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Facilitator Notes for Fraction Workshop

Day 1

Table of Contents

Section Page #

Chart of Files to accompany these activities 1

Introduction 2

Section 1 Meanings of Fractions, Developmental Work, and Misconceptions

Part 1: Fraction Meanings 4

Part 2: Partitioning and Iterating 7

Part 3: Fraction Misconceptions 8

Part 4: EMLA Common Errors 9

Section 2 Review Grade 7 fraction work

Activity 1: Warm-up and Questions 10

Activity 2: Fraction Hunt 11

Activity 3: Generalizations 11

Activity 4: Equivalent Fractions 12

Activity 5: Estimating the Sums 12

Activity 6: Which Fraction is Larger? 13

Activity 7: Fractions, Decimals, Percents 14

Section 3 Conceptualizations for Fractions 15

Section 4 Mental Math 16

Section 5 Bank of Questions 17

Section 6 Other Resources 18

Chart of Files to Accompany these Activities:

Section 1

|Number |Title / Description |File Name |

|01 |Fraction Meanings PowerPoint Presentation |01_fraction_meanings_presentation.ppt |

|01A |Meanings of Fractions |01A_meanings_of_fractions.doc |

|01B |Divide the Figure into Parts |01B_divide_figure_into_parts.doc |

|01C |Fraction Parts not Adjacent |01C_parts_not_adjacent.doc |

|01D |Number Line |01D_number_line.doc |

|01E |Chocolate Block Task – Instructions |01E_chocolate_block_task_instruct.doc |

|01F |10 Practical Tips for Making Fractions Come Alive and Make |01F_10_tips_fractions_come_alive.pdf |

| |Sense | |

|01G |Creating, Naming, and Justifying Fractions |01G_creating_naming_justifying_fractions.pdf |

|01H |Partitioning Activities |01H_partitioning_activities.doc |

|01I |Partitioning and Iterating – animated Geometer’s Sketchpad |01I_part_iter_animated.gsp |

| |file | |

|01J |Partitioning Items in Different Ways |01J_different_partitioning.doc |

|01K |Iteration – Proportional Reasoning |01K_iteration_proportional_reasoning.doc |

|01L |Activities with Fractions Greater than 1 |01L_fractions_greater_than_1_act.doc |

|01M |Fractional Parts – which are correct? |01M_fractional_parts.doc |

|01N |Fractions Greater than 1 |01N_fractions_greater_than_1.doc |

|01O |Common Errors seen in the2010–11 EMLA |01O_2010-11_EMLA_common_errors.doc |

Section 2

|02A |Warm-up question (PowerPoint) |02A_warm_up_question.ppt |

|02B |Name that Number (PowerPoint) |02B_name_that_number.ppt |

|02C |Fraction Hunt (PowerPoint) |02C_fraction_hunt.ppt |

|02D |Equivalent Fractions (PowerPoint) |02D_num_denom.ppt |

|02E |Naming Shaded Sections |02E_naming_shaded_sections.doc |

|02F |Estimating the Sums of Fractions |02F_estimating_sums_of_fractions.doc |

|02G |Which Fraction is Larger? |02G_which_fraction_is_larger.doc |

|02H |Principles for Comparing Fractions |02H_principles_for_comparing_fractions.doc |

|02I |Fraction, Decimal, Percent Cards |02I_frac_dec_percent_cards.doc |

Section 4

|04A |Mental Math Video, Disk 3, Segment 1 |04A_MMvideo_D3_seg1.doc |

|04B |Mental Math Video, Disk 3, Segment 1 (Answers) |04B_MMvideo_D3_seg1_with_answers.doc |

|04C |Fractions in Mental Math PowerPoint |04C_fract_mentalmath_presentation.ppt |

|04D |Mental Math Video, Disk 3, Segment 2 |04D_MMvideo_D3_seg2.doc |

|04E |Mental Math Video, Disk 3, Segment 2 (Answers) |04E_MMvideo_D3_seg2_with_answers.doc |

Section 5

|05A |Fractions Question Bank |05A_Gr8_fractions_question_bank.doc |

|05B |Fractions Question Bank with Answers |05B_Gr8_fractions_question_bank_answers.doc |

The purpose of this workshop is to give Grade 8 teachers an overview of the important fraction ideas from P-6 and help them teach the following Grade 8 outcomes for conceptual understanding.

B5 – add and subtract fractions concretely, pictorially, and symbolically

B6 – add and subtract fractions mentally when appropriate

B7 – multiply fractions concretely, pictorially, and symbolically

B8 – divide fractions concretely, pictorially, and symbolically

B9- estimate and mentally compute products and quotients involving fractions

B11 – model, solve and create products and quotients using fractions

The focus will be to use models, contexts, and activities that promote the development of conceptual understanding of fractions. Research says that

▪ Models help students understand the “size” of a fraction.

▪ Models help students clarify ideas that are often confused when they just use symbols.

Materials:

1. Indicated handouts

2. chart paper, markers, tape

3. pattern blocks

4. rulers

5. Fraction Factory

This workshop has been set up as sections.

1. Through activities, have teachers explore the meanings of fractions, the terms “partitioning” and “iteration”, and misconceptions with fractions. This is mainly a review of some of the work in P-6. This section has 4 parts. It is recommended that this entire section be done.

2. Through activities, have teachers review some of the fraction work from Grade 7. These activities complement the Get Ready work found in Chapter 2, pages 48-55, of the text. You may choose to do all or just some of these activities as time permits.

Sections 1 and 2 require the major allotment of time for this workshop.

3. Go over, with teachers, chapter 2 in the textbook to see how it promotes conceptual understanding and addresses the outcomes. The intent is not to do the section with teachers but have them see the layout of the chapter, the use of models, the conceptual understanding found in the DTMs, etc. Also go over Chapter 2 in the Teacher’s Resource that accompanies the text. Point out that each section of the chapter has Teaching Suggestions (e.g. page 44) that has suggestions on how to approach the section and especially the DTM.

4. Mental Math and Fractions in Grade 8. Explore the expectations, found in Grade 8, for mental computations with fractions. The Yearly Plan and PowerPoint and GSP files created from the yearly plan will be used. There is further support in the mental math videos for Grades 7-9.

5. Go over the included item bank of questions that can supplement the material in the text and other resource materials. There are 2 files – one just has the questions (05A_Gr8_fractions_question_bank.doc) and one has the questions and answers (05B_Gr8_fractions_question_bank_answers.doc)

6. In this section is a list of the resources that were used to prepare the workshop.

Section 1:

Through activities, have teachers explore the meanings of fractions, the terms “partitioning” and “iteration”, and learn about misconceptions and errors with fractions. This is mainly a review of some of the work in P-6. (There are 4 parts to this section.)

Materials:

1. Indicated handouts

2. chart paper, markers, tape

3. pattern blocks

4. rulers

Part 1: Fraction Meanings

To help students with the Grade 8 fraction work, it is important that junior high teachers review the various interpretations of a fraction. A. Have teachers do a Think-Pair-Share on the different meanings that fractions represent.

1. Put teachers in pairs and ask them to record the different meanings of a fraction. Tell them to include a picture.

2. At each table have the teachers combine their work and record on chart paper. Post the chart paper.

3. Have each group go over their meanings. While this is happening have another blank sheet of chart paper posted where the facilitator can record and summarize the teacher’s different meanings.

4. Pass out a copy of 01A_meanings_of_fractions.doc template for teachers to use as you go over the meanings using the information that follows. The template is to be filled in as explained below.

|Meanings of Fractions |

|Meaning |Example |

|Record the meaning here | |

| | |

| |Draw an example here |

| | |

|Record special notes about the meaning here – found in bullets below | |

A PowerPoint of the meanings in this section is provided: 01_fraction_meanings_presentation.ppt Using the PowerPoint, go over the following meanings with teachers using the bullets as discussion points. Have teachers do the indicated activities.

It is recommended that you first do the activity connected with the meaning and then ask what meaning is being explored. Then go over the meaning and the information contained in the bullets for that meaning.

Meanings or Interpretations of Fractions:

1. Part of a Whole – the result when the whole or unit is partitioned into equal-sized parts.

▪ Sharing tasks promote developmental understanding for part of a whole. Have teachers do and discuss this task – “Illustrate 2 different ways that 6 students can share 4 chocolate bars.” An alternate task: Ask teachers to divide a sheet of paper in fourths. Ask them to do it in more than one way.

▪ The parts into which the whole is divided are congruent but do not need to be identical. Activities found on page 29 in Proportional Reasoning are good to illustrate this.

01B_divide_figure_into_parts.doc

▪ Fraction parts do not have to be adjacent.

01C_parts_not_adjacent.doc

2. Parts of a Set – the result when a group or a collection of things is partitioned.

▪ For parts of a set, the members of the set do not have to be identical.

Put some pattern blocks on each table. Have teachers scoop a handful of pattern blocks and ask. “What fraction of the blocks you are holding in your hand are hexagons? Rhombi? etc

▪ Students find this more difficult as they have to view the entire set or group as one unit.

▪ Containing or enclosing the set helps students see the set as one unit.

3. Part of a Measure –this meaning involves associating marks on measuring devices such as rulers with fraction names. It is more sophisticated than the part of a whole meaning because each mark corresponds to a number.

▪ The number line is considered a measurement model.

▪ This is the most abstract of the first three meanings but the most useful for the teaching of computational skills.

▪ Pass out sheet 01D_number_line.doc to teachers. There are two lines on the page but just one is needed to do this activity. The directions below are to be read aloud, one at a time, allowing each teacher to work individually before moving on to the next step.

“For the number line below

i) divide the segment between 0 and 2 into 2 equal parts and label the point you made

ii) divide each new segment into 2 equal parts and label the new points

iii) divide each new segment into 3 equal parts and label the new points

iv) check with your partner to see if you labeled the points the same way.

Since teachers will have different names for some of the marks, you may wish to have a discussion here about equivalent fractions or wait till Section 2, activity #4.

4. Can be used to name a Ratio – a comparison between two quantities that may or may not involve different units.

▪ A ratio is a numerical relationship

▪ A fraction as a ratio does not involve wholes, groups or the name for a point (measure).

▪ The bottom half of page 147 in the text can be used here.

5. An indicated division - the symbolic form of a fraction is seen as the quotient of 2 integers with the fraction bar as a signal to divide.

▪ is an article by Doug Clarke that explains a fun activity you can do here

▪ 01E_chocolate_block_task_instruct.doc is a summary of the instructions for this activity.

I recommend you read the article or at least enough of it to get the idea and then go to the instructions. I summarized Doug Clarke’s task, as he wrote it. Although he uses chocolate bars, you may wish to use other items that will still satisfy the intent of the activity and fit our food policy.

6. A fraction can also be used as an operator to operate on a unit (e.g. [pic] of 12).

▪ In Grade 7 students mentally multiply whole numbers by fractions.

▪ A misconception that often occurs and teachers need to be aware of is that students think that multiplication “always makes bigger” and division “always makes smaller”. Using fractions as operators, contexts and visual images will help address this misconception.

▪ A task for this meaning is to ask teachers to create a context where a proper fraction is an operator.

NOTE: After the 6 meanings have been gone over, teachers could be given this follow-up activity:

Write about or draw [pic] to illustrate each of the meanings.

Have teachers look at the article 10 Practical Tips for Making Fractions Come Alive and Make Sense and read it before they start their unit on fractions. 01F_10_tips_fractions_come_alive.pdf

Part 2:

One of the words in the last section was “partitioning”. The article Creating, Naming, and Justifying Fractions (01G_creating_naming_justifying_fractions.pdf) discusses the two powerful images, partitioning and iterating, that have long been recognized as important to the understanding of and operating on fractions.

B. Present teachers with the following two problems, have them work individually and then discuss with a partner or at their table.

01H_partitioning_activities.doc

1. Niko orders a 24-piece pizza for his party. If [pic]of the pizza was eaten, how many pieces were eaten? Draw a diagram to support your answer.

2. Monique still has [pic][pic]of her piece of licorice shown below. Sketch her original piece of licorice.

A. When you discuss these questions, look for the following:

▪ in the first problem was the pizza divided into 4 equal parts first? This is partitioning – the whole is divided into 4 equal parts. After dividing into fourths, [pic] is then thought of as 3 of the four equal parts or 3 one-fourths

▪ in the 2nd problem, did they produce 4 more copies of the original piece ( [pic]) and put them together to get the one whole? This is iteration.

B. You can now use 01I_part_iter_animated.gsp which is an animated GSP version of page 396 in the article and will help you summarize partitioning and iterating. You will have to scroll down the screen in order to show the second example.

C. Here are 3 more activities you can do with the teachers to reinforce the ideas of partitioning and iterating. Discuss with teachers the visual images that are created when they think of the activities through the terms – partitioning and iterating.

▪ 01J_different_partitioning.doc – Students use the same whole set of cookies and do different partitioning

▪ 01K_iteration_proportional_reasoning.doc – several of the questions taken from pages 33 and 34 in Proportional Reasoning reinforce iteration

▪ 01L_fractions_greater_than_1_act.doc – the activity on page 395 of the article which extends to fractions greater than 1.

Part 3:

C. Fraction Misconceptions:

As Grade 8 teachers work with students and the fraction outcomes for Grade 8, they should be aware of the following misconceptions some students may have.

1. Students may look at a fraction and see two whole numbers instead of one relationship.

▪ Indicators of this problem are

a) if a student looks at [pic] and says “three-fours”

b) when asked to place [pic] in the correct place on a number line, they place it between 3 and 4 instead of between 0 and 1.

▪ Seeing a fraction as two whole numbers is a serious problem as the student does not yet understand the meaning of a fraction. Teachers should review the fraction progression maps for P-6 or the PRIME materials to help plan lessons that will help the student.

2. Students may not recognize or may not understand the importance of the whole.

▪ Students should be encouraged to circle the whole or if given a context use language to identify the whole.

3. Students may not realize the whole, whether it be a region or a measure, has to be divided into equal parts.

▪ Students could be shown the following diagram and asked which figures are correctly divided into fourths.01M_fractional_parts.doc

▪ Students need many opportunities to work with diagrams and contexts to correct this misconception

4. Students may not understand that parts of a set can be different. They may be confused when the members of the set are not all the same. Refer to the pattern block task done in the Meaning of Fractions section.

▪ Having students circle the set, given the image of one whole, will help.

▪ Using contexts: Jasmine has a selection of books on her shelf – 5 comics, 7 mysteries and 3 biographies. What fraction of her books are mysteries?

5. Fractions are never bigger than 1.

▪ Use a context :e.g.- 1 ½ hours to mow the lawn

▪ Present students with a number line with points marked that are greater than one and ask students to name the points using a fraction.

Use the activity 01N_fractions_greater_than_1.doc to reinforce this.

6. Multiplying gives a larger number and dividing gives a smaller number.

▪ Directly attending to “partitioning” and giving contexts where students need to partition to solve will help here.

Part 4:

D. The following comments are found in 01O_2010-11_EMLA_common_errors.doc which is a report that was sent out following the June 2011 EMLA. These comments are concerning problems with fractions. Go over the following comments with the teachers. This file is also found in the EMLA section of the website. You may wish to go over the document or parts of it at another time.

• Many students did not seem to know the meaning of ‘product’ or ‘difference’.

▪ Interpreting points on the number line representing integer values or improper fractions was a challenge

▪ Placing a fraction between 0 and [pic] or [pic]and 1 was a challenge.

Section 2:

Through activities, have teachers review some of the fraction work from Grade 7. These activities complement the Get Ready work found in Chapter 2, pages 48-55, of the text. You may choose to do all or just some of these activities as time permits.

These activities were chosen based on teacher feedback as to what students have troubles with, such as benchmarks, estimation etc. There is a warm-up question and 7 activities, each chosen to review a specific topic.

Materials:

1. handouts

2. Fraction Factory

3. Number line

Warm-up question to do with teachers: 02A_warm_up_question.ppt

For the following statement

The sum of two proper fractions is less than 1.

decide if each of the following statements is always true sometimes true or never true.

a) Each fraction is less than one-half

b) One fraction is greater than one-half

c) Their difference will be less than one-half

Activities:

1. Name that number: An open-ended question to help clarify understanding. 02B_name_that_number.ppt PowerPoint presentation – each question above is on a separate slide.

Ask teachers questions such as:

a) Name a fraction between [pic] and one.

b) Name a fraction between [pic] and [pic], other than [pic].

c) Name a fraction between 0 and [pic] whose numerator is not one.

d) Name an improper fraction between 2 and 2[pic].

e) Name a mixed number between 3[pic] and 4.

f) Name two fractions, both close to 1, whose sum is close to 2.

g) Multiply 24 by a proper fraction to get a product close to 17. Name that fraction.

(Teachers could have a number line in front of them. This activity can easily be differentiated.)

2. Fraction Hunt: An activity where students are given a list of fractions and clues to help them select the secret fraction. Students could be asked to generate their own list of clues

02C_fraction_hunt.ppt PowerPoint presentation: all parts are on one slide, and clues appear on mouse click

a) Ask the teachers to determine which fraction in the list is the one you would select from these 4 clues. After each clue they need to determine which one(s) are appropriate and then eliminate as each new clue is revealed.

Fraction List

[pic] [pic] [pic] [pic] [pic] [pic] [pic]

Clue 1: I am greater than 0.5.

Clue 2: I am not equal to 75%.

Clue 3: If you multiply me by 2, you get a number less than 2.

Clue 4: My denominator is a prime number.

b) Have teachers create their own set of clues for the same number list and share several of them. Encourage them to use decimals, fractions and percents.

3. This activity asks students to make generalizations about fractions by comparing the numerator and denominator.

a) If a, b, and c represent whole numbers different from 0, and a > b > c, what can you say about each of these fractions?

[pic] [pic] [pic] [pic] [pic]

b) If possible, tell which is larger. Justify your thinking.

[pic] or [pic] [pic] or [pic]

02D_num_denom.ppt PowerPoint presentation: each part of the slide will appear one at a time with a click of the mouse.

This task could be differentiated by allowing students to use values at the beginning. However generalization and being able to give a written explanation are the main purposes.

4. Equivalent Fractions: This activity reviews what should be known about equivalent fractions (fractions that name the same amount in different ways).

a) To start this activity, go back to Section 1, part 1, Parts of a Measure – 3rd bullet. Teachers were asked to name certain markings on a number line. If different names are given for the same mark, this is an opportunity to talk about equivalent fractions.

Record the different names for the same mark.

Ask teachers why a mark could have more than one name?

Have a discussion about the multiplicative relationships between the 2 fractions

b) After this teachers could work on 02E_naming_shaded_sections.doc

and/or

c) use Fraction Factory to create pairs of equivalent fractions.

Pass out a set of Fraction Factory to each pair of teachers. Go over the names of each of the different pieces in a set. Ask teachers to

determine 4 different pairs of equivalent fractions and justify with Fraction Factory

determine a set of 3 fractions that are equivalent and use FF to justify.

Discussion should follow whichever parts of this activity you decide to do.

5. Estimating the Sums: Have teachers work on 02F_estimating_sums_of_fractions.doc

This sheet is divided into three parts.

Have teachers do #s 1-5 and then discuss.

Have teachers do #s 6-14. Remind them that the answers should be “’About 1 ½”, “About 2” etc. Discuss.

Have teachers do #s 15-18 and discuss.

6. Which Fraction is Larger?: Give teachers a copy of 02G_which_fraction_is_larger.doc

a) Teachers are to do each comparison in their head and record their answer and strategy. No extra paper is to be used.

b) Go over each question, asking teachers to share their strategy.

c) Show the entire group of teachers the Principles for Comparing Fractions. 02H_principles_for_comparing_fractions.doc

Ask them to compare their strategies to these to see how many they used. I think you will find they intuitively used a lot of these strategies.

|Principles for Comparing Fractions |

|1. Fractions can be compared only if they are parts of the same whole. |

|2. If two fractions have the same denominator, the one with the greater numerator is greater. For example, [pic] |

|3. If two fractions have the same numerator, the one with the greater denominator is less. For example, [pic] |

|4. Some fractions can be compared by relating them to benchmark numbers such as 0, 1, and [pic] |

|5. Fractions can be compared by renaming them with common denominators, or by renaming them with common numerators. |

|6. No matter what two different fractions are selected, there is a fraction in between. For example, [pic]is between [pic]and[pic] |

7. You have been sent a set of cards to use for this activity. The purpose of this activity is to help students make connections among fractions, decimals and percents.

02I_frac_dec_percent_cards.doc

a) Give each student a card and have them line up in front of the classroom arranging themselves in order from smallest to largest. This activity is found on page 377 (#8) of the article 10 Practical Tips for Making Fractions Come Alive and Make Sense

b) There are 2 cards that you may not want to include if this is a Grade 7 activity (250% and ¼%) but could be included when doing this activity with Grade 8 students.

Section 3:

Go over, with teachers, chapter 2 in the textbook, pages 46-103, to see how it promotes conceptual understanding and addresses the outcomes.

The intent is not to do the section with teachers but have them see the layout of the chapter, the use of models, the conceptual understanding found in the DTMs, etc.

Also go over Chapter 2 in the Teacher’s Resource that accompanies the text. Point out that each section of the chapter has Teaching Suggestions (e.g. page 44) that has suggestions on how to approach the section and especially the DTM.

Things you might want to include.

▪ Page 46 – each chapter in the text starts with the outcomes for the chapter and Key Words found in the chapter.

▪ Page 47 introduces a chapter problem which is added to at various points – pg 67, 81, 95, 101

▪ Get Ready – pages 48-53- show that prior to Grade 8 students have done a lot of work on fractions. A lot of the material mentioned here has been addressed in the first 2 sections of the workshop.

▪ Each section of the chapter has

✓ a DTM-which uses conceptual understanding to develop the fraction operations

✓ worked examples that have been created to show something specific. Many worked examples show more than one solution.

✓ questions in the CKI which summarize the key learning in the DTM. All students should be doing these.

✓ Practice questions (CYU)

▪ The use of models, manipulative and contexts are very apparent in the chapter

▪ There is Chapter Review and a Practice Test.

▪ Chapter 2 in the Teacher’s Resource , pages 36-82, gives support in the following ways.

✓ Has outcomes, key words and planning charts, pages 36/37.

✓ Has teaching suggestions for every sections. These mainly deal with how to approach the DTM.

✓ Indicate common errors

✓ Has some solutions

✓ Indicate the level of each question in the CYU (e.g. page 48)

Section 4:

In this section, we explore the expectations, found in Grade 8, for mental computations with fractions. The Yearly Plan, and PowerPoint and GSP files created from the yearly plan will be used. There is further support in the mental math videos for Grades 7-9.

Mental math is mandated as five minutes per day. In Grade 8 we create questions that apply learned strategies and encourage teachers to

▪ do mental math at the start of the lesson

▪ solve appropriate questions mentally within the lesson.

1. Fraction Operations is found on pages 5-11 in the Mental Math in Mathematics 8 Yearly Plan. To introduce this section of the yearly plan, a Power Point was created using 14 questions. 04C_fract_mentalmath_presentation.ppt

Allow teachers to do the 14 questions individually and then discuss the strategies used to get an answer.

2. Go over the Fraction Operation Section, pages 5-11, in the Mental Math yearly plan mental_math_gr8.pdf. Teachers could be asked to look for questions that can be solved using

▪ a previously learned strategy such as make-one

▪ the distributive property (or another property)

▪ multiplicative reasoning

▪ benchmarks

It is important to remind teachers that there might be more than one way to do a question and students should be allowed to use their own method.

3. There are 2 segments on Disc Three of the Mental Math Video that could be used here.

Segment 1: Gr.8, Reinforce Addition and Subtraction of Fractions by Rearrangement

Segment 2: Gr. 8, Introduce Addition of Using the Make –One Strategy

Segment 2 is an introduction; start this session by showing this segment to teachers. The before, during after questions and answers that have been created to go with this segment are found on

Pass out 04A_MMvideo_D3_seg1.doc to teachers. Discuss the “before” questions. Go over the “during” questions, show the video and discuss the “during” questions. Then go over the “after” questions. A separate file with the answers is also provided for you. (04B_MMvideo_D3_seg1_with_answers.doc)

If time permits, Segment 1: Gr.8, Reinforce Addition and Subtraction of Fractions by Rearrangement, could now be shown.

Files for the Mental Math video - Disc 3, Segment 2 are also here: 04D_MMvideo_D3_seg2.doc and 04E_MMvideo_D3_seg2_with_answers.doc

Section 5:

Go over the included item bank of questions that can supplement the material in the text and other resource materials. There are 2 files – one just has the questions (05A_Gr8_fractions_question_bank.doc) and one has the questions and answers (05B_Gr8_fractions_question_bank_answers.doc)

1. The file 05A_Gr8_fractions_question_bank.doc has 23 questions that can be done when the unit on fractions is being taught. Go over these questions with the teachers. Depending on time, you may decide just to focus on a selection of these.

Teachers could be asked when in the unit these could be done. Which ones could be done before operations are started? After addition and subtraction are taught? After multiplication? After division? As a review of the chapter?

2. This entire workshop has a lot of material. You may decide to do some more work on fractions on Day 2. Some of these questions could be designated as ones for which you will collect student work to go over on Day 2. If you decide to do this, it would be helpful if the teacher could identify several solutions that show understanding and misunderstanding. Have the teachers send them to you ahead of your Day 2 so you can plan what you want teachers to see in the student solutions. You could also select a couple that you can ask teachers to score and then go over the scoring.

Section 6:

In this section is a list of the resources that were used to help prepare this workshop. Also there are some recommended pages from some of these resources that can be used for extra practice when teaching the conceptual understanding of fraction operations. These can complement Chapter 2 in the text.

Resources used:

1. Mathematics Curriculum: Grade 8

2. Mathematics 7: Focus on Understanding

3. Mathematics 8: Focus on Understanding and Teacher Resource

4. Developing Number Sense, Addenda Series, Grades 5-8, NCTM

5. Making Sense of Fractions, Ratios, and Proportions, NCTM, 2002 Yearbook

6. Making Math Meaningful for Canadian Students, K-8

7. Teaching Student Centered Mathematics, Grade 3-5 and 5-8

8. Principles and Standards for School Mathematics, NCTM

9. Good Questions to Differentiate Mathematics Instruction

10. 10 Practical Tips for Making Fractions Come Alive and Make Sense

11. Creating, Naming, and Justifying Fractions

12. Proportional Reasoning from AIMS

13. Actions with Fractions from AIMS – this resource has good activities for conceptual understanding of fraction operations.

14. Computational Estimation Grade 6

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