STUDENT PROBLEMS - CLASSROOM/WORKSHOP ACTIVITIES

Here is an explicit list of activities where students engage in solving maths problems designed to promote mathematical thinking.

• Most problems are from a branch of discrete mathematics called combinatorics- which is usually taught in high school or college as part of probability.
• Each Stage 1-3 activity is designed for one or two hours duration (depending on student level and weather on the day)
• Duration of Kindy and Year 1 activities depend on discretion of the teacher on the day.
• Each activity can be extended or repeated - both within and across grades/stages.
• The activities can be carried out in any order, depending on capabilities of children.
• A list of example student activities/problems follows…

Some example students activities for K-6 are explained in more detail in their own web pages:

1. TRAINS
2. SHIRTS-AND-PANTS
3. TOWERS-FOUR-HIGH
4. CUPS-BOWLS-PLATES
5. BOWLS-AND-CONES
6. TOWER OF HANOI
7. TAXI STAND

• STUDENTS naturally develop their own strategies, with guidance from a facilitator who who shows that teachers should not need to suggest strategies or provide answers

• TEACHERS should undertake each of the activities before attempting to work with students students and/or before watching the workshop videos.

PROBLEM - Stephen wears a different outfit every day and in his wardrobe, Stephen has:

• One white shirt, a blue shirt, and a yellow shirt.
• One pair of blue jeans and a pair of white jeans.

How may different outfits can he make?

Make a prediction/estimate and write it down (draw a circle around it)

Find out for yourself, how many different outfits can he make?

Develop a way to convince others that you have found all possible outfits and have included no 'duplicates'.

Write down the number of outfits that found, and draw a square around that number.

Share and compare your solution and justification with others in your group.

PROBLEM: Suppose Stephen was given an extra pair of pants, a black pair. How many different outfits can he make now?

Make a prediction/estimate and write it down (draw a circle around that number)

Develop a way to convince others that you have found all possible outfits and have included no 'duplicates'.

Write down your answer and draw a square around it.

Share and compare your solution and justification with others in your group.

PROBLEM 1: Suppose Stephen's mother gave him two new belts (a black belt and a brown belt)

1. Remember, that Stephen now has a white shirt, a blue shirt, and a yellow shirt.
2. He has a pair of blue jeans and a pair of white jeans.
3. Stephen has now been given a brown belt and a black belt.
4. His mother has told Steven that, to make sure his his pants do not fall down, he must always wear one of his two belts,
5. How many different outfits can he make now?

PROBLEM 2: Suppose Steven's mother has $10. She can only afford to buy only one of these items:

• One new shirt
• One new pair of pants
• One new belt

Which item should Stvene's mother buy so that Steven can make the most number of new outfits that are all different.

Make a prediction/estimate and write it down (draw a circle around that number)

Develop a way to convince others that you have found all possible outfits and have included no 'duplicates'.

Write down the number of outfits that you found, and draw a square around the number.

Share and compare your solution and justification with others in your group.

PROBLEM - Stephen's brother (Billy-bob) has exactly 16 different outfits.

• Think about how this might be possible?
• What pieces of clothing Billy-bob might have, so that he can make exactly 16 different outfits?

Make a prediction/estimate and write it down (draw a circle around it)

Write down your answer and draw a square around it.

Share and compare your solution and justification with others in your group. Develop a way to convince others that you have found all possible outfits and have included no 'duplicates'.

MATERIALS: Each group is given a total of 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

PROBLEM: Work together and make as many different towers four cubes tall as is possible when selecting from two colours.

See if you and your partner(s) can plan a good way to find all the towers four cubes tall.

Make a prediction/estimate and write it down (draw a circle around it)

Find out for yourself, how many different towers you can make?

Write down your answer and draw a square around that number.

Develop a way to convince others that you have found all possible towers and have included no 'duplicates'.

MATERIALS: Each group has the same number of cubes as previous activity; 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

Before you start to build towers, make a prediction/estimate about the number of towers you think you will find. Write down that number (draw a circle around it)

If you use all the brick that you used to build the four-tall towers with two colours, how many 3-tall towers could you make when selecting from the same cubes with two colours.

Write down your estimate before you start to build towers.

• Starting with the same number of cubes, could you build more 3-tall towers than 4-tall towers, or less 3-tall towers than 4-tall towers?

Write down the number of towers that you found and draw a square around that number.

Develop a way to convince others that you have found all possible combinations and have included no 'duplicates'.

MATERIALS: Each group has the same number of cubes as previous activity; 100 Unifix cubes comprised of two colours (50 cubes of each colour = total 100 cubes).

Write down your estimate and draw a circle around it BEFORE YOU START to build towers.

PROBLEM: Work together and make as many different towers five cubes tall as is possible when selecting from two colours.

See if you and your partner(s) can plan a good way to find all the towers five cubes tall.

Write down your answer and draw a square around it.

Develop a way to convince others that you have found all possible towers, and have included no 'duplicates'.

AFTER YOU HAVE COMPLETED THE ABOVE CHALLENGE: What do you estimate is the total number of cubes of each of the two colours needed to build all possible combinations of towers 5-high.

Write down “Total number of cubes for towers 5-tall +”, then write down the number that you estimate, and draw a circle around it.

Develop a way to convince others that you have found all possible towers, and have included no 'duplicates'.

Write down your answer and draw a square around it.

PROBLEM - Solve the problem below for yourself and develop a way to convince others that you have found all possible combinations of cups, bowls and plates and have no 'duplicates':

Pretend that there is a birthday/class party in your class today. It’s your job to set the places with cups, bowls, and plates. The cups and bowls are blue or yellow. The plates are either blue, yellow, or orange.

1. Is it possible for 10 children at the party to each have a different combination of cup, bowl, and plate?
2. Is it possible for 15 children at the party each to have a different combination of cup,

bowl, and plate?

Make a prediction/estimate and write it down (draw a circle around it)

Develop a way to convince others that you have found all possible combinations and have included no 'duplicates'.

Write down your answer and draw a square around it.

PROBLEM: Next Friday there will be a 500 metre relay race at the school.

• Each team that participates in the race must have a different uniform (a uniform consists of a solid coloured shirt and a solid coloured pair of shorts).
• The colours available for shirts are yellow, orange, blue, or red.
• The colours for shorts are brown, green, purple, or white.

How many different relay teams can participate in the race?

Make a prediction/estimate and write it down (draw a circle around it)

Develop a way to convince others that you have found all possible relay teams and have not included any team more than once.

Write down your answer and draw a square around it.

MATERIALS: Your group has two colours of Unifix cubes.

PROBLEM: Work together and make as many different towers five cubes tall as is possible when selecting from two colours.

Make a prediction/estimate and write it down (draw a circle around it)

See if you and your partner can plan a good way to find all the towers five cubes tall.

Write down your answer and draw a square around it.

MATERIALS: Your group has three colours of Unifix cubes.

PROBLEM: Your group has three colours of Unifix cubes. Work together and make as many different towers four cubes tall as is possible when selecting from three colours.

Make a prediction/estimate and write it down (draw a circle around it)

See if you and your partner can plan a good way to find all the towers four cubes tall.

Write down your answer and draw a square around it.

PROBLEM: How many pizza combinations that can be made when selecting from among five different toppings.

Make a prediction/estimate and write it down (draw a circle around it)

Develop a way to convince others that you have found all possible combinations and have included no 'duplicates'.

Write down your answer and draw a square around it.

You have been invited to participate in a TV Quiz Show and the opportunity to win a vacation to Disney World.

The game is played by:

1. choosing one of four possible winning combinations…
2. then picking a tower out of a covered box.

If the tower pick draw out of the covered box matches your choice, you win.

You are told that the box contains all possible towers that are three tall that can be built when you select from cubes of two colours, red, and yellow.

You are given the following possibilities for a winning tower:

• All cubes are exactly the same colour.
• There is only one red cube.
• Exactly two cubes are red.
• At least two cubes are yellow.

Which choice would you make and why would this choice be better than any of the others?

Make a prediction/estimate and write it down (draw a circle around it)

Develop a way to convince others that you have found all possible combinations and have included no 'duplicates'.

Write down your answer and draw a square around it.

IF YOU WON, then you can play again for the Grand Prize which means you can take a friend to Disney World… But now your box has all possible towers that are four tall (built by selecting from the two colours yellow and red). You are to select from the same four possibilities for a winning tower.

Explain which choice would you make this time and why would this choice be better than any of the others?

Write down your answer and draw a square around it.

A local pizza shop offer a basic cheese pizza only (no other sauce or topping).

The pizza shop owner has asked us to help design a form to keep track of certain pizza choices.

On this basic pizza, one or two toppings could be added to either:

• half of the basic cheese pizza
• the whole of the basic cheese pizza.

How many choices do customers have if they could choose from two different toppings (sausage and pepperoni) that could be placed on either the whole pizza or half of a cheese pizza?

Make a prediction/estimate and write it down (draw a circle around it)

List all possibilities. Show your plan for determining these choices.

Convince us that you have accounted for all possibilities and that there could be no more and that you have included no 'duplicates'

Write down your answer and draw a square around it.

A local pizza shop offer a basic cheese pizza with tomato sauce.

Customers have asked for more choice and so the pizza shop owner has asked us to help design a form to keep track of certain pizza choices.

A customer with a basic pizza can then add one or more of the following toppings:

• peppers
• sausage
• mushrooms
• pepperoni.

How many different choices for pizza does a customer have?

Make a prediction/estimate and write it down (draw a circle around it)

List all the possible choices.

Find a way to convince each other that you have accounted for all possible choices.

Write down your answer and draw a square around it.

The pizza shop was so pleased with our help on the other problems that they have asked us to continue our work.

Remember that they offer a basic cheese pizza with tomato sauce.

A customer with a basic pizza can then add one or more of the following toppings:

• peppers
• sausage
• mushrooms
• pepperoni.

The pizza shop now wants to offer each customer a choice of crusts:

• Regular (thin crust) or…
• Sicilian (thick crust).

How many choices for pizza does a customer have?

Make a prediction/estimate and write it down (draw a circle around it)

List all the possible choices.

Find a way to convince each other that you have accounted for all possible choices.

Write down your answer and draw a square around it.

At customer request, the pizza shop has agreed to fill orders with different choices for each half of a pizza.

Remember that they offer a basic cheese pizza with tomato sauce.

A customer with a basic pizza can then add one or more of the following toppings:

• peppers
• sausage
• mushrooms
• pepperoni.

The customer has a choice of crusts:

• Regular (thin crust) or…
• Sicilian (thick crust).

How many different choices for pizza does a customer have?

Make a prediction/estimate and write it down (draw a circle around it)

List all the possible choices.

Find a way to convince each other than you have accounted for all possible choices.

Write down your answer and draw a square around it.

What's the Difference? In English we use the word 'combination' loosely, without thinking if the order of things is important. In other words: * 'My fruit salad is a combination of apples, grapes and bananas' We don't care what order the fruits are in, they could also be “bananas, grapes and apples” or “grapes, apples and bananas”, its the same fruit salad.

• 'The combination to the safe is 472'. Now we do care about the order; '724' won't work, nor will '247'. It has to be exactly 472 (in that order).

So, in Mathematics we use more precise language:

• When the order doesn't matter, it is a Combination.
• When the order does matter it is a Permutation.

A 'combination lock' is really a “permutation lock'

BOWLS & CONES - On-Screen Math Activities (Ice Cream Problems)

• Bowls: There are six flavours of ice cream. If the ice cream is served in bowls that can hold up to six scoops, how many different ways can the ice cream be served?
• Cones: In a variation of the problem, the ice cream can be served in cones stacked up to

four scoops high. Given that the order of stacking matters, how many different cones could be served?

PROBLEM: The new pizza shop has been doing a lot of business. The owner thinks that it has been so hot this season that he would like to open up an ice cream shop next door.

He plans to start out with a small freezer and sell only six flavours of ice cream:

1. vanilla
2. chocolate
3. pistachio
4. boysenberry
5. cherry
6. butter pecan.

BOWLS: The cones that were ordered did not arrive in time for the grand opening so all the ice cream was served in bowls.

• How many choices for bowls of ice cream does the customer have?
• Find a way to convince each other that you have accounted for all possibilities.

CONES: The cones were delivered later in the week.

The owner soon discovered that people are fussy about the order in which the scoops are stacked. on the cone. One customer said “After all eating chocolate then vanilla is a different taste than eating vanilla then chocolate.”

The owner also quickly discovered that she couldn’t stack more than four scoops in a cone.

How many choices for ice cream cones does a customer have?

Find a way to convince each other that you have accounted for all possibilities.

PROBLEM I: How many feet and toes are there?

PROBLEM II: How many different two-digit numbers can be made from the digits 1, 2, 3, and 4?

Each of four cards is labelled with a different numeral: 1, 2, 3, and 4.

How many different two-digit numbers can be made by choosing any two of them?

Make a prediction/estimate and write it down (draw a circle around it)

List all the possible choices.

Find a way to convince each other that you have accounted for all possible choices.

Write down your answer and draw a square around it.