Multiplicative thinking
Download
1 / 36

Multiplicative Thinking - PowerPoint PPT Presentation


  • 96 Views
  • Uploaded on

Multiplicative Thinking. Workshop 2 Situations for Multiplication and Division. Learning intentions for this workshop. Know the range of situations in which multiplication and division are used. Analyse strategies that students might use in solving these problems.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Multiplicative Thinking' - jana-bennett


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Multiplicative thinking

Multiplicative Thinking

Workshop 2

Situations for Multiplication and Division


Learning intentions for this workshop
Learning intentions for this workshop

  • Know the range of situations in which multiplication and division are used.

  • Analyse strategies that students might use in solving these problems.


In workshop one we used equal groups problems to look at the properties of multiplication and division.

  • Let’s recap:

  • Here we have a model of 4 x 3 = 12.

  • Remember that 4 and 3 are the factors and 12 is the product.

  • Four is the multiplier and three is the multiplicand.


The next two animations will show two different types of division
The next two animations will show two different types of division.

  • For each animation write an appropriate story situation.




Both of the stories you made up for the animations can be recorded as 12 ÷ 3 = 4, but the numbers refer to quite different things.

  • The first animation showed 12 ÷ 3 as “twelve shared among three.”

  • In this case the answer, four, tells how many are in each share.

  • This is called partitive division (sharing).


  • The second animation showed recorded as 12 12 ÷ 3 as “twelve separated (measured) into sets of three.”

  • In this case the answer four tells the number of sets that could be made.

  • This is called quotative division (measurement).



  • You have 36 marbles. You put them into bags of six marbles. How many bags can you make?

  • If you have 35 apples and 5 children, how many apples can you give each child?

  • You need 1.3 metres of fabric to make a sports top. How many tops can you make with 17 metres?

  • 2.5 litre bottles are filled from a 50 litre barrel. How many bottles are filled?


People who understand division recognise that sharing and measurement problems can be answered with the same calculation and attach the objects last.

  • For example:

  • It takes eight pizzas to feed a soccer team.

  • How many teams can you feed with 256 pizzas?

  • Is the problem sharing or measurement division?

  • How has the student solved it?

256 ÷ 2 = 128,

128 ÷ 2 = 64,

64 ÷ 2 = 32.

32 teams can be fed.


  • The strategy was sharing: measurement problems can be answered with the same calculation and attach the objects last.

  • 256 shared into eighths.

  • The problem was measurement:

  • How many eights in 256?

128

32

64

256



18 answer.

÷ 6

= 3

Each time you share one object to each person you take away a set of six!

How many times can you do that?


All of the problems you are about to solve involve multiplication and division. As you encounter new problem types we’ll build up a matrix like this:


Solve each of these problems then discuss what they have in common
Solve each of these problems then discuss what they have in common.

  • You can wash 4 windows in 20 minutes. How many windows can you wash in 60 minutes?

  • You can wash 12 windows in 60 minutes. How many windows can you wash in 20 minutes?

  • How long does it take you to wash 4 windows if you clean 12 windows in 60 minutes?


All three of the problems involved a rate
All three of the problems involved a rate. common.

  • A rate is a multiplicative relationship between two measures, in this case windows and minutes.

  • Here are some other examples of rates.

  • What are the measures in these rates?

  • The bullet train travels at 300 kilometres per hour.

  • An elephant eats 600 kilograms of food per day.

  • The average family has 1.7 children.

  • The tax rate is 33 cents in the dollar.


The windows to minutes rate could be shown as a table in this way

Minutes common.

Windows

The windows to minutes rate could be shown as a table in this way:

  • By changing which of the four numbers is missing you can change the nature of the problem, as happened with the three problems before.

4

20

12

60



Solve each of these problems then discuss what they have in common1
Solve each of these problems then discuss what they have in common.

  • Henry has six times as many marbles as Sarah. Sarah has 12 marbles, how many marbles does Henry have?

  • Henry has six times as many marbles as Sarah. Henry has 72 marbles, how many does Sarah have?

  • Henry has 72 marbles and Sarah has 12 marbles. How many times more marbles does Henry have than Sarah?


The marbles problems could be shown as a table in this way

Sarah’s common.

Henry’s

The marbles problems could be shown as a table in this way:

  • Again each of the different problems reflected a different unknown in the table.

  • Which of the numbers is left out of this problem?

  • Henry has 72 marbles and Sarah has 12 marbles. For every six marbles that Henry has, how many does Sarah have?

6

1

72

12


All three of those problems involved multiplicative comparison
All three of those problems involved multiplicative comparison.

  • Here are some other comparison problems. For each problem identify the things being compared.

  • New Zealand’s population is about seven times that of Fiji. There are 4.2 million people in New Zealand. How many people live in Fiji?

  • A kiwi egg weighs about one fifth of the mother’s body weight. A female brown kiwi weighs about 2.8 kg. How heavy is its egg?

  • A fully stretched piece of elastic is 3.3 times as long as its original length. How long is a 24cm length of elastic at full stretch?



Solve each of these problems then discuss what they have in common2
Solve each of these problems then discuss what they have in common.

  • For every three boys in the class there are four girls. There are 35 students in the class. How many are girls?

  • For every three boys in the class there are four girls. There are 20 girls in the class. How many boys are in the class?

  • In a class of 35 students, 15 are boys. What is the ratio of boys to girls?


The boys to girls problems could be shown as a table in this way

Girls common.

Boys

The boys to girls problems could be shown as a table in this way:

  • Again each of the different problems reflected a different unknown in the table.

3

4

15

20

= 35 in total



Solve each of these problems then discuss what they have in common3
Solve each of these problems then discuss what they have in common.

  • Simon has four T-shirts and three pairs of shorts. How many different T-shirt and shorts outfits can he make?

  • Simon can make twelve different outfits of a T-shirt with a pair of shorts. He has three T-shirts. How many pairs of shorts does he have?



Shorts the number of possible combinations is determined by multiplication.

Red

Yellow

Blue

Blue

Yellow

Red

Green

T-Shirts

  • Array

  • Tree diagram


Create a representation for each of these cross product problems
Create a representation for each of these cross product problems:

  • Lana is a courier. Each day she drives from Auckland to Rotorua to deliver her parcels. There are five different routes from Auckland to Hamilton and four different routes to go from Hamilton to Rotorua.

  • How many different ways can Lana travel between Auckland and Rotorua via Hamilton?

  • In a game of chance two dice are rolled. The numbers are multiplied together. If the product is thirteen or more you win. If it is twelve or less you lose.

  • What are all the possible outcomes of the dice toss?

  • Is the game fair?



The cross product problems could be represented by an array which is a special case of area
The cross product problems could be represented by an array which is a special case of area.

  • In area problems the side length can be whole numbers or fractional numbers.

  • Have a look at the following area problems.


What is the area of each yellow rectangle in square units

4.6 which is a special case of area.

6.7

What is the area of each yellow rectangle in square units?

  • Notice that the answer to each multiplication is in square units but the factors are lengths. It is important to include these referents in describing the operation.


By changing the unknown in an area problem you can change the required operation to division

250 000 m which is a special case of area.2

By changing the unknown in an area problem you can change the required operation to division:

  • Farmer Carla’s rectangular paddock is 25 000 square metres in area.

  • One side is 250 metres long.

  • How long is the other side?

250m


Adding to our matrix4
Adding to our matrix: which is a special case of area.


This completes workshop two about different types of multiplication and division problems
This completes workshop two about different types of multiplication and division problems.

  • Go onto workshop three to learn about how multiplicative properties are applied to fractions, algebra patterns, measurement, geometry and statistics.


ad