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Teach Like An MVP MATHEMATICS VISION PROJECT 2013 WORKSHOP SALT LAKE CITY, UTAH
Who Are We? Sco% Hendrickson Joleigh Honey
Barbara Kuehl Travis Lemon
Janet Sutorius
Vision “The most necessary task of civiliza5on is to teach people how to think. It should be the primary purpose of our public schools . . . The trouble with our way of educa5ng is that it does not give elas5city to the mind. It casts the brain into a mold. It insists that the child must accept. It does not encourage original thought or reasoning, and it lays more stress on memory than observa5on.”
-‐-‐ Thomas A. Edison
Page 128
Teach Like an MVP
The MVP classroom experience begins by confronBng students with an engaging problem and then allows them to grapple with solving it. As students’ ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and exploraBons towards a focused mathemaBcal goal. As conjectures are made and explored, they evolve into mathemaBcal concepts that the community of learners begins to embrace as effecBve strategies for analyzing and solving problems.
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Two Different Types of Learning Experience
Classroom tasks
Ready – Set – Go
Page 129
Shopping for Cats and Dogs
Clarita is upset with Carlos because he has been buying cat and dog food without recording the price of each type of food in their accounBng records. Instead, Carlos has just recorded the total price of each purchase, even though the total cost includes more that one type of food. Carlos is now trying to figure out the price of each type of food by reviewing some recent purchases. See if you can help him figure out the cost of parBcular items on the receipts, and be prepared to explain your reasoning to Carlos. (For each of the following scenarios, assume that these are the purchase prices without sales tax.)
One week Carlos bought 3 bags of Tabitha Tidbits and 4 bags of Figaro Flakes for $43.00. The next week he bought 3 bags of Tabitha Tidbits and 6 bags of Figaro Flakes for $54.00. Based on this informaBon, can you figure out the price of one bag of each type of cat food? Explain your reasoning.
One week Carlos bought 2 bags of Brutus Bites and 3 bags of Lucky Licks for $42.50. The next week he bought 5 bags of Brutus Bites and 6 bags of Lucky Licks for $94.25. Based on this informaBon, can you figure out the price of one bag of each type of dog food? Explain your reasoning.
Carlos purchased 6 dog leashes and 6 cat brushes for $45.00 for Clarita to use while pampering the pets. Later in the summer he purchased 3 addiBonal dog leashes and 2 cat brushes for $19.00. Based on this informaBon, can you figure out the price of each item? Explain your reasoning.
One week Carlos bought 2 packages of dog bones and 4 packages of cat treats for $18.50. Because the finicky cats didn’t like the cat treats, the next week Carlos returned 3 unopened packages of cat treats and bought 2 more packages of dog bones. A`er being refunded for the cat treats, Carlos only had to pay $1.00 for his purchase. Based on this informaBon, can you figure out the price of each item? Explain your reasoning.
Carlos has noBced that because each of his purchases have been somewhat similar, it has been easy to figure out the cost of each item. However, his last set of receipts has him puzzled. One week he tried out cheaper brands of cat and dog food. On Monday he purchased 3 small bags of cat food and 5 small bags of dog food for $22.75. Because he went through the small bags quite quickly, he had to return to the store on Thursday to buy 2 more small bags of cat food and 3 more small bags of dog food, which cost him $14.25. Based on this informaBon, can you figure out the price of each bag of the cheaper cat and dog food? Explain your reasoning.
Summarize the strategies you have used to reason about the price of individual items in the problems given above. What are some key ideas that seem helpful?
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Can You Get to the Point, Too?
Part 1 In “Shopping for Cats and Dogs,” Carlos found a way to find the cost of individual items when given the purchase price of two different combinaBons of those items. He would like to make his strategy more efficient by wriBng it out using symbols and algebra. Help him formalize his strategy by doing the following:
For each scenario in “Shopping for Cats and Dogs” write a system of equa5ons to represent the two purchases.
Show how your strategies for finding the cost of individual items could be represented by manipulaBng the equaBons in the system. Write out intermediate steps symbolically, so that someone else could follow your work.
Once you find the price of one of the items in the combinaBon, show how you would find the price of the other item.
Page 108
Can You Get to the Point, Too
Part 2 WriBng out each system of equaBons reminded Carlos of his work with solving systems of equaBons graphically. Show how each scenario in “Shopping for Cats and Dogs” can be represented graphically, and how the cost of each item shows up in the graphs.
Part 3
Carlos also realized that the algebraic strategy he created in part 1 could be used to find the points of intersecBon for the “Pet Si%ers” constraints. Use the elimina5on of variables method developed in part 1 to find the point of intersecBon for each of the following pairs of “Pet Si%er” constraints.
Start-‐up costs and space constraints
Pampering <me and feeding <me constraints
Any other pair of “Pet Si%er” constraints of your choice
Page 108
Taken Out of Context Write a shopping scenario similar to those in “Shopping for Cats and Dogs” to fit each of the following systems of equaBons. Then use the eliminaBon of variables method you invented in “Can You Get to the Point, Too” to solve the system. Some of the systems may have interesBng or unusual soluBons. See if you can explain them in terms of the shopping scenarios you wrote. Three of Carlos’ and Clarita’s friends are purchasing school supplies at the bookstore. Stan buys a notebook, three packages of pencils and two markers for $7.50. Jan buys two notebooks, six packages of pencils and five markers for $15.50. Fran buys a notebook, two packages of pencils and two markers for $6.25. How much do each of these three items cost? Explain in words or with symbols how you can use your intuiBve reasoning about these purchases to find the price of each item.
€
3x + 4y = 235x + 3y = 31⎧ ⎨ ⎩
€
2x + 3y =144x + 6y = 28⎧ ⎨ ⎩
€
3x + 2y = 209x + 6y = 35⎧ ⎨ ⎩
€
4x + 2y = 85x + 3y = 9⎧ ⎨ ⎩
1.
2.
3.
4.
Page 109
Learning Progressions or Trajectories
“Trajectories involve hypotheses about the order and nature of the steps in the growth of students’ mathemaBcal understanding, and about the nature of the instrucBonal experiences that might support them in moving step by step towards the goals of school mathemaBcs.”
Learning progressions cannot be derived solely from “the disciplinary logic of mathemaBcs itself.”
◦ Phil Daro, Learning Trajectories in MathemaBcs: A FoundaBon for Standards, Curriculum, Assessment and InstrucBon
Page 131
What did you noBce about the differences in the three tasks that you just did?
Page 131
The Flow of Learning Experiences
Develop Understanding
Solidify Understanding
PracBce Understanding
The Learning Cycle
Page 131
Learning Cycles as Trajectories
Thinking Through a Learning Cycle: ◦ Ideas: What do we want students to know? ◦ Strategies: What do we want students to be able to do? ◦ RepresentaBons: How do we want students to make their thinking visible?
How do ideas, strategies and representaBons differ from the beginning to the end of a learning cycle?
Develop Understanding
Solidify Understanding
PracBce Understanding
Page 132
Develop Understanding Tasks ◦ Low threshold, high ceiling (easy entry, but extendable for all learners) ◦ Contextualized (problemaBc story context, diagrams, symbols) ◦ MulBple pathways to soluBons or mulBple soluBons ◦ Surface student thinking (misconcepBons and correct thinking) ◦ Purposeful selecBon of the vocabulary, numbers, etc. to reveal rather than obscure the mathemaBcs ◦ IntroducBon of a number of representaBons
Page 132
Solidify Understanding Tasks ◦ Features of the task (context, scaffolding quesBons, constraints) focuses students’ a%enBon on: • looking for pa%erns and making use of structure • looking for repeated reasoning and expressing regulariBes as generalized methods
• a%ending to precision in language and use of symbols • construcBng viable arguments and criBquing the reasoning of others
• using representaBons and tools strategically for the purpose of developing deeper levels of understanding of mathemaBcal ideas, strategies, and/or representaBons
Page 132
Practice Understanding Tasks
Prac5ce tasks focused on refining understanding ◦ Task allows student to use reasoning habits to contextualize (symbolic to real-‐world) and decontextualize (real-‐world to symbolic) problems and situaBons. ◦ Tasks involve sufficient complexity to refine mathemaBcal thinking beyond rote memorizaBon ◦ The task requires a high level of cogniBve demand because students are required to draw upon mulBple concepts and procedures, make use of structure and recognize complex relaBonships among facts, definiBons, rules, formulas and/or models
Page 133
Practice Understanding Tasks
Prac5ce tasks focused on acquiring fluency ◦ Task involves either reproducing previously learned facts, definiBons, rules, formulas or models; OR drawing upon previously learned facts, definiBons, rules, formulas or models; OR commiong facts, definiBons, rules, formulas or models to memory ◦ An appropriate vehicle of pracBce is selected (e.g., rouBnes, games, worksheets, etc.) which allows for reproducing, drawing upon, or commiong to memory previously examined mathemaBcs ◦ Task focuses on a broad definiBon of fluency: accuracy, efficiency, flexibility, automaBcity
Page 133
Mathematical Practices and Learning Cycles
Show and Tell
• Teacher’s Role: • Provide examples, definiBons and properBes, procedures, and models
PracBce
• Students’ Role • Replicate, drill, pracBce, and memorize
Develop
• Teacher’s Role: Provide experiences, orchestrate discussions using the 5 pracBces
• Students’ Role: NoBce pa%erns; make conjectures; create arguments
Solidify
• Teacher’s Role: Provide experiences, orchestrate discussions using the 5 pracBces
• Students’ Role: See structure; see regulariBes; a%end to precision; create and criBque
PracBce
• Teacher’s Role: Provide a vehicle for pracBce, provide feedback • Students’ Role: Reason quanBtaBvely; work towards efficiency, flexibility, accuracy; Apply (model with mathemaBcs)
Comprehensive MathemaBcs InstrucBon Model InformaBon Transmission Model
Prob
lem Solving, Reasoning and
Proving,
Mod
eling
Page 133
Something to Talk About A Develop Understanding Task Cell phones o`en indicate the strength of the phone’s signal with a series of bars. The logo below shows how this might look for various levels of service.
Figure 1 Figure 2 Figure 3 Figure 4
1. Assuming the pa%ern conBnues, draw the next figure in the sequence.
2. How many blocks will be in the size 10 logo?
3. Examine the sequence of figures and find a rule or formula for the number of Bles in any figure number.
Page 5
Solidify Understanding Work the task: What mathemaBcal ideas are students asked to draw upon? What representaBons were used in your group? Preparing your chart paper: Title of task Main mathemaBcal purpose or goal Show your work on the task including the representaBons you used—tables, graphs, equaBons, or diagrams
Pages 14 -‐ 60
Making Your Chart Read the teacher notes: What is the purpose of the task? What standards are addressed in the task? What notaBon, vocabulary, or procedures are used?
How Does It Grow? A Practice Understanding Task For each relaBon given: IdenBfy whether or not the relaBon is a funcBon; Determine if the funcBon is linear, exponenBal, quadraBc or neither;
Describe the type of growth Create one more representaBon for the relaBon.
Page 60
Functions Learning Progression Linear Func5ons
Secondary MathemaBcs I
Secondary MathemaBcs II
Concepts and Defini5ons Constant rate of change Exchange rate between variables ArithmeBc sequences are discrete linear funcBons Comparing linear and quadraBc funcBons Comparing linear and exponenBal funcBons
Module 3 Module 2 Module 4 Modules 3 and 4
Module 1
Procedures Solving systems of linear equaBons and inequaliBes WriBng equaBons of lines given various informaBon Changing the form of a linear equaBon
Module 2 Modules 2, 3, and 4 Module 4
Tools Graphs of linear funcBons Story contexts for linear funcBons Recursive formulas for arithmeBc sequences Point/slope, slope/intercept, and standard form of explicit equaBons Tables (including using the first difference)
Modules 2, 3, and 4 Modules 2, 3, and 4 Module 3 Module 4 Modules 2, 3, and 4
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Functions Learning Progression Exponen5al Func5ons
Secondary MathemaBcs I
Secondary MathemaBcs II
Concepts and Defini5ons Constant raBo between terms or growth by equal factors over equal intervals Geometric sequences are discrete exponenBal funcBons Comparing linear and exponenBal funcBons Comparing exponenBal and quadraBc funcBons
Module 3 and 4 Module 4 Module 4
Module 1
Procedures WriBng equaBons of exponenBal funcBons Solving basic exponenBal equaBons (without using logarithms) Recognizing equivalent forms and using formulas
Modules 3 and 4 Module 4 Module 4
Tools Graphs of exponenBal funcBons Story contexts for exponenBal funcBons Recursive formulas for geometric sequences Explicit equaBons for exponenBal funcBons Tables (including using the first difference)
Modules 3 and 4 Modules 3 and 4 Module 3 Modules 3 and 4 Modules 3 and 4
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Functions Learning Progression Quadra5c Func5ons
Secondary MathemaBcs I
Secondary MathemaBcs II
Concepts and Defini5ons Linear rate of change Product of two linear factors
Module 1 Module 1
Procedures Factoring CompleBng the Square Graphing using transformaBons
Module 2 Module 2 Module 2
Tools Graphs of quadraBc funcBons Story contexts for quadraBc funcBons Recursive formulas for quadraBc sequences Factored, vertex, and standard form of explicit equaBons for quadraBcs Tables (including using the first and second differences)
Modules 1 and 2 Modules 1 and 2 Module 1 Module 2 Modules 1 and 2
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Functions Learning Progression Other Func5ons
Secondary MathemaBcs I
Secondary MathemaBcs II
Concepts and Defini5ons RelaBonship between variables such that each input has exactly one output Domain Range
Module 4 Modules 3 and 4 Modules 3 and 4
Module 4 Module 1, 2, 4 Module 1, 2, 4
Procedures TransformaBon of funcBons Combining funcBons
Modules 5 and 7 Module 5
Modules 2 and 4 Module 4
Tools Graphs of funcBons Story contexts for funcBons Recursive formulas for funcBons Explicit equaBons using funcBon notaBon Tables
Modules 2, 3, 4, 5 Modules 2, 3, 4, 5 Module 3 Module 4, 5 Modules 2, 3, 4,5
Modules 1, 2, 4 Modules 1, 2, 4 Module 1 Modules 1, 2, 4 Modules 1, 2, 4 Page 136
Teach Like an MVP DAY TWO
The Teaching Cycle
Launch
Explore
Discuss
Page 136
Comprehensive Mathematics Instruction
Develop
Solidify
PracBce Launch
Explore
Discuss
Launch
Explore
Discuss
Launch
Explore
Discuss
Page 137
The Teaching Cycle: Launch
How will you . . .
hook and moBvate students;
provide schema (the problem seong, the mathemaBcal context, and the challenge) for the mathemaBcal task;
provide tools, informaBon, vocabulary, convenBons and notaBons, as necessary; and
describe what the expectaBons are for the finished task without giving away too much of the problem and leaving the potenBal of the task intact?
Page 137
The Teaching Cycle: Explore
How will you organize and encourage students to explore, invesBgate, experiment, look for pa%erns, make conjectures, collect and record data, parBcipate in group discussions, and revisit and revise their thinking relaBve to the mathemaBcal ideas intended to be elicited by the task?
What will you look for and listen for as you observe students? What will you accept as evidence of student understanding?
What quesBons will you ask to sBmulate, redirect, focus, and extend the students’ mathemaBcal thinking?
Page 137
The Teaching Cycle: Discuss
How will you select which students will present and discuss their soluBons and strategies?
How will you determine what ideas to pursue in depth and what to defer for another Bme?
How will you decide whether to contribute to the discourse by providing addiBonal informaBon (e.g., vocabulary, convenBons, notaBon), suggesBng other models, demonstraBng alternaBve strategies, clarifying difficult issues; or to allow students to conBnue to struggle to make sense of an idea or concept?
Page 138
Geometry from a Transformational Perspective: Developing Definitions In your own words, define each of the following transformaBons:
The rigid moBon transformaBons: • TranslaBon • RotaBon • ReflecBon
The similarity transformaBon: • DilaBon
Page 138
Geometry from a Transformational Perspective: Developing Definitions As you work on your assigned tasks, consider the following quesBons: • Are our definiBons explicit enough to carry out the work we are being asked to do?
• How might we refine our definiBons to make them more precise? • How did your choice of a tool and/or strategy impact what you could “see” relaBve to the definiBon? What did the tool or strategy reveal? What did it obscure?
Page 138
The Tasks: Leaping Lizards! A Develop Understanding Task Each statement below describes a transformaBon of the original lizard.
Do the following for each of the statements:
• Plot the anchor points for the lizard in its new locaBon • Connect the pre-‐image and image points with line segments or circular arcs, whichever best illustrates the relaBonship between them.
Lazy Lizard ◦ Translate the original lizard so the point at the Bp of its nose is located at (24, 20), making
the lizard appear to be sunbathing on the rock.
Lunging Lizard ◦ Rotate the lizard 90° about the point (12, 7) so it looks like the lizard is diving into the
puddle of mud.
Leaping Lizard ◦ Reflect the lizard about the given line so it looks like the lizard is doing a back flip over the
cactus.
Page 112
The Tasks: Photocopy Faux Pas A Develop Understanding Task
Burnell has a new job at a copy center helping people use the photocopy machines. Burnell thinks he knows everything about making photocopies, and so he didn’t complete his assignment to read the training manual.
Mr. and Mrs. Donahue are making a scrapbook for Mr. Donahue’s grandfather’s 75th birthday party, and they want to enlarge a sketch of their grandfather which was drawn when he was in college. They have purchased some very expensive scrapbook paper, and they would like this image to be centered on the page. Because they are unfamiliar with the process of enlarging an image, they have come to Burnell for help.
“We would like to make a copy of this image that is twice as big, and centered in the middle of this very expensive scrapbook paper,” Mrs. Donahue says. “Can you help us with that?”
“Certainly,” says Burnell. “Glad to be of service.”
Burnell taped the original image in the middle of a white piece of paper, placed it on the glass of the photocopy machine, inserted the expensive scrapbook paper into the paper tray, and set the enlargement feature at 200%.
In a moment, this image was produced.
“You’ve ruined our expensive paper,” cried Mrs. Donahue. “Much of the image is off the paper instead of being centered.”
“And this image is more than twice as big,” Mr. Donahue complained. “One fourth of grandpa’s picture is taking up as much space as the original.”
Page 114
The Tasks: Triangle Dilations A Solidify Understanding Task
Given Δ ABC, use point M as the center of a dilaBon to locate the verBces of a triangle that has side lengths that are three Bmes longer than the sides of Δ ABC.
Now use point N as the center of a dilaBon to locate the verBces of a triangle that has side lengths that are one-‐half the length of the sides of Δ ABC.
Page 116
Geometry from a Transformational Perspective: Developing Definitions As we discuss the tasks, consider the following quesBons:
• Were our definiBons explicit enough to carry out the work we are being asked to do? Do we need to refine our definiBons to make them more precise?
• How did your choice of a tool and/or strategy impact what you could “see” relaBve to the definiBon? What did the tool or strategy reveal? What did it obscure?
• What other thinking is supported by doing this transformaBonal work on a coordinate grid?
Page 140
Students’ Understanding of Transformations: What does the research show? one of the sides of a reflected image must coincide with the line of reflecBon
the center of a rotaBon must be located at a point on the pre-‐image (e.g., a vertex point) or at the origin
a pre-‐image point and corresponding image point do not need to be the same distance away from the center of the rotaBon or the line of reflecBon
Page 70
Symmetries of Quadrilaterals A Develop Understanding Task
For each type of quadrilateral, find: • any lines of reflecBon, or • any centers and angles of rotaBon
that will carry the quadrilateral onto itself.
Describe the rotaBons and/or reflecBons that carry the quadrilateral onto itself. (Be as specific as possible in your descripBons.)
Page 119
Quadrilaterals— Beyond Definition A Practice Understanding Task
What do you noBce about the relaBonships between quadrilaterals based on their symmetries and highlighted in the structure of the following chart?
In the following chart, write the names of the quadrilaterals that are being described in terms of their features and properBes, and then record any addiBonal features or properBes of that type of quadrilateral you may have observed.
Page 124
The Nature of Proof: Ways of Knowing How do we know something is true?
AccepBng it based on authority AccepBng it based on experience or experimentaBon
AccepBng it based on logical reasoning
Reasoning and sense making in mathemaBcs:
Page 141
Proof by Transformation A Develop Understanding Task
How do you know that the sum of measures of the interior angles of a triangle is always 180°? AccepBng by authority—How many Bmes have we stated or been told that fact?
AccepBng it based on experimentaBon:
AccepBng it based on logical reasoning
Page 141
Proof by Transformation A Develop Understanding Task
How do you know that the sum of measures of the interior angles of a triangle is always 180°? Here are some interesBng quesBons we might ask about this diagram:
• Will the second figure in the sequence always be a parallelogram? Why or why not?
• Will the last figure in the sequence always be a trapezoid? Why or why not?
Proof by transformaBon:
Page 141
The Nature of Proof: Underlying Assumptions 8.G.1. Verify experimentally the properBes of rotaBons, reflecBons, and translaBons:
• Lines are taken to lines, and line segments to line segments of the same length.
• Angles are taken to angles of the same measure.
• Parallel lines are taken to parallel lines.
MathemaBcs I Note for G.CO.1-‐G.CO.5:
Build on student experience with rigid mo<ons from earlier grades. Point out the basis of rigid mo<ons in geometric concepts, e.g., transla<ons move points a specified distance along a line parallel to a specified line; rota<ons move objects along a circular arc with a specified center through a specified angle.
Page 142
The Nature of Proof: Parallel Postulates Will the second figure in the sequence always be a parallelogram? Why or why not?
Will the last figure in the sequence always be a trapezoid? Why or why not?
Just like Euclid, we need parallel “postulates” for transformations!
Page 142
Parallelism Protected and Preserved A Develop Understanding Task
Which word best completes each statement? Give reasons for your answer. If you choose “someBmes” be very clear in your explanaBon how to tell when the corresponding line or line segments before and a`er the transformaBon are parallel, and when they are not.
• AMer a transla<on, corresponding line segments in an image and its pre-‐image are [never, someBmes, always] parallel.
• AMer a rota<on, corresponding line segments in an image and its pre-‐image are [never, someBmes, always] parallel.
• AMer a reflec<on, corresponding line segments in an image and its pre-‐image are [never, someBmes, always] parallel.
• AMer a dila<on, corresponding line segments in an image and its pre-‐image are [never, someBmes, always] parallel.
Page 76
Fundamental Definitions: Congruence 8.G.2. Understand that a two-‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotaBons, reflecBons, and translaBons; given two congruent figures, describe a sequence that exhibits the congruence between them.
Page 143
Fundamental Definitions: Similarity 8.G.4. Understand that a two-‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotaBons, reflecBons, translaBons, and dilaBons; given two similar two-‐dimensional figures, describe a sequence that exhibits the similarity between them.
Page 143
Productive Theorems: Triangle Congruence Criteria Zac’s Argument
“I know what to do,” said Zac. “We can translate point A unBl it coincides with point R, then rotate segment AB about point R unBl it coincides with segment RS. Finally, we can reflect ΔABC across line RS and then everything coincides so the triangles are congruent.”
[Zac and Sione’s teacher has suggested they use the word “coincides” when they want to say that two points or line segments occupy the same posiBon on the plane. They like the word, so they plan to use it a lot.]
Page 144
Productive Theorems: Triangle Congruence Criteria Sione isn’t sure that Zac’s argument is really convincing. He asks Zac, “How do you know point C coincides with point T a`er you reflect the triangle?”
How do you think Zac might answer Sione’s quesBon?
While Zac is trying to think of an answer to Sione’s quesBon he adds this comment, “And you really didn’t use all of the informaBon about the corresponding congruent parts of the two triangles.”
“What do you mean?” asked Zac.
Sione replied, “You started using the fact that angle A is congruent to angle R when you translated ΔABC so that vertex A coincides with vertex R. And you used the fact that segment AB is congruent to segment RS when you rotated segment AB to coincide with segment RS, but where did you use the fact that angle B is congruent to angle S?”
“Yeah, and what does it really mean to say that two angles are congruent?” Zac added. “Angles are more than just their vertex points.”
How might thinking about Zac and Sione’s quesBons help improve Zac’s argument?
Page 144
Productive Theorems: Transversal and Parallel Lines Examine the tessellaBon diagram, looking for places where parallel lines are crossed by a transversal line.
Based on several examples of parallel lines and transversals in the diagram, write some conjectures about corresponding angles, alternate interior angles and same side interior angles.
Page 144
Conjectures and Proof A Practice Understanding Task
Using this diagram, make some conjectures about lines, angles, triangles and other polygons.
How might you use transformaBons to prove your conjectures?
What other geometric definiBons or theorems did you draw upon in your proofs?
Page 78
Learning Cycles Across the Years: Congruent Triangles Criteria Develop Understanding (Math 7)
7.G.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given condiBons. Focus on construcBng triangles from three measures of angles or sides, noBcing when the condiBons determine a unique triangle, more than one triangle, or no triangle.
Solidify Understanding (Secondary I) G.CO.8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definiBon of congruence in terms of rigid moBons.
Prac<ce Understanding (Secondary II)
G.CO.9, G.CO.10, G.CO.11. Prove geometric theorems about lines and angles, triangles, and parallelograms. Congruent triangle criteria are o`en at the heart of these proofs.
Page 145
Learning Cycles Across the Years: Transversal and Parallel Lines Develop Understanding (Math 8)
8.G.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal . . . For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Solidify Understanding (Secondary II) G.CO.9. Prove theorems about lines and angles. Theorems include . . . when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent . . .
Prac<ce Understanding (Secondary II)
G.CO.11. Prove geometric theorems about parallelograms. Parallel lines cut by a transversal are o`en at the heart of these proofs.
Page 146
Learning Cycles Across the Years: Symmetry of Quadrilaterals and Regular Polygons Develop Understanding (Secondary I) 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotaBons and reflecBons that carry it onto itself.
Solidify Understanding (Secondary I) G.CO. 12 Make formal geometric construcBons with a variety of tools and methods (compass and straightedge, string, reflecBve devices, paper folding, dynamic geometric so`ware, etc.).
Note: Build on prior student experience with simple construc<ons. Emphasize the ability to formalize and defend how these construc<ons result in the desired objects. Some of these construc<ons are closely related to previous standards and can be introduced in conjunc<on with them.
Prac<ce Understanding (Secondary II) G.CO.11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Page 146
The Nature of Proof: What is the “intuition” you start from? Answer #1: Start in the same way the MathemaBcians of the 20th century did: turn Geometry into an axiomaBc system with undefined terms and accepted postulates. Geometry works because we created the right set of rules and the right set of definiBons, and it just so happens that these choices let us model some real-‐world situaBons. Our jusBficaBons are theorems and proofs derived from our choice of definiBons. In other words, Geometry works because I can prove it.
◦ h%p://mathymcmatherson.wordpress.com/2011/08/19/intuiBon-‐in-‐geometry-‐the-‐common-‐core-‐standards/
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The Nature of Proof: What is the “intuition” you start from? Answer #2: Start in the same way Euclid did: construct everything. If you’ve ever read (or skimmed) The Elements, everything is proved by construcBon – I can assert this theorem because I can describe a general procedure to construct the object which proves the theorem. Everything we do is the result of an algorithm which will create the object we’re looking for with the properBes we’re looking for. Geometry works because I can construct it.
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The Nature of Proof: What is the “intuition” you start from? Answer #3: Start in the same way the Common Core Standards do: Base everything on the noBon of transformaBons in the coordinate plane. “The concepts of congruence, similarity, and symmetry can be understood from the perspecBve of geometric transformaBon”. Two shapes are congruent if and only if there is a sequence of rigid transformaBons which superimposes one shape precisely on top of the other. Symmetries in polygons are used to ‘discover’ many of the properBes of triangles, quadrilaterals, etc. In other words, Geometry works because I can draw shapes and move them around.
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Teach Like an MVP DAY THREE
Tracking Student Thinking
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Ready, Set, Go! Each task in every module has a corresponding Ready, Set, Go! The RSG’s are intenBonally designed and broken down into three secBons: Ready, Set, and Go.
Write down what you think the purpose is for each of the three secBons: Ready
Set
Go
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Closing