Multiplication

We are beginning our unit on multiplication.

Today’s problem (Jan 31):

To activate our brains, the students were asked to represent 5X3 in as many ways as they could.  Then we read today’s problem:

Some students used repetitive addition to solve the problem:

Some students combined repetitive addition and doubling:

Some students used repetitive addition, but only after they broke up the number (48) into friendly numbers (40 + 8):

Some students broke the number up into friendly numbers, and then did small multiplication (times 3, twice):

Some students used doubling:

Some students used the traditional method of multiplication:

Wow!  What a wide range of thinking! Awesome!

Consolidation:  We focused our discussion on one strategy; the one that repeatedly added the same number.  We discussed how it compared to other strategies.  The students decided to name this strategy “MULIPLADDITION”.  They were given some problems to solve (using this strategy) for homework.

Today’s problem (Feb 1):

To activate our brains, we had a discussion about the rules used when multiplying by 1, 10 or 100.  Then we looked at today’s problem:

We first read the question and pulled out the essential information.

Some students solved the problem using yesterday’s strategy of Multipladdition (repetitive addition):

Some students broke up the number being multiplied into friendly number before adding (20 + 9):

Some students broke up 29 into it’s expanded form first (29=20+9) and then did the multiplication (20×9  +  9×9):

Some students rounded off 29 to 30 (a friendly number), did the multiplication (30×9) and then subtracted the extra 9 at the end:

Some students used the traditional method of multiplication:

Consolidation:  We focused on this solution…

…and had a discussion about why it is easier to break up the number this way when multiplying, especially when you are trying to work out the answer in your head.  The students were asked to come up with a name for the strategy.  The called it “Expandiplication” because you expand the number first (or write the number in it’s expanded form), and then do the multiplication.  It looks like this:

Homework:  to do questions 11 to 15, from yesterday’s sheet, using the Expandiplication strategy.

Today’s problem (Feb 3):

To activate our thinking we talked about Mr. Wendler’s magic numbers (1,2,5,10,100) and how they are easy numbers to multiply (friendly)

Today’s problem:

Some students used the Multipladdition strategy, but first broke up the number (168) into friendly numbers (100,60,8):

Some students used Multipadditon:

Some students tried to use addition more efficiently, so they added 2 recesses (336), and then used that number for their repetitive addition:

Some students used Expandiplication:

Some students used a rounding strategy:

Some students used the traditional strategy:

Consolidation:  We focused our discussion on the rounding strategy (168×7).  If you round off the number 168 to 170 (a friendlier number) by adding 2, then you can multiply 170×7, which = 1190.  But because you added 2 to each of the 7 168’s, you have added (2×7) 14 to the answer.  So that 14 needs to be removed, 1190-14=1176.  What a great idea!

The students named this strategy “Round the Nation” (as a play on Roundiplication).  It looks like this:

For homework, the students are asked to do #16-19 on the math sheet handed out the other day, using the “Round the Nation” strategy.

Today’s problem (Feb 6):

To activate our brains, the students were asked:

The connections between area and multiplication were made again.  Then the question 12×11 was represented using base 10 blocks to reinforce the connection between area and multiplication:

Today’s problem:

Some students used Multipladdtion to solve the problem:

Some students used repetitive addition, but used larger chunks when adding (for example adding 44’s – 2 groups of 22 – or 88’s – 4 groups of 22 – at a time):

Chunks of 5 22’s:

Chunks of 2 22’s:

Chunks of 4 22’s:

Some students used Expandiplication (22 x 17 – the 22 was expanded to 10, 10, and 2 – and each number was multiplied by 17):

Some students used a long traditional strategy:

Some students used the traditional strategy:

Consolidation: We focused on the addition strategy that chunked together groups of numbers to make the addition more efficient.  The students named the strategy “Double Trouble” (although, sometimes it can be more like Triple Trouble, or even Quadruple Trouble).  It looks like this:

Homework:  Try these 5 questions using the Double Trouble strategy.

1/ 21 x 6 =
2/ 35 x 4 =
3/ 11 x 8 =
4/ 15 x 9 =
5/ 23 x 8 =

Feb 7:

Today we learned about Magic Squares.  Magic Squares are a strategy that focuses on multiplication as area.  But it also has the affect of breaking up numbers into friendly numbers making it easier to do the work mentally.  Here is what it looks like.  Ask you children to explain it to you 🙂

Today’s problem (Feb 8):

To activate our brains we reviewed Magic Squares.  We had several students do this question on the board, several different ways:

Today’s problem:

It was a tough one!

Some students used Multipladdition:

Some students used variations of Double Trouble:

Some students used Expandiplication:

Most students used the Magic Squares:

Some students used the traditional method:

Consolidation:  We discussed the answer (Mr. W’s class: 32×9=288; Mrs. K’s class: 22×15=330; the difference being 42)  Then we looked at just the one question (22×15=330) done in the traditional method and using Magic Squares, and the students were asked to make connections between the two strategies. 

They realized that the numbers 220 and 110 were found in both.  The students were asked to think about (for homework) why that connection happened.

Today’s problem (Feb 9):

To activate our brains we reviewed the strategies explored so far:

Today’s problem:

Some students solved the problem using Multipladdition:

Some students used Expandplication:

Some students used Magic Squares:

Some students used the Traditional (long) strategy:

Some used the traditional (short) strategy:

Consolidation:  We focused on making connections between the pattern you follow when multiplying using the traditional strategy (fist ones x ones, then ones x tens), and the numbers that show up when you use Magic Squares:

Grade 4’s: Traditional (long) Strategy:

Grade 5’s: Traditional (long) Strategy:

The grade 5 homework is on the picture above.  They are to be done using the Traditional Long Strategy.  The grade 4’s got a homework sheet with the questions on them.  They are also to use the Traditional Long Strategy.

Today’s problem (Feb 10):

To activate our brains we reviewed the Traditional Long strategy:

Today’s question (Grade 4’s: 83×8; Grade 5’s: 83×34):

Some students solved the problem using Multipladdition:

Some students used the Double Trouble strategy:

Some students used Magic Squares:

Some used the Traditional Long strategy:

Some used the Traditional Short strategy:

Consolidation:  As a group we compared the Traditional Long and Traditional Short strategies, both for the grade 4’s…

… and for the grade 5’s

Their homework was to do yesterday’s questions again using the Traditional Short strategy and see if they get the same answers.

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.

Here it what it looks like:

Today’s problem (Feb 16th):

To activate our brains the students made connections between 4 different solutions to the same math question:

Today’s question:

Most groups had great starts, focusing on the hotel rooms.   Most groups tried to break up the problem into three areas, the hotel rooms, the conference rooms and the lobby:

A few groups managed to find their way through a very complicated problem and found the correct answer:

Consolidation: Our discussion focused around the idea that when you are not focusing on how to do the calculation, all your energy goes into trying to understand the problem.

Today’s problem (Feb 21):

To activate our brains we reviewed our strategies and the last consolidation (focus on understanding the problem):

Today’s question:

There were a number of different solutions, some focusing on the use of addition:

Some focusing on using almost entirely multiplication:

Consolidation:  The students did a great job working through the problem.  There were a few “aha” moments, however, where students were simply taking the numbers from the question and multiplying them, with little regard for what the question was asking (because we are currently learning about multiplication).  These “aha” moments helped lead the discussion in the direction of not seeing this as a MULTIPLICATION question, but rather, just a question.

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One Response to Multiplication

  1. angelina says:

    were is naper’s bones?

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