Bansho in the Class

Here is an example of what Bansho can look like in the class. This is a unit on multiplication from a grade 4/5 class. For more examples visit the Math in the Class section at: www.mrwendlersclass.wordpress.com or
www.mrwendlersblog.wordpress.com

The Activation component of the lesson was initially not not posted or discussed, but it always took place before the lesson. It can be as simple as taking up the last day’s homework, or it can be very specific to the lesson. I have begun to include the activation in more recent blog updates (see www.wendlersblog.wordpress.com)

Alright, here is an example of a unit on multiplication done in the class…

We are now beginning our unit on multiplication. There are numerous strategies for multiplication, which vary in efficiency and complexity.

A fun trick to learn and practise your 6 to 10 times tables:

Today’s question (Jan 31):

The students discussed and decided upon the important information:

Some students used a strategy that involved repeatedly adding the same number (adding 46 six times):

Some students used a similar strategy, but organized their information using a table:

Some students combined repetitive adding with ‘chunking’ (repetitively added larger chunks of information – in this case 98 (2 groups of 48)), which is more efficient than adding 46 six times:

Some groups broke the number 48 into ones and tens, and then multiplied each place value separately, and added the answers together (more efficient still):

Some students used a more traditional approach, but done in two steps:

Some groups used a traditional strategy:

Consolidation: One strategy the students developed was a strategy that involved repetitive addition. They called it the “Adding Attack” strategy.

Students were asked to give this strategy a try on the following homework questions:

Today’s question (Feb 3):

The strategies used have been ordered from least to most efficient.
Some students used the ‘Adding Attack’ strategy:

Or they used the ‘Adding Attack’ strategy as a table:

Some student used a ‘doubling’ or ‘chunking’ strategy:

Some students broke up the larger number (35) into friendly numbers (10, 10, 10 and 5) and then multiplied:

Some students broke up the larger number (35) into its place value columns (30 and 5) and then multiplied:

Some students rounded off the 9 (35 x 9) to 10 (35 x 10) and then multiplied. Then subtracted one necklace (35 x 1):

Some students used the traditional algorithm:

Consolidation: Our discussion centred around one strategy. The students named it the ‘Friendly Break-Away’ strategy. We discussed the reasons behind breaking up a number into friendly numbers (35 = 10 + 10 + 10 + 5) being that friendly numbers are much easier to multiply. Also using multiplication is more efficient than the Adding Attack strategy. The students were given 4 questions for homework to practise this strategy.

Another interesting discussion that began was the similarities found between the strategy that broke up 35 into its place value columns (30 and 5) and the traditional strategy. In this strategy the tens are multiplied first, and then the ones; and in the traditional strategy the ones are multiplied first, and then the tens. To be continued…

Today’s question (Feb4):
I was at a workshop today, so a variety of strategies are not posted, only the strategy that was focused on during the consolidation:

The question:

Some students solved the problem this way:

Consolidation: (278 x 8 =) In continuation of yesterday’s discussion, the strategy focused on today was one that broke up the number (278) into its place values (200 + 70 + 8).

Each place value was then multiplied by eight (200×8=1 600, 70×8=560, 8×8=64).

The totals were then added together (1 600 + 560 + 64 = 2224).

This students talked about this strategy being effective because it is efficient, and because it uses easy (friendly) numbers. They decided to call it the “Chunky Cheese” strategy (because the large number is being broken into easy to use chunks) or the “Place Value” strategy, because it breaks up the large number into its place values.

Today’s homework – using the “Chunky Cheese” strategy, answer the following questions:

1. 128 x 4 =
2. 252 x 5 =
3. 376 x 6 =
4. 497 x 7 =
5. 1 286 x 8 =

Today’s question (Feb 7) – 2 digit by 2 digit:

In order to activate the students thinking, they were shown a base 10 block, and asked what multiplication question it represents…

They were correct in answering 10 x 10. The connection the students noticed was that multiplication is area. They were shown another block and asked what question it represented:

Again they were correct in answering 10 x 11.

Then they were given today’s question:

Some students used the Adding Attack strategy:

Some students used a Chunking or Doubling strategy:

Some students used the Friendly Break-Up strategy:

Some students decided that multiplying by a 2 digit number was too difficult (32 x 16), so they split 16 in half, and multiplied it twice (32 x 8) x2:

Some students used the Chunky Cheese (Place Value) strategy, breaking up 32 into 30 and 2:

Some students realized that 32 x 16 means multiplying each place value by the other (30 x 10 plus 30 x 2 plus 2 x 10 plus 2 x 6):

Some students used the activation idea to look at the question as area:

And some students recognized that area could be calculated using more friendly numbers (much like the activation idea):

Some students used the traditional algorithm:

Consolidation: We discussed multiplication as area, and how, when looking at area, we could break up both two-digit numbers into friendly numbers, or into their place values.

32 x 16 broken up into friendly numbers (10 + 10 + 10 + 2) x (10 + 5 + 1):

32 x 16 broken into place value (30 + 2) x (10 + 6):

We called the use of area and friendly numbers to do multiplication: the ‘Magic Squares’ strategy.

Today’s question (Feb 9):

The students pulled out the important information:

There have been a number of strategies explored so far including: Adding Attack, Avalanche (doubling), Friendly Break-Away, Chunky Cheese (place value), and Magic Squares.

Here are some of the solutions the class came up with:

Consolidation: In our consolidation, we compared 2 of the answers given by the students. We focused on their similarities.

The students noticed that when multiplying 2 digit by 2 digit numbers (in a more traditional way),

83
x34
—-
12 (4×3) First, you multiply the ones by the ones;
320 (4X80) Second, you multiply the ones by the tens;
90 (30×3) Third, you multiply the tens by the ones;
2400 (30×80) Lastly, you multiply the tens by the tens;
——
2822 Then you add up the numbers to get the answer.

This is very similar to the magic square using place values:

So the students found a strong connection between the Magic Squares and the traditional algorithm.

Today’s question (Feb 10):

The students pulled out the important information:

Students used a number of strategies to solve the problem:

Consolidation: Our focus was on the traditional algorithm and how it works. For the grade 4’s we focused on a two-digit by a one-digit multiplication; for the grade 5’s we focused on two-digit by two-digit multiplication.

Grade 4 (including hw questions):

Grade 5’s (including hw questions):

Today’s question (Feb 15):

Today there were 2 questions and the students could choose either. They were asked to solve one of the problems using at least 4 different strategies. They were then asked to choose 2 strategies and compare them.

Here were the 2 questions:

We made a list of the strategies we have discovered so far:

The students solutions looked like this:

And here are some comparisons…

Taditional vs Magic Squares:

Chunky Cheese vs Magic Squares

Magic Squares vs Adding Attack:

Traditional Long vs Traditional Short:

Consolidation: Our discussion centred around the vocabulary and explanations surrounding how strategies are similar and how they are different, in order to develop a deeper understanding of multiplication.

Today’s lesson (Feb 16):

Napier’s Bones

Today we learned about Napier’s Bones. John Napier, a Scottish mathematician in the early 1600’s, invented a unique method of doing multiplication. It allows the students to multiply large digits, without using the “place value holders” needed when using the traditional strategy. Using a chart (similar to a multiplication chart), the first number is written at the top (one digit per column) and the second number is written on the right hand side (one digit per row). This allows each digit of one number to be multiplied by each digit of the second number (similar to the place value strategy). The answers are written in each box. This method automatically lines up the numbers on a diagonal in their correct place value column (ones, tens, hundreds, etc…). The answers in each box are then added up diagonally (remember to carry if necessary) and the answer is found at the bottom.