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 Name01 / 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 Sense / 01F_10_tips_fractions_come_alive.pdf
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 file / 01I_part_iter_animated.gsp
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.ppt02B / 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.doc04B / 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.doc05B / 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.
Section 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.
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.
Sections 1 and 2 require the major allotment of time for this workshop.
Section 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.
Section 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.
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)
Section 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 FractionsMeaning / 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.
§ http://findarticles.com/p/articles/mi_6932/is_3_11/ai_n28433379/ 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. 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 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 of the pizza was eaten, how many pieces were eaten? Draw a diagram to support your answer.
2. Monique still has 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, 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 ( ) 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.