1. Multiplication Made Meaningful Grade 4 October 4, 2010
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4. Do not assume that other people understand things the same way you do.
In fact, you should assume that they don't!
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5. If we want to understand other people properly, we must see what their behaviours or words or concepts mean to them, not what they would mean to us.
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6. Quickdraw helps break down barriers:
We all see different things, but we can make sense of each others perspective
Builds Vocabulary
Builds Confidence
Build Flexibility in Thinking
Engages the Brain
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7. How does Quickdraw link to multiplication?
Taking apart and putting together: part whole relationships is the heart of factoring
Naming parts as new units then putting back together: Distributive Property
Moving the parts around does not change their properties or equality: Commutative Property
Number is embedded in number, parts and wholes are related
Breaking apart in equal parts vs, breaking apart leads to thinking about fractions and divisions.
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8. What is multiplication?
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9. If the first thing you said was repeated addition.....
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10. I Think We have a Problem....
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11. Multiplication is not repeated addition....
It is a whole new operation...
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12. There are (at least) two basic things you can do to numbers:
You can add them and you can multiply them.
Multiplication is the mathematical operation of scaling one number by another.
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13.
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14. Arrays and Areas: Dimensional Growth Multiplicative Comparisons:
Equal Groups
Arrays and Areas: Dimensional Growth
Multiplicative Comparisons:
Times as Many
Rates and Ratios
Scaling up and Scaling Down
The green rod is 3 times as long as the pink rod.
The truck is twice as heavy as the car
Your age is one third of mine.
The GST is 10% of the pre-GST price.
The density of iron is 7.8 times that of water
Combinations: Cartesian Products
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15. Related Concepts: Place Value Fractions Decimals Area & Perimeter
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16. Demonstrate an understanding of multiplication to 5 x 5 by:
Grade 3
Demonstrate an understanding of multiplication to 5 x 5 by:
representing and explaining using equal grouping and arrays
creating and solving problems in context
modelling concrete and visual representations & recording the process symbolically
relating multiplication to repeated addition
relating multiplication to division [C, CN, PS, R]
Demonstrate an understanding of division by:
representing & explaining division using equal sharing and equal grouping
creating & solving problems in context that involve both
modelling both concretely, visually then recording
relating division to repeated subtraction
relating division to multiplication. (limited to 5 x 5) [C, CN, PS, R]
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17. skip counting from a known fact using doubling or halving
Grade 4
Describe & apply mental mathematics strategies, such as: to determine * & related ÷ facts to 9 x 9.
skip counting from a known fact
using doubling or halving
using doubling/halving, adding/subtracting one more group
using patterns in the 9s facts
using repeated doubling [C, CN, ME, PS, R]
Demonstrate an understanding of multiplication (2-or 3-digit by 1-digit) to solve problems by:
personal strategies with & without concrete materials
using arrays
connecting concrete representations to symbolic representations
estimating products
applying the distributive property [C, CN, ME, PS, R, V]
Demonstrate an understanding of division (1-digit divisor & up to 2-digit dividend) to solve problems by:
using personal strategies for dividing with & without concrete materials
estimating quotients
relating division to multiplication [C, CN, ME, PS, R, V]
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18. Build Explain Represent Compare Synthesize Self Evaluate
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19. Representations for Multiplication include: Equal Groups
Build 4 groups of 3 tiles, represent what you have on paper
The whole gets lost: it is hard to see the 12 without counting.
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20. Representations for Multiplication include: Equal Groups
Now build 3 groups of 4 tiles, represent what you have on paper
Same problem. The whole is lost, now we see parts but what about the whole they came from???
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21. 3 x 4 = 4 x 3 but the image does not encourage it
The commutative property is lost in this image.
3 x 4 = 4 x 3 but the image does not encourage it
These are equal? 4 3
These are equal?
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22. Representations for Multiplication include: Arrays or Areas
Put your 3 sets of 4 into equal rows and push your rows together to make a rectangle 4 3 4 3
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23. The commutative property is not lost.4 3 4 3
And you can still see the 4 groups of 3 or the 3 groups of 4. They cover the same area or fill the same space...
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25. Array (Area) Representations
The 4 x 3 is actually a representation of 4 repetitions of a set or unit of 3 or 3 repetitions of a set or unit of 4
This is 4 sets of threes
Or 3 sets of fours
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26. So I want to explain the 3 x 4 by taking it apart in its groups...
We need students to build, explain and represent in terms of groups or units.....
So I want to explain the 3 x 4 by taking it apart in its groups...
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27. This seeing the array pull apart in both directions is critical
This seeing the array pull apart in both directions is critical.... Practise it with the students... Link it to a representation 12 4 3
3 x 4 = 12
12 is made up of 3 sets of 4 or 4 sets of 3
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28. We want them to explain in terms of multiplication and division.
As students make sense of the arrays we want them to build with tiles, but build using units.
We want them to explain in terms of multiplication and division.
We want them to represent with images that allow them to see the commutative property as well as the link to division. 12 4 3
3 x 4 = 4 x 3
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29. Use your facts to build other facts....
Cut out the 3 x 4. Label it.
3 x 4 = 12 or 4 x 3 = 12 or both.
Use it to make another array.
Build it.
Explain what you did.
Represent the new array on paper.
Compare it to the 3 x 4. What can you say or explain about the relationship???
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30. Build the multiplication matrix with meaning: Relate your facts
Use your facts to build other facts....
Build the multiplication matrix with meaning:
Relate your facts
Add another group or set each time.
(property of one)
Doubles, halves
Distribute
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31. Use your facts to build other facts....
Build the 3’s and use them to build the 6’s
Build the 4’s and use them to build the 8’s
Let students in on the outcomes... The goal is to build strategies for solving problems...
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32. This powerpoint is not for public distribution
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33. This powerpoint is not for public distribution
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34. This powerpoint is not for public distribution
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35. This powerpoint is not for public distribution
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36. This powerpoint is not for public distribution
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38. Use your facts to build other facts....
Let’s Practise !!!!
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39. Double Sidewalks Mysteries
Sidewalks: A prefabricated concrete company makes double-row sidewalks.
They make two sizes of block:
singles, which are $3 each
sevens, which are $10.
How much would a double-row sidewalk of length 20 cost?
Can you design more than one?
Double Sidewalks Mysteries
“I had an interesting day at work today,” said Akira.
“Every double sidewalk order we filled required no cutting!”
Show me.
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40. Penny flowers and Mickeys
Seven pennies pushed together into a circle make a
“penny-flower”. A table covered with pennies can be rearranged into penny-flowers, so that counting by sevens will tell how many pennies there are.
Alternately, two pennies are placed with a nickel, like Mickey Mouse’s two ears. A table covered with pennies and nickels is arranged into Mickeys, so that counting by sevens, well you can take it from here.
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41. If I had 23 pennies, that would be 3 pennyflowers and 2 seeds
If I had 23 pennies, that would be 3 pennyflowers and 2 seeds. If I combined my pennies with someone else, we might not have any seeds left.
Show me.
Two people had the same number of pennies each. When they put their pennies together, will they have any seeds left over?
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42. PennyFlower Mysteries
I am thinking of a person who used some nickels to get pennies. he made pennyflowers with his pennies. he said, “If I can get 2 more pennies, I’ll have only pennyflowers.”
Show me.
Many people make pennyflowers out of pennies. I once saw a person make a pennyflower out of pennyflowers!
Mickeys Mysteries
Johan had some nickels. he traded ____ of them for pennies. When he made Mickeys, he had __ pennies left over.
Mahatma has 14 pennies. he asked the storekeeper for enough nickels to make Mickeys with his pennies. he had to give the storekeeper ___ dimes.
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43. This powerpoint is not for public distribution
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44. This powerpoint is not for public distribution
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45. On the planet ISME there are two sorts of creatures
On the planet ISME there are two sorts of creatures. The ZIOS have 3 legs and the ZIPPETS have 7 legs.
Famous space Ranger, Buck Rogers was exploring the planet when he came upon a herd of ZIOS and ZIPPETS. he managed to see that there was more than one of each kind of creature before they saw him. Suddenly they all rolled over onto their backs and put their legs in the air.
He counted 52 legs. How many Zios and how many Zippets were in the group?
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46. In fact, it has never happened any other way. Be the change!!!
Never doubt that a small group of thoughtful and committed (teachers) can change the world.
In fact, it has never happened any other way.
Be the change!!!
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47. This powerpoint is not for public distribution
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48. This is gonna hurt.....
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49. This powerpoint is not for public distribution
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50. But what’s the difference for your students????
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51. This powerpoint is not for public distribution
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52. This powerpoint is not for public distribution
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53. Math is about ideas. How do we represent ideas
Math is about ideas. How do we represent ideas? In images, words, notations, models. If you think you understand something, model it, don’t just write about it. The algorithm may not help. If we do not share the language we still can share the images, that lead to understanding.
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