Count Around
The Room
Make a circle (or go around the dinner table!). Decide
who goes first. Choose a number 1-20. The first person starts at 0. The next
person adds the chosen number. The next person says the next number in the sequence.
For example, if the counting number is 7, the first person says 0, the next
person says 7 and the next 14 and so on. When you finish the circle, then do
the counting sequence backwards. Don't worry about going too fast. Stop once
and a while and let people explain how they calculated their number. Other
variations: one person in the circle has to say their number silently (name
them “silent Bob” for fun); go around the circle multiple times instead of
going backwards; before doing the count, choose a person and predict what their
number will be; start on a number other than zero. Journal the numbers you generate and think
about the patterns you saw that help you predict the next number.
Choral
Counting
This task needs pencil/pen and paper. In class we
chanted the sequence altogether. Choral Counting is a lot like Count Around the
Room, in that you choose a number, repeatedly add and you can start at any
number. However, the numbers said in the sequence need to be written into an
array as you figure them out. The arrays can be 4 in each row, or even 3 or
larger. The goal is to look for patterns in the sequence. The same numbers put
in a different array reveal a different pattern, but some arrangements are more
interesting than others. Here are some examples with add 6.
This is different than just skip counting in several
ways: 1) encourages patterns finding which is the basis for algebraic thinking
and number theory, 2) it is pretty fun to just see how far you can go and even
find new patterns (most skip counting activities stop at 12 times the number),
3) starting at a number that is not a multiple of the added number creates a
related sequence (y = 2 x + b). While you are playing with these arrays of
sequences, ask yourself how you calculated from one number to the other. For
example, some kids in the class added 4 to 28 by breaking the four into 2+ 2.
So they thought, “28+ 2 = 30 and 2 more makes 32”.
Diffy Boxes
Diffy Boxes have been around
for a while and may have actually been invented in WWII, and introduced as an
activity for elementary students in Texas in the 80’s. Since that time mathematicians
have also studied them! So what begin as an activity to practice subtraction
ends as an interesting mathematical problem. The basic idea is that you draw a
box and put a number at each corner. Find the difference between each pair of
numbers (corners), write that number on the line between the two numbers, then
connect those numbers to make another box inside the first one. Keep doing this
process until all of the numbers (differences) are zero.
Open Number Sentences
Most math problems in the textbook look like this: 3 + 4 = ____
Open number sentences also have a blank, but it can be
anywhere, like the examples below. The other feature about this task is that
open number sentences encourage the problem solver to think about relationships
between the numbers to solve the problem rather than computation. For example
to solve 2+ 3 = ______ + 4, you might say, “well 2 + 3 is 5, so I know the
answer is 1, since 4 + 1 equals 5”. But, some might see the relationships
between the numbers: “well I am adding one more because 4 is one more than
three, so the other number has to be one less so the 2 will now be a one.”
You can see that this way of think could be very
helpful in deciding what goes here: 28+ 34 = 27 + ______. Seeing that you begin
with 27 instead of 28 provides a clue that the number added must be one more to
make up for the difference, so the answer is 35. No need to add the left side
and then find the difference. Below is a list of open number sentences that
provide a good place to start, but parents and kids can make up their own just
to see how this works. Open number sentences can be created using the other
operations (subtraction, multiplication and division), but the relationships
don’t work exactly the same, right?
5 + 5 = 6 + ___
9 + 6 = 10 + ____
_____+ 8 = 10 +
7
25 + 25 = ____+ 24
True-False
Number Sentences
True-False Numbers sentences are designed to bring out
relational thinking just like the open number sentences, except all of the
numbers are filled in, and the problem solver has to determine whether it is true
or false and provide an explanation. Your child can make these up for you and
then you have to explain why. The challenge here is justification. Saying why a
sentence is right or wrong is the focus. Just like the open number sentence
task, these can be solved by computation or number relationships. The questions
for each one are: “Is this statement true or false?” and then the sentence is
read: “Is five plus six the same as six plus 5? Why or why not?”
5 + 6 = 6 + 5 (true)
**This statement in particular also invites a
generalization that when adding two numbers the order does not matter (the
commutative property). If your child
mentions that order does not matter, you can ask is this true for all addition?
Then what about subtraction? Or multiplication?
Other True-False examples:
7 + 8 = 7+ 9 (false)
22 + 4 = 23 + 3 (true)
20 + 3 + 40 = 23 + 40 (true)
Math in the
World
This year I have invited the kids to look at things in
their world with a mathematical eye. So when looking at a crowd of people on the
4th of July, you might ask, “How many people do you think are
here?”;“How many kids?” ;“How many families?”
And then perhaps take a photo of a group or gathering and ask, “Can we
use this picture to help us make a good prediction of how many people are
here?” There are some photos on my blog
that are examples of Math in the World. One journaling idea is to take a photo
and put it in the journal with a few math questions and possible answers. A famous physicist, Fermi popularized this type of mathematical activity and you may hear them called Fermi questions or problems. A famous one is "How may piano tuners are in New York City?". The process of answering this question is called dimensional analysis. For young kids, encouraging them to make a plan to address a counting question in which the actual answer probably will not be found is one way to get kids to think in this way--So ask them how they came up with their estimate to get them started!
Complementary
Numbers
The Complementary Numbers task can make a good travel
game. Just state a target number. The one person says a number less than the
target. The other person has to say the number that needs to be added to that
number to make the target. Again part of the emphasis is saying how you know
the numbers work, not just the right answer. For example if the target number
is 20, and the first person says 12, the other person says 8 because 12 + 8 =
20. The justification might be a fact: “I just know that 12 + 8 = 20”, or “I
counted up from 12 to 20 and got 8”, or some sort of relational thinking like,
“I know if I add 10 that will make 22, so I add 2 less which is 8.” Young kids
may not be as articulate as I say here, but the reasoning is the same. I like
to use other benchmark numbers as target numbers like 10 or 100 or 50.
99 Plus
Anything
People in general are pretty smart about numbers.
Often everyday strategies people use to make computation easier are not
recognized as mathematical but they are! For example when adding or subtracting
some numbers, people will use a shortcut. Like 9 plus anything is just like 10
plus a number minus one. Example: 9 + 6 is equal to 10 + 6 -1 because 9 is
equal to 10-1. This strategy called compensation can be use with larger numbers
as well. One question is can you use this strategy with any number? For example
9 + 45?
99 Plus Anything is a game where you ask a person to
mentally solve problems that add 99. So 99 + 12 is 111, right? This can be done
by performing a paper and pencil calculation:
1
99
+12
111
OR thought of as 99+1+11 = 100+11=111.
The object is not to directly tell the learner this
method, but to let them reason through the numbers to understand why this
works. Of course compensating works with all numbers, but playing around with
99 is fun because it sounds so impressive to add such a large number to any
other number in your head. Doing lots and lots of examples and talking about
how it was done can help kids understand that it is the relationship of 99 to
100 (a difference of 1) that makes these calculations easy. After you get used
to this idea it is fun to ask 999 plus anything, or even 98 + anything and so
on.
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