Proportions & Ratios Workshop Lisa Heap and Alison Howard Mathematics Facilitators
Objectives • Understand the progressive strategy stages of proportions and ratios • Understand common misconceptions and key ideas when teaching fractions and decimals • Explore equipment and activities used to teach fraction knowledge and strategy
Revising the Framework: • Sort the addition/subtraction framework. • Align the multiplication/division framework. • How do they fit?
Solving a Division Problem: A sheep station has eight paddocks and 296 sheep. How many sheep are there in each paddock?
Reversibility 8 x 30 = 240 8 x 7 = 56 Place Value 240 ÷ 8 = 30 56 ÷ 8 = 7 30 + 7 = 37 296 ÷ 8 296÷ 8 = 148 ÷ 4 = 74 ÷ 2 = 37 Proportional Adjustment: Rounding and Compensating Tidy Numbers 4000 ÷ 8 = 500 500 - (320 ÷ 8)= 500 - 40 = 460 320 ÷ 8 =40 40 - (24 ÷ 8)= 40 - 3= 37 Algorithm
Multiplication Division Share Back: Did you try… • A knowledge check? • Diagnostic snapshot? • Recording in your modelling book? • Some Equal grouping/Sharing?
Importance of Place Value • What is place value? • Where does place value start? • What place value equipment have you currently got in your school? • Order the equipment from least abstract to most abstract.
Place Value Ideas: • 100 Day Party. • Place Value Hats. • Large Numbers Roll Over Page 43.
Place Value • Read, Say, Do x2 • Write the number as 63 • Write the number as sixty-three • Say the numeral one way, 63 is sixty-three • Say the numeral another way, 63 is six tens and three ones • Model the number as ones, 63 individual ones • Model the number in the PV form as 6 tens and 3 ones
Fraction Rope Game The Rope Game Find 2/5
Fraction Knowledge Test: • Write the symbols for one half, one eighth, one quarter, one third and one fifth • Put those fractions in order (smallest first) • Draw 3 pictures to represent one quarter • 7 is one third of what number? • 12 is three fifths of what number? • What is 3 ÷ 5? • On a number line from 1 – 5 show where five halves live. • Show me one half as a ratio.
1/2 1/2 1/2 1/2 1/2 5 children share three chocolate bars evenly. How much chocolate does each child receive? 3 ÷ 5 What are these pieces called? 2/12 !! What do you think they have done? 1/2 +1/10 =
A more sophisticated method for 3 ÷ 5 1/5+1/5+1/5 =3/5 Y7 response: “3 fifteenths!” Why?
A B C D E F 0 1 2 3 Which letter shows 5 halves as a number?
Ratios (Introduced at Stage 6) 1:1 Write 1/2 as a ratio 3: 4 is the ratio of red to blue beans. What fraction of the beans are red? 3/7 Think of some contexts when ratios are used.
Framework Practice Match the strategy stages to their definitions and assessment task(s) from GloSS.
What does this mean? 3 over 7 3 ÷ 7 3 out of 7 3 : 7 3 sevenths
2 3 1 2 3 5 8 6 2 3 The problem with “out of” + = “I ate 1 out of my 2 sandwiches, Kate ate 2 out of her 3 sandwiches so together we ate 3 out of the 5 sandwiches”!!!!! x 24 = 2 out of 3 multiplied by 24! = 8 out of 6 parts!
The Problem with Language Use words first before using the symbols e.g. one fifth not 1/5 How do you explain the top and bottom numbers? 1 2 The number of parts chosen The number of parts the whole has been divided into
Models of Fractions: • Complete the activity on discrete and continuous models of fractions. • In your thinking groups discuss the meaning of continuous and discrete.
0 1 Continuous Model: • Models where the object can be divided in any way that is chosen. e.g. ¾ of this line and this square are blue.
Discrete Model: • Discrete: Made up of individual objects. e.g. ¾ of this set is blue
Whole to Part: • Most fraction problems are about giving students the whole and asking them to find parts. • Show me one quarter of this circle?
Part to Whole: • We also need to give them part to whole problems, like: • 5 is a quarter of this number. What is the number?
Existing Knowledge & Strategies Using Imaging Using Number Properties Using Materials Using Materials New Knowledge & Strategies The Strategy Teaching Model
Developing Decimal Place Value Understanding Decipipes, candy bars, or decimats can be used to demonstrate how tenths and hundredths arise and what decimal numbers ‘look like’ and to compare decimals numbers.
Using Decipipes Explore the Decipipes. What is each piece called? How did you know? • Build 0.365 and 0.37 • Which is bigger? Why? • Add 0.4 and 0.25 on your decipipes. Discuss what you did and what you found out. Book 7: Pipe Music with Decimals, p.38
Using candy bars (and expressing remainders as decimals) 3 ÷ 5 3 chocolate bars shared between 5 children. 30 tenths ÷ 5 = 0 wholes + 6 tenths each = 0.6
Solve this problem… Standard written form (algorithm) Place Value I had 1.6m of ribbon. I used 0.98m for my gift wrapping. How much ribbon do I have left? Reversibility Equal Additions
Ratios • In the rectangle below, what is the ratio of blue cubes to green cubes? • What is the fraction of blue and green cubes? • Can you make another structure with the same ratio? What would it look like? • What confusions may children have?
More on Ratios • Divide a rectangle up so that the ratio of its blue to green parts is 7:3. • What is the fraction of blue and green? • If I had 60 cubes how many of them will be of each colour?
A Ratio Problem to Solve • There are 27 pieces of fruit. The ratio of fruit that I get to the fruit that you get is 2:7. How many pieces do I get? • How many pieces would there have to be for me to get 8 pieces of fruit? • What key mathematical knowledge is required?
The 4 Stages of the P.D Journey: Organisation Organising routines, resources etc. Focus on Content Familiarisation with books, teaching model etc. Focus on the Student Move away from what you are doing to noticing what the student is doing Reacting to the Student Interpret and respond to what the student is doing
Final Evaluation Complete your initial evaluation and mark on your progress. Thank you!