1e. Working mathematically - Problem solving

2. We need to teach children the strategies and language they require to tackle problems

There are three documents below which support us in building an understanding of the best strategies to use when problem solving. We need to teach each of the strategies to students over time so that they have their own repertoire or toolbox of strategies.

(Polya, 1945) describes a four-step approach to problem solving, including: 

• Understanding the problem:  it is impossible to solve a problem if you do not know what the problem is. 

What is known or unknown? 

Is there enough information or is more information needed? 

What do the terms mean?

• Devising a plan: the way we solve the problem. Possible strategies: 

(a) Draw pictures; 

(b) Use a variable and choose helpful names for variables or unknowns; 

(c) Be systematic; 

(d) Solve a simpler version of the problem; 

(e) Guess and check, Trial and error; Guess and test (Guessing is okay); 

(f) Look for a pattern or patterns; and 

(g) Make a list 

• Carrying out the plan: 

If the plan does not seem to be working, then start over and try another way. 

Often the first approach does not work. 

Do not worry just because an approach does not work. It does not mean you did it wrong. You actually accomplished something, knowing a way does not work is part of the process of elimination

• Looking back: 

Did you answer the question? 

Is your result reasonable? 

Is there another way of doing the problem which may be easier? 

Problem solving skills help students:

• gain more skills and knowledge to solve problems of different kinds

• increase reasoning skills. 

Activity: I do (demonstrate using this model to solve a problem)

I want to make enough cupcakes for each child to have one. The mixture makes 10 cupcakes and they take 10 minutes to prepare and 15 minutes to cook. There are 24 children, but 3 are gluten free. How many batches of cupcakes will I need to make to ensure all children get at least one cupcake. How long will it take me?

Activity: We do

Use the model above and as a class work through this problem. 

Jade is having a party. They ae going to order pizzas and need to know how many to order. There will be 11 people attending and they are going to allow 4 slices of pizza per person. Each pizza will have 8 slices. How many pizzas will they need to buy?

Activity: You do

Use the model above with one of these problems. Work with a partner.
Discuss which strategy worked best for you.

1. For a party, Jaime had 6 bottles of cola for her and her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left? 

2. During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. On Friday they ran twice as many as on Wednesday. How many kilometres did the students run in the three days? Write your answer as a fraction. 

3. Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional. 

4. Now try these problems with a partner or by yourself...which strategy works best for you?

You might use this framework to help you ... https://drive.google.com/file/d/0B7YM6iJAeDKVbklYM1pKLTF2Njg/view 

A Hole New Board Game Puzzle

How do we turn the problem below into numeracy?  Look at the problem below called A Hole New Board Game Puzzle and then think about how you could word that as a problem for students.

How about this?

I have a garden bed which needs to be covered in weed matting from edge to edge. The garden bed measures 2m x 12m.  You can make one cut only in the 3m x 8m weed matting so that we do not have too many joins. Where will you cut it?

OR

I have a sand pit which needs to be covered by a board to stop the cats and other animals using it. The sand pit measures 2m x 12m.  Bunnings only sell masonit in 3mx8m lots. You can make one cut only in the 3m x 8m masonite to make it fit the sand pit as it will be stronger with only one cut. Where will you cut it?