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Graphic Organizers in Math

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The use of a graphic organizer can start as early as first or second grade and can continue to be useful for some learners all the way through high school. In subjects such as math, that grow increasingly complex as students get older, these tools can be especially helpful in maintaining organized work habits and enhancing problem-solving skills. If used correctly and consistently as students develop, the concepts of strategic thinking graphic organizers instill will likely have reached the point that many learners no longer need them by the time they reach high school.

How To Use Graphic Organizers in Math

Using graphic organizers has been a proven effective problem-solving strategy for helping young learners think and process information more efficiently by allowing them to both visualize and organize the information they need to solve problems. Creativity and careful attention to detail can be greatly enhanced via the use of visual maps—which is exactly what a graphic organizer is. A graphic organizer aids in organizing thought processes as well as creating a framework to collect and compare the information that's being gathered. That's why, in addition to structuring information, organizers can be used to improve students' abilities to comprehend and process that information by seeing it separated it into categories of what is more important and what less important. 

​Over time, graphic organizers help learners become strategic problem solvers. Provided they're used effectively and consistently as an integral part of the  problem-solving  process, graphic organizers can also improve test scores. 

How Graphic Organizers Work for Math

A typical graphic organizer has the problem printed on it. The paper is divided into four quadrants with the problem appearing at the top, although sometimes, it can be found in the middle of the page. 

The first quadrant is used for the student to determine what the problem is actually trying to solve for. The second quadrant is used to determine what strategies are needed to solve the problem. The third quadrant is used to show the steps involved in order to solve the problem. The fourth quadrant is used to answer the question that is initially being asked and to indicate why the answer the reasoning behind how the answer was arrived at, and why the answer is correct. 

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Home > ETD > 157

Theses and Dissertations

Using the four square strategy to enhance math problem-solving.

Andrea Gerrard

Date Approved

Embargo period, document type, degree name.

M.A. Learning Disabilities

Language, Literacy, and Special Education

College of Education

Kuder, S Jay

Mathematics--Study and teaching

• Disciplines

Elementary Education and Teaching

This study examined whether the use of the Four Square problem-solving strategy would improve the math problem-solving performance of students who receive supplemental services for language arts literacy and mathematics from a basic skills teacher or a special education teacher. The Four Square strategy was used in conjunction with the enVision math series. It was also used during all problem-solving lessons. Students' growth in math problem-solving was measured using the enVision problem-solving pre- and post-test and the Measure of Academic Progress Test (MAP). Students showed growth on the enVision test. The students showed work when solving problems instead of just selecting one of the available answer choices and were able to increase the number of questions correct on the assessment. Students did show growth on the MAP test. They were able to close the gap between other students in the sixth grade who receive supplemental services for language arts literacy and mathematics from a basic skills teacher. The results indicate that the Four Square strategy can be a useful way to enhance students' problem-solving skills. The Four Square strategy allowed students to take ownership of their problem-solving skills and to improve these skills throughout the study.

Recommended Citation

Gerrard, Andrea, "Using the Four Square strategy to enhance math problem-solving" (2013). Theses and Dissertations . 157. https://rdw.rowan.edu/etd/157

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Completing the Square (Examples)

More examples of solving quadratic equations using completing the square.

In my opinion, the “most important” usage of completing the square method is when we solve quadratic equations. In fact, the Quadratic Formula that we utilize to solve quadratic equations is derived using the technique of completing the square. Here is my lesson on Deriving the Quadratic Formula .

Applications of Completing the Square Method

Example 1 : Solve the quadratic equation below using the method of completing the square.

Move the constant to the right side of the equation, while keeping the [latex]x[/latex]-terms on the left. I can do that by subtracting both sides by [latex]14[/latex].

Next, identify the coefficient of the linear term (just the [latex]x[/latex]-term) which is

Take that number, divide by [latex]2[/latex] and square it.

Add [latex]{{81} \over 4}[/latex] to both sides of the equation, and then simplify.

Express the trinomial on the left side as a square of binomial.

Take the square roots of both sides of the equation to eliminate the power of [latex]2[/latex] of the parenthesis. Make sure that you attach the plus or minus symbol to the constant term (right side of the equation).

Solve for “[latex]x[/latex]” by adding both sides by [latex]{9 \over 2}[/latex].

Find the two values of “[latex]x[/latex]” by considering the two cases: positive and negative.

Therefore, the final answers are [latex]{x_1} = 7[/latex] and [latex]{x_2} = 2[/latex]. You may back-substitute these two values of [latex]x[/latex] from the original equation to check.

Example 2 : Solve the quadratic equation below using the method of completing the square.

Subtract [latex]2[/latex] from both sides of the quadratic equation to eliminate the constant on the left side.

Divide [latex]8[/latex] by [latex]2[/latex] and square it.

Add [latex]16[/latex] to both sides of the equation.

Express the left side as square of a binomial.

Take square roots of both sides.

Example 3 : Solve the quadratic equation below using the technique of completing the square.

Eliminate the constant [latex] – 36[/latex] on the left side by adding [latex]36[/latex] to both sides of the quadratic equation.

Divide the entire equation by the coefficient of the [latex]{x^2}[/latex] term which is [latex]6[/latex]. Reduce the fraction to its lowest term.

Identify the coefficient of the linear term.

Divide this coefficient by [latex]2[/latex] and square it.

Add this output to both sides of the equation. Be careful when adding or subtracting fractions.

Express the trinomial on the left side as a perfect square binomial. Then solve the equation by first taking the square roots of both sides. Don’t forget to attach the plus or minus symbol to the square root of the constant term on the right side.

Finish this off by subtracting both sides by [latex]{{{23} \over 4}}[/latex]. You should obtain two values of “[latex]x[/latex]” because of the “plus or minus”.

The final answers are [latex]{x_1} = {1 \over 2}[/latex] and [latex]{x_2} = – 12[/latex].

Example 4 : Solve the quadratic equation below using the technique of completing the square.

Step 1: Eliminate the constant on the left side, and then divide the entire equation by [latex] – \,3[/latex].

Step 2: Take the coefficient of the linear term which is [latex]{2 \over 3}[/latex]. Divide it by [latex]2[/latex] and square it.

Step 3: Add the value found in step #2 to both sides of the equation. Then combine the fractions.

Step 4: Express the trinomial on the left side as square of a binomial.

Step 5: Take the square roots of both sides of the equation. Make sure that you attach the “plus or minus” symbol to the square root of the constant on the right side. Simplify the radical.

Step 6: Solve for [latex]x[/latex] by subtracting both sides by [latex]{1 \over 3}[/latex]. You should have two answers because of the “plus or minus” case.

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3 Ways to Solve Magic Square Math Puzzles

Last Updated: July 2, 2024 Fact Checked

What is a magic square?

• Solving a 3 x 3
• Solving a 4 x 4

Singly Even Magic Square

This article was co-authored by wikiHow staff writer, Hunter Rising . Hunter Rising is a wikiHow Staff Writer based in Los Angeles. He has more than three years of experience writing for and working with wikiHow. Hunter holds a BFA in Entertainment Design from the University of Wisconsin - Stout and a Minor in English Writing. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,127,466 times. Learn more...

If you’re a fan of math and logic puzzles like Sudoku or Kenken, then trying to solve a magic square is the perfect little challenge to try out. A magic square is a grid of numbers where each row, column, and diagonal add up to the same sum. How you solve your magic square depends on the size of the puzzle, but they each have easy instructions for filling them in. Keep reading and we’ll walk through our solving strategies step by step to finish any odd- or even-numbered magic square.

Things You Should Know

• A magic square is a square grid of numbers where each row, column, and diagonal add up to the same total.

• Use a solving technique based on the size of the magic square and how many boxes are in each row or column.

• On a non-normal magic square, the lowest number may be higher than 1. A non-normal order 4 magic square with a magic constant of 87 could be: 21 – 24 – 28 – 14 27 – 15 – 20 – 25 16 – 30 – 22 – 19 23 – 18 – 17 – 29

3 x 3 or Odd-Numbered Squares

• All rows, columns, and diagonals must add up to 15.
• Normal magic squares always have the same magic constant. If solving for a non-normal square, add all the numbers together and divide by the number of rows to find the magic constant.

• Example: If you have a 3 x 3 magic square, put 1 in the second box in the top row.
• Example: For a 15 x 15 magic square, put the 1 in the eighth box of the row.

• Example: In a normal 3 x 3 square, the 2 goes in the bottom right corner and the 3 goes in the left column in the center row.
• If the movement takes you to a box that is already occupied, go back to the last box that has been filled in, and place the next number directly below it. For example, when you place the 4 in a 3 x 3 magic square, it fits in the bottom left corner below the 3. Then, following the same up-one, right-one pattern, place the 5 in the center of the puzzle to continue.

4 x4 or Doubly Even Magic Square

• All rows, columns, and diagonals must add up to 34.

• In a 4 x 4 square, only mark the 4 corner boxes as your highlights.
• For an 8 x 8 magic square, the Highlights become 2 x 2 mini squares in the corners.
• In a 12 x 12 square, each Highlight is a 3 x 3 area.

• In a 4 x 4 square, the Central Highlight would be a 2 x 2 area in the center.
• In an 8 x 8 square, the Central Highlight would be a 4 x 4 area in the center.

• 1 in the top-left box and 4 in the top-right box
• 6 and 7 in the second and third boxes in Row 2
• 10 and 11 in the second and third boxes of Row 3
• 13 in the bottom-left box and 16 in the bottom-right box

• 15 and 14 in the second and third boxes in Row 1
• 12 in the leftmost box and 9 in the rightmost box in Row 2
• 8 in the leftmost box and 5 in the rightmost box in Row 3
• 3 and 2 in the second and third boxes in Row 4

• The smallest possible singly even magic square is 6 x 6 since 2 x 2 magic squares are impossible to make.

• Hence, the magic constant for a 6×6 square is 111 and all rows, columns, and diagonals will add up to this number.

• For a 6 x 6 square, each quadrant is 3 x 3 boxes.

• In a 6 x 6 square, Quadrant A contains the numbers 1-9, Quadrant B has 10-18, Quadrant C has 19-27, and Quadrant D ends with 28-36.

• In a 6 x 6 square, Highlight A-1 is the top right box, Highlight A-2 is the center box in the middle row, and Highlight A-3 is the bottom right corner of Quadrant A. The D highlights are in the same position in Quadrant D.
• If you tried to add up your columns, rows, and diagonals right now, the square isn’t considered magic since they don’t add up to the same magic constant yet.

• Example: In a 6 x 6 magic square, 8 swaps positions with 35, 5 swaps with 32, and 4 swaps with 31.

• For a 10 x 10 magic square, only swap the rightmost column in quadrants B and C.
• For a 14 x 14 magic square, swap the 2 rightmost columns instead.

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• ↑ https://www.researchgate.net/publication/315457341_A_STUDY_ON_MAGIC_SQUARES
• ↑ https://mae.ufl.edu/~uhk/MAGIC-SQUARE.pdf
• ↑ https://math.hmc.edu/funfacts/making-magic-squares/
• ↑ https://faculty.etsu.edu/stephen/matrix-magic-squares.pdf
• ↑ https://www.researchgate.net/publication/332496561_Algorithm_for_Doubly_Even_Magic_square
• ↑ https://support.sas.com/resources/papers/proceedings09/201-2009.pdf
• ↑ https://www.1728.org/magicsq2.htm
• ↑ https://www.math.wichita.edu/~richardson/mathematics/magic%20squares/even-ordermagicsquares.html
• ↑ https://www.rsisinternational.org/journals/ijrsi/digital-library/volume-5-issue-6/91-94.pdf
• ↑ https://www.1728.org/magicsq3.htm

About This Article

To solve an odd-numbered magic square, start by using the formula n[(n^2+1)/2] to calculate the magic constant, or the number that all rows, columns, and diagonals must add up to. For example, in a 3 by 3 square where n=3, the magic constant is 15. Next, start your square by placing the number 1 in the center box of the top row. Then, arrange the rest of the numbers sequentially by moving up 1 row, then 1 column to the right. To learn more, including how to solve singly even magic squares and doubly even magic squares, read on. Did this summary help you? Yes No

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