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

2. 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

3. 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.

4. 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

5. 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

6. 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.

7. 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.

8. 3-Part Lesson – An Algebra Example

9. 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’.

10. 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

11. 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

12. Part 1

13. 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.

14. 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.

15. 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

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

17. Assessment FOR Learning: Observation and Interview Template

18. 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.

19. 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.

20. 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

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

22. 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)

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

24. 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.

25. 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

26. 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?

27. 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.

28. 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/ • 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.

29. 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

30. 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

31. 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

32. Open Questions

33. 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

34. 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 – y = 6x - 4

35. 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?

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

37. 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¢

38. 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?

39. 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

40. 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.

41. 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.

42. 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.

43. 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.

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

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

46. 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.

47. 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

48. Gizmos ExploreLearning
Gizmos ExploreLearning.com offers the world's largest library of interactive online simulations for math and science education in grades In order to receive an username and password,