no matter how you slice it lessonplan · 2015. 1. 9. · SERP 2014! ! No Matter How You Slice It -...

SERP 2014 No Matter How You Slice It - Sixth Grade Poster Problem 1

No Matter How You Slice ItSixth Grade Poster ProblemThe Number System

This poster problem creates the need for fraction division using a realistic setting Through repeatedly solving problems about slicing blocks of cheese using different fractional thicknesses students will be motivated to use division to make calculations easier

Understanding fraction division can take a significant amount of time and it may be valuable to plan enough time for you and your students to get comfortable with the slicing context using other operations (eg repeated subtraction) before using division Teaching this poster problem may take more time than other poster problems but we think this will be time well used

MaterialsConsider using a hand-operated slicer (or a mandoline) to demonstrate how the thickness of each slice and the dimensions of the object being sliced determine the number of slices that can be made

Learning Objectivesbull Interpret and compute quotients of fractionsbull Solve word problems involving division of fractions by

fractions eg by using visual fraction models and equations to represent the problem

Common Core State Standards for Mathematics6NSA1

Understanding or ComputationIn this poster problem students need to understand some ldquoslicingrdquo situations Students will probably develop strategies for solving the problems without division but we want to go beyond that First they need to recognize that division is a good operation to use in similar situations Then they need to learn how to perform the division-by-fractions computation

There are several ways for students to do such a calculation using common denominators multiplying by one strategically and so forth The danger comes when we tell students to ldquoinvert and multiplyrdquo too early and they start doing this without (a) understanding why division is the right operation or (b) understanding why the procedure gives the right answer See the tune-ups below for more details

Teacher Tune-Ups bull What are the multiple models for whole-number

division bull About Division of Fractionsbull Delaying ldquoInvert and Multiplyrdquobull Multiplication makes things bigger and division

makes things smaller right Well not so fast

POSTER PROBLEMS

The way this works one lesson in six phases

LAUNCH Teachers set the stage by leading an introductory discussion that orients students to the context of the problem

POSE A PROBLEM

Teachers introduce a mathematical way of thinking about the context and engage students in a preliminary approach that opens the door to the workshop phase

WORKSHOP

The workshop starts with a more challenging and more open-ended extension of the problem In teams students plan and produce mathematical posters to communicate their work

POST SHARE COMMENT Teams display their posters in the classroom get to know other teamsrsquo posters and attach questionscomments by way of small adhesive notes (or similar)

STRATEGIC TEACHER-LED DISCUSSION

Teachers then compare contrast and connect several posters In the process they highlight a progression from a more basic approach to a more generalizable one By doing this teachers emphasize standards-aligned mathematics using student-generated examples

FOCUS PROBLEM SAME CONCEPT IN A NEW CONTEXT

Serving as a check for understanding this more focused problem gives teachers evidence of student understanding

Day 1

Day 2F

LE

XIB

LE


Directions for teacherStart with a question Has anyone been to the deli and seen how they make slices of meats and cheeses How does the butcher make the slices so thin and all equally thick She uses a deli slicer Letrsquos watch a video to see how a deli slicer works

Show Slide 1 (video)

Questionsbull Can you explain how the slicer works [answers will vary]bull According to the video what is the thinnest slice this slicer

can make [132 of an inch]bull What is the thickest slice this slicer can make [12 inch]bull If you know the length of a block of cheese can you

determine how many slices it can make [Answer You need more information than just the length You also need to know the thickness of each slice Given a certain size block of cheese if you make thicker slices then you will get fewer slices If you make very thin slices then you can make more slices]

bull Suppose you get a new block and you know how thick you want your slices What do you need to know in order to tell how many sandwiches you can make [The length of the block and how many slices go in a sandwich]

Slide 1 (video)

1 LAUNCH

video also viewable at httpmathserpmediaorgassetsslicerhtml

POSTER PROBLEMS - NO MATTER HOW YOU SLICE IT SLIDE 1


Directions for teacherStart this phase of the lesson with a warm-up problem to cue students to think about division Show Slide 2 and ask

ldquoWithout solving this problem what operations or steps would you use to find the number of sandwiches that the chef can make Whyrdquo

Have students talk in pairs

Many students will recognize that this is a division problem but they might struggle to explain why One common ldquotrickrdquo students use to select the correct operation to solve these problems is to reason that there must be fewer sandwiches than slices so they should divide 570 by 3 (they might even deduce that subtracting 3 is not small enough) This ldquodivision makes smallerrdquo reasoning works when dividing whole numbers However as we will see it is not necessarily true for dividing fractions

Now letrsquos solve a similar problem that uses fractions The chef is using a slicer to make thin slices of cheese for sandwiches The slicer has settings for different thicknesses ranging from 132 to 12 Consider this How many 112 slices can the chef make with a block of cheddar that is 2 inches long

Show Slide 3

Ask ldquoHow would you start thinking about this problem What operation will you use to find the answerrdquo Again have students work in pairs

After giving students time to discuss ideas (one or two minutes should suffice) ask two or three students to share how they thought about this problem Some students might not have a complete solution that is OKmdashpartially formed ideas can still contain useful information and it is helpful to show students it is OK to share an incomplete idea

Some students will start by noticing that there are 12 copies of a 112 inch slice of cheese in each inch of the block so therefore they can get 12 slices per inch or 2 times 12 = 24 slices total from the 2-inch long block of cheese Slide 4 shows how 1 inch of the block can be partitioned into 12 equally thick slices of cheese

Slide 2

2 POSE A PROBLEM

Slide 3

Slide 4

Each grilled cheese sandwich uses 3 slices of

cheese If the chef has 570 slices of cheese how many grilled cheese sandwiches

can she make


This is a block of cheese that is two inches long

0 1 2

This a 112 slice of cheese


0 1 2

12 slices 112 thick fit into one inch




Some students might notice that this problem parallels the previous question with whole numbers and say that you should divide 2 by 112 If no students make this claim you should do so

At this point do not introduce the standard algorithm for doing fraction division (ldquoOurs not to reason why just invert and multiplyrdquo) Nevertheless connecting these two solutions is key to understanding this poster problem dividing 2 by 112 has to be the same as multiplying 2 by 12 Let students discuss this and try to explain why both answers are correct

Letrsquos consider one way of thinking about the division Division finds the missing number in multiplication 15 divide 5 = is the same as 5 ⨉ = 15 So 2 divide 112 is the same as (112) ⨉ = 2 But 2 is 2412 so I want (112) ⨉ = (2412) And thatrsquos clearly 24

Note Some students might solve the problem by saying that 2 divided by 12 is 24 These students might be thinking ldquoTwo [inches where each inch is] divided [into] 12 parts is 24 [parts]rdquo That is the student understands the situation and that it involves dividing but he or she is confused about how the numbers in the situation take up their roles in the calculation So check for understanding and praise itmdashbut ask if itrsquos really true that 2 divide 12 = 24

Now pass out Handout 1 and remind students to spend ample time solving each question The problems are ordered in increasing difficulty

Discussing Handout 1Question 4 should prompt a discussion of the meaning of the remainder in division of fractions In this problem there is 120 of an inch of cheese left over after making 3 slices that are 25 thick each Some students might argue that the answer should be 3 and 120 This is a good moment to highlight the distinction between calculating the number of inches left over and the number of slices left over Stress that both ideas are correct but that 3 120 (written as a mixed number) is not because the 3 is in slices and the 120 is in inches The answer is 3 14 slices or 3 whole slices with 120 inch left overrdquo We will be focusing on the first representation because thatrsquos the result you get when you step up to doing division

Completing this worksheet and discussing the answers can easily take a full class period This is OKmdashfraction division is one of the most challenging topics in the school curriculum and it is worth the time to delve deeply into the ideas in this section

2 POSE A PROBLEM CONT

Handout 1 (two sides)

Answers to Handout 11 242 163 44 a 3

b 120 inch c 14 slice

5 13 136 see diagram on Answer Key

An extramdashstudent reality check Ask What do you think are realistic thicknesses for slices of cheese you would put in a sandwich

No Matter How You Slice It Handout 1

2014 httpmathserpmediaorgdiagnostic_teaching

The chef was busy making sandwiches for school lunches this morning While she was making cheese slices she often had to figure out how many slices she could get out of a block of cheese For each of these questions draw a picture and write an explanation of how you answered the question

Instructions

Student Name ____________________________________________________

1 How many 112 slices can the chef make with a block of cheddar that is 2 inches long

2 The chef is now making slices of American cheese that are 38 thick If

she has a large block of cheese that is 6 inches long how many slices can she make

3 How many 316 slices will she get from a small block of Gruyegravere cheese that is 34 long

4 Now the chef is making much bigger slices 25 thick She has a block of cheese 114 long

a How many complete slices will she get

b How thick is the leftover piece (measured in inches)

c What fraction of a slice is left over

Handout 1 continued on the next page



Student Name ____________________________________________________

5 How many 38 slices will she get from a block of Swiss cheese that is 5 inches long

Bonus Puzzler Can you cut a wheel of cheese into 8 equal parts with three straight slices


3 WORKSHOP

Directions for teacherArrange students in pairs or groups Pass out Handout 2 and explain that they will now create their own own cheese-slicing problems and create a poster showing how to solve them

While students work in groups to create posters encourage students to consider describing the problems using division rather than repeated addition or subtraction or multiplication guess-and-check

Handout 2



Student Name ____________________________________________________

Make up and solve two of your own slicing problems In problem A you should not have any cheese left over and in problem B you one must have some cheese leftover

For each problem you need to determine how much cheese you start off with how long is your block of cheese You also need to say how thick you want the slices of cheese to bemdashor you can decide how many slices you will need in total Keep in mind that the thickness of each slice should be between 132 and 12 inches thick

After you create your problems make a poster showing each problem and its solution Each solution should include an explanation at least one calculation and a diagram

Problem A - Slicing problem with no cheese left over

Problem Statement

Solution

Problem B - Slicing problem with cheese left over

Problem Statement

Solution


4 POST SHARE COMMENT

Directions for teacherTeams display their posters in the classroom get to know other teamsrsquo posters and attach questionscomments by way of small adhesive notes (or similar)

Sample Posters In Poster A for Problem A students chose a fraction with a numerator of 1 These fractions are called ldquounit fractionsrdquo If the length of the block of cheese is a whole number then dividing by a unit fraction will not leave a remainder In this example 3 divide (14) = 12

In Poster B (Problem B) the group used repeated addition to solve their problem It would help if they clearly labeled their solutionmdashbut you can still see how they did it (their answer is 27 slices) Their remainder is in inches not slices

Poster C shows a group that did their entire problem using a diagram

Finally Poster D shows a group that was able to correctly set up and solve a cheese slicing problem using non-trivial numbers (Note that this problem does use a slice that is more than 12 inch but the setup and solution are sound)

A

B

C

D


Directions for teacherSelect a sequence of posters to use during the teacher-led discussion that will help move all students from their current thinking (often Levels 1ndash3 below) up to 4 or 5

Level 1 The group has defined a block of cheese and a thickness and successfully use a diagram or number line to find the number of slices There is no calculation

Level 2 Here the group uses calculation (possibly in addition to a diagram) but the calculation may be guess and check repeated addition (as in Poster B) or repeated subtractionmdashbut not division

Level 3 The group uses division but its use might be muddled They might state division incorrectly eg stating that 2 divide 12 = 24 (instead of 2 divide 112) Numbers in the problem may not correspond with a diagram The remainder might be larger than the thickness of a slice

Level 4 The group uses more challenging fractions (eg 38) for the thickness but the block length is still a whole number Remainders are in terms of slices The group describes their computation in terms of division The diagram corresponds to the problem and solution

Level 5 The group uses challenging fractions for the slices and non-whole numbers for the block length The group describes the calculation in terms of division and shows how to perform the calculation When asked students can also show how the division calculation corresponds to less-sophisticated techniques

Connecting across groupsHere are a few things to highlight when you discuss this problem

bull Students will choose different numbers some ldquofriendlierrdquo than others Be sure to highlight some students who used non-unit fractions

bull Students may struggle to interpret the remainder Highlight that the remainder could be in inches or in slicesmdashbut you have to be clear Be sure students understand that if you use a calculator or a division algorithm your remainder is in slices

Other DirectionsDepending on your class you may want to consider extensions to the discussion like these

bull Remember the example with two inches of cheese and 112 inch slices Is it easier to think of this as 2 divide (112) = 24 or as 2 ⨉ 12 = 24 (The latter) Why does the second one work What would be a situation where thinking about it as division makes it easier rather than more confusing (Harder numbers that donrsquot match up eg a block 6 14 inches long divided up into slices 215 thick)

bull Before division always made numbers smaller But now sometimes division makes the answer bigger than the numbers you started with What has to be true for that to happen (Divisor is less than 1)

bull How can you check your work with a calculator (Find decimal equivalents but watch out because yoursquoll be working with approximations)

5 STRATEGIC TEACHER-LED DISCUSSION


Show Slide 5 to the class Have the students work on the question in pairs

Slide 6 can be used as a discussion guide after students have worked with the problem for a while

Slide 7 contains two optional ldquoExplore Morerdquo questions

Slide 8 contains the answers to the ldquoExplore Morerdquo questions on Slide 7

6 FOCUS PROBLEM SAME CONCEPT IN A NEW CONTEXT

Slide 5 Slide 6

Slide 7 Slide 8

Sylviarsquos bedroom is 2112 feet wide She is going to lay wood flooring throughout her entire bedroom Each strip of wood flooring is 34 foot wide How many strips will Sylvia needDoes one of the following approaches help solve the problem More than one of them All of them Which one do you think is most helpful Why

2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n

Discuss with a partner and then

be ready to share your thinking



2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n

All four approaches are valid

This approach is most similar to cheese slicing and

challenges students to make sense of an equation that uses

division of fractions

This approach is the same as the one above but it converts

the mixed number to an improper fraction and also adjusts to use a common

denominator Relation to whole number division in most apparent this approach

This approach is also fine but it is ambiguous in the meaning of

the multiplication problem some might think that

n x 34 = 21 12 is a better equation for the situation

This approach is commonly used and convenient Changes the mixed number to an improper

fraction and uses the reciprocal However connection to the

situation less apparent


Explore MoreThere are other ways to think about division of fractions Try these two questions They both use division but why And how do you know what to divide by what 1 The water level in the reservoir has gone down 2frac12 feet in

the last month and a half How fast is the water level going down per month

2 Farmer Schmidt owns frac34 of a square mile of land Her field

is a rectangle One side is ⅔ of a mile How long is the other side


Explore More (answers)There are other ways to think about division of fractions Try these two questions They both use division but why And how do you know what to divide by what 1 The water level in the reservoir has gone down 2frac12 feet in the last month and a

half How fast is the water level going down per month

2 Farmer Schmidt owns frac34 of a square mile of land Her field is a rectangle One side is ⅔ of a mile How long is the other side

212 is 52 One and a half is 32 So we have (52) divide (32) which is 53 (or 123) Answer One and two-thirds feet per monthOne great way to decide what to divide is to restate the problem with friendly numbers Suppose the reservoir went down 6 feet in two months Clearly itrsquos 3 feet per month and thatrsquos 6 divide 2

The area is 34 Length times width is area so width is area divide length Thatrsquos (34) divide (23) or 98 The other side of the field is 98 (118) miles long



Directions for teacherStart with a question Has anyone been to the deli and seen how they make slices of meats and cheeses How does the butcher make the slices so thin and all equally thick She uses a deli slicer Letrsquos watch a video to see how a deli slicer works

Show Slide 1 (video)

Questionsbull Can you explain how the slicer works [answers will vary]bull According to the video what is the thinnest slice this slicer

can make [132 of an inch]bull What is the thickest slice this slicer can make [12 inch]bull If you know the length of a block of cheese can you

determine how many slices it can make [Answer You need more information than just the length You also need to know the thickness of each slice Given a certain size block of cheese if you make thicker slices then you will get fewer slices If you make very thin slices then you can make more slices]

bull Suppose you get a new block and you know how thick you want your slices What do you need to know in order to tell how many sandwiches you can make [The length of the block and how many slices go in a sandwich]

Slide 1 (video)

1 LAUNCH

video also viewable at httpmathserpmediaorgassetsslicerhtml








Show Slide 3




Slide 2

2 POSE A PROBLEM

Slide 3

Slide 4



can she make



0 1 2



0 1 2





















Instructions

Student Name ____________________________________________________












Student Name ____________________________________________________




3 WORKSHOP



Handout 2



Student Name ____________________________________________________





Problem Statement

Solution


Problem Statement

Solution








A

B

C

D






















Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n
































Show Slide 3




Slide 2

2 POSE A PROBLEM

Slide 3

Slide 4



can she make



0 1 2



0 1 2





















Instructions

Student Name ____________________________________________________












Student Name ____________________________________________________




3 WORKSHOP



Handout 2



Student Name ____________________________________________________





Problem Statement

Solution


Problem Statement

Solution








A

B

C

D






















Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n











































Instructions

Student Name ____________________________________________________












Student Name ____________________________________________________




3 WORKSHOP



Handout 2



Student Name ____________________________________________________





Problem Statement

Solution


Problem Statement

Solution








A

B

C

D






















Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n



























3 WORKSHOP



Handout 2



Student Name ____________________________________________________





Problem Statement

Solution


Problem Statement

Solution








A

B

C

D






















Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n

































A

B

C

D






















Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n















































Slide 5 Slide 6

Slide 7 Slide 8


2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n





2112divide 34= n 3

4timesn= 211

2

864divide 34= n 43

2times 43= n


























no matter how you slice it lessonplan · 2015. 1. 9. · SERP 2014! ! No Matter How You Slice It -...

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Transcript of no matter how you slice it lessonplan · 2015. 1. 9. · SERP 2014! ! No Matter How You Slice It -...