multiplication and division: the inside story
DESCRIPTION
Multiplication and Division: The Inside Story. A behind-the-scenes look at the most powerful operations. http://elementary-math-resources.wiki.inghamisd.org/home. Three sessions. Today: Multiplication and Division Mar. 27: Fractions and Decimals Apr. 23: Geometric Shapes and Volume. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/1.jpg)
Multiplication and Division: The Inside Story
A behind-the-scenes look at the most powerful operations
http://elementary-math-resources.wiki.inghamisd.org/home
![Page 2: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/2.jpg)
Three sessionsToday: Multiplication and DivisionMar. 27: Fractions and DecimalsApr. 23: Geometric Shapes and
Volume
![Page 3: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/3.jpg)
TodayHow children learnMultiplication and division
problem-solvingMultiplication and division
combinationsMulti-digit multiplication and
divisionConnections with area and
perimeter
![Page 4: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/4.jpg)
How many ways?With a partner, make as many rectangles as you can using all the tiles at your table. Record each rectangle as a multiplication sentence.
6 = 2 x 3
![Page 5: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/5.jpg)
The first way we teach children to think about multiplication:
x x x x x 5x x x x x 1
0x x x x x 1
5x x x
x x 2
0Skip-counting of rows in an array.An example is 4 rows of 5 chairs lined up in a room.
![Page 6: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/6.jpg)
Is 5 rows of 4 the same number?
x x x x 4x x x x 8x x x x 1
2x x x x 1
6x x x x 2
0Make up 3 drawing that show jumps of 2-3-4.
![Page 7: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/7.jpg)
Key Strategies1. Use visual representations
![Page 8: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/8.jpg)
The second way we teach children to think about multiplication:
Equal groups. This is a generalization of equal-size rows of objects in an array.An example is 5 bags with 4 cookies in each bag. Make up 3 more examples using 10-11-12.
4 4 4 4 4
![Page 9: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/9.jpg)
The third way we teach children to think about multiplication:My dog can run 5 times as fast as your
rabbit.Your rabbit can jump 3 times as far as my
dog.My dog eats 10 times more food than your
rabbit.Your rabbit is 1/4 the height of my dog (or my
dog is 4 times taller than your rabbit).Your rabbit is twice as old as my dog.My dog can bark 100 times louder than your
rabbit!Multiplicative comparison.
![Page 10: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/10.jpg)
Key Strategies2. Teach the underlying structures
of word problems
Three blended goals of math education:
Conceptual understandingFluencyProblem solving
![Page 11: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/11.jpg)
Related problem typesRate Price CombinationThe Friendly Old Ice Cream Shop has 3 types of ice cream cones. They also have 4 flavors of ice cream. How many different combinations of an ice cream flavor and cone type can you get at the Friendly Old Ice Cream Shop?
![Page 12: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/12.jpg)
Here’s how I would do it:
sugar conechocolatemint choc. chiprocky roadvanilla swirl
4
safety conechocolatemint choc. chiprocky roadvanilla swirl
4
waffle conechocolatemint choc. chiprocky roadvanilla swirl
4
4 x 3 = 12
![Page 13: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/13.jpg)
Key Strategies3. Explain your thinking carefully
when showing proceduresRequire students to explain their thinking also.
If one lamp costs $45, how much do 12 lamps cost? Explain how you figured this out.
![Page 14: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/14.jpg)
Why is it important to recognize types of multiplication problems?The fixed costs of manufacturing basketballs in a factoryare $1,400.00 per day. The variable costs are $5.25 perbasketball. Which of the following expressions can beused to model the cost of manufacturing b basketballs in one day?A. $1,405.25b
B. $5.25b − $1,400.00C. $1,400.00b + $5.25D. $1,400.00 − $5.25b
E. $1,400.00 + $5.25b
![Page 15: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/15.jpg)
Number TalkWhat number do you think will go in the blank to make the equation true? Try to solve this by reasoning, without doing the calculations.4 x 9 = 12 x ___How did you think about this?
![Page 16: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/16.jpg)
The most powerful way of thinking about multiplication:
This is powerful because it connects multiplication to the area of a rectangle.8 x 7 = 56 8 in. x 7 in. = 56 sq. in.
![Page 17: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/17.jpg)
The most powerful way of thinking about multiplication:
Plus, it gives us insight into the process of multiplication, and new ways to compute:8 x 7 = (8 x 5) + (8 x 2) This is the distributive property (3.MD.7)
![Page 18: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/18.jpg)
The most powerful way of thinking about multiplication:
Now you can multiply bigger numbers in your head. Try 56 x 5. Try 8 x 23.
![Page 19: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/19.jpg)
Find a way to multiply 38 x 6 by representing 38 as a subtraction.Try 3,426 x 5 by decomposing into thousands, hundreds, tens and ones.
![Page 20: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/20.jpg)
Number Talks book and DVDNumber Talks:
Helping Children Build Mental Math and Computation Strategies, Grades K-5, by Sherry Parrish (DVD)
Watch Array Discussion
![Page 21: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/21.jpg)
FactorsThe word “factor” is an academic vocabulary term that is essential to understanding multiplication.6 x 1 = 63 x 2 = 6Which are the factors and which are the products in your rectangles?Watch 16 x 35
![Page 22: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/22.jpg)
Rectangle multiplication What does this visual representation tell you about multiplication? (knees to knees, eyes to eyes)
http://nlvm.usu.edu
![Page 23: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/23.jpg)
The Factor GameCommon Core Collaboration CardsWith your team member, see if
you can figure out a strategy for winning.
Also linked from our Elementary Math Resources wiki: Go to inghamisd.org, then click on Wiki Spaces.
![Page 24: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/24.jpg)
How to help a child become fluentAcquisition – Fluency – Generalization Concepts, strategies, procedures
Practice, then DrillExtensions
This learning progression is true for single digit “math facts” and for fluency with multi-digit procedures.
![Page 25: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/25.jpg)
Math facts, if not already knownMath fact strategy:1) Only work on unknown combinations2) Ensure knowledge of meaning of
multiplication (acquisition)3) Learn strategies through repeated
problem-solving (practice)4) Drill in game situations (fluency)5) Use in division situations
(generalization)
![Page 26: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/26.jpg)
IISD Fluency PacketResources for helping those
students who still need work on combinations.
![Page 27: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/27.jpg)
Fluency – Practice and Drill“Practice” refers to lessons that are problem-based and that encourage students to develop flexible and useful strategies that are personally meaningful.“Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.
From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally
![Page 28: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/28.jpg)
4. Practice, practice, practice
And use cumulative review often.
Daily Double!!!
![Page 29: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/29.jpg)
The Product GameGood practice for children who
don’t have all their combinations from memory yet.
A combination game from PhET
![Page 30: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/30.jpg)
Research RecommendationInterventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.Provide about 10 minutes per session of
instruction to build quick retrieval of basic arithmetic facts. Consider using technology, flash cards, and other materials for extensive practice to facilitate automatic retrieval.
For students in kindergarten through grade 2, explicitly teach strategies for efficient counting to improve the retrieval of mathematics facts.
Teach students in grades 2-8 how to use knowledge of properties, such as commutative, associative, and distributive laws, to derive facts in their heads.
![Page 31: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/31.jpg)
Box and Books of Factsfrom ORIGO Mathematics
![Page 32: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/32.jpg)
Procedures… The C-R-AConcrete-Representational-AbstractConcrete: Multiply 16 x 12 using base 10 blocks.
![Page 33: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/33.jpg)
Procedures… The C-R-AConcrete-Representational-AbstractRepresentational:
National Library of Virtual Manipulatives nlvm.usu.edu
![Page 34: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/34.jpg)
Procedures… The C-R-AConcrete-Representational-AbstractAbstract:
![Page 35: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/35.jpg)
Learning Progression
![Page 36: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/36.jpg)
![Page 37: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/37.jpg)
![Page 38: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/38.jpg)
![Page 39: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/39.jpg)
Problem-solving with area and perimeterTable for 22: Real-World Geometry Problem
![Page 40: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/40.jpg)
How many rectangles…?What does this represent?
Watch Associative Property 12 x 15
![Page 41: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/41.jpg)
How is division tied to multiplication?List several ways the two are connected…
![Page 42: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/42.jpg)
Two types of divisionPartitive (fair shares)We want to share 12 cookies equally among 4 kids. How many cookies does each kid get? How would you solve this with a picture?The number of groups is known; the number in each group is unknown.
![Page 43: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/43.jpg)
Measurement (repeated subtraction)For our bake sale, we have 12 cookies and want to make bags with 2 cookies in each bag. How many bags can we make?How would you solve this with a picture?The number in each group is known; the number of groups is unknown.
![Page 44: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/44.jpg)
Partial quotient method6 )234 -120 20 114 -60 10 54 -30 5 24 -24 4 0 39
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value. 4.NBT.6
This type of division is called repeated subtraction
![Page 45: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/45.jpg)
You try it
24)8280
Now the standard algorithm
Keep in mind that 8280 = 8000 + 200 + 80 + 0 or 8200 + 80 or 82 hundreds + 8 tens
![Page 46: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/46.jpg)
Division by Partitioning354 photos to share among 3 children
![Page 47: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/47.jpg)
Work with manipulatives also translates to procedures354 ÷ 3(300 + 50 + 4) ÷ 3 = 100 + 10 + 1 r 21
100 + 10 + 1 + 7
Try this with 251 ÷ 8. Partition base 10 blocks, then write a corresponding algorithm.
![Page 48: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/48.jpg)
What about remainders?The remainder is simply left over and not taken into account (ignored)It takes 3 eggs to make a cake. How many cakes can you make with 17 eggs? The remainder means an extra is needed20 people are going to a movie. 6 people can ride in each car. How many cars are needed to get all 20 people to the movie?
![Page 49: Multiplication and Division: The Inside Story](https://reader035.vdocuments.net/reader035/viewer/2022062316/56812d39550346895d923c06/html5/thumbnails/49.jpg)
The remainder is the answer to the problemMs. Baker has 17 cupcakes. She wants to share them equally among her 3 children so that no one gets more than anyone else. If she gives each child as many cupcakes as possible, how many cupcakes will be left over for Ms. Baker to eat? The answer includes a fractional part9 cookies are being shared equally among 4 people. How much does each person get?