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UCI MATH CEO - WINTER 2019 MEETING 6 1
Meeting 6: Meet & Math!
● Tuesday 9:00 AM - 9:50 AM ○ Place: UCI NS 2 1201 (Marco Forester comes)
● Tuesday 2:45 PM - 3:45 PM ○ Place: SANTA ANA : Carr Intermediate School
● Wednesday: 2:00 PM - 2:45 PM ○ Place 1: UCI, NS2 1201 (Lathrop comes) ○ Place 2: UCI, PSLC 1400 : (Villa comes)
Tuesday Morning (50 minutes) February 12
● Crash course: 8:45 - 9:00 in the same room ● Activity 1 : 45 minutes ● Weekly Youth Survey : 5 minutes
Wednesday Afternoon (80 minutes) February 13
● Activity 1 : 40 minutes ● Activity 2 : 20 minutes ● Stock market Game : 20 minutes ● Weekly Youth Survey : 5 minutes
Start at 3:35 Tuesday Afternoon (50 minutes) February 12
● Activity 1 : 45 minutes ● Weekly Youth Survey : 5 minutes
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ACTIVITY 1: CHOCOLATE Time: 40 minutes
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ACTIVITY 1: CHOCOLATE
Description In this task, students take on a project of designing chocolate boxes with certain goals in mind. During this project, students will develop spatial reasoning and finding area strategies (including predictions and estimations), develop counting strategies for pairs and triples, and will explore combinatorics of different shapes that can be made, recognizing when two shapes that seem different are really “the same” (under a rigid motion such as a rotation).
Learning Goals
● I can successfully estimate how much of the area of a region is filled out by a certain part of it, and explain my reasoning.
● I can count the number of objects using strategies such as counting simple objects and multiplying.
Materials ● Student Workbook ● Tokens for flavors
Set-up ● Have students read the problem individually. ● Once this is done, ask one or more students to explain the problem using their
own words . ○ Guide them to be precise in their explanation (but that does not mean
using the same words as the statement, in fact, encourage students to use their own words).
● Encourage kids to work in groups of 2 or 3. If desired, and depending on your group, you may also do part of the activity all together, leading with questions. If that is the case, make sure to ask questions to all kids, and not just 1 or 2.
My solution In this space, write your solution to the problem (working out details, not just the final answers). Use as many visual representations as possible! Also, write discussion questions: these are questions that help students, at the end, consolidate the math learning.
My solution
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My discussion questions (some examples are included)
● How would this problem change if we had only five numbers 1-5? How about numbers 1-10? Explain your thoughts.
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Productive discussion
This section gives you examples of prompts, cues and questions that you may ask students during or at the end of the problem solving process. Before you continue, please watch:
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Communication in the Teaching and Learning of Math More Math 192 Series Videos: ( www.math.uci.edu/mathceo/teachingvideos.php )
● If some groups are not able to “start” (overwhelmed)
○ “Can you give some examples of superflavors that you can think of? How can you represent them?”
■ It is important that students understand the objects that we are counting in the problem and how to express them. They should generate plenty of examples.
● If you see two students who seem shy or are working in isolation ○ “Hey Alan and Bianca, I see that you are working alone, maybe you want
to work together for a while? I think you can learn a lot from each other” “Melissa, I think you can give Nora great advice in this part!”
■ Don’t force them to pair up: instead, you should invite them to do so and provide at least one reason for it .
● If you see a student working in isolation who seems quite comfortable figuring out the problem
○ “Linda, would you like to present (all or part) of your solution to these students and take questions from them” ; “I see that you have the answers, but it’s also important that you can talk and convince others”
■ This can be especially useful to spark communication skills in students who do not see themselves as “good communicators” but are confident in math.
● Scaffolding / testing for understanding ○ “Is CVO the same as VCO, as superflavors? Why or why not? Convince
me!” ■ It’s important that students realize that they are choosing a set of
2 numbers, and so that the order does not matter.
● If you see a wrong solution ○ “I’m curious why you got this many options.
■ Notice the positive language, non-judgemental, but critical in a good way. It’s important to inspect the process and not just say that the answer is wrong and correct it (which is tempting but will not result in meaningful learning from the student, since you will not reach the “source of the mistake”).
Teaching tips
● It’s a good idea to begin this activity with some discussion about the problem, in which every student can contribute a little bit. Challenge students to explain the problem in their own words, and to define the key concepts such as superflavor, and give the restrictions for placing chocolates in the boxes.
● In part B, you may also encourage kids on using the result in A to find the answer to how many superflavors in total there are. You may ask: from the computation in A, which superflavors we did not count? (those NOT having C
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nor V). What is a good way to count them? (Students should realize that there are not too many; there are only 4 of them.)
Solutions (1 CHOCOLATE)
A) Find the number of superflavors that contain cinnamon or vanilla, and list them. Don’t forget to list
the superflavors that contain both cinnamon and vanilla. Solution: recall that there are 6 ingredients: C : Cinnamon V : Vanilla O : Orange H : Hazelnut P : Pistachio S : Strawberry Each superflavor has 3 such ingredients (and the order in which we use them is not important: for example CVO is the same superflavor as VOC or COV). To list the total number of superflavors that contain C or V, we can create the following 3 groups: GROUP 1: Superflavors that have both C and V GROUP 2: Superflavors that have C but NOT V GROUP 3: Superflavors that have V but NOT C Since these 3 groups cannot share a common superflavor and these 3 groups are really all options of superflavors according to the question, then we can find the number of superflavors in each group, and add them, to find the answer. GROUP 1: since we choose C and V, there is now a choice of 1 last ingredient from 4 options (O, H, P, S). This gives 4 superflavors: CVO, CVH, CVP and CVS. GROUP 2: since we choose C, there is now a choice of 2 ingredients to complete the superflavor. Since V is forbidden, we have a stock of 4 ingredients: O, H, P, S. We have 6 choices: OH, OP, OS, HP, HS and PS. Which gives: COH, COP, COS, CHP, CHS and CPS. GROUP 3: This is the same reasoning as in GROUP 2: there are 6 options. So there are 16 (4+6+6) superflavors that contain C or V (or both).
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B) i) Do you agree with Lucy’s idea? Why or why not? Solution: Yes, Lucy’s idea is valid and will work well. The reason why this works is that every superflavor will either be in the first list (if it contains vainilla) or in the second one (if it does not) and no superflavor can be in both lists (since if its in one list, then it won’t be in the other). This way, when we do the math and add, we are counting all superflavors, and we are not counting them twice. Note: It is important to let students elaborate an explanation of this in their own words. Don’t allow them to just say. Yes, it is obvious. Even if students seem to genuinely believe that Lucy is right, the skill here for students is to critique the reasoning of others and find a way to support arguments. This is not necessarily an easy task, and requires practice. ii) Use the strategy behind the conversation, to find the total number of superflavors. Solution : X = number of superflavors that contain vainilla: 10 (from the solution in A) Y = number of superflavors that do not contain vainilla: this can be found in several different ways: one can think that we now have only 5 ingredients available (as “vanilla has been forbidden”. To choose a superflavor one can, instead of saying which 3 ingredients will be used, say which 2 ingredients will not be used (from C, S, P, O, H). There are 5x4/2 = 10 ways to do that: (CS, CP, CO, CH, SP, SO, SH, PO, PH, OH). So Y = 10
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So adding, one gets that there are 20 superflavors in total.
C) What is the maximum number of chocolates that we can fit in a box? Solution : recall that chocolate sizes range from 1 to 4 hexes. By counting, we see that there are 30 complete hexes inside the chocolate box. So the maximum number of chocolates that can fit the box is 30 (all as small as possible).
D) How many ways are there to fill the box completely if we only use chocolates of size 2-hexes and 3-hexes? Solution : To answer this question, we can list all options in terms of how many size-2 chocolates, in increasing order:
● Zero 2-size chocolates: this means 10 3-size chocolates ● 3 2-size chocolates: this means using also 8 3-size chocolates ● 6 2-size chocolates and 6 3-size chocolates ● 9 2-size chocolates and 4 3-size chocolates ● 12 2-size chocolates and 2 3-size chocolates ● 15 2-size chocolates and zero 3-size chocolates
So there are 6 ways to do this (in terms of the quantities of each size, and not on the geometric configuration, of course). Note that the number of 2-size chocolates must be a multiple of 3, because we need to fill out a total of 30 hexes and so the number of hexes to be filled out by 3-sized chocolates must be a mutiple of 3. For example: if we chose 5 2-sized chocolates, that would yield 10 hexes (not a multiple of 3), and so 30 - 10 = 20 (also not a multiple of 3), which cannot be covered by an integer number of 3-sized hexes.
E) Find at least two different ways to fill out the box completely using a total of 8 chocolates. Indicate how many chocolates of each size you use. Solution : Since there are 30 hexes in total and we need to fill out the chocolate box completely, essentially we are looking at finding four positive integers adding up to 8, to fill the parentheses in the following expression, so that it equals 30: 1( ) + 2( ) + 3( ) + 4( ) There are several ways to do this, which we can list as rows of the table that appears in the problem:
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# of chocolates of size 1 hex
# of chocolates of size 2 hexes
# of chocolates of size 3 hexes
# of chocolates of size 4 hexes
0 0 2 6
0 1 0 7
Note that having 5 chocolates of size 4 (or less) is impossible here, since you would then need 3 chocolates of size 3 or less to cover 10 hexes, which is impossible since 9 < 10. So only these two ways are possible
F) If we fill the box completely with chocolates, approximately what percentage of the area of the box will be filled? Estimate and justify your answer.
Solution : There are several ways to make this estimation. One such way is the following: for each incomplete hex inside the box, we give a value of 0, 1, 2, 3, 4 or 5 depending on whether we estimate it to be close to 20%, 40%, 60%, 80% or 100% of the hex. In other words, how much of the hex is complete. For example, a hex that has very little of it inside the box would get a 0 (or maybe a 1). This is not “exact science” so to speak, but it does not matter. Values: > Top center hex: 5 > Bottom center: 3
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> Right hexes (12 of them): 2, 1, 2, 1, 2, 1, 4, 5, 5, 6, 2, 0 > Left hexes (12 of them): by symmetry, we get the same values: 2, 1, 2, 1, 2, 1, 4, 5, 5, 6, 2, 0 We add these 26 values: 5+3+ 2x(2+1+2+1+2+1+4+5+5+6+2+0) = 8 + 2x(31) = 70. So the joint area “contribution” of these incomplete hexes is 70/5 (we divide by 5 since that was the scale we chose), which is 14. Since there are 30 full hexes, then an area of 30 out of 44 is filled up, which is close to 30/45 or 2/3. So 65% would be a good estimation.
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ACTIVITY 2: SIDEWALK STONES Time: 20 - 30 minutes
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ACTIVITY 2: SIDEWALK STONES
Description In this task, students are given the first few instances of a geometric pattern that is growing, having white and black squares. The area is also growing. Students need to analyze this pattern and find expressions, for the general case, for the number of white and black squares.
Materials ● Student workbook
Set-up You may keep the same groups as before. But this time, allow for 2 minutes of individual work for question A).
My solution In this space, write your solution to the problems (working out details, not just the final answers). Use as many visual representations as possible! Also, write discussion questions: these are questions that help students, at the end, consolidate the math learning.
My solution
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My discussion questions (some examples are included)
● What are some ways in which we can create new “tied” situations, from the original ones? Explain.
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Productive discussion
This section gives you examples of prompts, cues and questions that you may ask students during or at the end of the problem solving process.
● If some groups are not able to “start” (overwhelmed) ○ “Describe the first figures. What do you see? What do you notice?
■ This can help you find out to what extent the students comprehend the situation and the pattern growth, and go from there.
● If you see two students who seem shy or are working in isolation ○ “Hey Alan and Bianca, I see that you are working alone, maybe you want to
work together for a while? I think you can learn a lot from each other” ■ At this point, you may choose to keep the same groups from the
previous activity, but you may choose to form new groups strategically (to prevent some distractions, or to pair students of different mathematical backgrounds).
● If you see a student working in isolation who seems quite comfortable figuring out the problem
○ “Linda, would you like to present (all or part) of your solution to these students and take questions from them” ; “I see that you have the answers, but it’s also important that you can talk and convince others”.
○ I see that you made some interesting discoveries related to how the number of squares increases. Please share it!
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■ Insist on this, but still be very gentle about it. If the discussion in activity 1 was “dominated” by only a few students, focus on other students. Don’t let anyone behind!
● Scaffolding / testing for understanding ○ “Come up with some values of ‘forces’ of each animal that would balance
the first picture” ■ Although trial and error may not be the ideal way to solve the
problem, using this strategy in the beginning may be helpful for students.
● If you see a wrong solution ○ “I’m curious why you got this number of squares. Guide me through it! I
want to understand what you were thinking. Can you please check you formula in the first few cases?”
■ Notice the positive language, non-judgemental, but critical in a good way. It’s important to inspect the process and not just say that the answer is wrong and correct it (which is tempting but will not result in meaningful learning from the student, since you will not reach the “source of the mistake”.
Teaching tips ● It is really helpful to encourage students to describe the patterns in their own words, providing some help along the way. It will help you see any misconceptions students may have, and will help students develop mathematical language useful for describing patterns. You can provide some models for talking about patterns. Make sure students use precise terms, which may not be formal at first.
● It may be helpful to ask students to draw pattern 4 or even 5. Also you may ask students to sketch how, say, pattern 20 would look like (so not draw the exact picture, as it would take too long to draw each small square), but to describe the size of each corner square, and the size of the center square, sketching the picture.
● Encourage students to describe what they notice in the pictures, so that they can recognize the different attributes of the shape: the size or area of the corner squares, the side length of the center square, the total grey area inside, the total area of the corners, etc. This should lead to students making a table to keep track of these values and finding patterns.
○ When students make the table, press them for details on the relation between a row and the next one, and also to relate different columns, as the values in some columns will directly depend on the values on other columns.
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Solutions (2: SIDEWALK STONES)
2) FIGURE OUT THE NUMBER OF THE PATTERN WITH 841 GRAY SQUARES
Solution: We present two different possible methods: Method 1:
● Pattern 1 has a center gray square of side 3 and 4 corner gray squares of side 1 ● Pattern 2 has a center gray square of side 5 and 4 corner gray squares of side 2 ● Pattern 3 has a center gray square of side 7 and 4 corner gray squares of side 3 ● ...
We notice that the side of the corner gray squares grows by 1 square, and it is equal to the corresponding number of the pattern. In contrast, the side of the center gray square is always an odd number and it grows by 2 each time. (In other words, and using variables to be precise, it is of the form 2 n +1, where n is the number of the pattern). To put all this information together, we can make a table where we list the following values:
Number of pattern
Side of corner gray square
Side of center gray
square
Area of center gray
square
Total area of the four corner gray
squares
Total “grey” area
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1 1 3 32 = 9 4 × 12 = 4 9 + 4 = 13
2 2 5 552 = 2 64 × 22 = 1 25 + 16 = 41
3 3 7 972 = 4 64 × 32 = 3 49 + 36 = 85
So far, the information in the first 3 columns can be read off from the data. The information that appears in the other columns can be computed. To fill out new rows of the table (patterns 4, 5, 6, etc.), students must notice that columns 1 and 2 grow by 1 and column 3 grows by 2. This will help them compute the remaining columns. To get all the way to 841 total gray squares, they must arrive to pattern 10, where we get:
Number of pattern
Side of corner gray
square
Side of center gray
square
Area of center gray
square
Total area of the four
corner gray squares
Total “grey” area
10 10 21 1 412 2 = 4 004 0× 1 2 = 4 441 + 400 = 841
Of course students don’t really need to fill out all rows up to pattern 10; they can estimate that they will get to 841 in a large pattern, say 12, then see that they got a larger number, and go down. This tracking ability is important and should be stressed out: no need to go 1 by 1. Method 2 In method 2, we use algebra. Do not use or illustrate this method, unless the students are 8 graders or very advanced students that are prepared for this type of formal reasoning. Or, you may also explore this method after doing the table in method 1 Essentially, what we do here is generalize the table in method 1:
Number of pattern
Size of corner gray
square
Size of center gray
square
Area of center gray
square
Total area of the four corner gray squares
Total “grey” area
N N 2N + 1 2N 1)( + 2 4 × N 2 +2N 1)( + 2
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4 × N 2
We compute: (this computation can help if one wants to use N N (2N )+ 1 2 + 4 × N2
= 8 2 + 4 + 1 quadratic formula). Note that his answers Question 1. Set this expression equal to 841 and find for which positive integer N, the equation is true. Using trial and error or using the quadratic formula, we get N=10. 3) FIGURE OUT THE NUMBER OF WHITE SQUARES Method 1
● For Pattern 1, the side of the big square is 5 (1+3+1) ● For Pattern 2, the side of the big square is 9 (2+ 5+2) ● For Pattern 3, the side of the big square is 13 (3+ 7+3)
Notice how this number grows by 4 at each step. That way, we can predict these values: Pattern 4: 17 Pattern 5: 21 Pattern 6: 25 Pattern 7: 29 Pattern 8: 33 Pattern 9: 37 Pattern 10: 41 So the total area at step (pattern) 10 is .1 6814 2 = 1 We subtract the number of gray squares (841). We get 840. Method 2 For pattern n, the side of the big square is (2n+1) + n + n = 4n+1, so the total area is .4n )( + 1 2 When N=10, the total area is .1 6814 2 = 1 Again, subtract the number of gray squares (841). You get 840.
