Subtract ABFD from both Andreas Mueller rectangles, then DFGE and BCI F have the same area. AF and FH bisect ABFD and FIHG respectively. Overview
From So AFD ∪ DFGE ∪ FHG has the same content as Geometry to ABF ∪ BCIF ∪ FIH, both being half of rectangle ACHE (Note Numbers
Proving the that the union of the six figures is all of ACHE. field axioms
Interlude on Here, AEHF is properly contained in AHE and ACHF properly Circles contains ACH. This contradicts the 4th area axiom; hence F An Area function lies on AH.
Side-splitter
Pythagorean Theorem
Irrational Numbers Proof. Let the lengths of AB, BF , AC, CH, JK be represented by a, b, c, d, t respectively and let AJ be 1. Now ta = b and tc = d, which leads to b/a = d/c or ac = bd, i.e.(AB)(BG) = (AC)(CI ).
Crucial Lemma 2
Geometry, the Common Core, and Proof
John T. Baldwin, Claim Andreas Mueller If ABGE and ACID are as in the diagram (in particular, have the same area), then in segment multiplication Overview (AB)(BG) = (AC)(CI ). From Geometry to Numbers
Proving the Prove this. field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Crucial Lemma 2
Geometry, the Common Core, and Proof
John T. Baldwin, Claim Andreas Mueller If ABGE and ACID are as in the diagram (in particular, have the same area), then in segment multiplication Overview (AB)(BG) = (AC)(CI ). From Geometry to Numbers
Proving the Prove this. field axioms Proof. Let the lengths of AB, BF , AC, CH, JK be represented Interlude on Circles by a, b, c, d, t respectively and let AJ be 1. Now ta = b and An Area tc = d, which leads to b/a = d/c or ac = bd, function i.e.(AB)(BG) = (AC)(CI ). Side-splitter
Pythagorean Theorem
Irrational Numbers Diagram for Claim
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 By congruence, we have (AE)(AB) = (WZ)(XY ) as required. Side-splitter
Pythagorean Theorem
Irrational Numbers The area function
Geometry, the Common Core, and Proof Definition John T. Baldwin, Andreas The area of a square 1 unit on a side is (segment arithmetic) Mueller product of its base times its height, that is one square unit.
Overview From Theorem Geometry to Numbers The area of a rectangle is the (segment arithmetic) product of Proving the field axioms its base times its height.
Interlude on Circles Proof. Note that for rectangles that have integer lengths this An Area follow from Activity on multiplication as repeated addition. For function an arbitrary rectangle with side length c and d, apply the Side-splitter identity law for multiplication and associativity. Pythagorean Theorem
Irrational Numbers What is there still to prove?
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview From Do activity fndef.pdf Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Towards the area formula for triangles
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller Exercise Overview
From Draw a scalene triangle such that only one of the three Geometry to Numbers altitudes lies within the triangle. Compute the area for each Proving the choice of the base as b (and the corresponding altitude as h). field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Proof. Consider the triangle ABC is figure 1. The rectangles in figures 1,3, and 5 are easily seen to be scissors congruent. By the Claim, each product of height and base for the three triangles is the same. That is, (AB)(CD) = (AC)(BJ) = (BC)(AM). But these are the three choices of base/altitude pair for the triangle ABC.
Towards the area formula for triangles
Geometry, the Common Core, and Proof
John T. Baldwin, Theorem Andreas Mueller Any of the three choices of base for a triangle give the same
Overview value for the product of the base and the height.
From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Towards the area formula for triangles
Geometry, the Common Core, and Proof
John T. Baldwin, Theorem Andreas Mueller Any of the three choices of base for a triangle give the same
Overview value for the product of the base and the height.
From Geometry to Proof. Consider the triangle ABC is figure 1. The rectangles in Numbers figures 1,3, and 5 are easily seen to be scissors congruent. By Proving the field axioms the Claim, each product of height and base for the three Interlude on triangles is the same. That is, Circles (AB)(CD) = (AC)(BJ) = (BC)(AM). But these are the three An Area function choices of base/altitude pair for the triangle ABC. Side-splitter
Pythagorean Theorem
Irrational Numbers Diagram for the area formula for triangles
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 Geometry, the Section 5: The side-splitter theorem 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 Basic Tool
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller We have shown. Overview Assumption From Geometry to 1 Numbers The area of a triangle with base b and height h is given by 2 bh Proving the using segment multiplication. field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Proving the Sidesplitter
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller
Overview From side-splitter activity Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers But there may be lengths (e.g. π) which aren’t there. (E.g. if the underlying field is the real algebraic numbers). So CCSM was right to have both segment length and arc length as primitive terms.
Resolving Raimi’s worry
Geometry, the Common Core, and 1 Proof Using the formula A = 2 bh we have shown. John T. Baldwin, Theorem Andreas Mueller Euclid VI.2 CCSS G-SRT.4 If a line is drawn parallel to the Overview base of triangle the corresponding sides of the two resulting From triangles are proportional and conversely. Geometry to Numbers
Proving the The theorem holds for all lengths of sides of a triangle that are field axioms in the plane. Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Resolving Raimi’s worry
Geometry, the Common Core, and 1 Proof Using the formula A = 2 bh we have shown. John T. Baldwin, Theorem Andreas Mueller Euclid VI.2 CCSS G-SRT.4 If a line is drawn parallel to the Overview base of triangle the corresponding sides of the two resulting From triangles are proportional and conversely. Geometry to Numbers
Proving the The theorem holds for all lengths of sides of a triangle that are field axioms in the plane. Interlude on Circles But there may be lengths (e.g. π) which aren’t there. (E.g. if An Area function the underlying field is the real algebraic numbers).
Side-splitter So CCSM was right to have both segment length and arc
Pythagorean length as primitive terms. Theorem
Irrational Numbers But there may be lengths (π) which aren’t there. (E.g. if the underlying field is the real algebraic numbers). So CCSM was right to have both segment length and arc length as primitive terms.
Resolving Raimi’s worry
Geometry, the Common Core, and Proof
John T. Baldwin, Theorem Andreas Mueller If two triangles have the same height, the ratio of their areas equals the ratio of the length of their corresponding bases. Overview
From Geometry to The theorem holds for all lengths of sides of triangle that are in Numbers the plane. Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Resolving Raimi’s worry
Geometry, the Common Core, and Proof
John T. Baldwin, Theorem Andreas Mueller If two triangles have the same height, the ratio of their areas equals the ratio of the length of their corresponding bases. Overview
From Geometry to The theorem holds for all lengths of sides of triangle that are in Numbers the plane. Proving the field axioms But there may be lengths (π) which aren’t there. (E.g. if the Interlude on Circles underlying field is the real algebraic numbers).
An Area So CCSM was right to have both segment length and arc function length as primitive terms. Side-splitter
Pythagorean Theorem
Irrational Numbers Geometry, the Section 6 The Pythagorean Theorem 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 Pythagorean Theorem
Geometry, the Common Consider various proofs: Core, and Proof 1 Look at the Garfield paper. (Draw the diagram that got John T. Baldwin, blacked out by 137 years.) Andreas Mueller 2 Prove using similar triangles. (Hint: draw a perpendicular
Overview from the hypoteneuse to the opposite vertex; follow your
From nose. http://aleph0.clarku.edu/~djoyce/java/ Geometry to Numbers elements/bookVI/propVI31.html Proving the 3 The standard Euclid proof http://aleph0.clarku.edu/ field axioms djoyce/java/elements/bookI/propI47.html Interlude on ~ Circles 4 your favorite An Area function 5 98 proofs Side-splitter http://cut-the-knot.org/pythagoras/#pappa Pythagorean Theorem Which (proofs) are in your textbook? Irrational Note that each proof uses either area or similarity. Numbers Geometry, the Section 7 Irrational 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 CCSSM on irrationals I
Geometry, the Common Core, and Proof
John T. Baldwin, The Number System 8.NS Know that there are numbers that Andreas Mueller are not rational, and approximate them by rational numbers.
Overview The Number System 8.NS Know that there are numbers that From are not rational, and approximate them by rational numbers. 1. Geometry to Numbers Know that numbers that are not rational are called irrational. Proving the Understand informally that every number has a decimal field axioms expansion; for rational numbers show that the decimal Interlude on Circles expansion repeats eventually, and convert a decimal expansion An Area which repeats eventually into a rational number. function
Side-splitter
Pythagorean Theorem
Irrational Numbers CCSSM on irrationals II
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Expressions and Equations 8.EE Work with radicals and Mueller integer exponents. 1. Know and apply the properties of integer Overview exponents From Geometry to Use square root and cube root symbols to represent solutions Numbers to equations of the form x2 = p and x3 = p, where p is a Proving the field axioms positive rational number. Evaluate square roots of small Interlude on perfect squares and cube roots of small perfect cubes. Know Circles √ that 2 is irrational. An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers The Golden Ratio
Geometry, the Consider the diagram below where ABDC is a rectangle, Common Core, and BD = DE, CE = CG and GH is parallel to CD and Proof CD AC AF John T. BD = EC = AG = φ. Baldwin, Andreas Mueller
Overview
From Geometry to Numbers
Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Howard on Fibonacci http://homepages.math.uic.edu/~howard/spirals/
Irrationality of the Golden Ratio
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller CD m Suppose φ = BD = n for some integers m, n.
Overview How can you now express each of the lengths
From CD, BD, CE, AC, GH, AG? Geometry to Numbers Is the choice of such m, n possible? Proving the field axioms
Interlude on Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers Irrationality of the Golden Ratio
Geometry, the Common Core, and Proof
John T. Baldwin, Andreas Mueller CD m Suppose φ = BD = n for some integers m, n.
Overview How can you now express each of the lengths
From CD, BD, CE, AC, GH, AG? Geometry to Numbers Is the choice of such m, n possible? Proving the field axioms Howard on Fibonacci Interlude on http://homepages.math.uic.edu/~howard/spirals/ Circles
An Area function
Side-splitter
Pythagorean Theorem
Irrational Numbers