Entry: Unit Plan

EDCP 342A Unit Plan

Your name: Jacob Brunner

School, grade & course: Ladysmith Secondary, Grade 9, Math 9

Topic of unit: Understanding Whole Number Powers with Integer Bases (excluding 0)

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)

Exponents is an incredibly important and useful topic, inside and outside of the classroom. Without an understanding of it they would be severely restricted in performing any higher level mathematics. Exponents are used throughout our daily life, such as in any squared measurement (my room is 12 square feet) and is used prolifically in the metric system (eg. 3km is simply 3 x 103m). Through our knowledge of exponential growth we can further understand computer science (notably in base 2 or 16), finance (eg. compound interest), biology (eg. population growth), and especially physics (eg. properties of light or sound).

Just as multiplication and linearity are important concepts for someone to understand in elementary school, exponents are equally as important in high school, as they open up the world to be studied through realistic models instead of ideal models (as most physical changes are not linear). It is fascinating to see that an uncontrolled nuclear explosion (or pandemic, etc.) will grow in energy exponentially (à la “E = mc2”) and conversely nuclear waste decay will have an inverse square decrease. Nature is teeming with mathematical patterns and many are shown to us through exponents.

(2) What is the history of the mathematics you will be teaching, and how will you introduce this history as part of your unit? Research the history of your topic through resources like Berlinghof & Gouvea’s (2002) Math through the ages: A gentle history for teachers and others  and Joseph’s (2010) Crest of the peacock: Non-european roots of mathematics, or equivalent websites. (100 words)

Exponents do not have a rich history, as they are basically a notation for an idea that was expressible without the notation. Therefore, it wasn’t until recently (mathematically speaking) that exponents took a solid, consistent form that we are familiar with today. The first uses of exponentiation were from Euclid who thought to square a line and was expanded upon by Archimedes who discovered that 10a10b = 10a+b when manipulating powers of 10. In Persia in the 9th Century, Muhammad ibn Mūsā al-Khwārizmī used notation for squares and cubes, which Islamic mathematicians later changed into m and k. It wasn’t until the 16th Century that Europeans began using Roman Numerals for exponents and the use became more prevalent. Due to the stretched out and incomplete nature of its history, I would simply introduce it through Greek mathematics and say that, “Euclid started with a line and turned it into a square. He noticed that the length of the line, x, became the area of the square which was, x*x. This later became known as x2 and was the beginning of exponents.”

(3) The pedagogy of the unit: How to offer this unit of work in ways that encourage students’ active participation? How to offer multiple entry points to the topic? How to engage students with different kinds of backgrounds and learning preferences? How to engage students’ sense of logic and imagination? How to make connections with other school subjects and other areas of life? (150 words)

First I would write 5x5 on the board and ask students if there is a simpler way to write that. Once they remember that it equates to 52 I could ask them what they think 55 means. Then we would discuss why we use this notation (and emphasize the simplicity it adds to repeated multiplication). I will give them some more examples, as I think that this is the best way to show what the notation stands for and how we use it.

Now the difficult part is to explain why we need this notation and where it is used, so that they become interested in what we are doing. We can talk about population growth, nuclear decay, earthquakes, atomic bombs, or whatever examples that the students are interested in and see how they function exponentially. To have the students test it themselves you can ask them if it’s better to get paid a $100 for a month’s work, or $2 on the first day of the month doubled each day. Then show them how you can construct this with exponents and walk them through the process.

(4) 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. (100 words)

A fun project for students who are just learning about exponents would be to model a very simplistic population. I will explain that for this project they can calculate P = P0(1 + Growth Rate)t. I will explain that P is current population and P0 is the starting population and t is the duration of the growth.

Next the students will create their ‘creature’ and the creature’s starting population, where they will also need to draw a picture of it. Then they will need to do two things: think of growth rates and timeframes (maybe they have lots of food and are happy for 5 years and have good growth, then there is a disease for 3 years and the growth slows). The purpose of this is for them to graph the growth rate and write a story to go along with it. This should help them understand how exponents function.

(5) 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)

My assessment would include understanding of vocabulary (what is base, power, exponent, cubed), comprehension of ideas, and application. The bulk of the assessment would be on quizzes and a subsequent test which would contain practice questions (eg. 38*38*38*38*38 = 38x, -47 = x) and short answer questions (eg. What is an exponent?), as well as observation. Observation is a culmination of the students’ classwork and the students’ homework which will result in immediate feedback and support. Though the short answer questions make up a smaller portion of the test, they carry greater importance and a poor grade on them would result in greater time spent teaching this unit of the course.

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	A1.1 - Representing repeated multiplication using powers which is a recap of Math 8
2	A1.2 - Using patterns to show that a power with an exponent of zero is equal to one
3	A1.3 - Solving problems involving powers and integer bases
4	A1.3 - Solving problems involving powers and decimal/fraction bases
5	A1.3 - Solving problems involving negative exponents
6	A1.3 - Solving problems involving multiplication and division of exponents
7	A1.3 - Power Rule and simplifying expressions involving exponents
8	A1.3 - Scientific notation: convert and compare
9	A1.3 - Scientific notation: multiply and divide
10	A2.1 - Solve equations using order of operations (involving exponents)
11	Review of A1.1-A2.1

b) Write a detailed lesson plan for one of the lessons which will not be in a traditional lecture/ exercise/ homework format.  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

Class: Math 9	Duration: 66 minutes
Curriculum Organizer/Big Idea/Focus:	Develop number Sense: Powers with Integer Bases Operations on Powers
Prescribed Learning Outcomes (PLOs)	demonstrate the differences between the exponent and the base by building models of a given power, such as 23 and 32 explain, using repeated multiplication, the difference between two given powers in which the exponent and base are interchanged (e.g., 103 and 310) express a given power as a repeated multiplication express a given repeated multiplication as a power explain the role of parentheses in powers by evaluating a given set of powers (e.g., (–2)4 , (–24 ) and –24 ) demonstrate, using patterns, that a0 is equal to 1 for a given value of a (a ≠ 0) evaluate powers with integral bases (excluding base 0) and whole number exponents *Ministry of Education (2008). Mathematics 8 and 9. British Columbia*
Lesson	Review of A1.1-A2.1
Materials and Equipment Needed	Whiteboard and Marker or Powerpoint Blank paper for students Pencil Crayons for students

	Lesson Components	Learning Activities	Time Allotted
1.	Review / Introduction	We will start with a warm-up, which would be to review the basic principles that we have learned in the previous 2 weeks. We will spend a few minutes discussing the key ideas and doing a few examples on the board.	7 minutes
2.	Hook	I will show them a demonstration of the assignment that I have done or that a previous student has completed. Hopefully they will become excited after seeing this.	6 minutes
3.	Development	3.a. Teacher-Led Activities I will show them how to make the assignment. They are to pretend that they are meeting either an alien race or simply a new group of people. They are to create a brochure that explains to this new group of people the laws and properties of exponents. They can be as creative as they like, as long as the brochure demonstrates an understanding of the properties. They should include things such as a definition, examples, how to multiply/divide, negatives, etc. 3.a. Independent Activities The students will have the remainder of class to create their brochure. This will also serve as their study guide in the future for tests and review. I will walk around and answer questions, as well as ensuring that the students are filling the brochures will adequate and accurate information. If any students finish early, they would be encouraged to add examples or creativity to their brochure, otherwise they are free to simply work ahead if they are caught up. The students are expected to finish this for homework if they are incomplete.	Approx. 50 minutes (10-15 minutes for teacher-led and remainder for seat-work)
4.	Closure	I will ask the students some closing questions to review. What is an exponent? Where do we see exponentials in nature? What was the most difficult part of exponents to explain in your brochure?	3 minutes

Feedback/Assessment of Students’ Learning	For this lesson, the feedback will obviously be from their brochure. I will be looking at i) their creativity ii) their clarity iii) that the information is accurate and sufficient Feedback will be given immediately from my observation, and the rest will be given when I mark their assignment on the criteria listed above.
Modifications	If someone is struggling, they would simply be given additional time and encouraged to use resources to find the information (most likely their textbook).
Extensions	If there were additional time or some students were further ahead, you could easily have the brochure encompass another part of mathematics that you felt the students needed to review, so as to reinforce their learning, but also give them additional study tools.

Entry: John Mason

   Mason's ideas are very related to inquiry based learning. The main idea of the article is to ask guided questions and to challenge the students, as opposed to "dumb[ing] down the question[s]" with the purpose of motivating students. Another method that Mason talks about is letting the students ask  questions and helping them find the answer through their own inquiry. It is quite clear that his ideas connect themselves to inquiry-based learning.

   The goal is to start the long practicum (I'm at a semester school, so I will have a fresh batch of students) by asking the students leading questions towards the answer. This will be easier for me, as they start with review of last year's content, where I can give them a question and help them focus towards the answer (that they have prior knowledge of). Throughout the semester I will ask my questions less clearly, as they will already be familiar with the idea of inquiry and will therefore need to ask the question themselves. For example, if we were reviewing surface area of rectangular prisms I could initially ask "What are you trying to find?", "What formulas have you learned?", or "What are the properties of this shape?". However, as they get deeper into the course, the goal is for them to ask these questions to themselves, so I would shift to less explicit questions such as "What question do you think I would ask you?" or "How did I solve the question on the board yesterday?". This will encourage them to ask themselves questions.

 

Exit: Micro Teaching feedback

I taught Multiplication of Fractions with Julie. We had allocated the time fairly accurately, however the structure of the lesson was a bit poor. Our prior knowledge check was a bit rushed and awkward (as we were all just pretending). The teaching of the material was also quite rushed and there was no time at all for any development of skills, only an introduction (due to having about 6 minutes for it). I would add to this and remove part of the participatory activity if this was real students, and do the activity the next day.

The activity itself for the second half of the teaching was quite effective and fun. The students enjoyed it and it reinforced their skill-set. I would do it again, but I would do it later in the lesson.

Here is my feedback:

Entry: Micro Teaching in Pairs

Micro Teaching Criteria

Name: Jacob & Julie

Topic: Multiplying Fractions and Mixed Numbers

Objectives & Goals	The purpose of this session is to introduce the students to multiplication of various fractions, including mixed numbers. The subsequent lesson (which we won’t do) would teach them about division.
Hook	I will introduce the lesson by showing them a fraction of the board (eg. 5 x ⅓) and ask them what this means in math. I will then explain that it is the addition of ⅓ five times (ie. ⅓ + ⅓ + ⅓ + ⅓ + ⅓), just like regular multiplication.
Materials	To teach this lesson, it is suggested that you have a whiteboard and marker.
Prior Knowledge Check	I would check that students: a) Understand what fractions are and b) Know how to add and subtract fractions. If these are met, we can continue the lesson after a short refresher. If not, we would spend a bit of time on that a it is a required prerequisite.
Activities	2 Minutes: I will explain that multiplication is just addition. I will say that this is also true of fractions and that they know how to add fractions. 2 Minutes: I will do an example. (6 x 1) is just 6 units by 1 unit If you count the boxes, you will get the answer of 6. Now for (6 x ½) you can simply cut this in half either way: If you count the half boxes, you will have 6 of them, which equals 3. 2 Minutes: I will show them that it works for only having fractions as well, such as (½ x ½): 2 Minutes: I will show them the formulaic approach that they will apply, now that they see the methodology: You simply cross multiply. If there is a mixed number, just turn it into an improper fraction first. 6 Minutes: I will give them an activity on Kahoot and review at the end.
Ideas and Skills	They will learn the basics of multiplication of fractions. It will be a culmination of addition of fractions, multiplication, and mixed numbers/improper fractions. This will reinforce algebraic skills.
Closing	The closing section will be at the end of the Kahoot activity. We will check their learning progress and use it to prepare the subsequent lesson.
Assessment	I will assess the students’ learning by seeing if they understand the method during Kahoot. This assessment will be used to determine the amount of homework to assign as well as if the following lesson should advance to division of fractions or further development of multiplication.