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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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I am delighted to share some work that I have been doing in conjunction with Siobhan McKenna to build a new curriculum. This curriculum is being used in 2 Scottish secondary mathematics departments. It is a loose interpretation of Third Level, but straddles Second and Fourth level where appropriate. For colleagues in England this is links up with KS3.

* Disclaimer*: We do not own the copyright for some of the material in the curriculum booklets. We have created a significant amount of the material ourselves, but have borrowed from a vast array of places. We must acknowledge: Don Steward, Tony Gardiner, Craig Barton, John Mason, Malcolm Swan, Jo Morgan, Dave Taylor, Richard Perring, M.R. Heylings, SMILE, SMP and many others who have shared ideas which we have been able to utilise in this compendiums of tasks.

We will upload material here as it is developed. Part of the rationale for sharing is to improve the quality of the material. If you think you have a task which would be good to include or if you spot any errors please let either of us know via Twitter.
Phase 1 - Mixed Attainment Block
Used with pupils on initial arrival at Secondary school. This is a mixed attainment booklet which runs for a few weeks. We did the quizzes at the back on an ongoing basis to help determine the appropriate working level for pupils.

Area and Length (basics)

Order of Operations

Basic Fraction Work

Diagnostic Quizzes on basic number

Download Booklet Here Powerpoint for Day One Activites Phase 2 Phase 3

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I am delighted to share some work that I have been doing in conjunction with Siobhan McKenna to build a new curriculum. This curriculum is being used in 2 Scottish secondary mathematics departments. It is a loose interpretation of Third Level, but straddles Second and Fourth level where appropriate.

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In this activity we'll see an example of how to create a dynamic Systems of Equations activity in the Desmos Activity Builder. This activity uses the following components: 1. Graphs 2. Tables 3. Notes 4. Computational Layer

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Elizabeth and Lindsay's final project for VMI Systems for All.

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Desmos student activity with function tables

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Desmos student activity with function tables

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includes input output machines

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[Copy of] Katie Jarvis and Heidi Whipple's Linear Systems: Gym Membership

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In this activity, students analyze several gym membership plans in order to make a recommendation to a friend. The activity assumes some familiarity with linear equations, particularly how slope and intercept appear in equations and how they affect the graph. If your students need a quick refresher on these concepts, consider offering a brief review prior to starting this activity.

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Fast drawing for everyone. AutoDraw pairs machine learning with drawings from talented artists to help you draw stuff fast.

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7th Grade Math Games | Seventh Grade Math Skills | ST Math

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Free 7th grade math games from the ST Math curriculum. Play to explore concepts like expressions, multi-step percents, two-step equations, and more.

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Due to popular demand, we've created gameboards of Number Hive to be played offline! And even better yet, we've added a variety of number grids to test student skillsets at different levels.

These sample packs are perfect to play in class or at home. Perfect for the BTC classroom or collaborative learning. Either use with counters like in the picture, or laminate and use markers. Play with 2,3 or 4 players.

We recommend you start with these and then encourage students to build fluency on their own using the app. Or, if you prefer them to play off line, we now have Number Hive Game Board small group or class packs available to purchase here.

These pintables are a great way to engage students anywhere!

Download these free mathematics resources and you'll get access to a sample game board of:

Multiplication: 1-12

Multiplication: 1-10

Multiplication: 1,2,3,4,5 & 10

Multiplication: Simple factors

Multiplication: Negative and Positive Integers

Multiplication: Decimals

Multiplication: Fractions

Multiplication: Algebra

The Hive Maze

Addition: 1-12

Addition: Negative and Positive Integers

Addition: Dice

Print them out to use them in your classroom. They're a great way to master multiplication facts and engage your students in deep thinking about number sense.

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Number Hive Printouts | Mathematics Classroom Resources

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Download the Number Hive printouts and build mathematics mastery through meaningful play in the classroom or at home. Play the multiplication or addition math(s) version of Number Hive, and also get Grids, a tournament poster, and multiplication chart.

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Those you share this with will see the current array and settings.

Make a new rectangle

Change the size of the rectangle by entering whole numbers (up to 50) in the boxes. Then press the MAKE button.

Add and move the sliders

Drag the slider handles from the left or top to divide the rectange into smaller parts. Include up to 4 sliders on each axis.

View partial products

Watch how the equation changes depending on the position of the sliders. Point at any region to see the corresponding piece of the equation.

Show or hide answers

Toggle these buttons to show or hide the results of the equation.

Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination. The sides of the rectangle can then be divided into parts to reveal partial products that make up the total product. This helps students develop the flexibility and conceptual understanding required to build fluency with basic facts and strategies for multiplication of larger numbers.

We Welcome Your Feedback

Help us guide the ongoing development of this app by sharing your reactions or suggestions.

Partial Product Finder is brought to you by The Math Learning Center, a nonprofit organization serving the education community since 1976. Visit us at www.mathlearningcenter.org, where you can learn more about our organization, the Bridges in Mathematics PK-5 curriculum we publish, and the many free math education resources we provide, including a robust suite of free virtual manipulatives for use online or as downloadable apps for a variety of operating systems, including Apple iOS and Google Chrome.

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Partial Product, by The Math Learning Center

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Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination.

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Erica Moy's best resources for middle school

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Erica Moy's best resources for middle school

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Printable puzzles. Thousands of 'em. All free.

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These puzzle worksheets will have your 2nd grade, 3rd grade, or 4th grade students navigating through math operations of various difficulty to find their way to the end of a chain of math problems. The puzzles have varying degrees of difficulty and include problems that focus solely on addition, subtraction, multiplication or division, as well as a mix of the various math operations.

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These puzzle worksheets will have your 2nd grade, 3rd grade, or 4th grade students navigating through math operations of various difficulty to find their way to the end of a chain of math problems. The puzzles have varying degrees of difficulty and include problems that focus solely on addition, subtraction, multiplication or division, as well as a mix of the various math operations.

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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After a lot of thinking, I settled on compiling real images into Google Slides to make each digital math word wall feel as much like a real classroom as possible.

Each image from the home wall is clickable so that students can zoom in on the references they need. Teachers are sending the digital versions to students and even projecting them onto their white boards. They've proven to be handy on-the-go math vocabulary references.

If you are looking for written definitions of the vocabulary words, they won't be here. I have found math word walls showing concepts in action, instead of explicit written definitions, allow more students to access the math vocabulary. With 15-20% of our kids having reading difficulties ( and up to 80% in students with specific learning disabilities), showing the concepts in context allows more students to make the connections that are critical for conceptual understanding.

That last reason is probably the most important impact I have seen since adding a word wall to our classroom. The supports are there for kids who need them, allowing more students to feel that maybe math is for them after all.

Every math word wall comes in printable color, black & white and interactive digital in Google Slides. The Slides versions can be downloaded as PowerPoint files if you are not in a Google school.

Each word wall includes printable color, printable black & white and no-prep digital in Google Slides.

"This product keeps getting better and better! I colored a lot of them last summer, then the color versions came out! and now the digital versions! This resource has been incredibly helpful to have up and reference and add to during the year. I worked with students who struggled and students who accelerated, this was a great resource for all of them!" -Chelsie S

I've written a lot about math word walls on my blog over the years. The post 5 Ways Math Word Walls Have Changed My Teaching highlights the 5 major benefits I have seen in my teaching and my students' learning since adding math word walls to our classroom walls:

◉ Greater student independence
◉ Keeping the class on track
◉ They make a classroom inviting
◉ Connections to previous topics
◉ Low floor, high ceiling

Seeking out and finding answers to their questions on their own by using our classroom word walls is empowering. This is one of my favorite things about math word walls.

Even with lots of encouraging that their voices matter, some of my students wouldn't speak up when they had a question. Adding visual supports to our classroom walls has given all of my students, and maybe especially these students, the tools to help them take charge of their education.

I know that when a student is confused her learning stops. That student won't hear anything else I say until her question is answered. This is why I would always stop to answer student questions. With visual aids on the walls, I can now point at a reference and get that student over his hurdle a lot faster.

Whenever a student was confused, I used to stop class to catch up that student. This, of course, gave a sense of free time to the rest of the class, and reeling everyone back in became time-consuming and pretty frustrating.

Students can find the information they need more independently, which leads to greater feelings of confidence and success. To me, confidence in math is everything. In my own classroom, I can focus on the students who need more intense help while directing others to use the walls for help.

With pictures, examples and reminders, math is more accessible to more students.

I have found that presenting math vocabulary in this way increases math confidence and lowers math anxiety in students who struggle with reading, while still communicating the vocabulary and concepts we are learning in class.

My students reference our walls during independent work and I reference them during lessons, especially when needing to make a connection back to a previous topic or moving a student past confusion.

In my experience as a general education, inclusion and special education teacher, an effective word wall shows examples and pictures. If we are learning about shifting a quadratic function in vertex form, for example, I can quickly point to our wall to remind students which way the vertex will shift. This has supported my classroom management by helping me keep my students on track. To make each math word wall, I think hard about how best to remove all unnecessary words and make the visuals as clear and descriptive as possible.

This gives more students access to the math, especially our visual learners, students with reading disabilities and English Language Learners.

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fun math teaching ideas and activities for middle school and high school

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math pennants and free reference sheets.

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After a lot of thinking, I settled on compiling real images into Google Slides to make each digital math word wall feel as much like a real classroom as possible. Each image from the home wall is clickable so that students can zoom in on the references they need.

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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TPT School Access is a school-wide subscription from TPT that empowers educators with essential teacher-tested resources and interactive learning tools.

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consumer math curriculum  $100 investment

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consumer math curriculum  $100 investment

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This new personal finance curriculum is an approachable guide to building financial literacy for teens. It includes everything you need -- a teacher's book, student book, student notes, quizzes and more -- to make the course easy to teach for you and enjoyable to learn for your students.

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Lesson 14 | Linear Relationships | 8th Grade Mathematics | Free Lesson Plan

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Compare linear functions represented in different ways.

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Math Flips on google slides all 4 operations

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Math Flips on google slides all 4 operations

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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In Ontario our grade 7 students are introduced to solving simple equations in the form ax + b = c where the values of a, b and c are whole numbers. We think it's a good idea for them to start by having some sort of visual representation of each equation. In this activity, students are given 16 cards that correspond to 16 equations represented as strips (the top and bottom of the strips represent the left and right sides of the equations). Students solve for x given the strips and then rewrite the algebraic form equation.
[ Updated Mar 6, 2018 - now both the printable cards and the dynamic websketch have equations in the form 2x - 5 = 19. This puts it beyond the grade 7 expectation but could be an extension or just used for grade 8]

Grade 7 Patterning & Algebra - solve linear equations of the form ax = c or c = ax and ax + b = c or variations such as b + ax = c and c = bx + a (where a, b, and c are natural numbers) by modelling with concrete materials, by inspection, or by guess and check, with and without the aid of a calculator

Grade 8 Patterning & Algebra - as review

Each group gets a set of 16 cards (24 if you use the cards with minuses)

Make several copies of the cards on card stock and laminate them so they last longer. You may wish to copy each set onto a different colour so that if they get mixed up you know each set by their colour.

Cut out the cards so that each group gets a set of 16.

1. Each group of 2-3 students gets one full set of 16 cards.

2. Students are to determine the value of x for each card.

3. Once determining x then they should then determine the algebraic expression for each card

4. You can circulate with the solution card to check answers.

5. Once finished you can create your own cards using this web sketch. This allows you to change the coefficients of a, b & c and it generates all four possible configurations. This web sketch assumes that a, b & c will be whole numbers and will not allow any solutions that have x as negative. Once you put your coefficients in then take a screenshot, use the screen capture software of your choice to copy and paste the version you want to use (For Windows use the Snipping Tool, for Chromebooks use Shift CTRL F5, for Macs use Command Shift 4, or iPad use the Home and Sleep buttons together. You can then paste into the word processor of your choice.

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks

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In Ontario our grade 7 students are introduced to solving simple equations in the form ax + b = c where the values of a, b and c are whole numbers. We think it's a good idea for them to start by having some sort of visual representation of each equation.

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Going Up? | Adding and Subtracting Integers 3 Act Math Tasks & Practice

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Show students the following video: Then, ask students: Give students 60 seconds (or more) to do a rapid write on a piece of paper. Then, ask students to share with their neighbours for another 60 seconds. Finally, allow students to share with the entire group.

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This Make Math Moments Lesson was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, number talks and extensions to elicit and emerge strategies and mathematical models.

Dig into our other units of study and view by concept continuum, grade or topic!

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Percent lesson that builds fluency using various strategies in a curious Problem Based Math Lesson.

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Browse through over 60 complete K-12 math units of study chalked full of problem based investigations, 3 act math tasks, visual number talks, consolidation prompts and purposeful practice worksheets to Spark Curiosity & Fuel Sensemaking in every student.

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Real World Math Lessons | 3 Act Math Tasks | Math Worksheets & Practice

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Contextual problem based real world math lessons with rich math tasks, multi-day 3 act math tasks, math practice materials and math worksheets

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Shovelling The Driveway | Multiplying Fractions | Problem Based Math Unit

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Show students the following video: Then, ask students: Give students 60 seconds (or more) to do a rapid write on a piece of paper. Replaying the video and/or leaving a screenshot from the video up can be helpful here. Then, ask students to share with their neighbours for another 60 seconds.

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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Keeping track of math facts and introducing strategies

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Keeping track of math facts and introducing strategies

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Procedural fluency is described as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately". (Adding It Up: Helping Children Learn Mathematics, 2001)

It is not simply memorizing answers! The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense. (Boaler, 2015)

Delta Math visual fluency cards were inspired by Graham Fletcher's Multiplication Subitizing Cards and are available thanks to the leadership and partnership of the Berrien RESA.

From Memory and Memorization: There IS a Difference

This ignite talk by Graham Fletcher provides a great opportunity for educators to discuss pathways to basic fact fluency

Distributed/spaced practice after students have explored the targeted fluency strategy using manipulatives and drawings

Approximately 5 to 15 minutes per session

Math talk samples help students communicate fluency strategies

The facts provided in each deck were limited to the facts students generally find difficult.

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Procedural fluency is described as "skill in carrying out procedures flexibly, accurately, efficiently, and appropriately". (Adding It Up: Helping Children Learn Mathematics, 2001) It is not simply memorizing answers! The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense.

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

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Notes with videos from Mrs. Jarvis for how we teach and learn math here at BTMES

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Notes with videos from Mrs. Jarvis for how we teach and learn math here at BTMES

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The Coherence Map shows the connections between Common Core State Standards for Mathematics.

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Cuisenaire-Rods
Introduction, background, and activities to try to familiarize yourself with cuisenaire rods.
Simon Gregg: How I Teach Using Cuisenaire Rods (3-5) Big Ideas in Arithmetic (k-5)
How to use:

Learn more about the power of Cuisenaire Rods: Video Book Simon Gregg Buy Rods

Watch and ask: What do you notice? What do you wonder?

Eventually, students will notice that the color of the rod represents a quantity. They will notice that the orange rod represents 10, and we can see the groups of ten in each number. They will see the pattern of the ones repeating over and over and the groups of ten increasing. That will lead to the idea that the written digits represent place value: orange for the tens place, and the other colors for the ones place.

What do you notice(K-5)

Use with sticky notes or fill in the spots below. You can recreate this document on an anchor chart as well.

Copy the picture above for the staircases that you can copy for students
Make Your Own Meter Stick
Cuisenaire Environment/Picture Virtual Manipulative. Students can add the values, make equations, make fractions, etc. The page gives suggestions. (K-5)

IHaveWhoHasCuisenaireRods(K-5)
I have Who Has cuisenaire directions
Free Play Grab Bag(K-5) Lesson 1 out of 3 for an introduction

Counting the staircase, making yellows(K-5)

Making Colored Rods-Recording Sheet for the above lesson; Lesson 2 out of 3 for an introduction

Cuisenaire Rod Fill-ins Lesson 3 out of 3 for an introduction

Use this to notice patterns when adding and subtracting or even and odd.

Reaching for Rods (K-5)

cuisenaire-rod-blog-pp(K-5) This gives an introduction to Cuisenaire Rods and mentions several activities and literature connections.

Math and Literature June 17 2014 (K-2) A resource that gives lists of several math books and activities you can do with them using any kind of manipulative, including Cuisenaire Rods.

Cuisenaire Cards Card games (K-2) Print out 4 of these pages to make a deck of cards. If you plan on having more people involved in games, you can increase that number.

Snakey Rods(K-2) Game

Chutes and Ladders(K-2) Game

Little Pig Centers(K-2) Help Little Pig Build a House

Cuisenaire Counting (k-1) Composing and decomposing numbers; fact families; related facts; three addends

PDL-Number-Building-Playmats-for-Cuisenaire-Rods(K-1) Composing and decomposing numbers. The second page gives great suggestions for ways to extend the activity.

Train for Two (1-3) Addition and subtraction

Counting the staircase, making yellows(K-5)

Making Colored Rods-Recording Sheet for the above lesson
Using the picture book, "Grapes of Math" by Greg Tang
Click the picture above for the lesson "What's the Difference" (1-3)

Subtraction strategies

Which Line is Longer Comparing; repeated addition; subtraction. Can be whole group and then in a center.

Challenge Match Challenge Match Task Card (K-2) Composing and decomposing numbers to 10; unknown addends; addition and subtraction

cuisenaire capers (1-5) Adding up to 100; properties of operations; place value; writing expressions; multiplication; distributive property

Triangle Magic (1-5) Three addends that equal ten. The logic and reasoning needed makes is suitable for intermediate grades.

color match doubles addition(1-5) Adding doubles. The logic and reasoning needed makes is suitable for intermediate grades.

sums to ten (K-2) Sums to 10.

Number of the Day Cuisenaire Rods(K-2) Even/Odd, adding/subtracting, greater than/less than

odds and evens(1-2) Addition

Shorter or Longer cuisenaire spinner(K-2) Measurement, comparing problems (addition/subtraction), greater than, less than

Same Difference(K-2) Noticing patterns with differences and addition and subtraction strategies.

One more, one less, two more, two less (K-Beginning 2)

multiplication task card (3-5) Prime/composite or multiplication/division

Same Length Trains (k-3) Virtual

same length trains(2-3) Can use repeated addition or multiplication
Cuisenaire Squared Three Act Task (3-5)
Cuisenaire Squared Three Act Task Cuisenaire Squared Three Act Task
Using the picture book, "Grapes of Math" by Greg Tang
Cuisenaire Rod Squares (3-5) Students will notice patterns with square numbers.

Squares and Rectangles (3-5) Patterns with area and perimeter.

Which Line is Longer(2-3) Commutative property, symmetry, patterns, multiplication, repeated addition

Cuisenaire Rod Equations(3-5) Order of Operations, solving equations with unknowns

cuisenaire capers(1-5) Adding up to 100; properties of operations; place value; writing expressions; multiplication; distributive property

Gridlock Recording Sheet Gridlock Game (3-5) Multiplication and area

Cuisenaire Rod Equations(3-5) Order of operations, solving equations with unknowns

Click on the picture above for the online activity (3-5)

Show connection between fractions and multiplication

Click picture above to go to website and learn more about the lesson(3-5).

More less equal to a whole (3-5)

CPALMS Fraction Measuring (3rd) Fractions, Equivalent Fractions, Measuring with CM

CuisenaireRodsNumberLines (3-5) Fractions on a Number Line

Parts of a Whole Fraction Packet(3-5) Riddles

Adding and Subtracting with Unlike Denominators Fractions (4-5) Riddles

Naming rods (4-5)
Make Your Own Meter Stick
Estimation Station(K-5) Measurement and Estimation. Older grades can convert using the metric system.

CPALMS Fraction Measuring (3rd) Fractions, Equivalent Fractions, Measuring with cm

COLOR An Amusing Adventure Game Board An amusing adventure An Amusing Adventure Game Board(2-5) Measurement, with estimating first. There are a lot of suggestions on how to extend this activity on page 7 of the last PDF.

Estimate and Measure (K-5) Estimate how many centimeters the line is and then measuring with Cuisenaire Rods.

Shorter or Longer kindergarten ShorterorLongerMathStationFREEBIE cuisenaire spinner (K-2) Measurement, comparing problems (addition/subtraction), greater than, less than

Squares and Rectangles(3-5) Patterns with area and perimeter.

Race to Get to 50 or 100(2-5) Skills: Addition, Subtraction, Skip Counting, Decimals, Metric System, Measurement.

Cuisenaire Cards(K-5) Print out 4 of these pages to make a deck of cards. Versions of War, Go Fish, Memory, etc.

Cuisenaire Art(K-5) There are several extensions/variations that make this activity great for multiple skills and multiple grade levels. (Add, subtract, multiply, divide, properties of operations, order of operations, place value, line them up next to a meter stick to find value). Use this to notice patterns when adding and subtracting or even and odd.
Number Patterns Using Cuisenaire Rods (K-5)
Number Patterns with Cuisenaire Rods (K-5)

Race to Get to 50 or 100(2-5) Skills: Addition, Subtraction, Skip Counting, Decimals, Metric System, Measurement.

Tricky Path (1-5) Skills: One more, one less, number sequences, or number patterns. You can challenge your students by asking them to make it with the greatest or least amount of rods possible. Extend: Students can count the value of the rods and make equations.

Triangle Magic (1-5) How can you place the rods so that each side of the triangle adds up to an orange rod?

color match doubles addition(1-5)

Squeeze Play(K-5) Logic and reasoning game

Odd One Out examples (1-5) Examples-addition/subtraction, multiplication/division

Equations with cuisenaire rods(3-5) Order of Operations, solving equations with unknowns

Shorter or Longer cuisenaire spinner(K-2) Measurement, comparing problems (addition/subtraction), greater than, less than

Cuisenaire Rod Equations(3-5) Order of operations, solving equations with unknowns

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Cuisenaire-Rods Introduction, background, and activities to try to familiarize yourself with cuisenaire rods. Simon Gregg: How I Teach Using Cuisenaire Rods (3-5) Big Ideas in Arithmetic (k-5) How to use: Learn more about the power of Cuisenaire Rods: Video Book Simon Gregg Buy Rods Watch and ask: What do you notice?

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Prior Experience

Students are unlikely to have previous experience with using Cuisenaire rods since the use of these materials to teach early number has been abandoned. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems. However, other units in the Cuisenaire rod fractions collection are available at level 3 and level 4.

Session One

Use Cuisenaire rods or the online tool to introduce multiplication of fractions in the following way.

Set up this diagram:

"If the blue rod is defined as one then what fraction is the dark green rod?" (two thirds)
"What coloured rod is half of two thirds?" (light green)
"Relating back to the blue rod what fraction is one half of two thirds? In other words, what fraction is the light green rod?" (one third)

"How can we write one half of two thirds as an equation?" (1/2 x 2/3 = 1/3)

Note that your students might not relate multiplication of whole numbers, e.g. 3 x 8 = 24, with 'of' as a function so help them to realise that.

Create this diagram:

"Let's try a more difficult example. If the orange rod is one, what fractions are the dark green and crimson rods?" (dark green is six tenths or three fifths, crimson is four tenths or two fifths)
"So what fraction of the dark green rod is the crimson rod?" (four sixths or two thirds)
"How can we record two thirds of three fifths as an equation?" (2/3 x 3/5 = ?)
"What is the answer to the equation?" (2/3 x 3/5 = 2/5)

Record the two equations that you have results for:

1/2 x 2/3 = 1/3 and 2/3 x 3/5 = 2/5

"Can you see any pattern here? Two examples are not enough really."

Students might notice that the same digits are on the / diagonal or that the denominator of the answer matches the denominator of the second fraction. However, there is no clear relationship in the equations since equivalence is involved.

"Our goal today is to find a pattern that allows us to solve any problem like this, where a fraction is multiplied by a fraction."

Ask the students to work in small co-operative groups using Copymaster One. Each group will need access to a set of Cuisenaire rods or to the Cuisenaire applet on their digital devices.

Look for the following as you roam:

Can they identify and name the fractions for each rod when told the referent one rod? To do so they will need to partition the rods into smaller equal units using the other rods that are available.

Can they accept the third rod as a unit of measure? To do so they need to consider how many of that unit fit into the second rod. This idea is particularly challenging where the third rod is larger than the second rod.

Can they map back to the original one to name the result of the fraction multiplication?

After a suitable period of investigation bring the class back together to discuss how they solved the problems. Record the first three solutions:

"Is there a pattern in these three examples?" Students should notice that the numerators and denominators of the factors are multiplied to give the numerator and denominator of the answer.

"Does this pattern apply to our first two examples?"

Invite the students to multiply numerators and denominators to see if the pattern holds. They may notice that the answers are equivalent to the products, i.e. , 2/6 = 1/3 and 6/15 = 2/5. If your students have difficulty with equivalence it may be worthwhile taking some sessions from Cuisenaire Rod Fractions: Level 4.

Focus on examples 6a and 6b from the Copymaster. In both cases the multiplier factor is an improper fraction. This idea is challenging but important if students are to generalise multiplication of fractions. In 6a the diagram models; . If the brown rod is one then the crimson rod is one half and the blue rod is nine eighths since each unit square is one eighth of the brown rod. The difficulty is in recognising that the blue rod is nine quarters of the crimson rod. This relationship requires the students to use the blue rod as the measure of the crimson rod.

"Does the rule for multiplying fractions hold if one of the fractions is improper?" (Yes, e.g. 5/4 x 2/3 = 10/12 = 5/6, for 6b)

Session Two

The purpose of this session is to consolidate multiplication of fractions and to see if the properties of whole numbers under multiplication hold for fractions as well.

Create this diagram for the students:

"What fraction multiplication is represented in this diagram?" (2/3 x 3/4 = 6/12 = 1/2)

Give the students 5-10 minutes in their small groups to come up with their own multiplication of fractions examples. Ensure they record their examples using squared paper and an equation.

When you bring the class back together, focus on the properties of multiplication.

"Which of these equations are true? Explain how you know."

5 x 3 = 3 x 5 4 x 2 x 3 = 3 x 4 x 2 3 x 7 = 3 x 2 + 3 x 5

All three equations are correct as they are examples of the properties of whole numbers under multiplication. Look for students to explain that the first equation is the commutative property, the order of the factors does not affect the product. The second equation is less obvious as it applies the associative property which is about the pairing of three of more factors not affecting the product. Since multiplication is a binary operation only two factors can be operated on at once. Students may recognise that in the third equation seven is distributed into five plus two. Ask a student to draw a diagram of how this example works. For example:

Record the properties under each equation.

"Do these properties also hold for fractions when they are multiplied? Let's work on the example we started with."

"If the commutative property works for fractions what should be true about; 2/3 x 3/4 = 6/12 = 1/2?"

Students should connect to the whole number example and say that 3/4 x 2/3 = 1/2 as well. Begin with the same stripey rod for one (12 unit squares).

"Which rod is two thirds?" (brown)
"Which rod is three quarters of two thirds?" (dark green)
"What fraction of the one rod is the dark green rod?" (one half)

"So from this one example the commutative property seems to work. See if it works with the fraction multiplication you made up."

Let the students work out if the commutative property works for their example. When the class re-gathers choose a couple of further examples to share. Expect that some students will argue that the property obviously holds for this reason.

So if the commutative property holds for whole numbers under multiplication it must also hold for fractions.

Return to the original example of, 2/3 x 3/4 = 6/12 = 1/2.

"What fraction multiplication is this diagram representing?" 1/2 x 2/3 x 3/4 = ?

"What is the answer?" 1/2 x 2/3 x 3/4 = 6/24 = 1/4

"If the associative property holds what also will be true?" Students should offer several possible examples, e.g. 2/3 x 3/4 x 1/2 = 1/4. Ask them to check if the other examples of pairing factors first give one quarter as the answer.

"From what we know about the commutative property, why must this be true?" Look for students to argue that a similar generalisation is possible because:

The distributive property is harder to illustrate with fractions than with whole numbers. Start with the original example; 2/3 x 3/4 = 6/12 = 1/2. "Three quarters is made up of one quarter plus one half. If the distributive property works, what must also be true?" 2/3 x 3/4 = 2/3 x 1/4 + 2/3 x 1/2

Challenge the students to work together in small groups to see if this example is true. Look for them to work between the symbols and the Cuisenaire rod model.

Replacing the blue rod with the light green (one quarter) and dark green (one half) gives this:

Finding two thirds of each rod (light green and dark green) gives:

"Do two thirds of one quarter and two thirds of one half combine to make our original answer, one half?" (Yes, since red and crimson rods combined make a dark green rod).

"While one example is weak evidence it appears that the distributive property might work for fractions under multiplication." If time permits asks your students to try out the distributive property for other fractions. Note that the distributive property is most commonly applied when working with mixed numbers, e.g. 3/4 x 1 2/3 = 3/4 x 1 + 3/4 x 2/3.

Session Three

In this session the purpose is to find the quotient of two fractions by division. A measurement view of division is used rather than a sharing view.

Pose this simple problem: "A group of four students make $72 on a job. If they share the money equally how much should each friend get?"

Ask your students to solve the problem and share with their neighbour how they solved it. Students should recognise it as a division problem, 72 ÷ 4 = 18. There are many ways to solve the problem but the main focus is interpretation of seventy-two divided by four. Most students will ask themselves, "How many fours are in seventy-two?" which is measuring 72 in sets of four. Expect strategies like:

"If they got $80, that would mean $20 each. So they got $2 less each, that's eighteen dollars."

"If there were eight students they would get $9 each. There are half as many students so they should get twice as much."

Record 72 ÷ 4 = 18 on the board and write 72 ÷ 1/2 = ?, as well.

"Ninety-five percent of people get this problem wrong. Try to solve it yourself then talk to your partner to justify your answer."

In the discussion that follows try to connect their interpretation of 72 ÷ 4 as "How many fours are in seventy-two" with the meaning of 72 ÷ 1/2 as "How many one halves are in seventy-two?"

"Why is the answer, 144, larger than the dividend, 72?"

You may need to use a simple example to convince some students. Make a Cuisenaire rod model like this:

"If the crimson rod is one how many ones do I have?" (five)
"What fraction are the red rods?" (one half)
"How many halve do you get in five ones?" (ten)

Record this as 5 ÷ 1/2 = 10. "So the important idea is that we treat division as a 'How many goes into' operation."

Create this simple example of fraction division using Cuisenaire rods. Build up the model as students answer.

"If the crimson rod is one half, then what colour is the one rod?" (brown)
"So what fraction is the light green rod?" (three eighths)

The model should now look like this:

Now answer the question, "How many three eighths fit into one half?" Record 1/2 ÷ 3/8 = ?

Ask the students to attempt an answer themselves then talk to a partner to justify their answer. Some students will provide answers like "A bit more than one" but encourage them to be accurate.

An important point is that the crimson rod (one half) is being measured with the light green rod (three eighths) so the light green rod is the measurement unit the answer must be expressed in. Many students will notice that a white rod fits into the 'gap'.

"How much of a light green rod is the white rod?" (one third)

"So we get an answer of one and one third. What is that number as a fraction, not a mixed number?" (four-thirds)

Record; 1/2 ÷ 3/8 = 4/3. Mention that equivalence fooled us last time with the multiplication rule.

"Write down some equivalent fractions for four thirds." (8/6, 12/9, 16/12...)

"If I write 1/2 ÷ 3/8 = 8/6 do you notice a pattern from one example?"

Get the students to work on the problems in Copymaster Three which scaffolds their learning towards division of fractions. Ask them to work in small co-operative groups with a set of Cuisenaire rods, or the online tool, and squared paper to record their ideas.

Look for the following:

Are the students able to accept the second rod as the unit of measure and express the answer in terms of that rod?

Do the students generalise that 1 ÷ a/b = b/a?, for example 1 ÷ 7/10 = 10/7.

Can they explain why 1 ÷ a/b = b/a?

Do they recognise that if the fractions have a common denominator then the division operates like whole number division?, e.g. 5/8 ÷ 3/8 has the same answer 5 ÷ 3 as because the units (eighths) are the same.

Do they generalise that if 1 ÷ a/b = b/a then using any fraction as the dividend just 'scales' the answer? For example, if 1 ÷ 4/5 = 5/4 then, 1/2 ÷ 4/5 = 1/2 x 5/4.

After a suitable period of investigation bring the class back together to discuss the points about. Use selected examples, as above, to ask students to reflect on a general rule for division of fractions. The discussion should lead to two useful algorithms. Use this example:
Express both fractions as equivalent fractions:
5/10 = 1/2 (Yellow rod) and 8/10 = 4/5 (Brown rod) so 1/2 ÷ 4/5 = 5/10 ÷ 8/10.

Since the units are all eighths the problem is just like "How many lots of eight somethings fit into five somethings." Be aware that students might be unaware of the quotient rule for rational numbers, a ÷ b = a/b, for example 5 ÷ 8 = 5/8.
Invert and multiply:
The most common algorithm, and that needed for algebra, derives from, 1 ÷ a/b = b/a. So the answer is the reciprocal. Any change to the dividend, one, is just a scalar so in general, c/d ÷ a/b = c/d x b/a.

Session Four

The aim of this session is to consolidate students' connection between multiplication and division of fractions. To do so we keep the representation the same, Cuisenaire rods, but we expect a considerable amount of reunitising and norming with the units that are created.

Begin with this simple arrangement of three rods:

Ask a series of "Can you see...?" questions. Expect students to discuss the questions in 'think-pair-share' format. Write the equations as well as saying the question.

"Can you see five eighths of four fifths? (5/8 x 4/5 = ?)" The yellow rod is five eighths of the brown rod which is four fifths of the orange rod.

"Can you see how many one halves fit into four fifths? (4/5 ÷ 1/2 = ?)" The yellow rod is one half, the brown rod is four fifths.

"Can you see how many four fifths fit into one half? (1/2 ÷ 4/5 = ?)"

"Can you see eight fifths of one half? (8/5 x 1/2 = ?)"

All of the above questions treat the orange rod as one. Ask: "What 'Can you see?' questions could we ask if we treated the brown rod as one?"

For example; "Can you see two times five eighths? (2 x 5/8 = ?)" or "Can you see how many five quarters fit into five eighths? (5/8 ÷ 5/4 = ?)".

In small groups of two or three challenge your students to do the following:

1. Choose a set of three different Cuisenaire rods.

2. Create as many "Can you see...?" problems as they can with that set.

3. Produce a published set of their problems with answers hidden somewhere for other students to solve. Problems should involve pictures, words and equations

Students might publish their problems as a word document or PowerPoint so that the sets can be published. Swapping problems among groups is an excellent way for students to check if their reasoning is correct as well as providing practice at reunitising.

After sufficient time ask some groups to share their favourite "Can you see...?" problem. Look for other students to:

Identify which rod is being used as the referent one rod

Name the other two rods in terms of that one rod

Connect word stories to appropriate multiplication or division equations

Solve the equations and explain their answers with reference to the Cuisenaire rod model

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Prior Experience Students are unlikely to have previous experience with using Cuisenaire rods since the use of these materials to teach early number has been abandoned. Their lack of familiarity with the rods is a significant advantage as they will need to imagine splitting the referent one to solve problems. However, other units in the Cuisenaire rod fractions collection are available at level 3 and level 4.

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It's your lucky day proportional rich task

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It's your lucky day proportional rich task

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TpT 3rd Comparing Fractions Math Kit.pdf

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources. Learn how to bring TpT School Access to your school.

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources. Learn how to bring TpT School Access to your school.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources. Learn how to bring TpT School Access to your school.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources.

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Meet every instructional need with TpT School Access - the new, school-wide subscription that gives teachers immediate access to a comprehensive library of PreK-12 Teachers Pay Teachers (TpT) resources. Learn how to bring TpT School Access to your school.

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evaluate expressions teacher intervention guide

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evaluate expressions teacher intervention guide