Multiplication Made Meaningful Grade 4 October 4, 2010 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

Do not assume that other people understand things the same way you do. In fact, you should assume that they don't! This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

What is multiplication? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

If the first thing you said was repeated addition..... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

I Think We have a Problem.... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

Multiplication is not repeated addition.... It is a whole new operation... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

http://www.maa.org/devlin/devlin_06_08.html http://www.maa.org/devlin/devlin_0708_08.html http://www.maa.org/devlin/devlin_09_08.html http://letsplaymath.net/2008/07/01/if-it-aint-repeated-addition/ http://letsplaymath.net/2008/07/28/whats-wrong-with-repeated-addition/ http://www.emis.de/proceedings/PME28/RR/RR018_Outhred.pdf This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

Related Concepts: Place Value Fractions Decimals Area & Perimeter This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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] This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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] This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

Build Explain Represent Compare Synthesize Self Evaluate This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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??? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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??? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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 This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

Use your facts to build other facts.... Let’s Practise !!!! This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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!!! This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This is gonna hurt..... This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

But what’s the difference for your students???? This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

This powerpoint is not for public distribution This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day

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. This powerpoint is not for public distribution. It is for the sole use of participants in the Oct. 4 2010 pd day