# Introducing the Maths Toolbox - Edgalaxy

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```					Kevin Cummins

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• 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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• 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”.

• 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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• 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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• 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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•   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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• 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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• 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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• 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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• 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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Workbook 1
a      b    c

• 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.com
Solution is on next slide

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• 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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• 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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• 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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•   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 tickets
Judy = 4 tickets
Check
8 + 4 = 12
8 - 4 = 4 ( Amy sold 4 more tickets)
These numbers do not work!
Second Guess:
Amy = 7 tickets
Judy = 5 tickets
Check
7 + 5 = 12
7- 5 = 2 ( Amy sold 2 more tickets)
These numbers do work!
Amy sold 7 tickets and Judy sold 5 tickets.

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• •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

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

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• •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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• Answer: 8 cows, 14 chickens

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• 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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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    Mike
Jim     Mike     Karen
Karen     Jim     Mike
Karen    Mike      Jim
Mike     Karen     Jim
Mike      Jim     Karen

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

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