Mathematics/Numeracy, K-8 3-Part Lessons Jane Silva Instructional Leader Mathematics/Numeracy, K-8 SW

Teaching Through Problem Solving Using a Three-Part Lesson Model allows teachers to develop rich and engaging tasks naturally embeds the mathematical processes expectations leads to conceptual understanding and more meaningful connections Photocopy Process quick sheet

Teaching Through Problem Solving A good instructional problem: builds on students’ prior knowledge and skills; considers a key concept or big idea; has a meaningful context; has multiple entry levels (differentiation); solution is not immediately obvious; may have more than one solution; promotes the use of one or more strategies; requires decision making; may encourage collaboration.

Components of a 3-Part Lesson Part 1 (Before/Getting Started/Minds On) Part 2 (During/Working On It/Action) Part 3 (After/Reflect and Connect) Pass out templates

Identify the Curriculum Expectations What are/is: the content and process expectations this lesson addresses for your grade level; the prior knowledge and skills students would have learned in the previous grade; the overall and specific expectations this problem addresses for the next grade level; and expectations/connections to other strands. Ex: Gr. 8 - solve and verify linear equations involving a one-variable term and having solutions that are integers, by using inspection, guess and check, and a "balance" model

Determine the Big Ideas The broad, important understandings that students should retain long after they have forgotten many of the details of something they have studied. Ex: Any pattern, algebraic expression, relationship, or equation can be represented in many ways. The principles and processes that underlie operations with numbers and solving number equations apply equally to algebraic situations.

Determine the Learning Goal Learning Goals: Consider the curriculum expectations and big ideas Describe what students are expected to learn Provide students with a clear vision of where they are going Focus effective teacher feedback on learning Develop students’ self-assessment and self-regulation skills Ex: I will be able to create equations and use different strategies and representations to show that the equations are true.

3-Part Lesson – An Algebra Example

Part 1 - Before/Getting Started/Minds On relates to the day’s lesson goals and problem; activates prior knowledge; assesses students’ prior knowledge and skills; engages students/develops a context; checks for students’ understanding for ‘during’.

Part 1 - Before/Getting Started/Minds On 5-10 minutes Activating students’ mathematical knowledge and experience that is directly related to the mathematics in the lesson problem Use a smaller problem similar to the previous known problem Use student work responses for class analysis and discussion to highlight key ideas and/or strategies

Part 1 Sort and sequence the equation strips. Ex: Opportunity for Differentiation - Parallel Questions Only 1 equation in the envelope 2 equations in different colours 2 equations with different variables Partially completed solutions 3 or 4 equations in the same colour

Part 1

Part 2 – During/Action/Working On It Students are: actively-engaged in problem solving; making hypotheses and conjectures; choosing methods, strategies, and manipulatives; discussing mathematical ideas with others; constructing their own knowledge; and developing perseverance.

Part 2 – During/Action/Working On It Teacher is: scaffolding students’ learning; conferencing with small groups or individual students; observing and noting student/group strategies, mathematical language, and models of representation; engaging in Assessment FOR Learning.

Part 2 – During/Action/Working On It 15-20 minutes Understand the problem, make a plan, carry out the plan Students solve the problem individually, in pair, or in small groups The teachers support student understanding and assesses for learning

Part 2 Write an equation with 4 different numbers and the variable x. At least 2 numbers must be from this list: 4 13 100 1000 Show that your equation is true.

Assessment FOR Learning: Observation and Interview Template

Part 3 – After/Reflecting and Connecting/ Consolidation and Debrief Students are: reflecting on their own thinking (meta-cognitive skills) and the thinking of other students; communicating problem solving strategies, methods, and solutions to their peers; consolidating the learning of new concepts.

Part 3 – After/Reflecting and Connecting/ Consolidation and Debrief Teacher is: deciding which group’s strategies, methods, or solutions should be presented to highlight the mathematical thinking and to develop the mathematical understanding of all students related to the problem and lesson goals; facilitating the learning by annotating and labeling work samples; asking for clarification or having students summarize for partners or the whole group the thinking of the presenting group; and engaging in Assessment FOR Learning.

Part 3 – After/Reflecting and Connecting/ Consolidation and Debrief 20-25 minutes Teacher selects 2 or more solutions for class discussion and decides which solution to share Teacher organizes solutions to show math elaboration from one solution to the next, towards the lesson goal Student authors explain and discuss their solutions with their peers Teacher mathematically annotates solutions to make mathematical ideas, strategies, tools explicit

Part 3 What thinking did you use to create your equation? Which strategies did you use to show that your equation is true?

Teachers Can Differentiate Content Process Product According to Students’ Go through and talk about each green box. Content – What is being taught. You can differentiate the actual content being presented to students. Process – How the student learns what is being taught. For example, some students need to interact with the material physically, some might prefer to read a book. Product – How the student shows what he/she has learned. For example, students can write a paper or they can present information orally. Readiness – Skill level and background knowledge of child. We try to stay away from the word “ability” because you don’t always know the ability level of a child if their readiness level is low. Interest – Child’s interest or preferences – these can be interests within the curricular area (for example, they might be interested specifically in learning about folklore in a unit on volcanoes) or in general (for example, knowing a student’s favorite cartoon character could allow you to tie that into an example and might motivate the student) Learning Profile – This includes learning style (is the student a visual, auditory, tactile, or kinesthetic learner), as well as preferences for environmental (such as level of distraction, exposure to light or noise) or grouping factors (small group, large group, or individual) Readiness Interest Learning Profile Adapted from The Differentiated Classroom: Responding to the Needs of All Learners (Tomlinson, 1999)

Differentiation Strategies Tell participants they are now going to look at some strategies in more depth. Go to the next slide.

Goal The goal is to meet the needs of a broad range of students, but all at one time– without creating multiple lesson plans and without making students who are often labelled as strugglers feel inferior.

Differentiated Instruction Structures and Strategies Cubing Menus Choice Boards RAFTs Tiering Learning Centers Learning Contracts Open Questions Parallel Tasks Strategies Anticipation Guide Think-Pair-Share Exit Cards Venn Diagrams Mind Maps Concept Maps Metaphors/Analogies Jigsaw

Cube Powers Face 1: Describe what a power is. Face 2: How are powers like multiplying? How are they different? Face 3: What does using a power remind you of? Why? Face 4: What are the important parts of a power? Why is each part needed? Face 5: When would you ever use powers? Face 6: Why was it a good idea (or a bad idea) to invent powers?

Fractions, Percent and Decimals Menu Fractions, Percent and Decimals Appetizer (Everyone): What does the denominator and numerator tell you? Main dish (Choose 1): You want to estimate 20/30 as a percent. Describe your thinking. You want to estimate 0.3 as a fraction. Describe your thinking. Side dishes (Choose 2): Draw a picture to show why 0.4 and 6/15 are equivalent. Draw a flow chart to show how someone should proceed to convert a fraction to a percent. Dessert(if you wish) A decimal begins 0.24…. but then it continues. What do you know about the fraction it could represent. Alicia says that the only fractions that are whole numbers of percents have denominators of 2, 4, 5, 10, 20, 25, and 50. Do you agree? Explain.

Choice Board Fractions Complete question # …. on page …. in your text. Choose the pro or con side and make your argument: The best way to add mixed numbers is to make them into equivalent improper fractions. Think of a situation where you would add fractions in your everyday life. Make up a jingle that would help someone remember the steps for subtracting mixed numbers. Someone asks you why you have to get a common denominator when you add and subtract fractions but not when you multiply. What would you say? Create a subtraction of fractions question where the difference is 3/5. • Neither denominator you use can be 5. • Describe your strategy. Replace the blanks with the digits 1, 2, 3, 4, 5, and 6 and add these fractions: []/[] + []/[] + []/[] Draw a picture to show how to add 3/5 and 4/6. Find or create three fraction “word problems”. Solve them and show your work.

R.A.F.T. ROLE AUDIENCE FORMAT TOPIC Coefficient Variable Email We belong together Algebra Principal of a school Letter Why you need to provide more teaching time for me Students Instruction manual How to isolate me Equivalent fractions Single fractions Personal ad How to find a life partner

Tiers Fractions Tier 1: all fractions are proper; have common denominators; and can be modeled Tier 2: fractions are proper and improper; have different denominators, but all can be modeled with pattern blocks Tier 3: fractions are proper and improper and not all can easily be modeled

Learning Centers Surface Area Station 1: Simple “rectangular” or cylinder shape activities Station 2: Prisms of various sorts Station 3: Composite shapes involving only prisms Station 4: Composite shapes involving prisms and cylinders Station 5: More complex shapes requiring invented strategies

Open Questions

Strategies for Creating Open Questions Start with the Answer Closed: √64 = 8 Open: An irrational number is about 8. What might it be? √65 √64 2π + 2 8/3 π

Strategies for Creating Open Questions Ask for similarities and differences Closed: Describe each term in the equation y = 3x - 2 Open: How are these two equations alike? How are they different? y = 3x – 2 y = 6x - 4

Strategies for Creating Open Questions Replace a number with a blank Closed: A rectangle has a length 3cm and a width 4cm. What algebraic expression can describe features of the rectangle? Open: A rectangle has a length __cm and a width 4cm. What algebraic expression can describe features of the rectangle?

Parallel Tasks The idea is to use two similar tasks that meet different students’ needs, but make sense to discuss together.

Parallel Tasks - Examples Cell phone Plans Choose Plan 1 or Plan 2. How much would 250 minutes cost? Provide an equation. Per month Per minute Plan 1 $27 200 free; then 35¢ Plan 2 30¢

Parallel Tasks - Examples Numeration and Number Sense Task A: 1/3 of a number is 24. What is the number? Task B: 2/3 of a number is 24. What is the number? Task C: 40% of a number is 24. What is the number?

Parallel Tasks - Examples Find the equation of a line to complete this parallelogram: y = 8 y = -3x + 12 y = 2 Task 2: Find the equation of a line to complete this right triangle y = -2x + 8 y = 1/3 x

How to Create Parallel Tasks Think about the underlying big idea. Think about how it can be made more accessible to struggling students. Alter your original task to allow for that accessibility.

Resources Leading Math Success (Grades 7-9) The Leading Mathematics Search Tool for Instructors, Parents and Students (or LMSTIPS for short) was developed as part the Leading Math Success project in order to assist Teachers, Parents and Students in finding quality, relevant math resources on the Internet.

Edugains A dynamic site where Ontario educators involved in Grades K-12 teaching and learning can access a wealth of resources and information to support mathematics. http://www.edugains.ca/newsite/math2/index.html

Balanced Assessment in Mathematics From 1993 to 2003, the Balanced Assessment in Mathematics Program existed at the Harvard Graduate School of Education. The project group developed a large collection of innovative mathematics assessment tasks for grades K to 12, and trained teachers to use these assessments in their classrooms.

National Council Of Teachers Of Mathematics Designed to "illuminate" the new NCTM Principles and Standards for School Mathematics. (Activities, Lessons, Standards and Web Links)

Wired Math Free math games and resources for Grades 7, 8, 9 from the Department of Mathematics at the University of Waterloo.

National Council of Teachers of Mathematics Access to elementary to high school resources that include: articles, rich tasks and activities, problems, technology tips, and more.

Ontario Education Resources Bank Supported by the Education Ministry of Ontario. Includes lessons, units, assessments and more. Note: The content of the website is available to teachers and students. Teacher userid: tdsbteacher Teacher password: oerb

Gizmos ExploreLearning Gizmos ExploreLearning.com offers the world's largest library of interactive online simulations for math and science education in grades 3-12. In order to receive an username and password, email: evelyn.heath@tdsb.on.ca.