Tuesday, May 1, 2018

Rational numbers are like midrashim

I enjoyed reading Ilana Kurshan's memoir "If All the Seas Were Ink."

I especially appreciated how she compares midrash to rational numbers. Join the discussion here.

Monday, April 9, 2018

Lessons Learned: Graphing an Israeli flag on a coordinate plane

Last year, I debuted this "graph an Israeli flag on the coordinate plane" activity in my 6th grade classes. This activity was very successful! I would estimate it takes about 30 minutes to make one graph (I had 15, so two different groups of students made each graph.) Not every group finished, but I just needed a few good finished copies. The kids enjoyed the activity regardless, though some of them got frustrated when they started with markers, made errors and then could not fix them.

Supplies are essential! I did not have enough yardsticks (you need one per group) and blue markers. You need 1 thin blue marker per group and 1 thick blue marker per person. Crayola does not appear to sell boxes of thin markers that are all blue. You don't actually need to do much with the thin markers. They are primarily useful to number the axes.

It is important for visibility that the Magen David is made with a thick marker. You can outline with a thin marker, but you'll need to trace over with a thick one for it to be seen, as you can tell in the difference between these two graphs.

One last change: I used axes that were not blue, and I thought it looked good. The goal of contrast was that the axes not impede the visual of the Magen David.

This was fun. I'm hoping to do it again this year!

Monday, October 16, 2017

What's new?

This week's parsha is Parshat Noach. Check out my lesson from last year here.

I also recently wrote up some reflections for JOFA about my experience layning Torah for the first time. You can find it here.

Friday, April 28, 2017

Beta: Graphing an Israeli flag on the coordinate plane for Yom Ha'atzmaut

It's rough having math class on Yom Ha'atzmaut. It's supposed to be math class, but YH"A is supposed to be the most awesome day of the year. How do we make them both happen?

I've done a number of activities over the years for YH"A, some of which I will share in the future. This year's activity is quite beta. There are challenges inherent in doing it properly, which I'll explain later. I generated this set of instructions that utilize the skills my 6th graders have already mastered.

The skills are:
a) Plot points on a coordinate plane.
b) Identify points that are reflections over the x-axis and the y-axis.
c) *Challenge only: Calculate the area of the triangles that form the Magen David on the flag.

If you just want the worksheet, it is here in PDF and Word
Screenshots appear at the bottom of the post.

When they are done, their picture should look like an Israeli flag.

(Image from the Wikimedia Commons.)

I started by buying some of these big coordinate grid pads. The kind that are sticky on the back are much more expensive. These are cheaper, although I recognize coordinate grid paper might be outside of some school's budgets. (You can use this activity on a normal piece of graph paper, too.)

I started my making giant axes for the students. For my weakest students I'll label the axis with its numbers, but for the other students, I just made the x-axis and y-axis.

I went through several drafts of the coordinates. The triangles must be equilateral, which means that we use a 30-60-90 triangle to find the side lengths. (Each main triangle on the Magen David is made up of two triangles like the one that appears below.)

(Image from the Wikimedia Commons.)

In a 30-60-90 triangle, if the height is rational, the side length is irrational and vice versa. This means that we have to round if we want the height ( a*root 3) and the side length (2a) to be integers. I approximated by making the side length 14. That makes a=7, which works out to a height of ~12.12.

I stumbled upon this excellent choice - excellent because the height of 12 is divisible by 3, and each triangle is 3/4 of the height of the total Magen David. (As always, I owe this particular insight to my estimable colleague John Watkins-Chow.)

The way I generated my triangle was to choose a top point, use a protractor to center a 60 degree angle around that point, and then create a side length that was close to a lattice point.

Then I moved up or down to center appropriately. I redid it if I thought the star was too large for the page or if it was not centered. I eventually arrived at the points I chose here.

If I was doing this with 8th graders, I would change #3-6 to graphing linear inequalities.

I'll discover on Tuesday how this works, but I encourage anyone who is interested to send me a shot of their graphs that they create.

Chag Ha'atzmaut Sameach!

Edits: The first version of this post neglected to mention John Watkins-Chow's assistance.  

Friday, April 14, 2017

Last night, we counted 10,100 - or was it 202?

Dedicated l'ilui nishmat Dov Nachman ben Aharon Yonah, my paternal grandfather, whose 86th birthday is today. May his memory be a blessing.

Isn't this how you count to 20?

We all count Sefirat Ha'Omer in two base systems - base ten and base seven. For example, on day 20, we count "הַיּוֹם עֶשְׂרִים יוֹם שֶׁהֵם שְׁנֵי שָׁבוּעוֹת וְשִׁשָּׁה יָמִים לָעֹֽמֶר", which basically tells us that 20 in base ten is 26 in base seven - two weeks and six days. (In fact, last night we counted 3 (or 11 in binary.) I chose the title of this post because of the time a few years ago when I saw a 6th grader of mine had been practicing her binary Sefirat Ha'Omer in her locker. This was a serious kvell moment for me.*

My interest in base system math was sparked by two things. The first was a class I took at Harvard called "Math For Teaching Arithmetic" (with my amazing professor Bret Benesh) where we invented and used our own number system to model how kids learn arithmetic. If you think long division is tough, you should try it in binary. That's rough! But most math teachers don't find long division tough, so these activities really helped us get into the heads of struggling students. The second impetus to begin the Binary Sefira project was a math team my strongest students were on. Though I was not coaching the math team, they were assigned base system problems during their competitions and they didn't know what base systems were.

For example, 10,100 in binary is 202 in what base? (Hence the title.) This is a challenging problem because most of us think in base 10 and would naturally turn 10,100 into 20 first before trying to figure out in which base 20 would be written as 202.* This is definitely a higher-level problem, so first, let's examine how this material would be taught to middle school students who would benefit from enrichment.

Base system work is an ideal extension - it's intellectually demanding, reinforces understanding of place value and exponents, and, most importantly, it is an entirely separate topic not in most curricula. Essentially, you're enriching the students without teaching the material someone else is planning to teach them next year. It fits the criteria for high-quality challenge materials. It's not for every kid, though most of our students are very comfortable answering the question "What time will it be in 12 hours?" Telling time, also known as "clock math" is math in a different base system.)

How to introduce the topic
With my initial groups of students, they knew what different base systems were from their math team, so we just jumped right in. For the first 3-5 days after Pesach, I would write the Sefira on the board in various different base systems, increasing the number of base systems every day. At first, I put the 'binary Sefira' (my term for Sefira in multiple base systems) up on the board. After a few days, student volunteers would put binary Sefira up on the board. There was a gradual build to comfort and mastery - over a period of 49 days, you can say to students, "You'll be more comfortable next week."

Teaching Explicitly
After a few years, I started getting students who had not worked with other bases on the math team. I also always had a few weaker students in my class for whom it was not as obvious. I began to introduce the concept with explicit instruction. 

So how does it work?
The column you think of as the "ones" column could also be described as the 10^0 column. It tells you how many 10^0 (ones) you have. When  that column fills up, we need to regroup into the 10^1 column. If the tens column fills up (i.e., you have ten groups of 10), you need a 10^2 (or hundreds) column.

As I mentioned before, we count Sefira in two bases - base 10 and base 7. Base 7 is a big enough base that we only get to the 7^1 column filling up on the last day. There is a beautiful mathematical and religious synergy to the fact that we count 7 weeks of 7 days - perhaps worthy of a Shmita or Yovel post another day.

When turning base 10 numbers into much smaller bases, we have to regroup really quickly. For all the big bases, day 44 is still in double digits. But for binary and base 3 (below), it's another matter.

The procedure I teach kids is to think of the biggest power of two that can be subtracted from the target number (in this case, 44). The answer, 32, is in the 2^5 column. That tells us we're going to have to fill 6 columns. In my early years, I liked to write the exponential expression that was the size of the column below the blank line. It clarified which column was which but made them look too confusingly like fractions. 
The number above is 44 in base 10: There is a group of 32, no groups of 16, one group of 8 and one group of 4, then placeholder zeroes in the twos and ones column. 

The base 3 number is also quite rich. They have to figure out that the greatest power of 3 is 27, subtract that from 44 to get 17, figure out that 17 has only one 9, subtract that to get 8, and so on. 44 is what I call a "full count" kind of day in base 3, because the addition of one more day would fill the next two columns, leading day 45 in binary to be 1200.
This exercise allows for a lot of differentiation. The strongest students can be most involved in the early days, as I mentioned above. You can also have different kids wrote different numbers on the board. Stronger students can do the smallest bases (binary, base 3) and the largest (base 14, base 15, hexadecimal - more about this later.) Students who are comparatively weaker can do the bases in the middle, such as 5, 6 and 7.

I also love that these worksheets are really easy to make!

Large bases
When you get past base 10, you need additional digits. If you are working in base 11, you need a single digit that represents 10 ones. If you are working in base 12, you need a digit for 10 and a digit for 11. With my students, I've gone as far as hexadecimal (base 16).

Here's the full list for 44.

For bases 11 through 14, the numbers work out nicely. However, since 44 is 2 groups of 15 and 14 ones, in base 15 we need to write it as 2E, with the E representing 14. Similarly, 44 is 2 groups of 16 and 12 ones, so we write it 2C, with the C representing 12.

We generally use A for 10, B for 11, C for 12, D for 13, E for 14 and F for fifteen.

Challenge materials:
Once students have been practicing for a while (for example, on day 44), you can ask them to work backwards in a few ways. The first is turning a number in another base back into base 10. Below, note that 44 is written as a number in bases 5 through 14. You can't have the number 44 in any lower base because none of those have enough room in the 'ones column' or the "tens column" for four groups of anything.
The students are instructed to turn 44 in various bases into numbers in base 10. For example, 44 in base 8 means (4)8^1 + (4)8^0 or 32+4 or 36 in base 10.

Challenge type #2 would be to ask kids to figure out what 44 in base 13 would be in base 7, much like the problem we did right at the beginning. In this case, two base changes are required.

Error analysis is a third way students can be asked to think differently. In the picture below, base 3 is right and base 4 is wrong. You can ask your students:
a) What day is it?
b) What is the correct base 4?
(Answers in *** below.)

Challenge type #4, the hardest by far, is "guess the base". You give the student the number (let's pick 47 in this case) in base 10 and then you say "47 in base 10 is 43 in what base?" In this case, the answer is base 11, so it is somewhat guessable since the two bases are close. However, make it "... is 133 in what base?"**** and it is a lot harder to guess. It has to be reasoned out. (Using, I'll admit, a series of educated guesses.)

Other methods:
There are other methods to calculate numbers in other bases (often referred to as "mod" or "modulo": 3 mod10 = 11 mod2) that don't involve using exponents explicitly, the way I have done in this post. These methods have their own distinct benefits and drawbacks. My colleagues John Watkins-Chow and Dr. Steven Steinsaltz shared two such methods with me, which I may yet summarize in another post. (After all, there are many more days of the Omer ahead of us.)

In conclusion:
I don't know why more strong students are not taught to convert between bases. It strengthens their understanding of place value, exponents and makes their thinking more flexible. Try it in your classroom and let me know how it works!

* Day 20 is also Yom Ha'Atzmaut, though this year Y"HA is pushed off to the 6th of Iyar - stay tuned for a future post for this important holiday.
** It's base 3. (2)3^2 + (2)3^0 = 202 in base 3 or 20 in base 10.
*** (1)3^3 + (2)3^2 + (0)3^1 + (1)3^0 = 27+18+1=46. However, (2)4^2 + (3)4^1=32+12=44. To do 46 in base 4, you need to add 2 ones, so the correct answer would be 232.
****Base 5: 25+15+3

Thursday, December 22, 2016

Chanukah: Experimental vs. Theoretical Probability Dreidl Game

Everyone says "probability" when they hear you played dreidl in math class. I was surprised to wake up Wednesday morning and find I did not have an "Intro to Probability" with dreidls worksheet in my archive. So I made one.

I decided to dig into experimental probability. The main part of the lesson involves the students playing dreidl and recording the results of all spins for their group. I deemphasized the "winning" aspect of it by having them play with poker chips.

Having three groups of students provided wonderful variability in the data. It was fascinating to see how one group had a low yield of gimmels while another had a high yield. This gets into the heart of randomness and variability, and I ended up taking more time than I expected to discuss it.

I reinforced equivalent fractions and estimation by asking students to identify which results were close to 25% (which I arbitrarily defined as 20% to 30%) and which were not. We had two groups with results mostly not near 25% for each possible outcome and one group with really well distributed rolls. 

In class, I only had time to add up the totals for the gimmel column, and we ended up with a grand total of 19 gimmels in 79 class rolls, about 24% and very close to 1/4. 

If I add up the totals for the others, I get: 
(5+4+8)/79 for nun, which is 17/79 (a bit less than 22%); 
27/79 for hey which is a bit more than a third, 
and 16/79 which is about 20%.
So even though in small groups we had varied data, when we added them up the data came closer to the expected 1/4. 

We barely had time on day 1 to talk about multiple events. The next day would be to cover how to find multiple events using a tree diagram and a table (pictured below).

The answer key makes clear that my handout needs more space for a tree diagram. I like tree diagrams because they allow us to compute the probability of 3 or more events. (A table limits you to two.) That way we can ask awesome questions like "What is the probability of getting AT LEAST one gimmel on 3 spins?" 

Teaching them about 'at least' versus 'exactly' is a great extension for strong students.

The tree and table take us to the fundamental counting principle, which is what allows us to multiply the probabilities. Next we would do dependent events, but I don't have dreidl examples for those. 

This lesson is really a rough draft. It went really well; the hardest part was getting the kids to read the directions! Tips, feedback and suggestions always welcome.

Tuesday, November 1, 2016

Parshat Noach - Using populations to model exponential versus linear growth

At the 2009 NCTM Boston Regional Conference, I came upon the work of an organization called Population Connection. They were advocating teaching about population growth in math classrooms - to further their ultimate goal of population stabilization. Cheekily, I attended their session and got lots of great ideas about how to educate our students about our ultimate goal of growing the Jewish population, so as to better fulfill our mission of bettering God's world.

I debuted this worksheet the next day. (It was, fortuitously, Parshat Noach that Shabbat.) Since the conference was a Thursday, and my Windows Explorer tells me I finished the worksheet at 7:06 am the next morning, you will not be surprised to note that the first draft of the worksheet is a bit rough. I'll discuss some of the potential areas of improvement below.

What does this week's parsha teach us?
וַיְבָ֣רֶךְ אֱלֹהִ֔ים אֶת־נֹ֖חַ וְאֶת־בָּנָ֑יו וַיֹּ֧אמֶר לָהֶ֛ם פְּר֥וּ וּרְב֖וּ וּמִלְא֥וּ אֶת־הָאָֽרֶץ׃
God blessed Noah and his sons, and said to them, “Be fertile and increase, and fill the earth. (Breisheet 9:1, fulltext and translation courtesy of Sefaria.org.)

The math scenarios (Gemara Yevamot 62a)
"תניא רבי נתן אומר ב"ש אומרים
שני זכרים ושתי נקבות ובה"א
זכר ונקבה"
“We learn in a Beraita: Rabbi Natan says: [Each person is commanded to bear a certain number of children.]
            Beit Shammai says: Two boys and two girls.

            Beit Hillel says: A boy and a girl.”

If Noach and his kids left the ark and immediately began to repopulate the earth, how would the earth's population be affected by using the Beit Hillel strategy as opposed to the Beit Shammai strategy?

The first time my students attempted this answer key, they found it quite challenging. Many teachers who like to develop understanding will use an exploration of exponential growth to introduce it to 8th graders (or perhaps, strong 7th graders.) The difficulty of this worksheet is based on your students' understanding of how exponential growth works. If they have not been exposed to exponential growth yet, they will need to approach the problem by calculating each row independently. It is unlikely that they will manage to find the pattern in terms of an exponential function unless they are very experienced with exponential growth.

First I'll walk through the Torah ideas behind the answer key, and then I'll suggest an adaptation to improve matters. To come up with the numbers, I made some assumptions. They are explained below.

What do we assume?
Noach had 3 sons, and they each had wives. (See Breisheet 6:10, 18).

וַיּ֥וֹלֶד נֹ֖חַ שְׁלֹשָׁ֣ה בָנִ֑ים אֶת־שֵׁ֖ם אֶת־חָ֥ם וְאֶת־יָֽפֶת׃Noah begot three sons: Shem, Ham, and Japheth.....
וַהֲקִמֹתִ֥י אֶת־בְּרִיתִ֖י אִתָּ֑ךְ וּבָאתָ֙ אֶל־הַתֵּבָ֔ה אַתָּ֕ה וּבָנֶ֛יךָ וְאִשְׁתְּךָ֥ וּנְשֵֽׁי־בָנֶ֖יךָ אִתָּֽךְ׃But I will establish My covenant with you, and you shall enter the ark, with your sons, your wife, and your sons’ wives. After the flood, these three sons were in charge of repopulating the world. We are assuming Noach and his wife had no more kids. שְׁלֹשָׁ֥ה אֵ֖לֶּה בְּנֵי־נֹ֑חַ וּמֵאֵ֖לֶּה נָֽפְצָ֥ה כָל־הָאָֽרֶץ׃These three were the sons of Noah, and from these the whole world branched out.
...וַֽיְחִי־נֹ֖חַ אַחַ֣ר הַמַּבּ֑וּל שְׁלֹ֤שׁ מֵאוֹת֙ שָׁנָ֔ה וַֽחֲמִשִּׁ֖ים שָׁנָֽה׃Noah lived after the Flood 350 years.
וַיִּֽהְיוּ֙ כָּל־יְמֵי־נֹ֔חַ תְּשַׁ֤ע מֵאוֹת֙ שָׁנָ֔ה וַחֲמִשִּׁ֖ים שָׁנָ֑ה וַיָּמֹֽת׃ (פ)And all the days of Noah came to 950 years; then he died.
(See Breisheet 9:19, 28-29. The Torah tells us explicit genealogies in these chapters, beginning with these verses. If Noach had had additional children, it is reasonable to expect they would have been enumarated in perek 10 with his other descendents.)

If Noach himself did not procreate after the flood, the base of 'people who are procreating' after the flood is 6. If each has only two children (obviously, not the case as enumerated in Breisheet 10), then we add 6 new children per generation, removing those who "age out" of the table.

Your students will discover quickly that Beit Hillel's strategy leads only to population replacement, not to population growth.

Challenges and opportunities:

Students had a lot of trouble calculating the boxes as Beit Shammai's numbers got larger. The sheet, which is portrait, not landscape, does not have a lot of space. It is beyond most middle school students to write an exponential growth function that models the total in the right-hand column.

It occurred to me belatedly that this was a perfect opportunity to create a spreadsheet activity. (Former colleague Steven Steinsaltz suggested I tell students to work down, not across. Going down the columns is easy, because the population just doubles every time, once the pesky 2* from Noach and his wife 'falls off' the table.)

Once you have the numbers for generation 6, you are set.

If you set up a Sheet with header columns like in the handout, and input the numbers we came up with for each generation, you can teach students how to use the SUM function to come up with the totals.

Careful not to include the generation number in the total population!

Fill in the generations column by typing 6, 7, 8 and then using everyone's favorite little blue box on the bottom right corner of the cell (henceforth, 'the magic blue box') to fill down.

Then we get this:

Then you can show them how to use a formula for doubling in column B.

And use the magic blue box to drag the B formula (not A!) all the way to column G.

For H, grab the total in row 2 and drag down to row 3. Then you can drag down B3:H3 all at once - the spreadsheet can accommodate different formulas for different columns.

The whole spreadsheet I worked with is here. Make yourself a copy if you want to tinker.

I highlighted the total in Generation 10....
Because your students will have been told before that there are 10 generations from Noach to Avraham, I highlighted this number. Ask your students if they think it's too high ... or too low.

How many kids did the generations after Noach actually have? Are these males or males and females? (See Breishit perek 10, here.)
Could we attempt to estimate how many are in generation 3, and from that make new predictions?
(I.e., change both the initial inputs and ALSO change the rate at which the population grows?) Note that to do this, we need to either make assumptions about how many males and females are born, or find rabbinic sources that attempt to answer this question and use their model.

Every so often there are conversations about demographics in the Jewish community. How many kids do Israeli Jews have? How many kids do Charedi Jews have? How many kids do modern Orthodox families have? How will having 3 kids instead of 4 per family impact schools, political clout, consumer options, housing, etc. etc? I think these conversations are fascinating but need to be grounded in a proper understanding of the math.

My new Rav, Rabbi Uri Topolosky, pointed out in a d'var Torah that the pasuk uses the language of blessing instead of commandment: "וַיְבָ֣רֶךְ אֱלֹהִ֔ים אֶת־נֹ֖חַ" (Breisheet 9:1 again.)

The fact that it's a blessing reminds us that many people struggle to conceive and that families in our era can't always have as many children as they might want. But the ability to raise the next generation is a divine blessing (as well as a divine obligation). Through our children (and our fruitful multiplication), we make a lasting impression on the world. Weighty stuff for math class!

Shabbat shalom,