## Dividing by Decimals & Fractions: Ham & Ribs

I bought a ham. It was touch-and-go there for awhile. As I was picking up and putting down hams of various sizes, I was calculating baking times. My essential question was, can I have this on the table by six? Simultaneously, I was trying to remember if this was partitive or quotative division.

In partitive division problems, a.k.a division as (fair) sharing, the number of groups is known. This type of problem asks how many are in each group. In quotative division problems, a.k.a. division as measurement, the number in each group is known. This type of problem asks how many groups. For example: 6 ÷ 3 = 2 (partitive) means ♦♦  ♦♦  ♦♦; 6 ÷ 3 = 2 (quotative) means ♦♦♦  ♦♦♦. This distinction isn’t limited to collections of objects. Consider 6 ÷ 3 as cutting a 6 m rope into 3 parts (sharing) vs. cutting lengths of 3 m (measurement). Nor are these meanings limited to whole numbers. Which brings me back to my ham…

The directions read “bake approximately 15 minutes per pound (0.454 kg) or until internal temperature reaches whatever.” But here’s the thing: Kilograms, not pounds. I could have converted from kilograms to pounds by doubling then adding ten percent of that. Instead, I divided 1.214 by 0.454. I know, I know, this still gives me the weight of my ham in pounds. But at the time, I interpreted 2.67 as the number of repeated additions of 15 minutes in my baking time. Either way, I determined how many 0.454s there are in 1.214. Quotative division. By a decimal.

As a math task, this is clunky. The picture book How Much Does a Ladybug Weigh? by Alison Limentani is a more promising jumping off point for quotative division in the classroom. On each page, the weight of one animal is expressed in terms of a smaller animal. Using the data at the back of the book, we have 3.2 ÷ 0.53 = 6. We could ask children to make other comparisons (e.g., how many grasshoppers weigh the same as one garden snail?). In the past, I have struggled with partitive division by decimals (or fractions). But I found the following example at The Fair this summer: It’s not intuitive–at least to me–to think of 1/3 in 12 ÷ 1/3 as the number of groups. Take a step back and think about 26 ÷ 1 = 26. The cost, \$26, is shared between 1 rack of ribs; the quotient represents the unit price, \$26/rack, if the unit is a rack. This result should be… underwhelming.

Before we think about dividing by a fraction here, let’s imagine dividing by a whole number (not equal to one). What if I paid \$72 for 3 racks? (Don’t look for these numbers in the photo above–I’m making them up.) In 72 ÷ 3 = 24, the cost, \$72, is shared between the number of racks, 3; again, the quotient represents the unit price, \$24/rack. Partitive division.

So what about 12 ÷ 1/3? The cost is still distributed across the number of racks; once again, the quotient represents the unit price, \$36/(full) rack. The underlying relationship between dividend, divisor, and quotient hasn’t changed because of a fraction; the fundamental meaning (partitive division) remains the same.

We could have solved this problem by asking a parallel question, how many 1/3s in 12? And this quotative interpretation makes sense with naked numbers. But it falls apart in this context–how many 1/3 racks in 12 dollars? Units, man! If dollars were racks, a quotative interpretation would make sense–how many 1/3 racks in 12 full racks?

As a math task, this, too, is clunky. My favourite math tasks for partitive division by fractions are still Andrew Stadel’s estimation jams.

(Looking for a quotative division problem that involves whole numbers? See Graham Fletcher’s Seesaw three-act math task. For partitive, there’s Bean Thirteen.)

## Math Picture Book Post #7: Bean Thirteen

A few ago, I was invited to teach a lesson on division (Grade 3). First, I read Bean Thirteen aloud – once just for fun. About Bean Thirteen, from the author:

Ralph warns Flora not to pick that thirteenth bean. Everyone knows it’s unlucky. Now that they’re stuck with it, how can they make it disappear? If they each eat half the beans, there’s still one left over. And if they invite a friend over, they each eat four beans, but there’s still one left over! And four friends could each eat three beans, but there’s still one left over! How will they escape the curse of Bean Thirteen?

(A funny story about beans, that may secretly be about . . . math!) Next, we revisited several of the pages. I asked students to write an equation to match the picture. I modelled this using magnetic “bean counters.” For the page above, students suggested 2 × 6 + 1 = 13; I introduced 13 ÷ 2 = 6 R 1. We discussed and recorded the meaning of this: In pairs, students then chose their own number of beans (counters) and built different division as sharing stories for this number. They recorded (.doc) their stories using pictures, numbers, and words:  I called on students to share their stories with the class. They observed that some numbers gave remainders more so than others; Bean Thirteen can also be used to explore even/odd and prime/composite numbers.

This lesson served as the students’ introduction to division. I wrestled with the decision to introduce remainders at this time. An alternative problem – one consistent with both the prescribed learning outcomes and recommended learning resources – might be to start with 18 beans – a “nice” dividend – and share equally among 2, 3, 6, and 9 bugs – “nice” divisors. Note 15 ÷ 3 = 4 R 3 (and 15 ÷ 2 = 6 R 3) above. This mistake would not have happened had I not introduced remainders. I wonder if including remainders makes it more difficult for students to understand division and relate division to multiplication.

## Math Picture Book Post #6: Fika

For fans of arrays (and those with OCD), there’s much to like about Fika, the Ikea cookbook. Each recipe spans two pages: the ingredients on the first, the finished product on the second.

A sample:  My daughters and I have been talking skip counting, equal grouping, repeated addition, arrays, multiplication, etc. “How many? How do you know?”

We got in on the act:  ## Math Picture Book Post #5: 100 Snowmen

I’m not usually a fan of equations in math picture books. But I like 100 Snowmen by Jennifer Arena and Stephen Gilpin. On each page, students can use the mental math strategy of adding one to a double to determine basic addition facts to 19. Each number is represented as both a number to be doubled and one more than a number to be doubled. Take five. Here, students can double five and add one more to determine five plus six. 5 + 6 = (5 + 5) + 1 = 11

Here, five is not doubled, but one more than four, which is doubled. 5 + 4 = (4 + 4) + 1 = 9

Dot cards can be used to draw attention to the doubles plus one strategy. Ask “How many do you see? How do you see them?”

Doubles Plus One Cards

To practice this strategy, students can play a game.

Taking turns:

• Roll a ten-sided die
• Build the number
• Build one more than the number
• Cover the sum with a transparent counter

The first player to cover all of the sums wins. Snowmen Doubles Plus One

On the last page, every single snowmen is added. This suggests a different mental math strategy: making tens. (1 + 2) + (3 + 4) + (5 + 6) + (7 + 8) + (9 + 10) + (9 + 8) + (7 + 6) + (5 + 4) + (3 + 2) + 1 (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + (5 + 5) + (6 + 4) + (7 + 3) + (8 + 2) + (9 + 1) + 10

Previous Math Picture Book Posts: 1 2 3 4

## Math Picture Book Post #4: One Is a Snail, Ten Is a Crab

One of my favourite read alouds is One Is a Snail, Ten Is a Crab. In April Pulley Sayre’s “counting by feet book,” one is a snail, two is a person, four is a dog, six is an insect, eight is a spider, and ten is a crab.

The odd numbers to nine and multiples of ten to one-hundred are represented as combinations of animal feet. For example, three is a person and a snail; ninety is nine crabs or ten spiders and a crab.

Last week, Sandra and I visited a Grade 1 classroom in which we asked “How many different ways can you make ten?” Children read a number sentence (e.g., “six and four make ten”) to go with each of their drawings. Some students built the animals using muli-link cubes. Some students wrote addition equations (e.g., 6 + 4 = 10). There were multiple approaches to solving this problem. For example, this student skip counted by twos (I think). These two students used the ten-fact pair of eight and two to make ten. Ten is a crab and a person (8 + 2) but this can be partitioned further as two snails and two dogs (1 + 1 + 4 + 4) or two dogs and a person (4 + 4 + 2).  Another student (sorry, no photo) broke up ten as five and five and then five as four and one; he drew a dog and a snail twice (4 + 1 + 4 + 1).

These solutions reflect an understanding of “ten-ness.” These students are not (just) counting feet. Gotta be the ten-frames.

It is important to provide opportunities for children to think about numbers as compositions of other numbers. Breaking up numbers, into tens and ones or in other ways, makes computations easier in later grades.

Thanks to Ms. Long and the young mathematicians at Fraser Wood Elementary for inviting us into your classroom. Also, thanks to Pete Nuij and Lesley Tokawa for helping make this happen.

## Math Picture Book Post #3: Miss Lina’s Ballerinas

Miss Lina’s Ballerinas by Grace Maccarone is about “teamwork, making new friends, and the pleasures of ballet.”

In my previous post, I wrote about multiplication in terms of groups of and arrays. Both models can be explored in Miss Lina’s Ballerinas. Eight ballerinas–Christina, Edwina, Sabrina, Justina, Katrina, Bettina, Marina, and Nina–dance in four groups of two and four lines of two¹. What happens when a new girl, Regina, arrives? Spoiler alert: three rows of three. What if there were ten dancers? Eleven? Twelve?

If you are playing alongMiss Lina’s Ballerinas falls into my third category; the math concept is between the pages but the author did not intend to write a math concept book.

¹ This bugs me. Should it?

## Math Picture Book Post #2: Calvin Can’t Fly

In my first math picture book post, I suggested these may fall into three categories. In this post, I’ll take a look at a book from the third category. Calvin Can’t Fly by Jennifer Berne is the story of a young starling who reads while his brothers, sisters, and cousins learn to fly. Calvin uses his aquired knowledge to save his migrating family from a hurricane. Calvin Can’t Fly is about a love of books (and libraries!). It’s about being different. It’s not about math. That is, the author did not intend to write a book about mathematics. Nonetheless, we can find math if we look for it…

How many starlings are there in the picture below? Take a guess. It’s free! It helps students to use a referent–a group whose quantity they know–to estimate the quantity in a larger group. A group of ten can be used (see below). Students can visualize the number of starlings in terms of groups of ten. Making groups of ten helps students count– it’s a place value thing. There are several other pages where students could be asked to estimate the number of starlings. Do you want to change your estimate?

One of the problems with my three categories is that it requires guessing the author’s intent. I am arguing that Jennifer Berne did not write “the story of a bookworm birdie” with referents in mind. Of course, I may be wrong. If I ever interview Jennifer Berne, she may insist that there is hidden meaning in her art– kinda like some sort of children’s literature anti-Dylan.

Watch the first 40 seconds of the video below for more estimation fun. Also, you have to watch uber-intense hair hat guy as he asks Dylan about the hidden meaning in the t-shirt he wears on the cover of Highway 61 Revisited.

Note: Great Estimations & Greater Estimations by Bruce Goldstone provide more opportunities for students to practice using referents to estimate. And check out Andrew Stadel’s new blog, Estimation 180Day 7 nicely uses a referent established on Day 6.

Ain’t that somethin’?

## Math Picture Book Post #1: Cats’ Night Out

My background is in secondary, but I have spent the majority of the past two years in elementary. This blog hasn’t always reflected that shift. This year, I plan to blog more about my experiences teaching math in K-7.

Often, I use picture books to launch math lessons. Picture books allow teachers to leverage literature-based methodologies. The plan is to make this a series of posts.

I classify math picture books into three categories:

1. mathematics is explained
2. mathematics is weaved into the storyline
3. mathematics is hidden

Books in the first category are, by and large, horrible. The reader is told that learning a particular mathematical concept is important and this concept is explained. Sometimes, art imitates life and a teacher-like character explains a topic to student-like characters. That’s just cheating.

There are some great picture books in the second category. In these books, math (not the characters’ learning about math) is central to the story. For example, in Bean Thirteen by Matthew McElligott, divisibility is introduced when the characters don’t want to get stuck with the unlucky thirteenth bean. In If a Chicken Stayed for Supper by Carrie Weston, part-part-whole relationships are explored when each fox counts the others and concludes someone is missing. Often, these books provide more questions than answers.

Books in the third category are the most difficult (and most rewarding– think #anyqs) to find. In these books, the author did not set out to write a math book. You won’t find these books in the math section of your local independent bookstore. But the math is there if the reader looks at the story through a mathematical lens. (More on this later.)

This week’s math picture book is Cats’ Night Out by Caroline Stutson. I’d place it in the second category. It’s a counting book and that might stretch your idea of ‘storyline’. (That’s fine.) Counting by twos from two to twenty, each page is illustrated with cats dancing in the city. Here are the pages for eighteen: How did you see 18? I first saw 9 on each page (5 and 3 and 1). Students could draw their own pictures of doubles on folded paper. Also, on the two pages there are 9 white cats and 9 black cats. Kids will find two 9s in other places. There are 9 cats with bows and 9 cats without. Doubles can also be seen in rows across the pages. For example, double 5 can be seen across the bottom row. The use of doubles is a strategy for mastering addition (and multiplication) facts.

These 10 cats can be seen in another way. There are 6 white cats and 4 black cats across the bottom row. Students could be asked to find ways of making a different number of cats or different pages could be copied and students could look for different part-part-whole relationships. This, too, helps students master addition facts. For example, 9 + 3 can be thought of as 9 and 1 makes 10 and 2 more is 12; 6 + 7 can be thought of as double 6 makes 12 and 1 more is 13.

My love of card stock and the laminator has been well-documented. For teachers wanting to use pictures of these cats, here you go: Cats’ Night Out Cats (Large) & Cats’ Night Out Cats (Medium)