grid method problem solving year 3

Visual maths worksheets, each maths worksheet is differentiated and visual.

Multiplication Grid Method

Multiplication Grid Method Resource (Free Download)

Multiplication grid method resource description.

Here is a page of multiplication grids that can be used for numerical and algebraic topics. All grids are 2 x 1 so can be used for multiplying a single digit by a two digit number of for expanding brackets containing two terms.

The Cazoom Maths Teaching Tools Multiplication Grid Method is a vital educational resource designed to enhance the teaching and learning of multiplication in maths. This printable PDF tool is tailored specifically for school year groups, making it a highly relevant and practical aid for educators and parents alike. With a focus on the grid method of multiplication, it serves as an essential component in the mathematical toolkit for students in KS2 and KS3, facilitating a deeper understanding of complex multiplication problems through a structured approach.

What is the Cazoom Maths Teaching Tools Multiplication Grid Method?

The Multiplication Grid Method simplifies multiplication by partitioning numbers, multiplying parts in a grid, and summing results. It aids understanding, reinforcing place value and multiplication's distributive property. This resource offers a comprehensive guide to mastering the grid method of multiplication, a technique pivotal for breaking down larger multiplication tasks into more manageable parts. By providing a clear, visual representation of how numbers are split and multiplied, it aids in reinforcing the foundational skills necessary for more advanced maths concepts.

Why is the Topic of This Resource Important in Real Life

Multiplication is a fundamental skill that applies to various real-life contexts, from calculating expenses to understanding patterns and measurements. Mastery of the grid method equips students with the ability to tackle mathematical problems efficiently, promoting numerical literacy and problem-solving skills that are invaluable in everyday life.

Why This Teaching Resource is Helpful for Learning

The Cazoom Maths Teaching Tools Multiplication Grid Method is an excellent resource for learning because it offers a step-by-step approach to multiplication. The inclusion of a free PDF printable version allows for easy access and distribution, ensuring that learners can practice at their own pace. It supports differentiated learning by catering to different year groups, enabling personalised instruction that meets each student's unique needs.

In summary, this teaching tool is not just a method for learning multiplication; it is a pathway to enhancing numerical understanding and application for students. Its practicality, combined with the ease of access, makes it an invaluable resource for educators and parents striving to support their learners in years 5 to 8 to achieve maths success.

Also, have a look at our wide range of worksheets that are specifically curated to help your students practice their skills in multiplication . These teaching resources and worksheets are in PDF format and can be downloaded easily.

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Year 3 Grid Method Multiplication

Grid method multiplication 2

Grid method multiplication 3.

Grid Method

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Year 3 Maths Worksheets UK Hub Page

Welcome to our Year 3 Maths Worksheets area. Here you will find links to our Year 3 Math worksheets for your child will enjoy.

Come and take a look at our multiplication pages, or money flashcards. Perhaps you would prefer our telling the time worksheets, maths games or identifying 2d and 3d shapes in the geometry section?

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• This page contains links to other Math webpages where you will find a range of activities and resources.
• If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

Year 3 Maths Learning

Here are some of the key learning objectives for the end of Year 3:

• know and use Place value up to 1000
• Compare and order numbers up to 1000
• Counting on and back in 1s, 10s and 100s from different starting points
• Position numbers on a number line up to 1000
• know addition and subtraction facts to 20
• add or subtract 1s, 10s or 100s from a 3-digit number
• subtract 2- and 3-digit numbers in columns
• solve problems using addition and subtraction
• count from 0 in multiplies of 4, 8, 50 and 100
• recall and use multiplication and division facts from the 2, 3, 4, 5, 8 and 10 times tables
• multiply 2-digit numbers by a 1-digit number
• solve problems using multiplication and division
• count up and down in tenths
• understand tenths as decimals
• recognise and use fractions as numbers
• understand and use fractions of objects or shapes
• add and subtract amounts of money
• use £ and p notation correctly
• measure, compare, add and subtract lengths, mass and volume
• measure the perimeter of simple 2D shapes
• tell the time from an analogue clock
• convert between 12- and 24-hour clock
• measure and compare times
• Identify angles including right angles
• identify acute and obtuse angles
• identify horizontal, vertical, parallel and perpendicular lines
• recognise and draw 2D and 3D shapes
• interpret and present data in bar graphs, pictograms and tables
• solve 1-step and 2-step problems using data in tables, pictograms and graphs

Please note:

Our site is mainly based around the US Elementary school math standards.

Though the links on this page are all designed primarily for students in the US, but they are also at the correct level and standard for UK students.

The main issue is that some of the spelling is different and this site uses US spelling.

Year 3 is generally equivalent to 2nd Grade in the US.

On this page you will find link to our range of math worksheets for Year 3.

Quicklinks to Year 3 ...

• Place Value Zone
• Mental Math Zone

Word Problems Zone

Fractions zone.

• Measurement Zone

Geometry Zone

Data analysis zone.

• Fun Zone: games and puzzles

Coronavirus Stay At Home Support

For those parents who have found themselves unexpectedly at home with the kids and need some emergency activities for them to do, we have started to develop some Maths Grab Packs for kids in the UK.

Each pack consists of at least 10 mixed math worksheets on a variety of topics to help you keep you child occupied and learning.

The idea behind them is that they can be used out-of-the-box for some quick maths activities for your child.

They are completely FREE - take a look!

• Free Maths Grabs Packs

Place Value & Number Sense Zone

Year 3 place value charts.

Here you will find a range of Free Printable Number Charts for Years 2 & 3.

These printable charts will help your child learn to read and write numbers.

Some of the charts are partially filled to help your child learn their place value.

Using these sheets will help your child to:

• learn to count in tens and ones;
• learn to read and write numbers.
• Printable Number Charts 0-99
• Hundred Number Charts (100 Squares)
• Number Grid up to 200
• Number Grid up to 300

Place Value & Counting Worksheets

Below are our selection of Place Value, Ordering and Rounding Worksheets

Using these Year 3 Maths worksheets will help your child to:

• learn to count by hundreds, tens, fives, twos and ones;
• learn their place value to Hundreds;
• order numbers to 999.
• Place Value Models and Representations
• Place Value Worksheets 3 Digits
• Math Practice - Counting by 1s 2s 5s 10s 100s

Ordering and Rounding Worksheets (3-digits)

Once your child is confident with their place value to hundreds, they can start learning about ordering and rounding numbers.

The Year 3 Maths worksheets here will help your child learn to compare numbers to 999, and round numbers to the nearest 10.

• Ordering Numbers Worksheets up to 3-Digits

Year 3 Number Line Worksheets

The following webpages involve correctly placing numbers on a number line.

Using these Year 3 maths worksheets will help your child to:

• count by 1s 2s 5s 10s 50s and 100s;
• position numbers on a number line;
• understand place value up to 3 digits.
• Number lines up to 100 (easier)
• Number Line Worksheets 3 Digits

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Year 3 Mental Maths Zone

Here you will find a range of printable mental math sheets for Year 3 to enjoy.

Each quiz tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math test or math quiz!

• Year 3 Mental Maths Worksheets Series A
• Year 3 Mental Maths Worksheets Series B

Year 3 Addition Worksheets

• learn their addition facts to 20+20
• learn how to solve addition sums where one of the addends is missing, but the total is given;
• learn to add numbers to 999 in columns.
• Number Bonds to 20
• Math Addition Facts to 20
• 2 Digit Column Addition Without Regrouping
• 2 Digit Column Addition Worksheets With Regrouping
• 3 Digit Column Addition Worksheets

Year 3 Subtraction Worksheets

The following worksheets involve using the skills of subtracting numbers, and solving subtraction problems.

• know their subtraction facts to 20;
• know how addition and subtraction are related to each other;
• learn to do 3 digit column subtraction.
• Subtraction Facts to 20 Worksheets
• Two Digit Subtraction Worksheets Without Regrouping
• 2 Digit Subtraction Worksheets With Regrouping
• 3 Digit Subtraction Worksheets

Addition & Subtraction Worksheets

You can create your own addition and subtraction worksheets with our random generator.

All you need to do is to set your number size and let the generator do the rest!

• Addition Subtraction Fact Worksheets
• Column Addition and Subtraction Worksheets

Year 3 Multiplication Worksheets

The following worksheets involve using the Year 3 Maths skills of multiplying, and solving multiplication problems.

• learn their multiplication tables up to 5 x 5;
• understand multiplication as repeated addition;
• learn how multiplication and division are related to each other.

Online Times Table Practice

• Times Tables Online Practice Zone
• Understanding Multiplication Worksheets
• Printable Times Tables - 2 Times Table sheets
• Printable Times Tables - 3 Times Table sheets
• Times Tables for kids - 4 Times Table sheets
• Times Table Math - 5 Times Table sheets
• Learning Times Tables - 8 Times Table

Multiplication Word Problems

• Year 3 Multiplication Word Problems (2nd Grade)

Randomly Generated Multiplication Worksheets

• Times Tables Worksheets (randomly generated)

2 Digit by 1 Digit Written Multiplication

• Multiplication Worksheets for 3rd Grade 2-digits by 1-digit
• Year 3 Division Worksheets

The following worksheets involve using the skills of dividing, and solving division problems.

Using these Year 3 maths worksheets will help your child learn to:

• understand division as repeated subtraction or sharing;
• understand different models of division;
• Division Facts Worksheets (randomly generated)

Here are a range of problems solving sheets for Year 3 children. Most of the sheets contain 'real-life' problems related to animal facts.

Using the sheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

These sheets involve solving one or two more challenging longer problems. This link opens in a new tab.

• Year 3 Maths Challenges

These sheets involve solving many 'real-life' problems involving data.

• Year 3 Math Word Problems

These sheets involve solving a range of addition word problems within 100.

• Addition Word Problems 2nd grade

These sheets involve solving a range of subtraction word problems up to 100.

• Subtraction Word Problems 2nd grade
• 2nd Grade Addition and Subtraction Word Problems

These sheets involve solving a range of multiplication problems.

• Multiplication Word Problems Year 3

Year 3 Fraction Worksheets

• understand what a half is;
• shade in half of a shape;
• find different numbers involving 'halves' on a number line;
• find half of different numbers up to 20.
• Fraction Math Worksheets - Understanding a half
• Fraction Riddles for kids (easier)

Year 3 Geometry Worksheets

At a Year 3 level, children are introduced to a wider range of 2d and 3d shapes, and start to learn to reflect, flip and rotate shapes.

They are able to classify and sort two-dimensional shapes using different criteria, and they learn the properties of three-dimensional shapes.

The following worksheets will help your child to:

• Identify and name a range of 2d and 3d shapes;
• Draw 2d shapes;
• Know some of the properties of 2d shapes.
• Year 3 Geometry Worksheets (2nd Grade)

Symmetry Worksheets

• learn how to reflect simple shapes in a horizontal or vertical mirror line;
• learn how to reflect simple shapes in 2 mirror lines.
• Symmetry Worksheets - Block Symmetry

Measurement Zone, including Time & Money

Year 3 measurement worksheets.

These sheets involve reading scales going up in ones, fives, tens and hundreds to find the length, weight or liquid capacity.

Using these sheets will help children to consolidate their counting as well as learning to read a simple scale.

• Year 3 Measurement Worksheets (2nd Grade)

Counting Money Worksheets UK - £2, £1, 50p and 20p coins

Here you will find a selection of Counting Money sheets designed to help your child count different amounts of money using UK coins up to £2.

• count a range of coins up to £5
• compare money amounts
• apply their existing skills to puzzle out clues;
• understand money terminology;
• develop their thinking skills.

The free Year 3 Money worksheets in this section are suitable for children in Years 2 and 3.

• Free Money Worksheets UK Coins up to £1
• Year 3 Money Worksheets to £5
• Money Riddles for Kids (UK coins)
• Year 3 Money Challenges

Time Worksheets

Here are our collection of clock worksheets for Year 3. These sheets cover telling the time for quarter past and quarter to times.

• read times to the nearest 5 minutes;
• convert analogue to digital times;
• adding and subtracting hours from different times.
• Clock Worksheets - Quarter Past and Quarter To
• Telling Time to 5 Minutes Worksheets
• Add and Subtract Time Worksheets

Year 3 Bar Graph Worksheets

These sheets involve reading and interpreting a range of bar graphs and picture graphs with a scale going up in ones or twos.

Draw missing bars on a bar graph to the correct height.

Use the information in a bar graph to add data into a table.

Using the link below will open the 2nd grade Math Salamanders website in a new browser window.

• Year 3 Bar Graphs (2nd Grade)

Fun Zone: Puzzles, Games and Riddles

Year 3 math games.

All children like to play Math games, and you will find a good range of 2nd Grade Math Games here for your child to play and enjoy.

The following games involve different Second Grade Math activities which you and your child can enjoy together.

• Year 3 Math Games (2nd Grade)
• Year 3 Maths Puzzles

The puzzles will help your child practice and apply their addition, subtraction and multiplication facts as well as developing their thinking and reasoning skills in a fun and engaging way.

The Number Square Puzzles page will open a new browser window which will take you to the 2nd grade math salamanders website.

Using these puzzles will help your child to:

• learn their addition facts to 20+20;
• learn and practice subtraction facts to 20;
• pratice multiplying up to 5x5;
• develop thinking and reasoning skills;
• develop perseverance.
• Number Square Puzzles
• Math Puzzles at 2nd Grade Math Salamanders

Math Salamanders Year 3 Maths Games Ebook

Our Year 3 Maths Games Ebook contains all of our fun maths games, complete with instructions and resources.

This ebooklet is available in our store, use the link below to find out more!

• Year 3 Maths Games Ebook

Help & Support

Looking for a resource which is not here.

Take a look at our 2nd Grade Math Salamanders website.

On this site you will find a wider selection of Year 3 maths worksheets and free math resources.

• 2nd Grade Math Salamanders

Other UK Maths Worksheet pages

See below for our other maths worksheets hub pages designed for children in the UK.

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Solving Problems with Time: Stretch (Year 3)

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This resource is suitable for the White Rose Maths Scheme of Learning, covering Summer Term Block 3 (Time), Step 12 (Solve Problems with Time).

Children can practise solving problems with time with this worksheet. The word problems test their ability to convert between minutes and seconds. They can also try comparing lengths of time and calculating start and end times using durations.

Children can mark their own work using the answers provided.

This worksheet is available in different levels. Please see the drop-down menu to select another level.

This resource is also available in a handy Lesson Pack, containing a selection of brilliant worksheets for this small step.

View Lesson Pack    See all Year 3 White Rose

• Key Stage: Key Stage 2
• Subject: Maths
• Topic: White Rose Maths | Summer Term: Time | Small Step 12
• Topic Group: Measurement
• Year(s): Year 3
• Media Type: PDF
• Resource Type: Worksheet
• Last Updated: 09/04/2024
• Resource Code: M2WHT16988
• Curriculum Point(s): Tell and write the time from an analogue clock, including using Roman numerals from I to XII, and 12-hour and 24-hour clocks. Know the number of seconds in a minute and the number of days in each month, year and leap year. Compare durations of events [for example to calculate the time taken by particular events or tasks]. Estimate and read time with increasing accuracy to the nearest minute. Record and compare time in terms of seconds, minutes and hours. Use vocabulary such as o'clock, a.m./p.m., morning, afternoon, noon and midnight.

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3.3: Solving Systems with Gauss-Jordan Elimination

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Learning Objectives

• Write the augmented matrix of a system of equations.
• Write the system of equations from an augmented matrix.
• Solve a system of linear equations using matrices and a graphing calculator.
• Solve financial applications using matrices and a graphing calculator.

Prerequisite Skills

Before you get started, take this prerequisite quiz.

Enter the following matrices in your calculator and then perform the indicated operations.  If the operation cannot be done, give a reason.

\(A=\begin{bmatrix} 5 & 1 & -2\\2 & 6 & 7\\4 & 1 & −5 \end{bmatrix} \), \(B=\begin{bmatrix} 3 & -7\\0 & 1\\2 & −8 \end{bmatrix} \), \(C=\begin{bmatrix} 9 & 4\\6 & -5\\7 & −1 \end{bmatrix} \)

a. \(A \cdot B\)

b. \(B \cdot A\)

c. \(4B-2C\)

a. \(\begin{bmatrix} 11 & -18\\20 & -64\\2 & 13 \end{bmatrix} \)

b. Undefined since the number of columns in matrix \(B\) do not match the number of rows in matrix \(A\).

c. \(\begin{bmatrix} -6 & -36\\-12 & 14\\-6 & −30 \end{bmatrix} \)

d. Undefined since the dimension of matrix \(A\) do not match the dimensions of matrix \(C\).

If you missed any part of this problem, review Section 3.2 . (Note that this will open in a new window.)

Carl Friedrich Gauss lived during the late \(18^{th}\) century and early \(19^{th}\) century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.

Earlier in this chapter we explored methods of solving systems of equations. In this section, we will learn another technique for solving systems, this time using matrices.

Augmented Matrices

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix .

For example, consider the following \(2 × 2\) system of equations.

\[\begin{align*} 3x+4y&= 7\\ 4x-2y&= 5 \end{align*}\]

We can write this system as an augmented matrix:

\(\left[ \begin{array}{cc|c} 3&4&7\\4&-2&5\end{array} \right]\)

We can also write a matrix containing just the coefficients. This is called the coefficient matrix.

\(\begin{bmatrix}3&4\\4&−2\end{bmatrix}\)

A three-by-three system of equations such as

\[\begin{align*} 3x-y-z&= 0\\ x+y&= 5\\ 2x-3z&= 2 \end{align*}\]

has a coefficient matrix

\(\begin{bmatrix}3&−1&−1\\1&1&0\\2&0&−3\end{bmatrix}\)

and is represented by the augmented matrix

\(\left[ \begin{array}{ccc|c}3&−1&−1&0\\1&1&0&5\\2&0&−3&2\end{array} \right]\)

Notice that the matrix is written so that the variables line up in their own columns: \(x\)-terms go in the first column, \(y\)-terms in the second column, and \(z\)-terms in the third column. It is very important that each equation is written in standard form \(ax+by+cz=d\) so that the variables line up. When there is a missing variable term in an equation, the coefficient is \(0\).

How to: Given a system of equations, write an augmented matrix

• Write the coefficients of the \(x\)-terms as the numbers down the first column.
• Write the coefficients of the \(y\)-terms as the numbers down the second column.
• If there are \(z\)-terms, write the coefficients as the numbers down the third column.
• Draw a vertical line and write the constants to the right of the line.

Example \(\PageIndex{1}\): Writing the Augmented Matrix for a System of Equations

Write the augmented matrix for the given system of equations.

\[\begin{align*} x+2y-z&= 3\\ 2x-y+2z&= 6\\ x-3y+3z&= 4 \end{align*}\]

The augmented matrix displays the coefficients of the variables, and an additional column for the constants.

\(\left[ \begin{array}{ccc|c}1&2&−1&3\\2&−1&2&6\\1&−3&3&4\end{array} \right]\)

Exercise \(\PageIndex{1}\)

Write the augmented matrix of the given system of equations.

\[\begin{align*} 4x-3y&= 11\\ 3x+2y&= 4 \end{align*}\]

\(\left[ \begin{array}{cc|c} 4&−3&11\\3&2&4\end{array} \right]\)

Writing a System of Equations from an Augmented Matrix

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

Example \(\PageIndex{2}\): Writing a System of Equations from an Augmented Matrix Form

Find the system of equations from the augmented matrix.

\(\left[ \begin{array}{ccc|c}1&−3&−5&-2\\2&−5&−4&5\\−3&5&4&6 \end{array} \right]\)

When the columns represent the variables \(x\), \(y\), and \(z\),

\[\left[ \begin{array}{ccc|c}1&-3&-5&-2\\2&-5&-4&5\\-3&5&4&6 \end{array} \right] \rightarrow \begin{align*} x-3y-5z&= -2\\ 2x-5y-4z&= 5\\ -3x+5y+4z&= 6 \end{align*}\]

Exercise \(\PageIndex{2}\)

Write the system of equations from the augmented matrix.

\(\left[ \begin{array}{ccc|c}1&−1& 1&5\\2&−1&3&1\\0&1&1&-9\end{array}\right]\)

\(\begin{align*} x-y+z&= 5\\ 2x-y+3z&= 1\\ y+z&= -9 \end{align*}\)

Reduced Row-Echelon Form

In order to solve a system of equations, we want to convert its matrix to reduced row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position above and below the main diagonal as shown.

Reduced row-echelon form \(\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

The following augmented matrices are in reduced row-echelon form.

\(\left[ \begin{array}{cc|c}1 & 0 & -2 \\ 0 & 1 & 5\end{array} \right]\), \(\left[ \begin{array}{ccc|c}1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 2\end{array} \right]\)

The following augmented matrices are not in reduced row-echelon form.

\(\left[ \begin{array}{cc|c}2 & 4 & -6 \\ 4 & 0 & 7\end{array} \right]\), \(\left[ \begin{array}{ccc|c}0 & 2 & 3 & 3 \\ 1 & 5 & 0 & 2 \\ 0 & 0 & 1 & 0\end{array} \right]\)

Example \(\PageIndex{3}\): Matrices in Reduced Row-Echelon Form

Write the system of equations from each of the matrices in reduced row-echelon form from above.  What is the advantage of this form?

a.  \(\left[ \begin{array}{cc|c}1 & 0 & -2 \\ 0 & 1 & 5\end{array} \right]\)

b.  \(\left[ \begin{array}{ccc|c}1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 2\end{array} \right]\)

a. \(\begin{align*} x=-2\\ y=5 \end{align*}\)

b.  \(\begin{align*} x=4\\ y=3\\z=2 \end{align*}\)

The advantage of reduced row-echelon form is that the solution to the system of equations is given in the right column.

GAUSS-JORDAN ELIMINATION

The Gauss-Jordan elimination method refers to a strategy used to obtain the reduced row-echelon form of a matrix. The goal is to write matrix \(A\) with the number \(1\) as the entry down the main diagonal and have all zeros above and below.

\(A=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\xrightarrow{After\space Gauss-Jordan\space elimination} A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)

We can perform row operations on a matrix, such as addition, multiplication by a constant, and interchanging rows, to create reduced row-echelon form.  The process of performing these steps by hand is beyond the scope of this class.  However, you are welcome and encouraged to find more information about the Gauss-Jordan Method HERE .

Solving Systems of Equations with Gauss-Jordan Elimination

For the purposes of this course, we'll demonstrate how to find reduced row-echelon form in the graphing calculator.

How to: Given a system of equations, solve with matrices using a calculator

Press 2 nd MATRIX. The screen will display the Matrix menu. Use the right arrow key twice to select the EDIT menu. From the EDIT menu, use the down arrow to move the cursor to select the matrix name desired from the menu, and press ENTER. The matrix input screen will appear.

Enter the dimensions of the total matrix size as rows \(\times\) columns. Type the number of rows, press ENTER, type the number of columns, and press ENTER again. The shape of the matrix adjusts on the screen to show the requested number of rows and columns.  Check that the shape matches the desired matrix; if it does not, then return to the top row and adjust the dimensions. If the matrix is too large to fit the screen, use the arrow keys to scroll right or down to see the remaining rows and columns.

Input the matrix entries, press ENTER after each. The cursor scrolls through the matrix by moving across each row from left to right and then down to the next row. Using the arrow keys to move the cursor instead of pressing ENTER may result in a value not being stored in the calculator memory.

Press  2 nd  QUIT to complete the saving process and return to the home screen.

From the home screen, press 2 nd MATRIX. Use the right arrow once to go to the MATH menu.

Scroll down (or up) to rref(, being careful not to select ref(, and press ENTER.

Press 2 nd MATRIX again and use the down arrow (if necessary) to select the name of the matrix and press ENTER. 

Press ENTER to complete the operation.

• If the reduced row-echelon form of the matrix exists, the calculator will display it on the home screen. ×

Example \(\PageIndex{4}\): Solving Systems of Equations with Matrices Using a Calculator

Solve the system of equations.

\[\begin{align*} 6x+4y+3z&= -6\\ x+2y+z&=\dfrac{1}{3}\\ -12x-10y-7z&= 11 \end{align*}\]

Write the augmented matrix for the system of equations.

\(\left[ \begin{array}{ccc|c} 6&4&3&-6\\1&2&1&\dfrac{1}{3}\\-12&-10&-7&11\end{array} \right]\)

On the matrix page of the calculator, enter the augmented matrix above as the matrix variable \([A]\).

\([A]=\left[ \begin{array}{ccc|c} 6&4&3&-6\\1&2&1&\dfrac{1}{3}\\-12&-10&-7&11\end{array} \right]\)

Use the rref( function in the calculator, calling up the matrix variable \([A]\).

Use the MATH --> FRAC option in the calculator to express the matrix elements as fractions.

\[\begin{array}{cc} {\left[ \begin{array}{ccc|c} 1&0&0&-\dfrac{2}{3}\\0&1&0&\dfrac{5}{2}\\0&0&1&-4\end{array} \right] \rightarrow} & {\begin{align*} x+0y+0z &= -\dfrac{2}{3} \\ y+0z &= \dfrac{5}{2} \\ z &= -4 \end{align*}} \end{array}\]

Thus the solution, which can easily be read from the right column of the reduced row-echelon form of the matrix, is \(\left(-\dfrac{2}{3},\dfrac{5}{2},−4\right)\).

Exercise \(\PageIndex{3}\)

\[\begin{align*} 4x-7y+2z&= -5\\ -x+3y-8z&= -10\\ -5x-4y+6z&= 19 \end{align*}\]

\(\left[ \begin{array}{ccc|c} 4&-7&2&-5 \\ -1&3&-8&-10 \\ -5&-4&6&19 \end{array} \right]\)

\([A]=\left[ \begin{array}{ccc|c} 4&-7&2&-5 \\ -1&3&-8&-10 \\ -5&-4&6&19\end{array} \right]\)

\[\begin{array}{cc} {\left[ \begin{array}{ccc|c} 1&0&0&-2\\0&1&0&0\\0&0&1&\dfrac{3}{2}\end{array} \right] \rightarrow} & {\begin{align*} x+0y+0z &= -2 \\ y+0z &= 0 \\ z &= \dfrac{3}{2} \end{align*}} \end{array}\]

Thus the solution, which can easily be read from the right column of the reduced row-echelon form of the matrix, is \(\left(-2, 0,\dfrac{3}{2}\right)\).

Example \(\PageIndex{5}\): Applying \(2×2\) Matrices to Finance

Carolyn invests a total of \($12,000\) in two municipal bonds, one paying 10.5% interest and the other paying 12% interest. The annual interest earned on the two investments last year was \($1,335\). How much was invested at each rate?

We have a system of two equations in two variables. Let \(x=\) the amount invested at 10.5% interest, and \(y=\) the amount invested at 12% interest.

\[\begin{align*} x+y&= 12,000\\ 0.105x+0.12y&= 1,335 \end{align*}\]

As a matrix, we have

\(\left[ \begin{array}{cc|c} 1&1&12,000\\0.105&0.12&1,335\end{array} \right]\)

Enter this matrix as matrix variable \([A]\).  Use the  rref(  function, calling up the matrix variable \([A]\).

\(\left[ \begin{array}{cc|c} 1&0&7000\\0&1&5000\end{array} \right]\)

Thus, \($7,000\) was invested at 10.5% interest and \($5,000\) at 12% interest.

Example \(\PageIndex{6}\): Applying \(3×3\) Matrices to Finance

Ava invests a total of \($10,000\) in three accounts, one paying 5% interest, another paying 8% interest, and the third paying 9% interest. The annual interest earned on the three investments last year was \($770\). The amount invested at 9% was twice the amount invested at 5%. How much was invested at each rate?

We have a system of three equations in three variables. Let \(x\) be the amount invested at 5% interest, let \(y\) be the amount invested at 8% interest, and let \(z\) be the amount invested at 9% interest. Thus,

\[\begin{align*} x+y+z &= 10,000 \\ 0.05x+0.08y+0.09z &= 770 \\ 2x−z &= 0 \end{align*}\]

\(\left[ \begin{array}{ccc|c} 1&1&1&10,000\\0.05&0.08&0.09&770\\2&0&-1&0\end{array} \right]\)

\(\left[ \begin{array}{ccc|c} 1&0&0&3000\\0&1&0&1000\\0&0&1&6000\end{array} \right]\)

The answer is \($3,000\) invested at 5% interest, \($1,000\) invested at 8%, and \($6,000\) invested at 9% interest.

Exercise \(\PageIndex{4}\)

A small shoe company took out a loan of \($1,500,000\) to expand their inventory. Part of the money was borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount borrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three loans was \($130,500\). Use matrices to find the amount borrowed at each rate.

\($150,000\) at 7%, \($750,000\) at 8%, \($600,000\) at 10%

Access these online resources for additional instruction and practice with solving systems of linear equations using Gaussian elimination.

• Solve a System of Two Equations Using an Augmented Matrix
• Solve a System of Three Equations Using an Augmented Matrix
• Augmented Matrices on the Calculator

Key Concepts

• An augmented matrix is one that contains the coefficients and constants of a system of equations. See Example \(\PageIndex{1}\).
• A matrix augmented with the constant column can be represented as the original system of equations. See Example \(\PageIndex{2}\).
• We can use Gauss-Jordan elimination to solve a system of equations. See  Example \(\PageIndex{4}\).
• Many real-world problems can be solved using augmented matrices. See Example \(\PageIndex{5}\) and Example \(\PageIndex{6}\).

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a  Creative Commons Attribution License 4.0  license. Download for free at  https://openstax.org/details/books/precalculus .

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Grid Method for Multiplication

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

28 March 2018

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UPDATE: I’m so grateful for the positive feedback. Thank you to you all.

Here you will find a PowerPoint to introduce (or remind pupils) the Grid Method of Multiplication.

The overriding Lesson Objective is for pupils to be able to multiply two 2digit numbers, however the worksheet creator (which automatically changes the numbers and has answers) will challenge high flyers with multiplying two 3digit numbers.

Any constructive comments would be greatly received.

Created to use with a low ability Year 6 cohort, these resources should be apt for Y4 - Y6.

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louisemaylorpridmore

Fantastic resource! Thank you.

It's not often I find EXACTLY what I need on Tes... but this time I have! Thanks!

Thanks, MollyARC. Glad it's useful.

Exactly what I was looking for, for a lockdown lesson video.Thank you so much

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