How do you create true problem solving lessons that are aligned to specific mathematical knowledge and /or skills? Include a lesson idea and a reflection explaining how you would implement this planning idea. A good place to start if you are a teacher in the state of Washington is http://standards.ospi.k12.wa.us/Default.aspx?subject=7,PE After you understand the learning targets of the mathematics standards you must have a good resource of problem solving activities and ideas. Here is a good website for adapting textbook math problems to make them real problem solving activities http://www.homeschoolmath.net/teaching/problem_solving.php.
Please share with use your ideas for planning problem solving activities that are aligned to specific mathematics standards.
For my Standards-based Problem Solving lesson, I have chosen to have my students tackle the following question: “When is it financially beneficial to purchase a motor home for a long road-trip?” By asking the question in this form, i have left a lot to be questioned by my students. This was intentional. My plan is to challenge my student to think about important clarifying questions, information that is important/needed, and assumptions that we will need to make. After giving students a short while to ponder these things, i plan to lead a class discussion until we come to agreement on some of these. The reason this is Standards-based is because this question leads students to setting up and solving a system of equations. This is a Washington State Algebra 1 standard. The following link will direct you to my blog with more information (formal lesson plan, worksheet, and my reflection):
http://radmathteacher.edublogs.org/category/math-524/
(I couldn’t find a way to upload these items to this blog, so I uploaded them to my blog and have linked this above)
Oops, I forgot to sign my comment above again….that was Aaron
The following lesson is an attempt to apply Dave Meyer’s instructional strategy of encouraging patient problem solving through limited teacher intervention; i.e., being less helpful. Meyer suggests the following four steps to accomplish this: 1) Provide a basic visual, 2) Ask the Big Question, 3) Introduce the Math Structure, and 4) Develop the steps together. The students will work in pairs to determine the furthest possible location for a new mini-mart scheduled to be built in the neighborhood of the school; with the question being, ”Is the store going to be near enough for students to frequent the establishment during their 30 minute lunch break?”.
The students will be given a map of the area, including a scale, in addition to research tools such as the internet. They must use a verifiable rate of human travel, whether walking, jogging, etc., in conjunction with the formula d = rt to determine the furthest possible distance the store can be from the school. Then, using the map, the students will determine the possible locations for said mini-mart.
The outcomes of this problem require students to solve the problem, explain their reasoning, and verify that their solution is reasonable.
Lesson Title: Using d = rt to determine the location of a mini-mart.
Outcomes: Students will measure the distance an object travels in a given interval of time using d = rt.
Students will show their work, explain their thinking, and verify that the answer to the problem is reasonable.
Standards Addressed: 7.2 B: Solve single- and multi-step problems involving proportional relationships and verify the solutions.
Materials:
1. Enlarged map of the area near their school with scale.
2. Internet access.
3. Teacher consultation.
Procedures:
Students will be asked to work in pairs to determine furthest feasible locations for a new mini-mart based on the assumption that they would need to be able to walk to the store and return by the end of their 30-minute lunch. The only resources provided will be an enlarged map of the area around the school with scale in addition to internet access.
Students will use their knowledge of d = rt to determine the furthest possible distance the store may be from the school, and consequently possible locations. They must justify their reasoning using verifiable data (i.e. the average human walking speed is 3 mph).
Reflection – Problem Solving
I teach algebra I and geometry at Prosser High School. We adopted Holt McDougal textbooks for Algebra I and Geometry last year, which is very traditional. I found that the Holt curriculum contains a good problem solving handbook in the textbook and problem solving worksheets for every lesson found in its accompanying “Chapter Resources” workbook. The handbook includes problem solving examples and homework problems and the worksheets include homework problems, starting with free-response and ending with multiple-choice. Both the example problems and the homework problems need only a little tweaking to make them open-ended.
The problems are definitely real-world and are relevant to today’s students. I want to make sure I include standards-based problem solving regularly in my lessons to make the mathematics more relevant and more engaging and give the students a reason to learn the steps to solving mathematical problems. With practice at solving real-world problems, the students will add tools (strategies) to their mathematical toolbox and will be able solve any problem they encounter, including those on the End-of-Course Exam.
Although, this is an entire lesson that will introduce the unit on solving systems of equations, my goal for the next school year is to include standards-based problem solving in nearly every lesson. I plan on beginning each lesson with a motivating, relevant problem that the students need to solve to introduce the mathematical concept at the center of the daily lesson. I’m hoping this will also keep them engaged in the mathematics and help them find reason and purpose in learning how to use mathematics to solve problems. I also want to include a few interesting and relevant word problems to be solved on the daily homework assignment.
Lesson Plan – Standards-based Problem Solving
Title: Solving Systems of Equations
Outcomes: After doing this lesson, students will be able to solve an open-ended word problem involving systems of equations.
Standards addressed: “A1.1.C Solve problems that can be represented by a system of two linear equations or inequalities” (OSPI, Washington State 6-12 Mathematics Standards, 2008).
Materials: Problem Solving Strategies that is already pasted in their math journals and posted on the wall of the classroom
60 copies of Problem Solving worksheet (enough for 2 Algebra I classes)
Classroom copy of Holt text
Classroom set of graphing calculators
Procedures: The teacher will state her expectations of the students during group work. The teacher will remind the students to utilize the list of problem solving strategies that they have pasted in their math journals. The students will be working collaboratively in their groups of four (the desks are already put together in groups of four) on the Problem Solving worksheet. Following their work on the problems, the teacher will guide the students in a whole-class discussion about solving systems of equations.
Assessment: The Problem Solving worksheet must be completed for the students to get half their daily homework points. The other half of their points comes from participating in the discussion. Solving systems of equations is a concept tested on their unit test, revealing mastery of this topic. Formative assessment will take place as the teacher walks around the room, making sure everyone is engaged and using the math on the worksheet and in the discussions.
Problem Solving Strategies
Draw a Diagram or Graph
Make a Model
Guess and Check
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Problem Solving Strategies
Draw a Diagram or Graph
Make a Model
Guess and Check
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Problem Solving Strategies
Draw a Diagram or Graph
Make a Model
Guess and Check
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Problem Solving Strategies
Draw a Diagram or Graph
Make a Model
Guess and Check
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Problem Solving Strategies
Draw a Diagram or Graph
Make a Model
Guess and Check
Work Backward
Find a Pattern
Make a Table
Solve a Simpler Problem
Use Logical Reasoning
Use a Venn Diagram
Make an Organized List
Please show your work.
Problem Solving Worksheet
1. Bowl-o-Rama charges $2.50 per game plus $2 for shoe rental, and Bowling Pinz charges $2 per game plus $4 for shoe rental.
a) For how many games will the cost to bowl be the same at both places?
b) What is that cost?
c) When is it cheaper to bowl at Bowl-o-Rama and when is it cheaper to bowl at Bowling Pinz?
2. Long Distance Inc. charges a $1.45 connection charge and $0.03 per minute for a long distance call. Far Away Calls charges a $1.52 connection charge and $0.02 per minute for a long distance call.
a) For how many minutes will a call cost the same from both companies?
b) What is that cost?
c) When is it better to call using Long Distance Inc and when is it better to call using Far Away Calls?
d) What if Long Distance Inc. raised its connection charge to $1.50 and Far Away Calls decreased its connection charge by 2 cents. Now which company is better to use for calling long distance? Why?
Problem Solving Worksheet – Page 2
3. One high-speed Internet provider has a $50 setup fee and costs $30 per month. Another provider has no setup fee and costs $40 per month.
a) In how many months will both providers cost the same?
b) What will that cost be?
c) If you plan to cancel in 1 year, which is the cheaper provider?
4. Maribel has 8 coins in her pocket totaling $1.25. The money is in quarters and dimes. How many of each coin does Maribel have in her pocket?
5. Vincent grilled 21 burgers at a block party. He grilled the same number of pounds of turkey burgers as hamburgers. Each turkey burger weighed 1/4 pound and each hamburger weighed 1/3 pound. How many of each did Vincent grill?
Problem Solving Worksheet Key
1. a) 4 games
b) $12
c) If less than 4 games are bowled, Bowl-o-Rama is cheaper. If more than 4 games are bowled, Bowling Pinz is cheaper.
2. a) 7 minutes
b) $1.66
c) It is better to use Long Distance Inc. if the call is under 7 minutes. If the call is over 7 minutes, it is better to use Far Away Calls.
d) Far Away Calls is cheaper because it costs less per minute.
3. a) 5 months
b) $200
c) Provider 1: $410, Provider 2: $480; Provider 1 is cheaper
4. 3 quarters, 5 dimes
5. 12 turkey burgers, 9 beef hamburgers
My standards based problem solving lesson is to be used as a supplemental activity in a lesson about the Pythagorean Theorem. All materials can be found on my EduBlog (link below).
http://pierckat.edublogs.org/2011/07/11/standard-based-problem-solving-lesson-plan/
-Katelyn
To see my standard based lesson plan and reflection click the link below. This lesson involves students using actual data on obesity to form their own graphs.
http://jacobsod.edublogs.org/2011/07/11/standard-based-problem-solving-lesson-plan/
My Standards Based Lesson Plan is a follow-up lesson for Geometry Classes after the concept of similar triangles has been covered. To see my lesson plan and reflection visit my edublog.
http://jcoulson.edublogs.org/2011/07/12/hello-world/
My lesson is in Geometry in the Right Triangles Unit. The specific lesson is using special right triangles to find missing sides of a triangle in real life situations. Here is the link:
http://elenalala.edublogs.org/planning-standard-based-problems/
I am working in the design of a class plan about factorization. I want to use different representation in order to make conections amoung graphics, factors and root. I am new in blogs, so I am doing different mistakes. It is funny, specially because mistakes also is a way to learn. I promise to fix it.
Go ahead and give me your comment.
http://estradaa.edublogs.org
http://esesanalu.edublogs.org/
This standards based lesson problem solving lesson focuses on real life application of Mathematics. Students are asked to create a healthy dinner menu and cook for a minimum of four people, all for less than $5 a person.
http://birruetaf.edublogs.org/2011/07/17/dinner-project/
My Standards-based Problem-solving lesson encourages development of a student’s ability to visualize cross-sections of a 3-D solid. Students take 3-D solids, as represented by cheese, to create and explore cross-sections. The problem solving portion happens as students explore and generalize the types of cross-sections that can be created.
http://gplb.edublogs.org/2011/07/18/standards-based-problem-solving-lesson/
Lesson Plan
Grade: College Class: MPC 090 – Prealgebra
LESSON TITLE: “Which is a better buy?”
OBJECTIVES :
Students will be able to:
Use ratios and proportions
Use reasoning, problem solving, and communication skills
RATIONALE:
The purpose of this lesson is to introduce the concept of unit pricing, the multiple strategies to find out which is a better deal and to work on improving problem solving skills.
ASSESSMENT:
To assess students understanding I will walk around the room asking individual students questions. Each student will need to have explained what the solution is and what information was necessary to solving the problem and why and what strategy they used.
PROCEDURE:
Assign the student to begin working individually on the following problem: Exploration: Unit Pricing
After a few minutes of letting them struggle through some of this, I will start a class discussion to see if we can come up with multiple ways to solve the problem.
Lastly, I will then ask the students to begin solving another problem showing all of the ways to solve a unit pricing problem.
INSTRUCTION MATERIALS:
Copies of the worksheet
Worksheet: Unit Pricing
Unit pricing is now common. Underneath most items in the grocery stores is the unit price of the item. For example, if a 24-ounce jar of applesauce costs $2.79, the unit price is 11.6¢ per ounce. Many states now require grocery stores to show the unit price below each item. But, what if the items you are trying to compare are given different units, such as one is cents per ounce and the other is dollars per pound. How can you decide which is a better deal.
Let’s say you have pancake mix in two different sizes: 30 ounces and 50 ounces. The smaller box is marked at $2.89, and the larger is $4.69.
Determine which is a better deal and explain your process.
Jackie did the problem this way:
30/2.89=10.38 50/4.69=10.66
What do those two numbers mean?
Describe a situation when you might buy an item that has a higher unit cost.
What are the four strategies for solving this type of problem?
Standards-based Problem Solving Lesson Reflection
For this assignment I decided to use solving proportions to find unit prices because it is one of the “standards” I need to cover for MPC 090.
What I like about this question is that there are multiple ways to solve it and no one way is better than another. This allows the student to problem-solve and think critically about the information needed to answer this question and which method will work best for them. It is also great that this problem is relevant. Now, most students may not feel that this is a dilemma, but that is what many companies depend on, larger sized products are not always the better deal.
I would go about this lesson by first reviewing what proportions are and then posing the question to my students, and then giving them the handout. After giving my students some time to think through these questions, then I will have a quick class discussion to allow students to discuss the different ways they used to solve the problem. Through this discussion and possibly some prompting from me we should come up with the four strategies to solve the problem.
Next, I plan to let my students work individually to solve another problem using all four methods, and to reflect on their answers, and to decide on a strategy they prefer when solving these types of problems. I feel that the students seeing multiple strategies and trying each of them out will allow them to see that there is almost always more than one way to get to an answer and just because it might not be the strategy you were taught doesn’t always mean that it isn’t a valid strategy.