Mathematics Senior Level Capstone Course Unit Overview
Title of Unit: Squared Squares Unit Designers:
Erin Hopple and Judy Cain
Frederick County Public Schools Unit Editor:
Vickie Inge, UVA-SCPS Mathematics Outreach Context: An investigation into algebraic expressions and equations using the
characteristics of quadrilateral. The investigation includes linear and quadratic functions.
Number of Class Hours:
5- 6 hours Unit
Design: __X_Task Based ___Project Based Other Subject
Areas/Disciplines Addressed:
Construction/carpentry industry, writing
Driving Question: How can one measurement on a complex diagram be used to find the measures of all the others? Mathematics Content
Addressed:
Characteristics of quadrilaterals, adding and subtracting and multiplying polynomial expressions, creating and solving a linear equation, creating and solving a quadratic equation, using deductive reasoning to test and evaluate the reasonableness of the solutions
MPE Addressed:
Problem-solving, decision-making, integration, procedure and calculation, verification and proof
Assumption of Prior Knowledge:
Knowledge of the characteristics of quadrilaterals; ability to write an algebraic expression; ability to write and solve an algebraic equation; ability to multiply polynomials; ability to use the Quadratic Formula or the graphing calculator to find roots of a quadratic function
College and Career Readiness/21st Century Skills to be taught (T) during this unit or expectation (E) for student use during this unit and assessed (A):
Collaboration – Students will work in pairs (groups of three if needed)
E Research – Students will research the cost of flooring materials and installation costs
E
Communication (Written) – written explanation of process used to solve the last problem in the unit
T & A Technology – use of calculator if needed E
Critical Thinking/Decision Making E Other: (Describe)
Major Products and/or Performances:
Group – Five paragraph essay describing how the group solved the last problem in the unit and verified their solution; essay must include a discussion of how all problems in the unit relate to each other.
Presentation Audience: Class School Individual – Mathematician’s journal entries including work for
both squares questions and reflections on each
Expert Community X Other: Teacher
Launch: Event or experience used to engage the students interest and inquiry:
Hopple Flooring Proposal – Work in small groups to determine the area of a complex figure.
Evaluation: Formative Assessments
(During the Unit) InterviewMathematicians Journal XX Practice PresentationsNotes Preliminary
Plans/Outlines/Prototypes
Checklists
Rough Drafts Concept maps
Field Tests Other:
Summative Assessment (End of Project)
Written Products, with a checklist X Peer Evaluation, with a rubric
Oral Presentation with a rubric Self Evaluation, with a rubric
Other Product(s) or
Performance(s), with a rubric
Other:
Resources Needed: On-site people, facilities: Facilitator/ Teacher
Equipment/Technology: Internet access (at home or school), graphing calculators
Materials: Chart Paper
Community Resources: Employee at a home improvement store
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal X Small/Focus Groups Whole Class Discussions X Fishbowl Discussions
Survey Other: essay X
Material Adapted From:
Discovering Geometry: An Investigative Approach, Serra, Michael. Key Curriculum Press, 2010 – for Hopple Flooring Proposal www.exeter.edu/academics/72_6539.aspx - for in-class tasks
Template adapted from Buck Institute for Education: Project Based Learning for the 21st Century
Virginia’s Senior Level Capstone Course Instructional Plan
Unit Title: Squared Squares
Driving Question: How can one measurement on a complex diagram be used to find the measures of all the others?
Task: Students will be given a floor plan for an addition of a room and a deck to a house. Given the amount of decking, the task is to find the dimensions of the family room and decide if there are multiple solutions to the task.
ENGAGE How will students’ interest be peaked so they want to engage in the inquiry in this unit? Time required: 20 minutes
Present students with HO #1: Hopple Flooring Proposal Task. Ask students to work in small groups to determine what is needed to answer question 1. Have them work together to discuss this
solution. Have each group post their solution on large chart paper. Directions for Instructor:
Teacher should give students 5 – 10 minutes to complete a gallery walk of their solutions.
Teacher will lead class discussion about area calculation errors.
Materials and/or Resources Needed:
Student HO #1: Hopple Flooring Proposal Task attached Internet access or access to home improvement store nearby
Students can complete HO #1 for additional credit or homework and bring their proposals to the class during this unit to present to the group.
Included on HO #1B is a list of internet sites for students who need review of skills required for this unit.
Mathematician Journal Prompts:
What difficulties did you encounter while calculating the area of the floor plan? EXPLORE Teacher provides guidance for the explorations to prepare students with the knowledge and skills to engage in the task.
Exploration 1: HO #2 Squared Squares Task A
Provide HO #2 Squared Squares Task A to each pair of students. The activity consists of a rectangle cut into nine square pieces, no two being the same size. Given that the smallest square is 2 cm by 2 cm, the students must figure out the sizes of the remaining pieces. Students write algebraic expressions for each of the unknown square sizes and use those expressions to write and solve a linear equation. Students will describe their solution and verify that it is correct, showing their work in their Mathematician’s Journal.
Mathematician Journal Prompts:
Students will self-assess on the prior knowledge and skills assumed for the unit
Time required: 70 minutes
Some suggested teacher hints to get groups started:
If the first responses are based on relative sizes of the squares, students need time to realize that their answer is incorrect, and then proceed to find a correct solution.
It may take several adjacent squares before students see the pattern.
Pose questions that ask students to describe what they know about the squares that make up the diagram to help move them from known square size to the adjacent square size using an algebraic expression.
Whole group instruction may be needed but try to use questioning techniques to elicit student participation.
If giving aid, try not to proceed beyond a few adjacent squares before having pairs work independently on the task.
Exploration 2: HO #3 Squared Squares Task B
Students work with their partner to complete Task B, a rectangle cut into nine smaller square pieces, no two the same size, with a
different composition from Task A. The smallest square is 9 cm by 9 cm. This task requires students to write algebraic expressions for each of the unknown square sizes, to use those expressions to write and solve a linear equation, and to use the solution to determine if the composite quadrilateral is a square. Students will complete the task and record all work in the Mathematician’s journal.
Skills or knowledge needed:
knowledge of the characteristics of quadrilaterals; ability to write an algebraic expression;
ability to write and solve an algebraic equation.
Directions for Instructor: The teacher will monitor work and offer prompting and probing questions to advance student thinking as needed.
How did doing Task A help you complete this one?
How did you verify that your answer was correct?
EXPLAIN Teacher introduces the main task of the unit and prepares students to in small group independent work.
Prepare student pairs to work independently on HO #4: Squared Squares House Plans Task
Give each student a copy HO #4 with the given floor plan for an addition of a room and a deck to a house. Have students read the task and remind the class they are given the amount of decking and their job is to find the dimensions of the family room and determine if there are multiple solutions to the task.
Point out that once the pair teams have completed the task, each person in the pair (teachers may decide to have the pairs do this together) must write an essay describing the process they used to complete the task with some justification as to why their solution(s)
Mathematician Journal Prompt:
Time required: 20 minutes
will work. The essay must also include a discussion of the connections with the Hopple Proposal that started off the unit as well as the Squared Tasks A and B.
Review the Grading Checklist for HO #4B Squared Squares House Plans that also provides some guidance for students to develop their essay.
To help students determine their own effectiveness in working in a group go over HO #5 Peer/Self Evaluation Squared Squares House Plan and have some discussion about what each component of the rubric means and what it looks like when partners are working together. ELABORATE The student groups are working independently with teacher consultations. Time required: 90 minutes
Student pairs work independently on the task on HO #4 Squared Squares House Plan .
Before students begin their essay ask them to respond to the Journal Prompt.
After they have written and discussed their journal prompt with their partner they will write their essay as described on HO #4B in the Grading Checklist for the Squared Squares House Plan.
Students will complete HO #5 Peer/Self Evaluation Squared Squares House Plan and turn this in with their essay.
Mathematician Journal Prompt:
Describe how the four tasks in this unit connect to one another mathematically. EVALUATE Working groups submit products or make presentations Time required: 90 minutes
Directions for Instructor regarding final evaluation:
Each group should complete the Mathematician’s journal entries individually prior to writing the essay.
Map the Unit
What do students need to know and be able to do to complete the task/project/problem
successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.
Task:
An investigation into algebraic expressions and equations using the characteristics of quadrilateral. The investigation includes linear and quadratic functions.
KNOWLEDGE AND SKILLS NEEDED Assumed
already learned Students will self-assess Will be taught during the unit 1. Knowledge of the characteristics of
quadrilaterals X X
2. Ability to write an algebraic expression
X X
3. Ability to write and solve an algebraic
equation X X
4. Ability to multiply polynomials
X X
5. Ability to use the Quadratic Formula or the graphing calculator to find roots of a quadratic
function X X
6. Verify that a solution makes sense in the
context. X X
7. Ability to write a five-paragraph essay
X What project tools will student’s use?
Know/need to know lists Daily goal sheet
X Mathematician’s Journals Briefs/Memos
Task lists
Planning Calendar
□ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ HO #1
Hopple Flooring Proposal Task
The Hopples want to redo their kitchen and family room floors. Here is a floor plan for the two rooms. There is a counter in the upper corner of the kitchen. It measures 12 feet long by 3 feet wide.
1. Find the area that needs to be covered with new flooring.
2. Research appropriate flooring that could be used in both rooms.
3. Pretend you are a general contractor competing for this job. Generate a proposal including…
appropriate flooring and where you will purchase it (a picture of the flooring will help your bid) the cost per square foot of the flooring,
the total cost of the flooring (excluding installation)
reasonable installation fee (based on research for your choice of flooring)
Student Independent Review of Knowledge and Skills
Skills required for unit are below. If you think you will need extra practice in these skills, you may want to use the links below for tutorial and/or practice:
Writing algebraic expressions:
http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11_simp.htm Solving linear equations:
http://hotmath.com/help/gt/genericalg1/section_1_4.html Multiplying polynomials:
http://www.brightstorm.com/math/algebra/polynomials-2/multiplying-monomials-andor-binomials-and-foil - tutorial
Solving a quadratic equation with the Quadratic Formula:
http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadformula.htm - lesson/review http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadpractice.htm - practice problems
HO #2
Squared Squares Task A
HO #3
Squared Squares Task B
The diagram below shows a rectangle that has been cut into eleven square pieces, no two being the same size. Given that the smallest piece is 9 cm by 9 cm, figure out the sizes of the other ten pieces. The original rectangle also looks like it could be square. Is it? Be able to describe your thinking and justify your solution.
HO #4
Squared Squares House Plan Task
Pat and Kim own a small, pleasant house that is rectangular in shape, 50 feet by 30 feet. They want to add on a family room that will be square, and then fill in the space adjoining the new room with a deck. A plan of the setup is at the right. They have not decided how large a family room to build, but they do have 400 square feet of decking. If they use it all, and keep to the plan, how large will the family room be? Is there more than one solution to this problem?
House
Room Deck
50
30
HO #4B
Grading Checklist for Squared Squares Task
Your group must complete the task and write an essay. The essay must contain a description of how you found your solution, how you verified your solution plus a discussion of how all the tasks in the unit related to one another. Be sure to include diagrams and work where appropriate. Your product should look more like a technical paper than a traditional English essay. You will be graded by the checklist below. Please use it as you write your essay.
Introduction (first paragraph) ____ of 2 points
Task restated
What you are supposed to do with the task (be specific)
Body of the essay (3 paragraphs) ____ of 10 points Diagram labeled with variable expressions (2 points)
Equation and how you developed it (2 points) Process used to solve the equation (3 points)
Verification that your solution is reasonable (2 points) Solution of task with context (1 point)
Conclusion (final paragraph) ____ of 4 points
Connections are made between all four tasks in the unit
Total points ____ of 16 points
Grade Assignment
A = 15 – 16 points B = 13 – 14 points C = 11 – 12 points D = 9 – 10 points F = below 9 points
HO #5
Peer/Self-Evaluation form: Squared Squares Project
The following is a list of statements to be answered about yourself and each of your group members. Think carefully about assigning rating values for each of the statements.
1 = Strongly Agree 2 = Agree 3 = Neutral 4 = Disagree 5 = Strongly Agree Self Teammate: Teammate: Willingly accepted
Answer Keys
Squared Squares Task #1 (HO #2)
The smallest square is 2 x 2, so assign the next smallest square side lengths of (x + 2), continue labeling the sides of the squares using algebraic expressions.
Set 2 sides of the largest square equal to each other and solve for x. You will find that x = 2.
Plug x = 2 into the expressions and you will find the sizes of the squares. They are (from smallest to largest):
2 x 2 (given), 4 x 4, 6 x 6, 8 x 8, 14 x 14, 22 x 22, 24 x 24, 28 x 28, 32 x 32
Squared Squares Task #2 (HO #3)
The smallest square is a 9 x 9, so assign the next smallest square side lengths of (x + 9), continue labeling the sides of the squares using algebraic expressions. (Note that some subtraction will need to be used to find the expression for the smaller, out-of-order square.)
Set 2 sides of the largest square equal to each other and solve for x. You will find that x = 7.
Plug x = 7 into the expressions and you will find the sizes of the squares. They are (from smallest to largest):
9 x 9 (given), 16 x 16, 21 x 21(this is the “out-of-order” square), 25 x 25, 34 x 34, 41 x 41, 43 x 43, 57 x 57, 77 x 77, 78 x 78, 99 x 99
Finally, the problem asks if the original rectangle is a square. No, it is not. If you add up two of the sides, the side lengths are not equal to each other.
Squared Squares House Plan Task (HO #4)
Since we know the area of the deck, we can use the A = l*w to find the side length for the new family room.
In order for the dimensions of the deck to make sense based off of the original problem, x = 10 would be the best answer.
