Reversing the Question

Don Steward posted this on Sunday. Like Don, I really like this task and also think it has a certain Malcolm Swan je ne sais quoi about it.

I showed only the top part to my 6th graders, and I gave them 2 minutes to write down what they noticed.

In addition to noticing the given information, the students also mentioned:

It takes a lot of grams to make a sponge cake.

A kilogram must have a lot of grams in it.

The unit of measurement is changed in the cake and in the big bag.

This problem doesn't have a question.

The sponge cake has no price.

There is frosting on the cake.

You need to change the measurements first (kg - g)

The cake weighs more.

The cake is really small and the flour is really big.

Not much information and there's no question.

There is no question.

The weight of the bag is in kg but the flour it takes for the cake is in g.

You have to convert 24 kg to grams.

It uses only very little of the flour.

I gave them another 2 minutes to write down what they wondered.

How much the sponge cake costs.

How many sponge cakes can you make.

How big is the sponge cake.

How to convert from kg to g.

If the sponge cake is good.

24 kg is <, =, or > 150 g.

What we are going to have to solve.

Is the question going to be about if there's enough flour or is it going to make us change it from kg to g.

How much 24 kilograms is in pounds.

How many grams are in a kilogram?

How many krumkakes can you make with the bag of flour.

How long will it take for the cake to be ready.

What a sponge cake is and what the recipe is for the cake.

How much flour is left.

What flavor is the cake. (Sorry. I'm hungry!)

What you're wondering that I'm wondering about.

What the question is, and if there are more ingredients.

If we will have to find the price for the sponge cake, or maybe we have to find out how much half as much flour is worth.

Then we moved on to next part of providing questions that would go with the calculations. This was so very tough for my kids. While 21 of 31 kids could come up with the correct question for part (a), they were lost with what to write for the other three parts.

Because we didn't care what the numerical answers were, the kids didn't do any calculations, instead they were supposed to focus on the operation(s) in each problem and decide on the question that would prompt a specific operation.

The most common question for part (b) was, "How much does 1 kg of flour cost?" (I'd swapped out £ for $.)

Then my next step would be for them to go ahead and use a calculator to get the numerical answers. They will see that for part (b), 24 divided by 21.50 equals approximately 1.12. And if 1.12 were the cost for 1 kg of flour, then 24 kg of flour should cost more than $24. But, wait. We already know that the bag of flour costs $21.50. Hopefully they'll arrive at this contradiction on their own, and re-think their question.

We'll then attach the units to the numbers accordingly and let the conversations continue. When kids tell me that they will calculate two numbers using operation w, I always follow up with, What does your answer mean? What unit or units does it carry?

Too often kids have trouble with word problems. Too often they don't know what to do with two numbers let alone a bunch of numbers. They guess at division when one number is big and one is small. They add when they see two fractions. They multiply because that was how they solved the last word problem.

I will also do this with my 8th graders because I suspect they will have trouble too. And this is exactly the kind of trouble we need to get into. Now rather than later. This task gets them thinking about ratios — which is like the most important math thing in all of the math things.

Steward is exactly right about this task. And I'm thankful he shared.

It was used as a fine example of how reversing the question can often lead to a more challenging task.

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.