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Introducing the Maths ToolboxKevin Cummins

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What is the maths toolbox?• The maths toolbox is a set of strategies that

students can put into place to solve mathematical problems.

• The purpose of the maths toolbox is to demonstrate to students that there is generally more than one way to find a solution and simplify the process of mathematical problem solving.

• The maths toolbox is best used when you are facing a mathematical problem that you are unsure of what method of attack to use to solve it.

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Why a maths Toolbox?• The maths tool box is just a simple visual concept

that students can grasp but you might prefer your students to create a “Maths Utility Belt” Like Batman or a “Maths Survival Kit”. Whatever best engages your students.

• The maths toolbox is best utilised when students can see it in the classroom just as easy as a times tables chart and it should also be readily available within their own math books.

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Let’s Take a Look at the Toolbox

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Finding the right tool for the job.

• There may be a number of tools from the toolbox that fit the problem you are trying to solve.

• But in the next few slides we recommend some tools for specific maths problems.

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Draw a diagram• What Is It?• The draw a diagram strategy is a problem-solving

technique in which students make a visual representation of the problem. For example, the following problem could be solved by drawing a picture:

• A frog is at the bottom of a 10-meter well. Each day he climbs up 3 meters. Each night he slides down 1 meter. On what day will he reach the top of the well and escape? Why Is It Important?

• Drawing a diagram or other type of visual representation is often a good starting point for solving all kinds of word problems. It is an intermediate step between language-as-text and the symbolic language of mathematics. By representing units of measurement and other objects visually, students can begin to think about the problem mathematically. Pictures and diagrams are also good ways of describing solutions to problems; therefore they are an important part of mathematical communication.

•Source from teachervision

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Work Backwards• Working backwards is a handy strategy when you know a final result or number

and you need to determine how that was achieved. Below is a great example of this strategy sourced from mathstories.com

• Question: Jack walked from Santa Clara to Palo Alto. It took 1 hour 25 minutes to walk from Santa Clara to Los Altos. Then it took 25 minutes to walk from Los Altos to Palo Alto. He arrived in Palo Alto at 2:45 P.M. At what time did he leave Santa Clara?

Strategy:

1) UNDERSTAND:

What do you need to find?

You need to find what the time was when Jack left Santa Clara.

2) PLAN:

How can you solve the problem?

The Solution to this problem is on the next slide.

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Solution – Working Backwards• You can work backwards from the time Jack reached Palo

Alto. Subtract the time it took to walk from Los Altos to Palo Alto. Then subtract the time it took to walk from Santa Clara to Los Altos.

3) SOLVE:

Start at 2:45. This is the time Jack reached Palo Alto.Subtract 25 minutes. This is the time it took to get from Los Altos to Palo Alto.Time is: 2:20 P.M.

Subtract: 1 hour 25 minutes. This is the time it took to get from Santa Clara to Los Altos..

Jack left Santa Clara at 12:55 P.M.

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Look for a pattern• This involves identifying a pattern and predicting

what will come next.

• Often students will construct a table, then use it to look for a pattern.

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Look for a Pattern• Question: Carol has written a number pattern that begins with 1, 3,

6, 10, 15. If she continues this pattern, what are the next four numbers in her pattern?

Strategy:

1) UNDERSTAND:

What do you need to find?

You need to find 4 numbers after 15.

2) PLAN:

How can you solve the problem?

You can find a pattern. Look at the numbers. The new number depends upon the number before it. Source: Mathstories.com

• The Solution is on the next slide

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Look for a Pattern: Solution• 3) SOLVE:

Look at the numbers in the pattern.

3 = 1 + 2 (starting number is 1, add 2 to make 3)

6 = 3 + 3 (starting number is 3, add 3 to make 6)

10 = 6 + 4 (starting number is 6, add 4 to make 10)

15 = 10 + 5 (starting number is 10, add 5 to make 15)

New numbers will be

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 9 = 45

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Make a Table• Question: You save $3 on Monday. Each day after that you

save twice as much as you saved the day before. If this pattern continues, how much would you save on Friday?

Strategy:

1) UNDERSTAND:

You need to know that you save $3 on Monday. Then you need to know that you always save twice as much as you find the day before.

2) PLAN:

How can you solve the problem? Source:mathstories.comSolution is on next slide

Workbook 1  a b c

       

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Create a table - Solution• You can make a table like the one below. List

the amount of money you save each day. Remember to double the number each day.

Day Amount of money Saved

Monday $3

Tuesday $6

Wednesday $12

Thursday $24

Friday $48

The total amount of saved was $48 by Friday

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Guess and Check explained• This strategy does not include “wild” or “blind

"guesses.• Students should be encouraged to incorporate

what they know into their guesses—an educated guess.

• The “Check” portion of this strategy must be stressed.

• When repeated guesses are necessary, using what has been learned from earlier guesses should help

• make each subsequent guess better and better.

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Guess and Check Problem• Question: Amy and Judy sold 12 show tickets altogether. Amy

sold 2 more tickets than Judy. How many tickets did each girl sell?

Strategy:

1) UNDERSTAND:

What do you need to find?

You need to know that 12 tickets were sold in all. You also need to know that Amy sold 2 more tickets than Judy.

2) PLAN:

How can you solve the problem? – Solution on the next slide• Source: mathstories.com

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Guess and Check - Solution• You can guess and check to find two numbers with a sum of 12 and a difference of 2. If

your first guess does not work, try two different numbers.

3) SOLVE:

First Guess:Amy = 8 ticketsJudy = 4 tickets

Check8 + 4 = 128 - 4 = 4 ( Amy sold 4 more tickets)These numbers do not work!

Second Guess:Amy = 7 ticketsJudy = 5 tickets

Check7 + 5 = 127- 5 = 2 ( Amy sold 2 more tickets)These numbers do work!

Amy sold 7 tickets and Judy sold 5 tickets.

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Act it out• •Stress that other objects may be used in place of

the real thing.

• Simple real-life problems can posed to “act it out” in the early grades.

• The value of acting it out becomes clearer when the problems are more challenging.

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Try all possibilities• •This is often used with “look for a pattern” and

"construct a table.”

• The Farmer's Puzzle: Farmer John was counting his cows and chickens and saw that together they had a total of 60 legs. If he had 22 cows and chickens, how many of each did he have?

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Try all possibilities solution• Answer: 8 cows, 14 chickens

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Make a model / Make a list• Question: Judy is taking pictures of Jim, Karen and Mike. She asks

them, " How many different ways could you three children stand in a line?"

Strategy:

1) UNDERSTAND:

What do you need to know?

You need to know that any of the students can be first, second or third.

2) PLAN:

How can you solve the problem?

You can make a list to help you find all the different ways. Choose one student to be first, and another to be second. The last one will be third.

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Solution make a model / make a list

When you make your list, you will notice that there are 2 ways for Jim to be first, 2 ways for Karen to be first and 2 ways for Mike to be first.

First Second Third

Jim Karen MikeJim Mike Karen

Karen Jim MikeKaren Mike JimMike Karen JimMike Jim Karen

So, there are 6 ways that the children could stand in line.

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Seek an Exception• This is a tough one to explain but essentially it

means to find an irregularity. • This exception may stand out from a series of

numbers or objects within a maths problem.• When you identify the exception this will allow you

make conclusions as to why it is part of the problem and work towards solving it.

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Break a problem into smaller parts

• This one is very self explanatory and encourages students to work with concepts that they can deal with by dismantling the larger problem and recreating it in smaller parts.

• The major problem with a large project is where and how to start. Start by breaking it down into a "What TO DO" list and go from there.

• Be careful not to stray from the original question and remember BODMAS

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Solve a simpler Problem First• This one really flows on from breaking a large

problem into smaller parts and reinforces the dissecting of complex maths problems into simple steps.

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Final Thoughts?• Above all else make sure that you have downloaded

your free A4 and A3 math toolbox posters and they are visible when required.

• You can access them here.

• http://www.edgalaxy.com/journal/2011/3/2/must-have-maths-problem-solving-toolbox-posters-for-your-cla.html

• Regularly revisit the maths toolbox throughout the year to keep these concepts fresh in your students minds.