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More Everyday Math Activities: Oranges.
This page was daunting because so much math goes on in these deceptively simple Everyday Math Activities. Doing the lesson is no problem. Explaining it all on a web page took a lot of typing...and therefore a lot of reading for you. I can't wait until I can make short ten minute videos to go along with these pages. Meantime we read. These Everyday Math Activities pages will one day come with video because it's so much faster to just watch (or make) a video than to take stills and type. Cross teaching of concepts is the best feature of this method. Soon there will be a page of math mind maps to go along with the lessons so you can see where you are and where you are going and what is related to what. A short example is counting by twos. First we count, then we count by twos just skipping every other number, we call this skip counting oddly enough, which leads to dividing by twos, which makes completing the square easy which might help solve complex equations which helps with the calculus both integral and derivative when wanting instantaneous values on curves which leads to vectors...all starting with counting to two. People don't understand why I can "tear up" when I see my sons do simple math. It's not because of where they are but because I know where they are going. And they are going to get there unlike me who FAILED calculus with many thoughts of suicide, and uncounted tears...and then I come to find out it's EASY. But I digress. Remember all math is, is counting. We don't just teach addition; we can't help but teach subtraction (which is just "small addition") at the same time we teach multiplication which is addition and counting quickly, and division which is the inverse of multiplication...and fractions, which is changing the set and counting all the way to "one"...and the "one" can change. Any fruit will do, but we just happened to have an orange tree nearby so I turned picking oranges into an Everyday Math Activity. We counted them as we picked them. One plus another one plus two more plus another one etc...went back and made a big deal every time we got to ten. "There's three tens in that basket!" toddler very excited after counting so high. "We call three tens thirty!" "Wow!" Thinking ahead, that's ten for me and ten for each of the two little boys...can you see some fractions coming and some division? All I'm going to do is count
and direct their discovery
letting them do as much of the counting as possible. I also keep in mind the first three of
"the five basic concepts".
Little kids love that joke. Older kids? Not so much. Now lets get down to business: In the picture at the top of the page you can see the oranges are cut in half. FIRST we counted three oranges. Then cutting them in half we counted six halves...but for the little child we were just counting things: and there were six of them. At this age no discussion of fractions is needed. We did however spend a few moments with the first one I cut in half. "How many?" "TWO!" Very enthusiastic because he knows the answer. It makes the student very happy when they get it right. They want to please you and thus pleases themselves. To the older boy: "can you see I had one orange but now I have two halves?" Nods vigorously. "How many halves?" "Two..." looks at me like I'm stupid because his brother just answered the same question. "And two halves make..." "One whole one." he says triumphantly. No paper. No pencil. No symbols. Just talking about concepts. Later when we do get to drawings/pictures** and then symbols like 1/2 it makes sense and has meaning. If you start with symbols** meaning can get confused and lost. And then you get students who think that 1/9 is bigger than 1/2 because of the 9. And teachers who ask questions like "why is it when you divide fractions they get 'bigger' and when you multiply them they get 'smaller?'" A very common, earnest question from students and teachers alike from all across this fine innumerate nation of ours. I know: I did a tour of math teaching (teaching teachers and parents) lasting three years, 3 cities per week. (It was at this point when I realized I was going to be rich.) **TO BE CLEAR: drawing means pictures, symbols means numerals or numbers not pictures. Wait until you see how easy this method makes fractions! The little pictogram above is actually just eight O's arranged in a certain order but I bet your trained mind looked at that and thought "that's a representation of the eight oranges." If it didn't it is now. With proper training you could look at the symbols in a math expression showing area under a curve in the calculus and understand what those symbols represent too. Right now, most people just say "it's all greek to me". Even after spending 100,000's on a college education. That was a good deal for somebody. 
We have eight halves now. First, we focused on on just counting up to eight. Then we counted by twos. Then we looked at the factors: both 2 x 4 and 4 x 2. One side is four and the other is two the whole thing is eight. 4 across, 2 up. 4 x 2 = 8 Eight divided by four is two and eight divided by 2 is four. See the inverse functions? Multiplication and division are inverse functions and explaining what that means is a whole lot easier with the blocks. Do you also see that you get the most information with division: you get the whole rectangle and one side. Look at the picture below put your finger over the answers on the division problem on the right and realize that the question tells you the whole amount and one side and asks you for the other side. How easy is that? The other reason we LOVE math is that the answers are the same EVERYDAY. Four twos is always eight, whether the teacher likes you or not. All this was "discovered" by me asking questions...with visually obvious answers...directed discovery. They could see the whole thing was 8 and if they counted one side, it was 2 and the other 4. How much math is that? I once had the privilege of working with a little deaf girl in an odd and wondrous place called Omaha. I don't speak sign language so the mother of this little girl who was about 12 translated. The child could not understand how they were getting the answers to the division problems, the numbers and symbols made no sense and her special ed teacher couldn't explain it to her, nor could her mother or father or other teachers and they were banging their head against the wall. MANY tears. Frustration. I sat down with her, started small: they were trying to do three digits divided by two digit problems. It wasn't two minutes before the mother burst into tears and I saw the universal "ah-ha" on the childs face. "What? WHAT?" I said. "She just signed "That's all you wanted?'" Her mother gushed.
The sheet above is just a representation of the oranges: the mathematics is a language that expresses reality numerically. Once we speak the language we can begin to describe the physical world around us, using math. Click the picture for more. On another day we might, depending on the ages of the students, move to a drawing so they can see that blocks and drawings and symbols represent the physical reality around them. When the students are older this happens all in the same lesson. In and of themselves the symbols don't have much meaning when the children are very young. Sometimes this lasts all the way to college because no one ever explains what the symbols MEAN. Note: the blocks are in the center because I want YOU to see the relationship between the blocks and the symbols on the right and the pictures/drawings on the left even though we read left to right. It's for your comparison. The progression is: blocks to drawings to symbols; however, you may have to mix it up with students who haven't been exposed to this method and are used to symbol based teaching. Can you see that this simple lesson, Everyday Math Activities with oranges, teaches a lot of math and we are only just getting started. This took less than 5 minutes to talk about and count. The student gets familiar with lots of concepts not just addition or multiplication. Older students would start with oranges, (if you had some, otherwise use imagination and/or pictures; for fractions lessons have at leaste one orange on hand to start that you can cut in half) move to blocks, then to drawings, then to symbols. Can you see that multiplication and division really are inverse functions? The pattern for multiplication is to count over and UP and for division it's over and DOWN. The top drawing would be written 4 x 2 = 8, and spoken four taken two times and the bottom one would be 2 x 4 = 8 spoken two taken four times. If you wanted to be a stickler about it, the top picture could be drawn with a yellow crayon and use two fours...but you don't because then we'd have lemons and we are playing with oranges. Think like a kid. If you were going into fractions you could draw the halves and show that 8/2 = 4. NOTHING GOT BIGGER. We counted two's. Two is contained in eight 4 times. Can you see we use rectangles to facilitate counting: #3 in
the five basic concepts.
Childs play. This method really does make math child's play. People who ask why fractions get bigger when we divide them don't understand what they are counting. I have spent trainings where it took two days for the class to get the answer to the question "why invert and multiply." Not because I was cruel, but because I was directing their discovery. THEY figured out the answer for themselves they own it, they get the "ah-ha experience." And it drives the point home that directed discovery isn't just a cute alliteration. It's a CRUCIAL part of the methodology. CRUCIAL. One of the cruxes of it. Everybody knows the rule invert and multiply but they don't know WHY it works. And often after they "get it" they have a heck of a time explaining it. (Hint: it has to do with concept number two.) It's just magic otherwise, flip it over and multiply and you get the answer...what if you flip the wrong one over? But again I digress. More on this elsewhere. Lets move on, because to use the language of the child "bigger is funner."
Now we've cut 7 oranges. This gives us 14 halves. 14/2 = 7, and 14/7 = 2...note we talked about each one as we went up. By now the answers came quickly from the older boy; the younger boy was happy to count by ones and twos...all the way to 14! Stopping to celebrate 10 of course. How many is 14 halves? "Seven, cuz we cut up seven oranges in half!" "Most excellent my brilliant little boy!" Cut up another one. "How many now?" Of interest is that at this age they don't just add two more, they count from the beginning. They don't start at 14, they start over again at 1. Starting at 14 comes with age and some coaxing and practice. Counts under his breath by ones. "16!" Of course we have to arrange it into a 4 x 4 and show 16 is a type of number, a square number...and ask questions about it, "if one side is 4 and the other side is four, how many?" and "if the whole thing is 16 and one side is four, what's the other side?" This allows the little one to get in on the action because he can count to four. I have to restrain the older boy from doing it for him. Even I was a little surprised because when I asked the toddler said "Me, me, ME!" Indicating that he wanted to answer...and he proudly counted to four for me and his be brother. Look at the pictures. They are beaming. You can't fake or pose that. They are/were having FUN. Just an everyday math activity...VERY powerful. Now they need 18 to 20 more exposures to place the information into the long term memory with easy recall. Repetition is the mother of skill. Teachers often lament that the kids go home for the summer and forget everything they were taught. They haven't forgotten: it's in there. There just isn't a neuro-pathway to get it back out again. It's not about storage baby, it's all about retrieval. Repetition builds the pathway from the sub conscious to the conscious mind. That's why we do everyday math activities. Practice ANYthing a little bit everyday and you will get good at it. "The object of teaching a child is to enable him to get along without a teacher." ~Elbert Hubbard (1856-1915) American author, editor and printer. 
By now we've counted countless times, celebrated ten lots of times figured out many ways to arrange them, and are singing "Three is a Magic Number" at the top of our lungs. (I put the music on in the back ground.) We count by threes, we count by tens...we have fun counting...and we haven't made a drop of juice yet which is the reward for all this counting. I am mindful not to go past the fun part. They can get a little "antsy" because they want juice..."you get juice after we get done counting." Change the "problem", or cut more oranges and ask questions to distract them. Kids LOVE to count big numbers...as long as it isn't hard and they have plenty of help if it's needed. Everyday math activities also constitute "quality time" by anyone's definition. Some everyday math activities are more fun than others. If they involve food they are usually high on the list of fun things to do. 
The youngest is hamming it up for the camera but observe closely: the oldest boy is concentrating very hard and counting by three's. He is at nine. He is singing and relating the words of the songs (we also sang the Rover skip count song) to the physical reality in front of him. A HUGE amount of computing power is being used by his brain. Also if you look closely in the picture you can see a carton of eggs...it had 18 in it. We skip counted it by 3 and then by 6...and had to go back to the oranges because we ran out of eggs at 18 and we had to get to 30...because the song goes to thirty...eventually they will go to 60, and complete a 20 x 20 matrix, not just 9 x 9 or 12 x 12. If you don't know what I mean go here. The youngest boy found it fun to parrot "10, 20, 30" several times: counting by tens should be one of the everyday math activities you do. In fact several doesn't cover it...10 or 20 does. That is the stage his brain is in, repetition at this age is NATURAL, by the time they are ten it is more UN-natural...or should I say not as natural...they'll do it but they'd rather be doing something else. By the age of ten we want more thinking and puzzles than wrote memorization. 
Now it's the younger boys turn...he does NOT have a correspondence with the oranges and numbers and is just mimicking what he saw his brother do, WHICH IS GREAT. Later he won't just be going through the motions, he'll be counting by threes. For now he has the pattern down and is learning the words to the songs and most importantly: he's participating and having FUN. Note the older boy is keeping count and has his hand at 15 which was the right place allowing me to know that he knew where he was and what he was doing. Total time spent was not more than a half hour. Which would be a long lesson normally, but we didn't go beyond their attention spans because we had fun, broke it up into lots of counting activities, singing, a bit of dancing and generally just played with oranges while we made juice. Everyday math activities should be short emphasize FUN and then math.
Then, rather than just drink orange juice, we made a super delicious smoothie with all manner of ingredients in it of which the orange juice was one. Brains need good nutrition to function properly. One day a section of this site will be devoted to brain nutrition....at any rate: we can do a whole lot more with fractions out of this Everyday Math Activity lesson using oranges with just a minor shift in focus. Think of all the math you can do with the measuring cup. For Example: Each whole orange gives about 1/4 of a cup of juice...each half gives about X amount then. Find X. About how many oranges to fill a cup? If we have 7 people and each one wants a cup of juice how many oranges do we need...? More here. Too advanced for this crowd at the moment but not by much. Think of all the math you can do with an imaginary (or real) lemonade stand. Don't forget you can also arrange the oranges to count by fours and fives and so on... Everyday Math Activities with grapes; we just do basic operations, no fractions:My Favorite part of this vid is "Yes. Yes, you could."
And even less math. "They say that we are better educated than our parents' generation. What they mean is that we go to school longer. They are not the same thing." ~Douglas Yates "There is a great danger in the present day lest science- teaching should degenerate into the accumulation of disconnected facts and unexplained formulae, which burden the memory without cultivating the understanding." ~J. D. Everett [In the preface to his 1873 English translation of Elementary Treatise on Natural Philosophy by A Privat Deschanel. (D. Appleton and Co.)] And here we are today. |
