Build and label glorwaythinking 101 ca 2017 Build
Build and label glorway@thinking 101. ca 2017
Build and label 3 4 glorway@thinking 101. ca 2017
How do I know 3 x 4 = 12 3 4 glorway@thinking 101. ca 2017
If 3 x 4 = 12 then I also know 12 ÷ 3 = 4 3 4 glorway@thinking 101. ca 2017
12 ÷ 3 could mean: I have 12 things. I want to know how many sets of 3 I can make. The situation might be there are 12 candies and I want to package the in sets of 3. How many threes can I make? 3 3 glorway@thinking 101. ca 2017 4
12 ÷ 3 = 4 3 3 I made fair shares of 3. glorway@thinking 101. ca 2017
12 ÷ 3 could also mean I have 12 things and I want to divide them into 3 groups. I know how many groups, I want to know how may will go in each group. 3 4 I made 3 groups. 12 ÷ 3 = 4 glorway@thinking 101. ca 2017
If 3 x 4 = 12, 4 x 3 = 12 3 4 glorway@thinking 101. ca 2017
If 4 x 3 = 12 then I also know 12 ÷ 4 = 3 3 4 glorway@thinking 101. ca 2017
I could think 12 ÷ 4 means I want to know how many groups of 4 are in 12…. . 3 4 glorway@thinking 101. ca 2017
I see 3 groups of 4 are in 12…. . So 12 ÷ 4 = 3 3 4 glorway@thinking 101. ca 2017
I could think 12 ÷ 4 means I want to make 4 groups, how many will go in each? 3 4 glorway@thinking 101. ca 2017
I made 4 groups, there are 3 in each group …. . So 12 ÷ 4 = 3 3 4 glorway@thinking 101. ca 2017
3 4 If 3 x 4 = 12 and 4 x 3 = 12 then 12 ÷ 4 = 3 and 12 ÷ 3 = 4 glorway@thinking 101. ca 2017
What will happen with 4 x 5 glorway@thinking 101. ca 2017
Build and label glorway@thinking 101. ca 2017
Build and label glorway@thinking 101. ca 2017
Build and label 3 4 glorway@thinking 101. ca 2017
What’s the multiplication fact? 3 4 glorway@thinking 101. ca 2017
3 x 4 = 12. 3 4 glorway@thinking 101. ca 2017
What’s the other fact that is equal? 3 4 glorway@thinking 101. ca 2017
Can you explain why the labels still work? 3 4 glorway@thinking 101. ca 2017
They both represent 3 x 4 or 4 x 3 3 4 glorway@thinking 101. ca 2017
If 3 x 4 = 12 then I also know 12 ÷ 3 = 4 3 4 glorway@thinking 101. ca 2017
12 ÷ 3 could mean: I have 12 things. I want to know how many sets of 3 I can make. The situation might be there are 12 candies and I want to package them in groups of 3. How many groups of threes can I make? 3 4 glorway@thinking 101. ca 2017
12 ÷ 3 could mean: I have 12 things. I want to know how many sets of 3 I can make. The situation might be there are 12 candies and I want to package them in groups of 3. How many groups of threes can I make? 3 Do you see the 2 ways to think about 12 ÷ 3? 4 glorway@thinking 101. ca 2017 I could make 3 groups or I could Arrange into groups of 3
I made 3 groups, how many are in each group? 3 3 4 glorway@thinking 101. ca 2017 4
I made groups of 3. There are four 3 s in 12. 3 3 4 glorway@thinking 101. ca 2017 4
I could also think about 12 ÷ 4 3 4 glorway@thinking 101. ca 2017
12 ÷ 4 could mean made 4 groups, how many are in each…… Do you see it? 3 4 glorway@thinking 101. ca 2017
12 ÷ 4 could mean divide the 12 into groups of 3, how many groups of 3 are in 12 …… Do you see it? 3 4 glorway@thinking 101. ca 2017
Try this one glorway@thinking 101. ca 2017
What are the two facts? glorway@thinking 101. ca 2017
What are the two divisions? glorway@thinking 101. ca 2017
Explain using equal groups…. . glorway@thinking 101. ca 2017
Explain using equal shares…. . glorway@thinking 101. ca 2017
What will happen with 4 x 5 glorway@thinking 101. ca 2017
4 x 5 = 20 glorway@thinking 101. ca 2017
Area models make visible: • that multiplication is related to repeated addition glorway@thinking 101. ca 2017
Area models make visible: • that multiplication is related to division glorway@thinking 101. ca 2017
12 ÷ 3, I read I have 12 things and I want to put them into sets 3 of 3. I see 4 sets of 3. 4 glorway@thinking 101. ca 2017 12
3 4 12 12 ÷ 3 can be solved by thinking how many repetitions of 3 get me to 12. Four threes glorway@thinking 101. ca 2017
3 4 12 12 ÷ 3 can be solved by thinking what do I repeat three times to get to 12? I know 4, 8, 12. Three fours. glorway@thinking 101. ca 2017
12 ÷ 3, I read I have 12 things and I want to put them into sets 3 of 3. I see 4 sets of 3. 4 glorway@thinking 101. ca 2017 12
Push them back together to see 4 sets of three equals 12. 3 4 glorway@thinking 101. ca 2017 12
3 4 glorway@thinking 101. ca 2017 12 ÷ 3, I can also think I have 12 things and I share them into 3 equal groups
3 4 glorway@thinking 101. ca 2017 12 ÷ 3, I can also think I have 12 things and I want to put them into 3 equal groups 12 divided into equal sets of 4 s
? 4 glorway@thinking 101. ca 2017 12 12 ÷ 4 Think what times 4 = 12 ? X 4 = 12
3 4 glorway@thinking 101. ca 2017 12 ÷ 3 12 divided into equal sets of 3 s
12 ÷ 3 4 3 Turn it, it still works 12 divided into equal sets of 3 s glorway@thinking 101. ca 2017
Area models allow students to: Represent division using equal sharing and equal grouping glorway@thinking 101. ca 2017
Area models to 5 x 5 are small enough to hold in visual memory…. . Students can practice with them as ”flashcards” until they have automatic recall. (Lorway, 2017). tiles Stress moving in sets or groups of, not counting by ones. folding The relationships that emerge are related to multiplication relationships. When thirds cross fourths, twelfths emerge 3, 4, and 12 are related. grids Stress seeing the “groups of” or “units” in rows crossing columns. glorway@thinking 101. ca 2017 COMPARE rods 3 four rods cover the same area as 4 three rods. Cuisenaire rods remove the distraction of seeing individual ones. the focus is on thinking in units.
Multiplication represents a many to one correspondence. Learners must learn to think in units of units. Three fives, tens. Numbers are composed of equal units. There are 4 fives in 20. There are 6 hundreds or 60 tens in 600. There are 12 twelfths in 1, 24 twelfths in 2, 36 twelfths in 3. glorway@thinking 101. ca 2017
In order to understand multiplication as a way of thinking learners must understand connect three elements: • groups of equal size or equal size parts(Lorway, 2018) • numbers of groups or number of equal size parts (Lorway, 2018) • and the total amount of parts, groups or space that has been created Students who are able to construct and coordinate these elements in both multiplication and division problems before they carry out a count are thinking multiplicatively. When learners are able to think multiplicatively they can apply the commutative property, the associative property, the distributive property and inverse relations to solve problems. Kouba (1989), Steffe (1992), Lorway (2018), Mulligan and Mitchelmore (1997) and Mulligan & Watson (1998). glorway@thinking 101. ca 2017
When students arrive in Grade 4 with the facts to 5 x 5 stored in memory, they can apply reasoning strategies that involve NUMBER PROPERTIES to understand how to determine facts to 9 x 9. Area models make visible • the distributive property Grade 4 glorway@thinking 101. ca 2017 Grade 5
I know 3 x 4 and 3 x 5. Now I can solve 3 x 9. Area models make visible • the distributive property glorway@thinking 101. ca 2017
Area models make visible: • place value as a multiplicative function glorway@thinking 101. ca 2017
As grade 2 s learn to build, describe and compare 2 digit numbers, they place them in a hundred grid. The hundred grid is a multiplication image: 10 sets of ten. By placing tens into the grid they come to understand two digit numbers as a product of repeating tens. 3 tens 3(10) replaces 10 + 10. glorway@thinking 101. ca 2017
Area models make visible: • the multiplicative nature of fractions glorway@thinking 101. ca 2017
glorway@thinking 101. ca 2017
Mathematics is used to describe and explain relationships. Students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form. POS p 8 “Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding. . Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching” POS p 11 Learning through problem solving should be the focus of mathematics at all grade levels. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies. If students have already been given ways to solve the problem, it is not a problem, but practice. Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them. Visual images and visual reasoning are important components of number, spatial and measurement sense. reasoning involves: glorway@thinking 101. ca 2017
It is highly visual: “The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them. Visual images and visual reasoning are important components of number, spatial and measurement sense. Number visualization occurs when students create mental representations of numbers. ” Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p. 2 glorway@thinking 101. ca 2017
Relationships Matter: “Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form. Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p. 8 glorway@thinking 101. ca 2017
It is presented spatially: “Spatial sense involves visualization, mental imagery and spatial reasoning. These skills are central to the understanding of mathematics. ” Mathematics Kindergarten to Grade 9 Program of Studies , Update 2016, p. 8 glorway@thinking 101. ca 2017
The outcomes are highlighted in red contribute to the development of multiplicative reasoning. glorway@thinking 101. ca 2017
Specific imagery connects across the grades: When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant and integrated. “Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding. . Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching” (Caine and Caine, 1991, p. 5). glorway@thinking 101. ca 2017
Start with a pile of tiles. Push or pull 2 at a time to make rows of threes or 2 at a time to make rows of 2. glorway@thinking 101. ca 2017
Slide four sets into an array and look. 3 4 glorway@thinking 101. ca 2017
Here is my diagram. Can you explain the labels? 3 4 glorway@thinking 101. ca 2017
12 ÷ 4 I have 12 cans of soup and want to give 4 to each customer. How many customers will get soup? 1 person glorway@thinking 101. ca 2017 1 person 4 12
12 ÷ 4 I have 12 cans of soup, organized into 4 stacks. How many are in each stack. 4 stacks 3 in each stack glorway@thinking 101. ca 2017 4 12
12 ÷ 3 I have 12 cookies and want to put 3 in each box. How many boxes do I need? 1 box glorway@thinking 101. ca 2017 1 box 3 12
12 ÷ 3 I have 12 cookies and 3 boxes. If I share them equally, how many cookies go in each box? 3 boxes 4 in each box glorway@thinking 101. ca 2017 3 12
The results of folding paper in two directions create an image that is related to multiplication. Paper folding bridges the imagery of arrays with fractions. glorway@thinking 101. ca 2017
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