Ben Hartley Math8 Sqr Roots and Pyth Theorem Unit Plan UPDATED

 

EDCP 342A Unit planning: Rationale and overview for planning a unit of work in secondary school mathematics

Your name: Ben Hartley
School, grade & course: J.N. Burnett, Math 8

Topic of unit: Squares, Square Roots, Pythagorean Theorem

Preplanning questions:

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words) This unit builds some background geometry skills and introduces the concept of radicals and square roots with integers. The first reason why this topic is taught in secondary school is that these are mathematical skills that will be used regularly in subsequent math classes. The second reason is that these skills are generally useful in many real work applications. Many people will, at various points in their life, want to build something with their hands. They will likely need to construct a right angle or calculate an area. These skills are relevant for tasks like that. The third reason to learn about the Pythagorean Theorem is that it is a valuable bit of our collective human history. The fourth reason is that the Pythagorean Theorem is one of the first theorems that students might come across. Teaching it introduces students to the concept of theorems and potentially to the concept of proofs!

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words) Topic: Mathematical Poetry Aims: Motivate students to reiterate and recall how to solve a specific kind of mathematical problem (Pythagorean geometry problems), Give students opportunities to associate math and art as related fields, Students have fun! Process: At the beginning of the unit students are presented with the project: For one type of problem that we learn about during this unit students will have to write ‘instructions’ on how to solve such a problem in poem form. (The instructions can be fairly vague, especially for poem forms like haiku that are very short) However, the instructions do need to be accurate! This should be challenging linguistically! When writing the poem is challenging students will want to try to write a poem for each new topic when they think that it might be easier. Writing poetry is also meditative. Giving students time to clear their desks and work quietly in their own mind is a chance to step away from any other worries. Students should NOT use any AI in order to write their poems. In order to prevent AI usage students will work on this entirely during class, without devices. This can be achieved easily by primarily giving them time to work on this as an activity after completing a test or quiz (their devices should already be out of reach). Optionally, students will have a chance to share their poems aloud. Timing: Small amounts of work time spread out throughout the entire unit. Product: Students will produce a poem that captures how to solve a particular problem that they will want to solve. Assessment: The poems will be assessed based on the accuracy of the student’s poems (correctly describes the process of solving their chosen problem) and the inclusion of poetic features. Some poetic features include: - adherence to a specific poetic form such as Haiku, Sonnet, Acrostic, etc. - rhyming, assonance, consonance, alliteration, meter, etc. - other poetic features can be included if students are passionate about an idea and suggest it to the teacher.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) Though assessment methods have to be different for different learning standards, there are few different categories that are fairly similar. Formative assessments: Students are assessed daily in the form of “thumbs up / thumbs down” feedback when discussing new topics. Class engagement, and small quizzes, also factor into this kind of assessment. This data helps the teacher identify students that might struggle with particular concepts or to identify topics that need additional review as a class. Summative assessments: The students will have to option to take a written paper exam or to do an oral exam. In either case each student will demonstrate whether or not they meet the learning standards specified in the BC Curriculum. Some of the learning standards cannot be evaluated in these ways (learning standards related to collaborating in mathematics or to social skills) and instead are evaluated based on the products of group activities and projects and their behavior during.

Elements of your unit plan:

a) Give a numbered list of the topics of the 10-12 lessons in this unit in the order you would teach them.

Lesson	Topic
1	Perfect Squares
2	Perfect Cubes
3	Square Roots of Perfect Squares / Cube Roots of Perfect Cubes
4	Quiz 1 & Open Cubes Math Art
5	Squares (Fractions and Decimals)
6	Estimating Square Roots of Non-Perfect Squares
7	Estimating Cube Roots of Non-Perfect Cubes
8	Quiz 2 & Open Cubes Math Art
9	The Pythagorean Theorem I
10	The Pythagorean Theorem II
(11)	Quiz 3 & Open Cubes Math Art
(12)	Exam

b) Write a detailed lesson plan for three of the lessons which will not be in a traditional lecture/ exercise/ homework format. These three lessons should include at least three of the following six elements related to your mathematical topic. (And of course, you could include more than three!)

These elements should be thoroughly integrated into the lessons (i.e. not an add-on that the teacher just tells!)

a) History of this mathematics

b) Arts and mathematics

c) Indigenous perspectives and cultures

d) Social/environmental justice

e) Open-ended problem solving in groups at vertical erasable surfaces (“thinking classroom”)

f) Telling only what is arbitrary, and having students work on what is logically ‘necessary’

Be sure to include your pedagogical goals, topic of the lesson, preparation and materials, approximate timings, an account of what the students and teacher will be doing throughout the lesson, and ways that you will assess students’ background knowledge, student learning and the overall effectiveness of the lesson. Please use a template that you find helpful, and that includes all these elements.

Lesson Plan 1: History of Mathematics – Pythagorean Theorem

Lesson Plan 2: Open-ended Problem Solving – Estimating Cube Roots

Lesson Plan 3: Sol Lewitt’s Variations on Open Cubes – Arts and Mathematics

Lesson 1

Title	History of Mathematics – Pythagorean Theorem
Duration	1.5 classes – project
Description	This lesson introduces the Pythagorean Theorem by embedding it in the history of its use and discovery. Each student will practice their research skills by investigating how ancient people used the Pythagorean Theorem, and the different ways that it was discovered or expressed in different cultures. The product of the presentation can be a short presentation, a written report, or an art piece that summarizes their findings.
Preparation and materials	Project Handout - instructions / prompt - rubrik - due dates / other dates - useful starting points for students that are unsure where to begin Example project product produced by teacher and presented as an introduction to the project.

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
Two-Step equations with integer coefficients, constants, and solutions.	Solving an equation using the Pythagorean Theorem requires some algebraic manipulation skills.	Outcomes in prior units
Square and cube roots.	Taking square roots is necessary to solve using the Pythagorean Theorem.	Outcomes in prior units
Perfect squares and cubes.	Squaring numbers is necessary to understand Pythagorean Triples.	Outcomes in prior units

Learning Standards

Learning Standard	Relevance	Assessed by...
Model mathematics in contextualized experiences.	Students must investigate how ancient people’s used the Pythagorean theorem.	Analysis of project product and the work that went into it.
Visualize to explore mathematical concepts.	The Pythagorean theorem is related to geometry, which is very visual.	Analysis of project product and the work that went into it
Engage in problem-solving experiences that are connected to place, story, cultural practices relevant to local First Peoples Communities, the local community and other cultures.	The whole point of the project is to have students engage with mathematical history.	Analysis of project product and the work that went into it
Explain and justify mathematical ideas and decisions.	The students will have to consider how the ancient people’s discovered these equations, which involved justification.	Analysis of project product and the work that went into it
Communicate mathematical thinking in many ways.	The various allowed forms of the final product give students a chance to communicate their thinking in many ways.	Analysis of project product and the work that went into it
Represent mathematical ideas in concrete, pictorial, and symbolic forms.	Geometry and algebra are both very relevant to the Pythagorean Theorem.	Analysis of project product and the work that went into it
Pythagorean Theorem	Of course, the students will be engaging with the Pythagorean Theorem and investigating how it is used.	Analysis of project product and the work that went into it

Day 1 (65 min)

Duration	Title	Teacher will...	Students will...
5 min	greet	Take attendance and write the shape of the day on the board	Enter class and find their seats
15 min	hook	Get students hooked into the project by presenting the teacher’s version of the project product (about 10 minutes) be clear that the presentation is not something they need to take notes on.	Pay attention to the presentation and how it is constructed.
15 min	Project intro	Pass out the project description handout for the students to begin to review. Then go over key parts of the sheet making sure to emphasize due dates and the nature of the final product. Lastly assign groups	Ask clarifying questions about anything that is immediately unclear in the handout.
25 min	Project work	Circulate the classroom and check in with students that are unsure of how to start. Make sure that students know that they can talk to each other to bounce ideas off of friends.	Students will have time to begin their initial research into different ways that the Pythagorean theorem has been used throughout history and choose a specific instance.
5 min	Closing	Announce reminders for next class or for the upcoming week Their homework is to complete any part of the project that they did not finish in class	Listen to announcements and get ready to go to their next class.
5 min	Flex

Day 2 (80 min)

Duration	Title	Teacher will...	Students will...
5 min	greet	Take attendance and write the shape of the day on the board	Enter class and find their seats
10 min	hook	Remind students about the projects that they worked on the previous day. The presentations (for those who chose to present) are going to happen today. Ask for volunteers and create an order in which students will present.	Get ready to present or to engage with presenters.
25 min	presenting	Ensure a respectful presentation environment. Administer the switching between presentations as necessary. Ask questions for each group. Fill out rubriks as students’ present.	Students begin presenting their material to the class. Each presentation should be about (5-10 minutes)
5 min	break	Model good break taking by stretching and stepping away from work	stretch
25 min	presenting	Continue presenting after students have had a stretch break	Continue with the second half of the presenters
5 min	closing	Announce reminders for next class or for the upcoming week	Listen to announcements and get ready to go to their next class.
5 min	Flex

Lesson 2

Title	Estimating Cube Roots
Duration	0.5 classes – instructional
Description	This lesson is a chance for students to try to work out a solution to a tough problem without being presented with the answer. In prior lessons the students will have learned a method for estimating square roots (what amounts to a linear interpolation at each of the perfect squares). The students will then be given time to come up with a strategy for estimating cube roots of non-perfect cubes. There are many strategies they could come up with.
Preparation and materials	Gap notes for the estimating square roots lesson vertical spaces for students to work + markers

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
Square roots and cube roots	The students need to understand what it means to take a cube root before they can try to estimate it.	Outcomes in prior units
Perfect squares and cubes	In order to understand the concept of cube roots it is helpful to understand cubing.	Outcomes in prior units

Learning Standards

Learning Standard	Relevance	Assessed by...
Use reasoning and logic to explore, analyze, and apply mathematical ideas.	The students need to use reasoning in order to connect this estimation problem to other concepts they’ve learned or to come up with new ideas.	Observation during the activity work time along with the final product.
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving.	This is an inquiry and problem solving task.	Observation during the activity work time along with the final product.
Visualize to explore mathematical concepts.	Students will likely want to graph these functions in order to get a sense of what is happening.	Observation during the activity work time along with the final product.

Day 1 (80 min)

Duration	Title	Teacher will...	Students will...
5 min	greet	Take attendance and write the shape of the day on the board	Enter class and find their seats
10 min	hook	Remind students about what they learned recently about square roots of perfect squares. Today they will learn about square roots of non-perfect squares.
10 min	lecture	Show students the interpolation method of estimating the square root of a non-perfect square. Gap notes method for this with one or two “your turn” problems.	Students follow along the very brief lecture and fill out their gap notes work with table partners during “your turn” problems to check understanding.
10 min	transition	Announce that we are moving on to a different part of the lesson where they will learn quite differently. Put students into groups of 2 or 3. introduce the task: “Come up with a method of estimating the cube root of a non-perfect cube”	Ask questions about the task if any come up rearrange to be with group members at a vertical board space.
25 min	Activity time	Circulate the classroom asking prompting questions and checking in on groups to make sure that everybody is making some progress. Give more explicit hints where necessary.	Work with group members on coming up with a way to estimate the cube root of non-perfect cubes. Vertical spaces allow groups to get hints from each other’s progress. Students should know how to use this by this point in the semester.
15 min	sharing	Facilitate each group to share their strategies with the rest of the class	Students will share the strategy they came up with with the rest of the class and make comparisons and contrasts. The class gets to see different styles of thinking and different results.
5 min	closing	End of day announcements	Get ready for next class.

Lesson 3

Title	Sol Lewitt’s Variations on Open Cubes – Arts and Mathematics
Duration	3 classes – project (distributed in ½ classes throughout semester)
Description	This lesson is an ongoing project that students can work on throughout the semester. It is not explicitly related to any of the content learning standards, however, it does provide practice in creative thinking, spatial reasoning, and logic applied to mathematics. The project is modeled on an art piece by Sol Lewitt’s called “Variations on Open Cubes”. The project asks, what are all of the possible ways that you could make a 3D wire-frame cube with some edges missing (requiring that it remain 3D and that the remaining edges are connected.) Answering this question requires lots of experimentation, spatial reasoning, and discovery. Plus, it is something that is very easy for students to actually create with their hands. The final produce is a set of as many distinct cubes as the students can find along with some justification that those cubes are distinct.
Preparation and materials	Create a handful of “incomplete cubes” to bring to class as a starting point. Handout explaining the concept in words for students to refer to if they miss something or forget something. Space in the class to store the cubes students make and assemble the set without risk of breaking building materials. Toothpicks and marshmallows would work well but not store easily. (have students make the designs in toothpicks and marshmallows and then periodically ‘cement’ the new ones in a more durable material on your own time.)

Prerequisite Learning Standards

Learning Standard	Relevance	Assessed by...
n/a

Learning Standards

Learning Standard	Relevance	Assessed by...
Use logic and patterns to solve puzzles and play games.	This is a mathematical puzzle.	Observation during project work time and final product.
Use reasoning and logic to explore, analyze, and apply mathematical ideas.	The students need to use a lot of reasoning if they are going to find all of the incomplete cubes.	Observation during project work time and final product.
Apply multiple strategies to solve problems in both abstract and contextualized situations.	The students will likely have to use multiple strategies to work through this problem.	Observation during project work time and final product.
Visualize to explore mathematical concepts.	This is a very visual problem at its core	Observation during project work time and final product.
Develop, demonstrate, and apply mathematical understanding through play, inquiry, and problem solving.	The students will demonstrate their understanding of rotation and symmetry.	Observation during project work time and final product.
Use mathematical vocabulary and language to contribute to mathematical discussions.	Students will need to use some mathematical terminology in order to justify their work.	Observation during project work time and final product.
Represent mathematical ideas in concrete, pictorial, and symbolic forms.	The students will be encouraged to actually build models of the cubes they discover.	Observation during project work time and final product.
Construction, views, and nets of 3D objects.	The students will be working with cubes and thinking about the various views that you can look at a cube with.	Observation during project work time and final product.

First Day (80 min)

The art activity will be split across multiple days and is a fun puzzle activity to do after students have taken a quiz or exam.

Duration	Title	Teacher will...	Students will...
5 min	greet	Take attendance and write the shape of the day on the board	Enter class and find their seats
5 min	hook	Remind students that their test is today. Pass out test	Students prepare to take the test
35 min	test	Monitor class for respectful test taking behavior and provide help where needed / appropriate. Ensure that students are aware of the time constraints and finish the test on time. Remind students that this is just a snapshot and it will not make or break their grade.	Take the test respectfully
5 min	break	Model good break taking by stretching and stepping away from work	stretch
5 min	intro	Show students example of the art project they will be working on. The “incomplete cubes” project. These are actualy physical things you can bring in. hand out with written instructions.
20 min	Work time	Circulate class and help students reason about the art puzzle students check in the me to see if they have found one that adds to the class set and make an announcement when that happens	Students use drawings or preferably manipulatives such as toothpicks and marshmallows to create the open cubes. If they’ve found one that they think is different from the other’s that the class has found then they can bring it up to get incorporated.
5 min	closing	Announce reminders for next class or for the upcoming week	Listen to announcements and get ready to go to their next class.