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Math Facilitator Meeting January 2012

Welcome! Today we will begin with our break-out sessions. Please Report to the Room as indicated below

• 11:00-12:10 Breakout session Auditorium• 12:15 – 1:25 Breakout session 108A• 1:30 – 2:00 Auditorium, large group session

Breakout Session NotesK-2 Fraction Expectations in the Common CorePlease see ppt. on the “Math Facilitator Breakout Sessions” page of the wiki.Big Measurement Ideas in the Common CorePlease see ppt. on the “Math Facilitator Breakout Sessions” page of the wiki.

January 23rd Elementary Common Core PD DayThe plan for Elementary can be found on the Intranet under Common Core Resources. I have included a copy of the Math, Scavenger Hunt activities at the bottom of this page.

Large-Group Session Notes:Have the 1 st and 2 nd grade pacing been changed? Where can I find them?The opening pages for 1st and 2nd grade have been changed on the wiki. They now look like the “Year at a Glance” page of the planning guide. This way, teachers can easily view any changes/additions as they are highlighted in yellow.

When are the K-2 Pearson Common Core materials coming? They were supposed to be here January 6th. We have all labels ready and will get them out as soon as they arrive

Where are the 1st grade partitioning lessons?They are located on the 1st grade page of the Elementary Math Wiki. I have included a copy below as well.

CMS Elementary Mathematics January 23, 2012 Common Core Materials

“The BIG idea in 1st grade is that: Students understand for these examples that decomposing into more equal shares creates smaller shares.” North Carolina Department of Education, RESA Common Core presentation

CMS 1st Grade Mathematics Partitioning Lessons

The following lessons were designed around the Common Core State Standard 1.G.3 (Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.)

What is partitioning? Partitioning involves dividing an object or set of objects into parts. Partitioning into equal-sized (non-overlapping) parts is an important concept that lies at the heart of understanding fractions, percentages and decimals. Knowing that shapes, sets and quantities can be partitioned into equal-sized parts, and understanding the importance of equal-sized partitions is fundamental to recognizing the part-whole relationship between the numerator and denominator in fractions.

When partitioning, it is important to use a variety of representations of sets, shapes and quantities to ensure that students are thinking and responding to different issues, not simply memorizing images or procedures to solve problems. This helps students develop a more robust understanding of what a partition is.

Research shows the following actions may increase the likelihood of fraction misconceptions later in a student’s career.

Teaching fraction notation before ample concrete experiences hinders students’ development of conceptual understanding

Providing pre-drawn circles and rectangles; shaded and/or un-shaded takes away opportunities for students to create a shape and partition it themselves.

A student’s conceptual understanding is stronger when they can partition various shapes (e.g., circles and rectangles) to the language of “half of”, and “quarter of.”

Frequently Asked Questions:Why don’t we just use the Pearson Common Core lessons on partitioning (3.A1-3.A4) ?

- Connecting the language we use when telling time (quarter of, half past, etc.) is very abstract. Half of an hour is not the same as half of a circle. Students may want to link what they know about halves and fourths to work with clocks after they have had experiences partitioning circles and rectangles.- The Big Idea in Grade 1 is for students to notice that when we create more shares, the size of the share gets smaller. Students need work that allows them fold and cut circles and rectangles, then discuss what happens when they increase the number of cuts (or folds).

Should students be writing the fractions? Or just saying the names?- In Grade 1, students should focus on partitioning shapes and using the phrases “half of”, “a fourth of” or “a

quarter of.” Time need not be spent on writing fractions. - *Additional ideas can be located at

http://elementarymath.cmswiki.wikispaces.net/Partitioning+Circles+and+Squares

Resources:The Rational Number Project http://www.cehd.umn.edu/rationalnumberproject/default.html“Beyond Pizzas and Pies” Julie McNamara, Meghan M. ShaughnessyNew Zealand Council for Educational Researchhttp://arb.nzcer.org.nz/supportmaterials/maths/concept_map_fractions.php


http://arb.nzcer.org.nz/supportmaterials/maths/concept_map_fractions.php

http://www.cehd.umn.edu/rationalnumberproject/default.html

http://elementarymath.cmswiki.wikispaces.net/Partitioning+Circles+and+Squares

1st Grade Partitioning Circles and Squares Lesson #1

Common Core State Standard:1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Common Core Math Practices

3. Construct viable arguments and critique the reasoning of others.5. Use appropriate tools strategically6. Attend to precision (in communication)

Materials: You will need enough of the following materials so that your each pair of students will be able to create one whole sandwich, cookie, plate, and placemat- White paper squares” bread” - Round yellow or orange circles “cheese”- Tan rectangle squares “turkey” - Brown circles “cookies”- Construction paper (for placemats) - Paper plates

Additional materials needed: Scissors, crayons

Vocabulary: One-half, half of, equal share, fair share

Classroom Routine: Watch a section of the Discovery Education Video: The Number Crew: The Big Slide. The entire video is 9minutes; however, you can choose to watch a small segment at a time. Segments include: analog vs. digital clocks, telling time to the hour, half-hour, and quarter hour.After watching a portion of the video, write a time on the board (to the half hour) and ask students to show it on their clocks. Examples: 8:00, 8:30, half past 8,

Introduction/ Mini-lesson Tell students that for the next few days we are going to explore sharing fairly. Ask, “What does it mean to get

a fair share?” Turn and talk to your partner. Hold up a large paper square. Tell students you want to share it with another teacher. Ask them how we

could divide it so we both get a fair share. Students may say fold it, cut it. Explain that sometimes we can fold to show a fair share, and other times we can cut. Tell students that during workshop today, they will be folding and cutting paper to show their fair shares.

Workshop*Note: Students do not need to do the following activities in order. They can start with a placemat, cookie, etc. and move to the next activity. You may want to provide students a bag or folder to keep their pieces in as they move around the room. Explain to students that they will be going on a “fair share” picnic today. Before

getting students into partner pairs, explain each station: Making a sandwich. Tell students they will be making a sandwich with their

partner using: square bread, round yellow cheese, and brown rectangular turkey. Remind students that they are only making ONE sandwich. Then they should cut their sandwich so that each partner gets a fair share.

Half of a plate: Explain to students they will get ONE paper plate. Partners should fold it into two equal parts. Each partner designs his/her half of the plate.-make sure to students don’t spend too much time coloring (explain that they can color

Half of a placemat: Partners get a rectangular place mat (piece of construction paper). They decide how to fold their placemat and write their name on their half of the mat.

Half of a cookie: Partners choose a cookie (brown circle) and cut it in half.

Once partners have completed the activities at each station, have them sit next to each other and lay out their Fair Share Picnic.

While waiting for all partners to finish, partners can explore making halves of rectangles, circles, and squares on the Geoboard.

What to look/listen for:The discussion portion of this lesson is the most important part.

Observe students as they work so that you can tailor the discussion to meet the needs of your students.

As students work, watch for examples of the following:

-partners who incorrectly fold or cut the plate or placemat in half.-students’ explanations of how they divided up their object.


Discussion When all students have finished setting out their “picnic” tell students they are going to go on a gallery walk. Tell students, “As you walk, see what kinds of ways your classmates divided up their shapes.” This will allow students to see multiple representations of halves, as well as get a look at non-examples.

If there are incorrect examples, students may say things like, “This one isn’t right/fair/etc.” Stop students and ask for clarification. “What do you mean?” Why isn’t this one fair?” What do the rest of you think? If students agree each partner would NOT get half, ask students, “What could these partners do to fix it?” Allow students to change their representation if needed.

After discussing non-examples and incorrect examples, ask students to orally explain how they determined how to cut their objects in half. Make sure you have a few students talk and share their strategies.

***VIP! You will need to save these representations for tomorrow’s lesson!


1st Grade Partitioning Circles and Squares Lesson #2Common Core State Standard:1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Common Core Math Practices3. Construct viable arguments and critique the reasoning of others.5. Use appropriate tools strategically6. Attend to precision (in communication)

Materials: You will need enough of the following materials so that your each group of 4 students will be able to create one whole sandwich, cookie, plate, and placemat.- White paper squares” bread” - Round yellow or orange circles “cheese”- Tan rectangle squares “turkey” - Brown circles “cookies”- Construction paper (for placemats) - Paper platesAdditional materials needed: Scissors, crayons

Vocabulary: One-half, half of, equal share, fair shareOne-fourth, fourth ofQuarter of

Classroom Routine: Watch a section of the Discovery Education Video: The Number Crew: The Big Slide. Then, ask students to show what time we go to lunch, etc. Be sure the times you choose are to the hour and half-hour.

Introduction Ask students to recall how they shared their “picnic” yesterday. Tell them today we are going to do the picnic again, but this time with groups of four. Ask, “What do you think might happen?” Have students turn and talk to their partner to respond. At this point, you may want to show students a representation of a square and ask them to generate ideas on how to

cut or fold it so everyone in the group would have a fair share. However, if you think your students are able, you could just have students move to workshop.

Workshop*Note: Students do not need to do the following activities in order. They can start with a placemat, cookie, etc. and move to the next activity. You may want to provide students a bag or folder to keep their pieces in as they move around the room. Explain to students that they will travel in groups of four to create their “fair share picnic”.

You may want to explain each station before asking students to move: Making a sandwich. Tell students they will be making a sandwich with their group using:

square bread, round yellow cheese, and brown rectangular turkey. Remind students that they are only making ONE sandwich. Then they should cut their sandwich so that each student in the group gets a fair share.

A fourth of a plate: Explain to groups they will get ONE paper plate. Partners should fold it into four equal parts. Each student in the group designs his/her fourth of the plate.-make sure to students don’t spend too much time coloring (explain that they can color

A fourth of a placemat: Groups get a 1 rectangular place mat (piece of construction paper). They decide how to fold their placemat and write their name on their fourth of the mat.

Half of a cookie: Groups choose a cookie (brown circle) and cut it into fourths.

Once groups of 4 have completed the activities at each station, have them sit next to each other and lay out their Fair Share Picnic.

While waiting for all groups to finish, students can explore making halves of rectangles, circles, and squares on the Geoboard.

What to look/listen for:The discussion portion of this lesson is the most important part.

Observe students as they work so that you will be able to tailor the discussion to meet the needs of your students.

As students work, watch for examples of the following:

-groups who incorrectly fold or cut the plate or placemat in fourths.-students who notice that the pieces are smaller than yesterday.

Discussion When all students have finished setting out their “picnic” ask students to go on a gallery walk. This will allow students to see multiple representations of fourths, as well as get a look at non-examples. Students may say things like, “This one isn’t right/fair/etc.” Stop students and ask for clarification. “What do you mean?” Why isn’t this one fair?” What do the rest of you think? If students agree each student in the group would NOT get one fourth, ask students, “What could this group do to fix it?” Allow groups to change their representation if needed.

Bring a few samples from yesterday to the discussion. Ask, “how was sharing today different from the day before?” “What do you notice about the pieces? What do you notice about the size of the pieces?” “Compare the size of a fourth to a half, what do you notice?”


Elementary Mathematics Scavenger Hunt

Explanation for facilitators:

The materials in this packet contain everything you will need for the Elementary Mathematics Scavenger Hunt station.

None of the materials/concepts included have been used in any training before.

Suggestion for Including Special Area and Pre-K Teachers: Ask special area and pre-k teachers to participate on a vertical team. Items have been included for pre-k and special area teachers in the “grade-specific questions.

Materials to have in the Elementary Math Scavenger Hunt Station:

1. Copy of A Vertical Look at Measurement in the Common Core

2. Copy of the Wordle

3. Copy of Measurement Support Station

4. Copy of Measurement Scavenger Hunt Questions

Materials to have teachers bring: copy of standards and unpacking


A Vertical Look at Measurement in the Common Core

Measurement and Data K.MDDescribe and compare measurable attributes.1. Describe measurable attributes of objects, such as length or weight.Describe several measurable attributes of a single object.

2. Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.3

*Data standards have been omitted from this document*

Measurement and Data 1.MDMeasure lengths indirectly and by iterating length units.1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.

2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

Tell and write time.3. Tell and write time in hours and half-hours using analog and digital clocks.*Data standards have been omitted from this document*

Measurement and Data 2.MDMeasure and estimate lengths in standard units.1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

2. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

3. Estimate lengths using units of inches, feet, centimeters, and meters.

4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Relate addition and subtraction to length.

5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, and represent whole-number sums and differences within 100 on a number line diagram.

Work with time and money.7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?


Measurement and Data 3.MDSolve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

2. Measure and estimate liquid volumes and masses of objects usingstandard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.7

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

5. Recognize area as an attribute of plane figures and understand concepts of area measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area.b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

7. Relate area to the operations of multiplication and addition.a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.8. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.



A Vertical Look at Measurement in the Common Core

Measurement and Data 4.MD

Solve problems involving measurement and conversion ofmeasurements from a larger unit to a smaller unit.

1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Geometric measurement: understand concepts of angle and measure angles.

5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees

6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.


Measurement and Data 5.MD

Convert like measurement units within a given measurement system.1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.



Measurement Scavenger Hunt Questions

1. Look at the wordle (this was created by pasting all measurement text from the K-5 Common Core) what do you notice?

2. How do the measurement standards reinforce work with operations?

3. In what grade level does standard measurement begin? How does using a non-standard unit of measure help to build a deep understanding of the concept of measurement?

4. How do the standards provide students the opportunity to BUILD a deep understanding of volume throughout the grades?

5. Answer one of the following grade-specific questions below:

Pre-K: Look at the Kindergarten and 1st Grade standards. Describe an activity you will do with students using the words “taller, shorter, heavy, light, long, short”.

Kindergarten: What are measurable and non-measurable attributes? Give examples of each.

1 st Grade: Give an example of what students would do to measure an object using iteration.

2 nd Grade: How is number sense supported in the measurement standards? Provide evidence to support your answer.

3rd Grade: How does area relate to addition? Provide evidence to support your answer.

4 th Grade: Explain what “recognize angle measure as additive” would mean to a student. You may give an example of what it might look like in a word problem.

5 th Grade: How are volume and the associative property of multiplication linked in your measurement standards?

Special Area Teachers: Identify one “big idea” from the Common Core Measurement standards. Give a short example of an activity you will do in your content area to support student understanding of this measurement “big idea”.

Measurement Scavenger Hunt QuestionsAnd Answers


1. Look at the wordle (this was created by pasting all measurement text from the K-5 Common Core) what do you notice?

Sample Answer: varies

2. How do the measurement standards reinforce work with operations?

Sample Answer: Adding, subtracting, multiplying and dividing are embedded in the work students do. After they measure, they combine and compare measurements (+/-). As students work to find area and volume they add, multiply and divide.

3. In what grade level does standard measurement begin? How does using a non-standard unit of measure help to build a deep understanding of the concept of measurement?

Sample Answer: Second grade students need to understand the iteration of an object using any unit.

4. How do the standards provide students the

opportunity to BUILD a deep understanding of volume throughout the grades?

Sample Answer: Students begin to explore area by using by using concrete objects. They determine the formula for area and volume themselves by actually using square units and cubic units.

5. Answer one of the following grade-specific questions below:

K- What are measurable and non-measurable attributes? Give examples of each.

Sample Answer: Measurable attributes- length, width, weight Non-measurable attributes- color, texture.

1st – Give an example of what students would do to measure an object using iteration.

Sample Answer: If students were measuring the length of their desks, they could use index cards laid end-to-end with no gaps. They determine the desk is __ index cards long.

2- How is number sense supported in the measurement standards? Provide evidence to support your answer.

Sample Answer: Students use measurements to add and subtract. They also estimate measurements.

3rd – How does area relate to addition? Provide evidence to support your answer.

Sample Answer: The concept of area begins when students add the number of units necessary to fill a space. They then use repeated addition to find the area (such as, the area of a 4x5 rectangle could be found by adding 5+5+5+5).

4th- Explain what “recognize angle measure as additive” would mean to a student. You may give an example of what it might look like in a word problem.

Sample Answer: A skate-boarder wants to a 360. His first attempt is 180 degrees. How many more degrees does he need to accomplish his goal?

5th- How are volume and the associative property of multiplication linked in your measurement standards?

Sample Answer: The ultimate goal is for students to discover that they can multiply the height, base, and length in any order to find the volume of a figure.


MEASUREMENT SUPPORT STATION

Rectilinear

- DPI Unpacking document

Iteration

MEASUREMENT SUPPORT STATION

Additive

The sum (or the total measurement) is equal to the addition of all of the individual parts)

- DPI Unpacking document


Wordle

Wordle is a tool for generating “word clouds” from text that you provide. The clouds give greater prominence to words that appear more frequently in the source text.


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