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Lessons From the Classroom
Fractions Operations Using Rectangles
A few days ago, Mary had replied to Nat’s tweet.
@fawnpnguyen is the queen of this. Pretty sure she has a blog post on it.
— Mary Bourassa (@MaryBourassa) May 14, 2021
God, I love Mary. She sent me a decadent chocolate bar from 2,800 miles away. She remembers my birthday when 2/3 of my children did not.
I am queen only in my own head, and no, I did not have a blog post on adding fractions. I thought surely there must be a wide assortment of videos on adding fractions using rectangles. But the very first two that I’d clicked on - this and this - really astonished me. They both used grid papers without using the grids. Like, what the heck.
Say we want to add 2/3 and 4/5, same two fractions that popped in my head when I replied to Mary.
Draw two same-size rectangles using the denominators as dimensions.
Students will ask, “Why 3 by 5?” If they don’t, you ask why. And you answer them by asking them to shade in 2/3 for one rectangle and 4/5 for the other. Give them a few seconds to do this and they’ll understand why the 3 by 5 rectangles work pretty well here.
We’re adding the fractions, so let’s combine them.
Similarly with subtraction of fractions.
For multiplication, the word “of” is useful. Of course, the commutative property of multiplication applies too.
To take 2/3 of 4/5, we’d look along the height in this rectangle as we can see the thirds and just grab two of the sections.
Likewise, taking 4/5 of 2/3 is to look along the width of 5 and grab 4 sections of it.
I’ve written about division of fractions previously:
Solving an Equation With a Fraction
From CPM:
The Sutton family took a trip to see the mountains in Rocky Mountain National Park. Linda and her brother, Lee, kept asking, “Are we there yet?” At one point, their mother answered, “No, but what I can tell you is that we have driven 100 miles and we are about 2/5 of the way there.”Linda turned to Lee and asked, “How long is this trip, anyway?” They each started thinking about whether they could determine the length of the trip from the information they were given.
And I like both methods, especially Linda's.
Without using a visual, we may have students solve for x in the equation (2/5)(x) = 100 by multiplying both sides by 5/2.
But I notice two things: 1) Students don't always remember why they are multiplying by the reciprocal, and 2) Students have difficulty showing Linda's method with an equation like (9/2)(x) = 27.
So, I'm having the students think through the problem by answering these two questions:
If we know that nine halves of x is 27, then what is one half of x?
Now that we know what one half of x is, what is a whole x?
As we write the fractions, we can keep our focus on the whole number numerator and treat the denominator as if it were a thing, and that thing is not changing.
Another example,
This helps us go back to finding the unit rate in the first step via division, and then find a multiple of that unit rate via multiplication.
Once students make sense of these two steps and become fluent in solving for a whole x, then they can work on the not-so-friendly equations — such as (5/6)(x) = 4 — because they are more confident and trust the process.
Sure, multiplying by the reciprocal would have solved for x in one step, but there's something uniquely comforting to students when they can first find just one part of something.
Two Pizzas and Five People
I'm thinking a lot about how my 6th graders responded to a pre-lesson task in "Interpreting Multiplication and Division" — a lesson from Mathematics Assessment Project (MAP) .
MAP lessons begin with a set structure:
Before the lesson, students work individually on a task designed to reveal their current levels of understanding. You review their scripts and write questions to help them improve their work.
I gave the students this pre-lesson task for homework, and 57 students completed the task.
I'm sharing students' responses to question 2 (of 4) only because there's already a lot here to process. I'm grouping the kids' calculations and answers based on their diagrams.
Each pizza is cut into fifths.
About 44% (25/57) of the kids split the pizzas into fifths. I think I would have done the same, and my hand-drawn fifths would only be slightly less sloppy than theirs. The answer of 2 must mean 2 slices, and that makes sense when we see 10 slices total. The answer 10 might be a reflection of the completed example in the first row. "P divided by 5 x 2 or 5 divided by P x 2" suggests that division is commutative, and P here must mean pizza.
Each pizza is cut into eighths.
Next to cutting a circle into fourths, cutting into eighths is pretty easy and straightforward.
Each pizza is cut into tenths.
I'm a little bit surprised to see tenths because it's tedious to sketch them in, but then ten is a nice round number. The answer 20, like before, might be a mimic of the completed example in the first row.
Each pizza is cut into fourths.
I'm thinking the student sketched the diagram to illustrate that the pizzas get cut into some number of pieces — the fourths are out of convenience. The larger number 5 divided by the smaller number 2 is not surprising.
Each pizza is cut into sixths.
It's easier to divide a circle by hand into even sections, even though the calculations do not show 6 or 12.
Each pizza is cut into fifths, vertically.
Oy. I need to introduce these 3 students to rectangular pizzas. :)
Five people? Here, five slices.
Mom and Dad are bigger people, so they should get the larger slices. This seems fair. We just need to examine the commutative property more closely.
Circles drawn, but uncut.
I'm wondering about the calculation of 5 divided by 2.
Only one pizza drawn, cut into fifths.
9
Twenty percent fits with the diagram, if each person is getting one of the five slices. The 100 in the calculation might be due to the student thinking about percentage.
Only one pizza drawn, cut into tenths, but like this.
10
I wonder if the student has forgotten what the question is asking for because his/her focus has now shifted to the diagram.
Each rectangular pizza is cut into fifths.
Three kids after my own heart.
Five portions set out, each with pizza sticks.
I wonder where the 10 comes from in his calculation.
Five plates set out, each plate with pizza slices.
Kids don't always know what we mean by "draw a picture" or "sketch a diagram." This student has already portioned out the slices.
What diagrams and calculations would you expect to see for question 3?
There's important work ahead for us. The kids have been working on matching calculation, diagram, and problem cards. They're thinking and talking to one another. I have a lot of questions to ask them, and hopefully they'll come up with questions of their own as they try to make sense of it all. If I were just looking for the answer of 2/5 or 0.4, then only 12 of the 57 papers had this answer. But I saw more "correct" answers that may not necessarily match the key. We starve ourselves of kids' thinking and reasoning if we only give multiple-choice tests or seek only for the answer.
That's why Max Ray wants to remind us of why 2 > 4.
"Working Together" Problem
Heya, back-to-back post about a problem from Five Triangles mathematics.
When I tweeted how much I love this problem, a few people did not feel the same at all. Here are my reasons for appreciating this problem:
It's a notched up "working together" problem that I have not seen before.
It has percentages and fractions.
I can use rectangles to solve this. (I was asked on Twitter how I would solve this using rectangles, hence this post.)
I had to work on this problem. This is a big reason for me. We should assume that if we're teaching a particular math subject — Geometry, Statistics, or Calculus — that we're able to easily do all the exercises in the textbook. A set of exercises allows us to practice a particular skill. But a problem should require us to think. I hope I've encouraged problem-solving enough with my students that they value a problem more when they have to struggle with it, when they don't know immediately how to start it, when they get stuck and become frustrated, when they seek others for help, when they can leave the problem and come back to it another day.
While I'm at it, I also love the site Five Triangles in general for a couple of reasons:
The Geometry problems are simply stated and interesting. They make me pause and think, very few have been automatic gimmes.
The solutions are not posted. I really appreciate this because if they were, we might be tempted (mainly due to lack of time) to check the answers too early before we allow ourselves a chance to work through the problem and perhaps struggle with it. "Anticipating" is the first of 5 Practices that gives us insight on how students might solve the problem.
I did, however, retype the question above so it's easier to read and track information. I also numbered the paragraphs for quicker reference.
How we worked through this problem. Colors and all.
Draw a rectangle to represent the task. It has an area of 80 square units because that's the LCD of the three fractions in the problem.
Because this grid represents the task, we use it to fill in the amount of work done. Paragraph [3] is the first concrete piece of information that allows us to do this.
We continue to fill in the work done as described in paragraph [4].
Paragraph [5] is the first piece of information that allows us to figure out C's rate. Knowing that C can do 16 boxes in 8 hours means C can do the task — 80 boxes — in 40 hours.
With C's rate, we can now take on paragraph [2]. We know from the last step that C's hourly rate working alone is 2 boxes per hour or 10 boxes in 5 hours. But when working with A, C's rate is 40% faster, therefore instead of getting just 10 boxes done, C can get 14 boxes done in 5 hours when working with A.
From picture above in green, we know A and C did 24 boxes in 5 hours, and since C was responsible for 14 of those, the remaining 10 boxes were done by A.
Then A's hourly rate when working with C is 2 boxes per hour. Because this hourly rate represents a 20% increase than if A were to work alone, the math we need to do is 2 boxes divided by 1.2 to get 5/3 boxes. Solving for x in the proportion below gives us the answer that A completes the task in 48 hours.
Lastly we use paragraph [1] to figure out rate for B. We know A's alone rate is 5/3 boxes per hour, but when working with B, A's rate is 40% faster. Thus we multiply 5/3 by 1.4 to get 7/3. If A can do 7/3 in 1 hour, then A can do 35/3 in 5 hours when working with B.
The yellow boxes show that A and B can do 25 boxes in 5 hours, so subtracting 35/3 from 25, we see that B did 40/3.
To get B's alone rate, we divide 40/3 by 1.2 (because B is 20% faster when working with A) to get 100/9. Solving the proportion below gives us the answer of B completing the task in 36 hours.
Fraction Division via Rectangles
I appreciate Christopher Danielson's post on common numerator fraction division because it's important to examine how various algorithms work and how we can help our students become more flexible with their thinking. It's not surprising that I teach fraction division using rectangles, and I really believe the kids seem to grasp it better because it's visual.
I'll start with this problem: 3/4 ÷ 2/3. But before we do fraction division, I ask kids about whole number division. What is 8 ÷ 2? What is 15 ÷ 5? Eventually we settle on something like: asking what is 8 divided by 2 is the same as asking how many groups of 2 are in 8. Then we apply the same question to 3/4 ÷ 2/3 as "how many groups of 2/3 are in 3/4?" I guide them through this process:
Me: Let's draw out 3/4 and 2/3 on paper.
Half of them draw circles. Awful, drunk, ill-behaved circles.
M: Let's use grid paper instead to draw our rectangles. I think you can show 3/4 much more accurately on grid paper than on a circle. Please draw 2 rectangles of the same size.
(By doing this, we are really dividing two fractions using the common denominator strategy. Christopher writes about it here.)
Students: Any size?
M: What size do you think? Does it matter? Shade the first one to show 3/4 and the second one to show 2/3.
They mess up. They might draw a 1 x 4 rectangle, shade in 3 to show 3/4. But they don't quite know how to shade in 2/3 of a 1 x 4.
M: So maybe we should think about the size of the rectangle more carefully. Look at the problem again. Three-fourths divided by two-thirds. Hmmm... What dimensions should our rectangles have so it's easy to divide into fourths and thirds.
This prompt is enough for someone to say, Draw a 4 by 3 rectangle!
M: Bingo! I'm drawing these with you. Okay, so two rectangles of 4 by 3 — or 3 by 4 — doesn't matter. I'm shading in 3/4 on the first one and 2/3 on the second one. So our question is: How many groups of 2/3 are in 3/4? Because I colored mine in, can you help me ask the question again using colors instead?
Someone responds, How many pinks are in the greens?
M: Yeah. And how many little squares are pink? Okay, eight. So, I'm going over to the green here and round up 8 pink squares. I'm able to round up one group of 2/3 (pink) in the 3/4 (green).
Someone says, There's one left over.
M: How much is this one little green square left over worth? Right! 1/8 because we called 8 little boxes as one, so 1 little box must be 1/8. Our answer then is 1 and 1/8.
A few students say, I get it.
M: How do we know that our answer of 1 1/8 is correct? Okay, we'll use a calculator.
I purposely use an online calculator where I'm entering the fractions as they appear. I don't need to distract them right now with decimals or talk about parentheses. This is from CalculatorSoup.
M: Let's do this again. Now with a mixed number just for fun. Let's do 1 1/2 ÷ 2/5. How many rectangles are we drawing? What dimensions should they be? Oh, but we have more than 1 whole here, so...? We should have something like this then.
They say, How many groups of orange are in blue?
M: So let me round up the groups of orange that are in the blue. I got three. And the leftover is? Right, three. Three out of...?
More students say, Three-fourths! Three and three-fourths. I get this!
This online calculator from Calcul allows for entries of mixed numbers.
M: Okay, your turn to do one all by yourself. Please do 2/3 ÷ 1/2. (Same one Christopher used.)
I think these kids' papers show understanding.
While these are not there yet. I don't know. But it seems that drawing pictures and doing more visual stuff start to disappear in middle school.
Below is our textbook's treatment of "dividing fractions and mixed numbers" — Chapter 5, Section 7 — the full 3 pages before the Exercises.
Notice the two circles at the start of the section — that's pretty much it. And circles are great if you have denominators of 2, 4, and 8. I think if I can get my kids to first see the answer, then I can sell them the other algorithms — like multiply the reciprocal — and not come across as a fraud.
I also want to point out that I normally see this visual below for division of fractions. My way is different than this — I deliberately ask kids to draw 2 rectangles whose dimensions are the denominators.
[Update 01/07/2017]
Thank you to Rachel Emily Tabak for creating this accompanying worksheet, 18 - Frac division rectangles
Smallest and Largest
Most of my 6th graders struggled with this handout today. Only 4 of 36 students scored 8 out of 8 on it. I offered no help on this except my reading the instructions to them and reiterating that for each operation, the two numbers must be different.
I retyped this task from the Noyce Foundation.
If you teach 6th grade math or if you could share this with a colleague who does, I'd be very curious to learn how your kids might do on this. (Guess I'm now wondering how 7th and 8th graders fare on this too.)
I'm feeling disheartened and ineffective with this group of 37 kids. If I squinted really hard, then maybe I could spot just a few tiny specks of mathematical growth in them.
This teaching thing is tough.