We live in a terrific neighborhood with parks, playgrounds, a lake, a library, and a public pool and lots and lots of sidewalks to safely get to all those places on foot.
So one day we were returning from a playground, when Rocket Boy asked how many concrete sidewalk pads were between the park and our house. We guessed, then counted, then revised our estimates and counted some more. Rocket Boy’s estimate was surprisingly close to the actual count (he estimated 650 pads and we counted 657).
But then, I realized that we do a lot of estimating. We go to Costco and, before getting to the checkout, estimate the total of food in the cart. Same goes for other grocery stores. We estimate how long it would take us to get something done. And how far we have yet to drive. We estimate how much money’s in the piggy bank. And how many chocolates are on a tray. And, given Rocket Boy’s weekly allowance amount, how long would it take for him to save up for a particular toy.
Estimating is such a useful skill in the adult life! As Анатолий Гин wrote in his Фактор успеха (Success Factor) book,
… As a rule, school teaches to solve problems with precise calculations… In life, however, most situations require guesses, estimations, so that there is enough food on a camping trip, enough gas for a drive, enough money, enough flyers for an election campaign, buses for the city, etc.
And yet, I realized that, when our friends, homeschooled or not, talk about math, they hardly ever mention learning estimations. When I asked, some said that their kids are given estimation assignments that look like this:
Estimate the sum by rounding each number to the nearest ten and then adding: 91 + 67
or like this:
Is this all there is to estimating? Rounding up/down with a sole purpose of checking an existing answer?
What about estimating the length of a bridge, something Rocket Boy had to do a few days ago? He estimated it (correctly) to be two and a half times his height. No rounding.
What if you have a bottle of orange juice and you want to share it with a few friends in a way that’s fair? Can you use the rounding method to figure out whether your cat has enough food (assume dry food) until the next time you go by the pet store?
Ok, enough examples. Back to Rocket Boy and learning estimations. The big question for me right now is what smart, engaging, rich resources can I use to give Rocket Boy more practice with estimations.
There aren’t many children’s books on the subject. I think our local library has only two, both by the same author, Bruce Goldstone – Great Estimations and Greater Estimations. I got super lucky and picked up a copy of the Greater Estimations at the library book sale last week.
Other resources that I’ve come up with so far:
Estimation 180 – rounding up/down is not very helpful for estimating amounts of juice to pour for a friend. This task requires a trained eye (and a steady hand). Estimation180 shows photos and asks questions that let you practice the art of eyeballing it.
Estimation Trainer – Math is Fun has a nice primer on estimations. And since, “estimation is a skill” that “takes lots of practice”, they have a game-like trainer. Speed is emphasized, but the closer you get to an actual answer, the more points you earn. They have lots of categories – counting, measuring, fractions, multiplication, percentages – for different ages and levels of math knowledge.
Natural Math Citizen Science project – if you’d like to combine teaching your 4-9 year old how to estimate while also introducing her to early algebra AND helping advance scientific knowledge, this is a project for you. [yes, it’s me appearing briefly in the intro video]
38 попугаев – if you and your child speak Russian, this is a great cartoon to watch any time. And the episode called Как измерять удава might be a nice intro into estimating length. In it the characters – Monkey, Elephant and Parrot – get together to measure the length of their friend, the Boa Constrictor. But without rulers or measuring tape, they end up using themselves as measuring sticks. The big idea to pay attention to is that you can use anything to measure anything else – your feet, toothpicks, bananas, monkeys. The other big idea is the one about rounding. It comes up when the Parrot announces the length of the Boa to be “38 parrots and 1 parrot wing”.
A Million Dots and How Much is a Million – these are not books about estimating, but so what? There are quite a few books about very large numbers. Let’s use what we’ve got then. The key is to ask lots of questions, like how did they figure out that “364,800 cans of soup would fill more than 950 grocery carts?”
Ranger Counting Beads (aka Pace Counter Beads) – these are available on online and at many outdoors stores, but they are very easy to make. US Army soldiers (and not just Rangers either) use these to measure distances covered on foot. Note that it’s not really an estimations tool (but it’s so simple and cool and there’s so much great math in it, that I just had to sneak it in). Here’s a good explanation on how to use the beads. Why is this not an exact measurement? What happens when you don’t walk on a paved level surface? Can you still use your pace count to estimate the distance? What would be the distance at which using the beads would lead to too much of an error?
Mathematical Imagery Trainer – Proportions – finally, an app and it’s free! Developed by a team at UC Berkeley, it helps students better understand relationships between fractions. What does it mean to keep two values proportional? And how does it apply to real life need for estimations?
Here’s an example: my friend has two cats who eat different dry food. Their portion sizes are different. My friend tries to buy cat food on the same trip to a store meaning the bags of different food should last the same amount of time. To make things more complicated (real life is complicated), while she usually buys the same brands of cat food, my friend doesn’t always buy the same size bags of each brand (’cause sometimes there are sales or coupons on certain size bags). So she needs to quickly and accurately estimate proportions or else one of her cats will go hungry. If you download the trainer, I highly recommend reading the About and Info sections first, then playing with the Settings.
DIY Estimations Problems – take a camera with you everywhere. Snap pictures of things that would be good candidates for estimation problems. Here are some ideas:At the Farmers’ Market – Grandma is holding a 7lb pumpkin. How heavy is the big pumpkin?
At the library – Estimate how many books are on one shelf. What about a section of the bookcase? How many bookcases are in a row? How many rows are on this floor? How many books are in the library?
At the movies – how many popcorn kernels are in the big tub of popcorn? How would you estimate without making a big mess? How would you estimate after the lights go out
Play time – How many LEGO blocks do you have? How many do you need for a specific project? A million, you say? Ok, how much room would one million LEGO blocks take?
Snack time – pour an entire box of bunny crackers into a bowl. Now, take another box, this time of those yummy chocolate bunny crackers. Count 30 crackers and add them to a bowl. Mix well to evenly distribute crackers. Scoop out a handful (or be precise and use a measuring cup). You will have a few chocolate bunnies and lots of regular (still yummy) bunnies in front of you. Can you estimate how many regular bunny crackers are in the bowl? (Michael South’s idea)
Screen time – let’s watch Duck Tales. How much money DOES Uncle Scrooge have in his Money Bin (I like this video because it has, among other things, a depth gauge)?
More Money – would you rather get your allowance in nickels stacked to your height or in quarters laid next to each other to your height? (from the I Hate Mathematics! Book by Marilyn Burns)
Craft time – do we have enough plastic beads of each color to make this picture?
Bedtime story – in our house, I read to Rocket Boy for about half an hour before bedtime, covering a certain number of pages. Will I be able to finish this book in one week? If not, how should I adjust my reading time?
Just about any picture or situation can be turned into an estimation puzzle. And what about getting a totally off-the-wall answer, like “a billion”? Let’s accept it and figure out how much space would a billion of this something take. Or, if a billion of something takes up a given space, how big would each something have to be?