Math 247 Unit 2: Base 10 numbers and algorithms

Section 7: Multiplication and division algorithms.

Tech skill

screen capture
If you don't have another favorite screen capture tool, you should get to know the one that is built into Jing

Goal: The goals this week are for you to


SMART notebooks with manipulatives for base 10 blocks, base 5 blocks, Montessori stamp game stamps. Montessori checkerboard, and a Montessori bead frame/abacus*, and the ALAbacus*.  Here is a site where you can find printable grid paper

*Note that if you want to show me what you did on these abacuses, you can't just save the SMART Notebook, because the abacus is a Flash program that is embedded in the notebook, and not a graphic.  If you want to use this and show me a picture of what you did, you will need to use some sort of screen capture device.  The screen capture camera in the SMART Notebook toolbar works fine, as does the screen capture built into Jing.

Some applets you might find useful: the base 10 blocks you can glue together and break apart; the site that has the buiilding with blocks applet, and the site with the array multiplication applet (Rectangle Multiplication), also a Rekenrek, with an adjustable number of rows.

Assignments: Please scan as necessary, and put all assignments in the dropbox.


Mental math:

Watch: Associative law of multiplication and mental multiplication.

Read about other mental arithmetic strategies in the text book: 175-178 (3rd edition pg. 172-175) Mental Multiplication Investigation.

Assignment 35: In the textbook, p. 187 # 22 (3rd edition pg. 181 # 18) there are several multiplication problems to do by mental arithmetic.  Choose 3 of them which you can do mentally in different ways (other than the standard algorithm).  For each of the problems of your choice, tell the problem, the answer, and how you figured it out. July 9

Do first (base 5):

Do: Use your base 5 manipulatives to figure out these base 5 multiplication problems: 315 x 45;     315 x 105;    315 x 205;     315 x 245

Watch: An organized way of solving these base 5 problems

Assignment 36: Show with manipulatives and numbers how to multiply 435 x 325 (your solution should have a base 5 blocks diagram and the number work to go with it--if it were me, I would make the diagram in SMART Notebook, print it and add my writing to it, and then scan it back in). July 9

Paths to the standard algorithm:

Arrays and the expanded algorithm leading to the standard algorithm


Arrays with manipulatives and the expanded algorithm
Arrays with manipulatives and 10's grids (another example, optional)
Sketched arrays
Expanded algorithm with more digits
Going from the expanded to the standard algorithm: example 1, example 2
Comparing the US and UK algorithms
More arrays (optional)

Assignment 37: July 9
Arrays, the expanded algorithm, and the standard algorithm
Lattice multiplication:

Montessori checkerboard work
Checkerboard and lattice multiplication
Array and lattice multiplication
Stamp game multiplication
Stamp game larger lattice, and connecting the 3 main algorithms

Assignment 38: July 11

Lattice multiplication
Modeling and the standard algorithm:


Bead frame multiplication--a manipulative version of the standard algorithm
Another example of bead frame multiplication
(Array and the standard algorithm--optional)

Assn 39: Teaching the Standard Algorithm See also Guidelines for presenting explanations July 11

Error patterns July 11

Assignment 40: Figure out the error patterns and identify the alternate algorithms in the assignment D2L->Content->Assignments->Week 6: Multiplication error patterns and algorithms


Do first:

Do and watch: try out some problems

Assignment 41: Write about modeling multidigit division July 13

Alternate algorithms:

Repeated subtraction division
Scaffolding division example 1, example 2 with connection to the standard algorithm
(page 198 (3rd ed. pg. 194) in Bassarear also explains the scaffolding algorithm, though with the partial quotients recorded above rather than to the right  of the problem.  I prefer the version where one records partial quotients to the right, since then I don't need to guess how much space to leave above my problem when I start.  I have seen the version where one records to the right in several elementary textbooks.)
Short division; Galley algorithm division (this web site also explains short division, and is an optional resource)

Assignment 42: Show how to divide 3 ways July 13
The standard algorithm:

Connecting scaffolding division to the standard algorithm. Example of how to teach the transition (this example uses the long division version where you don't record all the 0's)
Modeling long division with a 2-digit divisor (stamp game), and connecting to the standard algorithm: example 1, example 2.

D2L->Content->Readings->Week 6: The Goal of Long Division.  Pay particular attention to the discussions under the subheading Conceptual Understanding.

Bassarear: Division Algorithms pgs 195-198 (3rd ed. 190-193)

Assignment: July 13

Assn 43: Video/detailed explanation assignment.  So far everyone has needed more than one Jing video for these, so you may want to use Screencast-o-matic instead, which has the advantage that it will let you record 15 minutes at a time rather than 5 minutes at a time.
1. Show how to divide 3743 by 26 using stamp game manipulatives.

2. Choose one of the two methods I have discussed for approaching long division: either scaffolding division or modeling using stamp game manipulatives.  Come up with an appropriate problem (one with a 3 digit quotient), and explain how to transition from your chosen approach (scaffolding or manipulatives) to the standard long division algorithm. By transition, I mean that you may assume that your audience knows how do do either scaffolding division or stamp game division, and you must teach that audience (using what they know about scaffolding or stamp game) how to do standard long division. See also Guidelines for presenting explanations

Why can't you divide by 0?


3 reasons you can't divide by 0
The problem with dividing 0 by 0

Read: Bassarear pg 193 (3rd ed. pg. 188)

Assignment: Jul 16

Assn 44: Write or video a good explanation of why you can't divide 10 by 0.
Error patterns

A common and recalcitrant error

Assignment 45: Do the assignment "division error patterns and algorithms" on D2L->content->assignments. (print, do, scan, dropbox) July 16

[Do the Unit 2 review, look at my solutions and comments and arrange to take exam 2 on or before July 18]