3-Part Lessons

Jane SilvaInstructional Leader

Mathematics/Numeracy, K-8SW

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

Teaching Through Problem Solving

• 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.

A good instructional problem:

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)

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

Identify the Curriculum Expectations

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 Big Ideas

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.

Determine the Learning Goal

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.

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

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.

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 – After/Reflecting and Connecting/ Consolidation and Debrief

Part 3

What thinking did you use to create your equation?

Which strategies did you use to show that your equation is true?

Content Process Product

According to Students’

Readiness Interest LearningProfile

Teachers Can Differentiate

Adapted from The Differentiated Classroom: Responding to the Needs of All Learners (Tomlinson, 1999)

Differentiation Strategies

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 studentswho are often labelled as strugglers feel inferior.

Goal

Differentiated Instruction Structures and Strategies

Strategies• Anticipation Guide • Think-Pair-Share• Exit Cards• Venn Diagrams• Mind Maps• Concept Maps• Metaphors/

Analogies• Jigsaw

Structures• Cubing• Menus• Choice Boards• RAFTs• Tiering• Learning Centers• Learning Contracts• Open Questions• Parallel Tasks

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?

CubePowers

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.

MenuFractions, Percent and Decimals

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.

Choice BoardFractions

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

Variable Students Instruction manual How to isolate me

Equivalent fractions Single fractions Personal ad How to find a life partner

R.A.F.T.

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

TiersFractions

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

Learning CentersSurface Area

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 + 28/3 π

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?

Strategies for Creating Open Questions

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

Parallel Tasks

• Cell phone PlansPer month

Per minute

Plan 1 $27 200 free; then 35¢

Plan 2 0 30¢

• Choose Plan 1 or Plan 2.• How much would 250

minutes cost?• Provide an equation.

Parallel Tasks - Examples

• 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?

Numeration and Number Sense

Parallel Tasks - Examples

Parallel Tasks - Examples

Task 1:Find the equationof a line tocomplete thisparallelogram:y = 8y = -3x + 12y = 2

Task 2:Find the equationof a line tocomplete thisright triangley = -2x + 8y = 1/3 x

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.

How to Create Parallel Tasks

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: tdsbteacherTeacher password: oerb

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.