The day started answering some questions following yesterdays session that dealt with Math in Contest. Yesterdays session make us partake in a group activity in mathematizing. This activity would take a whole class. Todays session was more about the "drill and practice" if you could call it that.
Strings are minilessons. They should take between 10 to 15 minutes of classtime. These activities focus on a students ability to do "mental math". That does not mean that they have to do all computation in their head. Mental math is giving students strategies to solve questions in a variety of ways.... the way mathematicians would.
Strings of computation problems where numbers are chosen for a reason to connect in giving a student a feel of numeracy.I jump around her a bit. I hope it is clear.
Questions and comments that arose out of string 1
10 x 17 = 170
Question is... How did you know?
The zero trick… Why does it work.
Why is it like magic?
Talk to your partner?
This discussion with partners within the learning community is important. Some prompts the facilitator can give are:
Did the person next to you say something interesting to you?
Did the person next to you say something that made you said WHAT?
Why does adding a 0 when x10 GENERAIZE?
Pull this information out of the community of learners.
Switch the question instead of 10 seventeens lets think about seventeen 10’s. Ah the good old communicative property. This is a rule that generalizes
17 x 10 = 17 x 10
2 x 17 (double)
12 x 17
12 is 10 and 2 so use this to help you multiply the more difficult question.
Use and array to illustrate this question.
Eventually you can use and open array to show the multiplication on all kids strategies.
(draw rectange with 10 by 17 then underneath it do a 2 by 17 show the two combined to show the new array 170 and 34 is 204
Mental arithmathic is to break the habits of pen and paper. Mental math is more to break the bonds of the traditional algorithm.
Get kids to build on their strategies instead of putting them into our strategies.
Mathematicians play with what they know about number and go with it. They are not restricted to just using the traditional number algorithm. Multiplication and the array is so important for understanding the multiplication of polynomials. (x+6)(x+3). Many of us learned the FOIL method. You can use the array to display the multiplication better.
When we teach the students the traditional method of multiplying what happens is the opposite to what kids need to know when multiplying polynomials. The traditional way of multiplying ends up being LIOF the opposite of FOIL that they will need in middle and high school.
Distributive property and the array much more effective.
Strings and how they built.
1.) 10 x 17
2.) 2 x 17
3.) 20 x 17 (doubled the 10 x 17 (draw a double array 10 x 17 twice) Show the array over again )
4.) 22x17 is the next part of the string
Next part of string is 19 x 17 Which strategies will kids choose? I chose 170+90+63 because I can multiply but you can see another strategy that would be 20x17-17….or -10-7
5.) 19 x 117 next part of string just 100x19 more than the previous string
Towards the end of the string take the strategies away
6.) 13x22 but the strategies you have been working on will be the helpers
Come in with a string but be prepared to go with the flow of the learners.
3.) 8x18 (doubles)
4.) 16x9 (double half or x10-16)
5.) 4x36 (4x30 4x6) strategies are honoured even if they are outside the 48x3 (could be 4x12x3 use the factors to make it easier) make an array
This string takes into place the associative property
When kids get the distribuive property and the associative property they get multipication.
6.) 4.8 x0.3 (make it 48 x 3)
7.) 3 1/2 x 14 (double and half so that it is 7x7)
8.) 3 1/3 x 150 (x3 divide by 3 becomes 10 x 150)
Now for some Division
1.) 130/13 (10’s rule)
2.) 26/13 (doubles)
3.) 52/13 (26 doubled answer doubles)
4.) 182/13 (52 and 130 from above using partial quotients)
Partial quotients uses the distributive property (inverse of multiplication.)
5.) 195/13 (just 13 more)
6.) 260/13 (doubles)
7.) 247/13 (260-13) one less
What we have been working on is Associative property.
Distributive property and how it relates to division and partial quotients
1.) 100/4 (four quarters)
2.) 200/4 (just doubles)
3.) 200/8 (divisor doubles but the divident remains the same it halves)
Partial quotients are helpful when there are no common factors or if one of the numbers is a prime
Simplifying are effective
4.) 400/16 (double double) equivalent fractions
5.) 800/32 (fractions again)
6.) 300/12 (use the first question and create equivalents)
7.) 1200/48 (simplify)
8.) 3.6 /0.9 make it simpler
These are great warm-up and mini lessons. Highly recommend that you go and buy the books buy these authors. This is the way math was meant to be taught. I am looking forward to the last day and the school year.