Unit Title: Transformations & the Pythagorean Theorem
Grade Level: 8
Timeframe: Marking Period 3
Unit Focus & Essential Questions
Unit Focus:
(1) Understanding that the entire 8th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.
(2) Understanding of the nine quantities and units associated with each of the quantities, specifically length of sides and angles.
(3) Understanding the relationship within polygons are mostly lengths of sides and angles
(4)Understand transformations as the relationship between polygons.
(5) Understand how to represent transformations physically, on a graph, narrative and symbolic form.
(6) Understand the types of questions we ask and answer about transformations.
(7) Continue to build fluency with the values (square root and 1 ½) and calculations that arise in transformation and Pythagorean relationships.
Rubric for Learning — Transformation / Rubric for Learning — Pythagorean Theorem
- Object (polygon)
- Representation (physically, graphical, narrative or symbolic)
- Transformation
~Rotation –center, angle of rotation and direction of rotation
~ Translation – distance (units) direction of translation
~ Dilation – scale factor & center of dilation
- Pre-image and image pair
- Object (Right Triangle)
- Distinguish between hypotenuse and legs
- Apply relationship
- Values
Essential Questions:
(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?
(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables, symbols, graphs, and narratives?
(3) Can we become effective and efficient at asking and answering typical mathematical questions?
(4) Can we become effective and efficient at talking about relationships given in any representation?
(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?
*Students should be able to ask and answer additional A- and B- type questions about the relationships.
Please note that the “A-type” and “B-type” references are non-standard identifiers which are used throughout this document to maintain simplicity and clarity. As often as possible, encourage the students to create their own A- and B- type questions from the relationships and measurements that they explore. Consider posting student work around the room, then having other students visit the work (Gallery Walk) and posting their questions and answers on sticky notes as they explore.
Please note the following uses and clarifications:
For an A-type question: Given the value of the quantity you varied, find the related value of the calculated quantity. This may be described as providing the “INPUT” value with the students calculating the “OUTPUT” value.
Later, students can expect to see the same skill phrased as:
- Evaluate the given expression for this value of x
- Given the value of x, find the related value of y.
- Given a value for the independent variable, find the related value of the dependent variable.
- Evaluate function f(x) when x = …
Later, students can expect to see the same skill phrased as:
- Given this “OUTPUT” value, find the “INPUT” value.
- Given the value of y solve for x.
- Given a value for the dependent variable, find the related value of the independent variable.
- In 8th grade: What is x when the value of function f(x) is given [inverse functions]
Easy: the answer is in the representation.
Medium: the answer could (reasonably) be in the representation.
Hard: the representation needs to be extended or generalized in order to find the answer.
Questions with a twist could include…
Comparative: A question that says: “Shamar measured 3 more than, twice as many as, a quantity already given.
Units: A question that gives the information in one unit, but either gives additional information or asks for the answer in a different unit.
Percentage: Malik has a percentage increase/decrease of a quantity already given.
Find the “other value”: A jar of 50 marbles contains only red and blue marbles. If 30 of them are red, what percentage areblue?
New Jersey Student Learning Standards
8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
8.EE.8 Analyze and solve pairs of simultaneous linear equations.
8.G.1 Verify experimentally the properties of rotations, reflections, and translations
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 rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
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 rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. 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.
8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Instructional Plan
Unit 3 Pre-assessment
- Using figures and manipulatives have students identify which are congruent, similar or neither and explain how they know.
- Using figures have students identify right triangles and explain how they know (what makes a right triangle different from other triangles, why is it a right triangle)
- Using a square or rectangle without a ruler ask students to find the length of the diagonal
- Using rulers have students draw and identify parallel lines
- Using pictures without graphs have students identify the transformations that have occurred be aware students may use vocabulary from earlier grades (flips, slides, turns, shrinking, growing
- Using triangles with one missing angle measure ask students to provide it.
Standards & Objectives SWBAT
8.G.1 Verify experimentally the properties of rotations, reflections, and translations
SWBAT:
CREATE & REPRESENT
- Create rigid transformations
- Create dilations
- Represent transformations
- Represent transformations in a narrative, graphically, and symbolically
- Discuss all characteristics: object, transformation, representation
- Compare dilations and rigid transformations in terms of congruence and similarity
- Interpret transformations in textbook/teacher created problems
- Communicate those transformations through other representations
FLUENCY
½ Start at whole numbers and count by dollars and cents(that are 50 cent intervals), begin at – whole numbers and count up by whole numbers and then count up by decimals to the tenths.Counting/combinations of values (fractions, percents and decimals) used in Touchpoints.
As the students begin to create their progressions of items in the classrooms, teachers need to gather the numbers students are creating and use them to count around the room.
CREATE & REPRESENT TRANSFORMATONS
Distribute shapes to represent real world objects and perform transformations. Work on one rigid transformation at a time. Trace the pre-image, perform the transformation, trace the image.
In discussion: introduce A A’ Review and note weaknesses in labeling/locating points for small group/center assignment
introduce symbolic representation forms
FLUENTLY TALK ABOUT TRANSFORMATIONS
Students describe their transformation in terms of the rubric for their transformation. Students include congruence their discussion of transformation (pre-image and image of rigid transformation are congruent) With physical objects this can be done by laying the object over the image and the pre-image to show they are the same object.
FLUENTLY READ TRANSFORMATIONS
Stress accuracy in tracing. Have students label points for practice. Reinforce quadrants. You want them to notice that the tracing “fits” on top of the image – congruence.
Along with this practice also embed counting units, labeling and locating points in the coordinate grid.
Students need to record what has occurred in their read activities in narrative and symbolic form to connect representations.
ELL: Use art images of transformations. Show short video clips of transformations. Have students keep vocabulary notebook with frayer model or other illustrated vocabulary page for them to refer to as they work. Consider interactive notebook use.
SPED Accommodations: Use art images of transformations. Show short video clips of transformations. Provide two different colored pencils for pre-image and image. Provide larger sized grids. Provide sentence scaffolds for narrative. Provide model. Provide multiple strategies to perform task. Some students who have difficulty visually may be able to perform task symbolically. Provide centers to support additional practice on plotting points or other prerequisite skills.
Gifted & Talented: once they become proficient with transformations, students may begin to perform composite motions (glide reflections) / RUBRIC FOR LEARNING— TRANSFORMATIONS
- Object (polygon)
- Representation (physically, graphical, narrative or symbolic)
- Transformation
~Rotation – center, angle of rotation direction of rotation
~ Translation – distance (units) direction of translation
~ Dilation – scale factor & center of dilation
- Pre-image and image pair
Use Rubric for Learning to check for each of the bulleted items as students create transformations and identify created transformations.
FLUENTLY TALK ABOUT TRANSFORMATIONS
Use Rubric for Learning to check for each of the bulleted items as students talk about the representations of the transformations
Move towards students using the rubric on each other’s representations, talking, and reading.
FLUENTLY READ TRANSFORMATIONS
Use Rubric for Learning to check for each of the bulleted items as student’s read/work on each of the rigid transformations.
DIFFERENTIATION
Providing feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.
As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.
Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything. / FLUENCY
Count around the room and combinations.
CREATE & REPRESENT TRANSFORMATIONS
Distribute objects for students to trace and physically perform transformations. (1 transformation at a time. Do for all Rigid Transformations)
FLUENTLY TALK ABOUT TRANSFORMATIONS
Students practice talking about the math in their own and their group member’s representations.
Studentstalk about the math in other group’s representations.
FLUENTLY READ TRANSFORMATIONS
• Distribute coordinate grids with pre-image and image already drawn and tracing paper. Demonstrate tracing pre- image, marking origin and transform to image.
• Distribute coordinate grid with pre-image on it and have students perform a given transformation. Pass it to a partner to check it.
• Distribute a blank coordinate grid to students have them draw a pre-image, pass it to a partner, who performs a transformation and passes it back. They must then determine what occurred.
• Distribute a blank coordinate group to students, have one draw a pre-image, pass to the next student who decides what the transformation should be, pass to a third who performs it and a fourth who checks it, around a table.
CLOSING ACTIVITY
Sum up for individual transformations stressing the features of each. Reflection over a particular line, translation along a line in a given direction for a certain number of units, and rotation around a given point (the origin in 8th grade) in a given direction focusing on the pre-image and image being congruent.
Reflection:
Standards & Objectives SWBAT
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 rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
SWBAT:
CREATE & REPRESENT
- Create rigid transformations
- Create dilations
- Represent transformations
- Represent transformations in a narrative, graphically, and symbolically
- Discuss all characteristics: object, transformation, representation
- Compare dilations and rigid transformations in terms of congruence and similarity
- Interpret transformations in textbook/teacher created problems
- Communicate those transformations through other representations
FLUENCY
½ Start at whole numbers and count by dollars and cents (that are 50 cent intervals), begin at – whole numbers and count up by whole numbers and then count up by decimals to the tenths. Counting/combinations of values (fractions, percents and decimals) used in Touchpoints.
As the students begin to create their progressions of items in the classrooms, teachers need to gather the numbers students are creating and use them to count around the room..
CREATE & REPRESENT TRANSFORMATIONS
Distribute shapes to represent real world objects and perform transformations. Work on one composition at a time. Trace the pre-image, perform the 1st transformation, trace the image, perform the 2nd transformation, trace the image.
In discussion: introduce A → A’ Review and note weaknesses in labeling/locating points for small group/center assignment; Continue to support symbolic notation forms and narrative.
FLUENTLY TALK ABOUT TRANSFORMATIONS
Students describe their transformation in terms of the rubric for their transformation. Students include congruence their discussion of transformation (pre-image and both images are congruent) With physical objects this can be done by laying the object over the images and the pre-image to show they are the same object and thus congruent
FLUENTLY READ TRANSFORMATIONS
Continue to embed labeling points, talking about quadrants, writing transformations symbolically and in narrative.
Focus always on the ending image being congruent to the pre-image
ELL: Show short video clips of transformations. Have students keep vocabulary notebook with frayer model or other illustrated vocabulary page for them to refer to as they work. Consider interactive notebook use.
SPED Accommodations: Use art images of transformations. Show short video clips of transformations. Provide three different colored pencils for pre-image and images. Provide larger sized grids. Provide sentence scaffolds for narrative. Provide model. Provide multiple strategies to perform task. Some students who have difficulty visually may be able to perform task symbolically. Provide centers to support additional practice on plotting points or other prerequisite skills as needed.
Gifted & Talented:Once students become proficient ask them to explore if order in composition matters. They should justify their response with examples using a variety of compositions. Working towards presenting to class their findings either via podcast or mini-lesson. / RUBRIC FOR LEARNING— TRANSFORMATIONS
- Object (polygon)
- Representation (physically, graphical, narrative or symbolic)
- Transformation
~Rotation – center, angle of rotation direction of rotation
~ Translation – distance (units) direction of translation
~ Dilation – scale factor & center of dilation
- Pre-image and image pair
Use Rubric for Learning to check for each of the bulleted items as students created progressions and graphs of the relationship.
Move towards students using the rubric on each other’s representations.
FLUENTLY TALK ABOUT TRANSFORMATIONS
Use Rubric for Learning to check for each of the bulleted items as students talk about the representations of the relationships of the object.
Move towards students using the rubric on each other’s talking.
FLUENTLY READ TRANSFORMATIONS
Use Rubric for Learning to check for each of the bulleted items as student’s read/work the representations of the relationships of the object.
Move towards students using the rubric on each other’s reading.
DIFFERENTIATION
Providing feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of
.
As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.
Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything. / FLUENCY