Sorcha Moran sorcha.moran@gmail.com Add1ng Num8er5 t0 L1fe Multiply the possibilities Sorcha Moran sorcha.moran@gmail.com

Education System failed Learners lack self-belief in ability to learn maths Perception of intelligence

Understanding is the key to successful learning

‘Rules’ are shortcuts – NEVER begin teaching using ‘rules’ No Shortcuts!

Nurture your learners Ensure they understand the basic concepts Visuals & manipulatives for understanding & acceptance Real-life material for recognition & affirmation Link to previous knowledge for making connections Present in many ways for confidence Link maths symbols to language for proficiency

Lost in Translation Maths is not just a jumble of numbers Order of numbers combined with symbols gives them meaning

Replace Fears with Confidence

What makes learning fractions so difficult?

Local v. Global The brain recognises the values of 3 and 4 (local values); but the fraction has a global value of 0.75 http://www.cne.psychol.cam.ac.uk/math-memory/what-makes-learning-fractions-so-hard-part-1

So many rules!! Fractions addition - common denominator Fractions addition - different denominators Finding equivalent fractions Simplifying fractions Comparing fractions Fraction multiplication Fraction division Convert a mixed number to a fraction Convert an improper number to a mixed fraction Converting fractions to decimals

Multiply the numerators and the denominators. Convert the fractions so they have a common denominator. Then compare the numerators Multiply the numerators and the denominators. Divide using long division or a calculator First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule number 1 Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number. Find the (greatest) common divisor of the numerator and denominator, and divide both by it. Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same. Multiply both the numerator and denominator by a same number. Add the numerators, use the common denominator Find the reciprocal of the divisor, and multiply by it

Students can become blind followers of the rules, getting answers without having any idea if they are reasonable or not

So, what can we do to help?

Use Visual or Manipulatives Fractions become something concrete to the student, and not just a number on top of another without a meaning

Think through pictures/exploration Discover the rules

By using visual models extensively at the beginning stages, the rules will make more sense

Some ideas...

Introduction - What are Fractions?

What are Fractions - Activity (Playdough) Other options – lego, lengths string, strips of paper, etc

Talking Fractions - Visual 1. Shape your playdough into any regular shape you like e.g. a rectangle, a ‘cake’, or a ‘sausage’. 2. Cut your playdough into as many equal size portions you like. 3. Take some portions away. 4. How do you describe to me what you have taken away, or what you have left?

Talking Fractions Begin to use Fractions Terminology Don’t correct their answer in simplified fractions Write replies on the board so they see the notation we use in fractions Bottom number describes how many pieces would be in the whole thing Top number describes how many of those pieces we have Same number on the top and bottom if we have 1 whole

Give plenty of opportunities to play around with the concept Express the shaded part of a shape as a fraction Draw and shade a shape to match a fraction Discuss where they see fractions every day - what do they mean? Change a sentence into a fraction *Language - ‘out of’ Give plenty of opportunities to play around with the concept Use correct language No need to simplify at this stage

Equivalent Fractions - Visual

Equivalent Fractions - Terminology When we have two fractions that are actually the same size portion, we call them equivalent fractions (equal). It is usually best to show an answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying, or Reducing the Fraction

Equivalent Fractions - Visual

Equivalent Fractions - Visual

Equivalent Fractions - Visual ...Learner creates some equivalent fraction shapes

Equivalent Fractions - Activity (Exploration)

Simplifying Fractions - Visual

Simplifying Fractions - Visual

Simplifying Fractions Equivalent Fractions Simplifying Fractions Equivalent fractions, but in the other direction

Simplifying Fractions - Visuals Using the visuals and giving the learners a chance to explore with the playdough, makes it clear why we have to make the same change above and below the line. Now they understand the concept instead of trying to remember a rule.

Fractions of Whole Numbers - Visual

Fractions of Whole Numbers - Activity 1. Make 20 small balls of playdough 2. Divide them into 4 equal bundles 3. Count how many balls you have in “3 out of 4” bundles (3/4)

of always translates to Maths is a language!

Mixed and Improper Fractions - Visual Improper: How many pieces? Mixed: How many pies?

Mixed and Improper Fractions - Activity Mark measurements on a ruler (in/cm) Measure a line Measure everyday items: pencil, eraser, finger, etc

Adding Fractions, Same Denominator

Adding Fractions, Different Denominators - Visual Back to the playdough! Present ½ of a ‘pie’, and ⅛ of a ‘pie’ Ask the learners how much pie you have The only way we can explain how much we have accurately is to make all the pieces the same size, and then count them.

Adding Fractions, Different Denominators - Explore If a learner asks why ½ + ⅛ is not just 2/10 (adding top, adding bottom). Ask them to explore it themselves with the playdough: Cut a shape the same as yours into 10 equal pieces Are 2 of those pieces a similar size to your ½ and ⅛ together?

Adding Fractions, Different Denominators - Visual

Multiplying Fractions Top by Top, and Bottom by Bottom Multiply the top numbers Multiply the bottom numbers Simplify if possible

Multiplying Fractions - Visual Use the playdough to represent ⅖ Remind the learners that ‘of’ in English translates to multiply in maths, and visa versa So, ½ x ⅖ is the same as “half of two fifths” Remove half of the pieces We are left with ⅕ - same as above - Magic! Point out to the learners that they can make life a little easier by dividing before they multiply - i.e. cancel a top with a bottom before they calculate.

Dividing Fractions ½ ÷ ⅙ = “one half divided by one sixth” Begin by doing division with integers (something familiar). Take, 20 ÷ 4. We can ask “What is 20 divided by 4?”, or we can ask the same question in another way - “How many 4s in 20?” This is the way we need to think about division with fractions: “How many ____ will fit in _____?”

Dividing Fractions - Visual Use the playdough to visualise: Now it’s not so odd that dividing two fractions can result in a whole number!

Dividing Fractions - Visual Try the same with dividing a small fraction by a larger one e.g. ⅙ ÷ ½ Use the playdough to visualise. Only part of the half will fit in to the sixth...how much of it? Here we are trying to fit a bigger piece into a smaller piece, so only a portion of the piece will fit in. Hence, we can expect our answer to be a fraction.

Dividing Fractions So, how do we do the same on paper? Turn the second fraction upside down and multiply Now, that’s just strange!

Dividing Fractions Again, bring it back to the familiar! 8 ÷ 2 If we want to share 8 sweets between 2 children, they get ½ of the 8 sweets each So, we can rewrite ‘8 ÷ 2’ as ‘½ x 8’ or ‘8 x ½’ See what just happened!! Turn the second fraction upside down and multiply

Dividing Fractions - Visual ½ ÷ ⅛ 1. Make a regular shape with your playdough e.g. a circle, a rectangle, etc 2. Divide your shape into 8 equal pieces, i.e. ⅛ ths 3. Check how many ⅛ ths are in half of your shape Check the answer mathematically

Making Sense of Algebra

Replace Fear with Confidence

Fill the Gaps

Learn the Language = x ’s & y’s / ()

No Shortcuts

Layout on Page

Success! Remove the fear Fill the gaps Learn the language Don’t skip steps Layout on the page When do we ever use algebra???

Fill the Gaps Order of operations Distributive property Divisibility/Factors Working with negative numbers

Learning the Language What does ‘=’ mean? 2+3=5 ☐ + 2 = 6 ☐ + ☐ = 6, why we use letters Operators in algebra Why we don’t use x for multiply

Learning through discovery

Trial and Error

Trial and Error

Removal of Blocks

Removal of Numbers

On Paper – Emptying the Box! 4x + 2 = 3x + 9 5x + 2 = 2x + 14 2x + 4 = 10

On Paper – Layout is Important! Always work down the page Only one change on a line Use colour 4x + 2 = 3x + 9 4x = 3x + 7 x = 7 - 2 - 2 -3x -3x

Remove Blocks as Part of Set-Up 5x - 3x + 2 = x + 5

Multiplying out Brackets 2(x + 3) = x + 8 Looking at 2(x + 3) we read it as “two times ‘x + 3’”, or, in other words, ‘x + 3’ twice. So we place it on the scales like this: “two times ‘x + 3’” is the same as “two times x” plus “two times 3”, or 2(x + 3) is the same as 2(x) + 2(3)

Multiplying out Brackets 2(x + 3) = x + 8 2(x) + 2(3) = x + 8 2x + 6 = x + 8 -x -x x + 6 = 8 - 6 - 6 x = 2

Mystery Solved!

Individuality

Sorcha Moran sorcha.moran@gmail.com Add1ng Num8er5 t0 L1fe Multiply the possibilities Thank You! Sorcha Moran sorcha.moran@gmail.com