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.

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. For each animation write an appropriate story situation.

Example 1: What story situation could be used to represent this model of division?

Example 2: What story situation could be used to represent this model of division?

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 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).

On the next slide there are four division problems. Classify them as either sharing or measurement division.

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 = teams can be fed.

The strategy was sharing: 256 shared into eighths. The problem was measurement: How many eights in 256?

Discuss why sharing and measurement give you the same answer. Use 18 lollies shared among six people as a starting point. 18 ÷ 6 =

18÷ 6 Each time you share one object to each person you take away a set of six! How many times can you do that? = 3

Problem TypeMultiplicationDivision by Sharing Division by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rates 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. 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. 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: By changing which of the four numbers is missing you can change the nature of the problem, as happened with the three problems before. Windows Minutes

Problem TypeMultiplicationDivision by Sharing Division by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rate 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? Adding to our matrix:

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: 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? Henry’s Sarah’s

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?

Problem TypeMultiplicationDivision by Sharing Division by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rate 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? Comparison 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? Adding to our matrix:

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: Again each of the different problems reflected a different unknown in the table. Boys Girls = 35 in total

Problem TypeMultiplicationDivision by SharingDivision by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rate 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? Comparison 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? Part-Whole 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? Adding to our matrix:

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?

The T-shirt problems involved a Cartesian product in which the number of possible combinations is determined by multiplication. Cartesian products are often important in probability. They can be represented graphically in the following ways.

Array BlueYellowRedGreen T-Shirts Blue Yellow Red Shorts Tree diagram

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

Problem TypeMultiplicationDivision by SharingDivision by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rate 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? Comparison 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? Part-Whole 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? Cartesian Product Simon has four T-shirts and three pairs of shorts. How many different T-shirt and short 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? Adding to our matrix:

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? 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.

250m m 2 By changing the unknown in an area problem you can change the required operation to division: Farmer Carla’s rectangular paddock is square metres in area. One side is 250 metres long. How long is the other side?

Problem TypeMultiplicationDivision by SharingDivision by Measuring Equal groups 5 taxis have 4 passengers in each. How many passengers are there altogether? 5 x 4 = passengers have to fit equally into 5 taxis. How many passengers go in each taxi? 20 ÷ 5 = 4 20 passengers travel 4 to a taxi. How many taxis are needed? 20 ÷ 4 = 5 Rate 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? Comparison 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? Part-Whole 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? Cartesian Product Simon has four T-shirts and three pairs of shorts. How many different T- shirt and short 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? Rectangular Area A rectangle has sides of length 8 and 7 units. What is the area of the rectangle? A rectangle has an area of 56 square units. One side is 8 units long. What is the length of the other side? Adding to our matrix:

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.