Numeracy+Plan

Professional Plan to Guide Numeracy Workshop Colleen Taylor

**Welcome and Introductions**

**Part A: The Big Ideas and Fractions**

1/ One large paper with each of the //big ideas// will be placed on each table in the room. The teachers at each table establishes general definitions for their //bid idea// and records them on the paper. The leader will guide them in this task using the following key points from the eworkshop.

Key Points

quantity: •Having a sense of quantity involves understanding the “howmuchness” of whole numbers, decimal numbers, fractions, and percents. •Experiences with numbers in meaningful contexts help to develop a sense of quantity. •An understanding of quantity helps students estimate and reason with numbers. •Quantity is important in understanding the effects of operations on numbers.

operational sense: •Operational sense depends on an understanding of addition, subtraction, multiplication, and division, the properties of these operations, and the relationships among them. •Efficiency in using the operations and in performing computations depends on an under- standing of part-whole relationships. •Students demonstrate operational sense when they can work flexibly with a variety of computational strategies, including those of their own devising. •Solving problems and using models are key instructional components that allow students to develop conceptual and procedural understanding of the operations.

relationships: •An understanding of whole numbers and decimal numbers depends on a recognition of relationships in our base ten number system. •Numbers can be compared and ordered by relating them to one another and to benchmark numbers. •An understanding of the relationships among the operations of addition, subtraction, multiplication, and division helps students to develop flexible computational strategies. •Fractions, decimal numbers, and percents are all representations of fractional relationships.

representation: •Symbols and placement are used to indicate quantity and relationships. •Mathematical symbols and language, used in different ways, communicate mathematical ideas in various contexts and for various purposes.

proportional reasoning: •Proportional reasoning involves recognizing multiplicative comparisons between ratios. •Proportional relationships can be expressed using fractions, ratios, and percents. •Students begin to develop the ability to reason proportionally through informal activities.

2/ The leader will instruct each group to brainstorm ways the study of fractions can support their //big idea//. I will use the following examples from the eworkshop to help the groups generate ideas.

**quantity**

Representing fractions using a variety of models (e.g., fraction strips, fraction circles, grids) develops students’ ability to interpret fractional quantities. Providing opportunities to solve problems using concrete materials and drawings helps students to understand the quantity represented by fractions.

**relationships**

Relating numbers to benchmarks is a useful strategy for comparing and ordering fractions. For example, to order 3/8, 5/6, and 7/8 from least to greatest, students might consider the relative proximity of the fractions to the benchmarks of 0, 1/2, and 1: •3/8 is less than 1/2 (4/8); •5/6 is 1/6 less than 6/6 and therefore is close to 1; •7/8 is 1/8 less than 8/8, and is also close to 1.

Students might realize that 3/8 is the only fraction in the set that is less than 1/2 and therefore it is the least. To compare 5/6 and 7/8, students might reason that sixths are larger fractional parts than eighths, and so 5/6 is less than 7/8.

In the junior grades, the focus should be on the use of models to show the relationship among fractions, decimal numbers, and percents, rather than on learning rules for converting among forms.

**representation**

Numerical symbols, and their position within a number or notation, are used to indicate quantity and relationships.

The symbolic representation of a fraction is a convention that extends our number system to infinitely smaller parts. Although understanding that a fraction describes a relationship to a whole or a part of a group has already been discussed, it is also important to understand that a fraction is a quantity. A fraction can be thought of as a single entity, with its own unique place on the number line – just as 3 is a single entity. As such, fractions, like whole numbers, can be compared and ordered (and operated on).

**proportional reasoning**

Proportional relationships can be expressed using fractions, ratios, and percents: for example, about 1/3 of all homes have a pet dog; the ratio of the team’s wins to losses is 2 to 3; sale items are 10% off the regular price. Students learn that the fractions, ratios, and percents used to express proportional relationships do not represent discrete quantities, but that they infer ideas such as “out of every”, “for every”, “compared with”, and “per”. The curriculum expectations outlined in the Number Sense and Numeration strand for each grade in The Ontario Curriculum, Grades 1–8: Mathematics, 2005 1 are organized around these big ideas.

Source: //A Guide to Effective Instruction in Math//, **Grades 4 to 6** Number Sense and Numeration: __Volume One: The Big Ideas__ Pages 11 to 45

**Part B: The General Principles for Instruction for Mathematics**

Each of the following principles will be posted at stations around the room. Teachers will circulate in pairs, or small groups, and comment on how they can, or how they do already, incorporate the principles in their instructional strategies. Each group records their ideas on large paper or sticky notes. As they circulate, they will also comment on the ideas of others.

foster positive mathematical attitudes; focus on conceptual understanding; involve students actively in their learning; acknowledge and utilize students’ prior knowledge; provide developmentally appropriate learning tasks; respect how each student learns by considering learning styles and other factors; provide a culture and climate for learning; recognize the importance of metacognition; focus on the significant mathematical concepts (big ideas)

Source: //A Guide to Effective Instruction in Math,// **//Kindergarten to Grade 6//**: __Volume One, Foundations of Mathematical Instruction,__ Chapter 2 Principles Underlying Effective Mathematics Instruction, pages 23-43

**Part C: The Website**

1/ Teachers will move to the computer lab. The leader will instruct the teachers to go to the following website: HYPERLINK "http://www.eworkshop.on.ca/edu/core.cfm" __www.eworkshop.on.ca/edu/core.cfm__

They will briefly explore //A Guide to Effective Instruction in Math, Grades 4 to 6 Number Sense and Numeration//. Here they will look through volumes one and five and compare the information from the resource with the ideas generated in the previous activity.

2/ The leader will then direct the students to the numeracy module on fractions. They will open the module and browse the contents.

3/ In pairs or small groups, teachers will complete the lesson plan using the template found here: HYPERLINK "http://www.eworkshop.on.ca/edu/pdf/Mod22_planning_template.pdf" __www.eworkshop.on.ca/edu/pdf/Mod22_planning_template.pdf__

4/ Teachers consider the materials they currently use for teaching fractions and brainstorm other materials which could be used.

5/ Finally, encourage the teachers to try the lesson posted here:

HYPERLINK "http://www.eworkshop.on.ca/edu/pdf/Mod22_lesson_summary.pdf" __www.eworkshop.on.ca/edu/pdf/Mod22_lesson_summary.pdf__

All good professional development should include some follow up and time for reflection. The teaches will meet again in a few months and reflect on how they have used the website to inform their teaching practice.