MATHEMATICAL CONTENT KNOWLEDGE -
Multiplicative thinking is indicated by a capacity to work flexibly with the concepts, strategies and
representations of multiplication and division as they happen in a wide range of contexts -
e.g. larger whole numbers, decimals, common fractions, ratio and percent. The key characteristics
include a capacity to work flexibly and efficiently with an extended range of numbers, an ability to
recognise and solve problems that involve multiplication or division including direct and indirect
proportion, and the means to communicate this effectively in multiple ways - e.g. words, diagrams,
symbolic expressions, written algorithms. Multiplicative strategies include formal equal groups,
perceptual multiples, figurative, repeated abstract composite units, flexible strategies for multiplication,
and flexible number properties.
proportion, and the means to communicate this effectively in multiple ways - e.g. words, diagrams,
symbolic expressions, written algorithms. Multiplicative strategies include formal equal groups,
perceptual multiples, figurative, repeated abstract composite units, flexible strategies for multiplication,
and flexible number properties.
PEDAGOGICAL CONTENT KNOWLEDGE -
An example of a student developing mathematical thinking will progress from a multiplication problem
such as:
- ‘If there were 4 boxes with 2 chocolate bars in each, how many are there altogether?’
to…
- ‘If you buy two pens you get a 50% discount on the second pen. If one pen costs $2.30, how much will 2 pens cost?’
When writing sentence problems, it is common to use real-world situations or items to help students
understand the relations of these concepts.
In multiplicative situations, students must construct and coordinate three aspects of multiplicative
situations when they are developing multiplicative thinking. These include groups of equal size, the
number of groups and the total amount. Students must then go from models and representations
that work for whole numbers to more general ideas accommodating rational numbers and algebra.
These more general ideas include ratio, proportion, multiplicative comparison, multiplication of
measures and the use of intensive quantities.
Reference: (Van et al., 2015), (Victoria State Government, 2018)
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Activity - How much does this cost?
Year Level - 6
Classroom Context -
For students with different learning styles, this activity includes visual, linguistic, logical, social and
solitary learning styles. Students have images to help make sense of the problem, but it includes
written words to help those who want to focus on determining the actual problem. It is logical as it
is based on real-world situations, and it caters to both social and independent students as they can
choose to work in groups or individually.
solitary learning styles. Students have images to help make sense of the problem, but it includes
written words to help those who want to focus on determining the actual problem. It is logical as it
is based on real-world situations, and it caters to both social and independent students as they can
choose to work in groups or individually.
Content Descriptor -
Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and
without digital technologies (ACMNA132 - Scootle )
without digital technologies (ACMNA132 - Scootle )
Learning Objectives -
This activity is designed for students to solve multiplication problems in relation to percentages,
using both mental and digital strategies (calculators to check answers).
using both mental and digital strategies (calculators to check answers).
Resources -
Key Mathematical Language:
Discounted by, decimals, fractions, hundredths, __% off, price, amount, total, percent
Discounted by, decimals, fractions, hundredths, __% off, price, amount, total, percent
Symbols: Using and understanding the percentage symbol ‘%’ correctly, as well as writing
decimals and fractions where appropriate.
decimals and fractions where appropriate.
Hands-on Manipulatives: Sale coupons/discount cards, writing books, pens or pencils
Use of ICT: Calculators
Use of calculators are are common in many teaching and learning situations. Introducing
the basic functions of the tool in primary years will be beneficial for them in older years where
calculators will be increasingly used.
the basic functions of the tool in primary years will be beneficial for them in older years where
calculators will be increasingly used.
Prior Knowledge -
Interpret percent as a meaning of ‘out of 100’
Recognise that 100% is a complete whole
Interprets a percentage as an operator
Uses percentage to describe and compare relative size
Represents relative size of percentage of an amount
Instructions
- With each sale coupon card, find the percent discount and change it to a decimal value or a fraction.
- E.g. 20% means 0.2 or 20/100 or ⅕.
- Remind students that 100% is always the full price amount, so to find a certain amount of that full amount, you must multiply the full amount by the decimal or fraction from the first step
- When students have calculated the discounted amount, find the final price by subtracting the discounted amount from the full amount to attain the answer
- The teacher will demonstrate 1 example for the students before they begin
- Students will write the sale coupon details in their books, followed by the steps or ‘working out’ they did, and their final answers.
- Students may check their answers with their calculators
Questions to ask -
How can they be represented?
What do we know?
How can we use what we know?
Which strategies are better and why?
How did you attain your answer?
How do you know it’s correct?
Enabling Prompts
Have these students practice representing a percentage as a fraction and a decimal
(e.g. writing 25% as 25/100 → 5/20 → ¼. And writing 25% as 0.25).
Extending Prompts - Prepare extended percentage problems (problem solving questions) for the fast finishers. E.g. Kate bought her dress for $75. Anna bought the same dress at a different store for 20% more. How much more did Anna pay for it?
Images -
Main resource for the activity - Discount/sale vouchers that use percentages
Student Example: Using the 'End of year sale' shoe voucher discount.
The shoes are $59 each.
It is 50% off all shoes.
50% is represented as 0.5 (decimal) or 1/2 as a fraction.
100% (whole) is the original price of $59.
59 (whole) x 0.5 (discount) = 29.50
$59 (whole) - $29.5 (discount) = $29.50
The sale price of this item is $29.50.
Students may use their calculators to check their answers
Reflection -
In week 4 our topic focused on rational number, mathematical thinking, and proportional reasoning.
My activity is based on multiplicative thinking, a concept that involves working with larger whole
numbers, decimals, fractions, ratios and percent. The activity focuses on percentages and links
to real-world contexts such as sales and discount problems. In order for students to solve these
percentage problems, they must use multiplicative thinking strategies, including making fraction-
decimal connections. It is important for students to understand that all percent problems are the
same as equivalent-fraction problems, involve a whole and a part, and are measured in hundredths
(Van et al., 2016).
My activity is based on multiplicative thinking, a concept that involves working with larger whole
numbers, decimals, fractions, ratios and percent. The activity focuses on percentages and links
to real-world contexts such as sales and discount problems. In order for students to solve these
percentage problems, they must use multiplicative thinking strategies, including making fraction-
decimal connections. It is important for students to understand that all percent problems are the
same as equivalent-fraction problems, involve a whole and a part, and are measured in hundredths
(Van et al., 2016).
A journal article by Hurst and Hurrell (2016) mentions that students must re-conceptualise their
understanding about number to understand multiplicative relationships. It is significantly different
to additive thinking, but learners often confuse additive thinking with multiplicative thinking. It is
necessary for teachers to ensure their students understand that the idea of counting (additive
thinking) differs from splitting or sharing (multiplicative thinking) in order to develop conceptual
understanding of this. Teachers must create activities that continuously link to splitting and sharing,
rather than setting ‘rules’ or teaching ideas that multiplication and division are two separate concepts.
Instead, they should focus on how the two are simply different ways of representing the same situation,
and why numbers behave the way they do when operating (reasoning).
understanding about number to understand multiplicative relationships. It is significantly different
to additive thinking, but learners often confuse additive thinking with multiplicative thinking. It is
necessary for teachers to ensure their students understand that the idea of counting (additive
thinking) differs from splitting or sharing (multiplicative thinking) in order to develop conceptual
understanding of this. Teachers must create activities that continuously link to splitting and sharing,
rather than setting ‘rules’ or teaching ideas that multiplication and division are two separate concepts.
Instead, they should focus on how the two are simply different ways of representing the same situation,
and why numbers behave the way they do when operating (reasoning).
References -
Hurst, C., & Hurrell, D. (2016). Investigating children's multiplicative thinking: implications for teaching.
European Journal of STEM Education, 1 (3).
European Journal of STEM Education, 1 (3).
Van, D. W. J. A., Karp, K. S., & Bay-Williams, J. M. (2015). Elementary and middle school mathematics:
teaching developmentally, global edition. Retrieved from
https://ebookcentral-proquest-com.ezproxy.library.uq.edu.au
teaching developmentally, global edition. Retrieved from
https://ebookcentral-proquest-com.ezproxy.library.uq.edu.au
Victoria State Government (2018). Scaffolding numeracy in the middle years: Multiplicative thinking. [online] Education and Training. Available at: https://www.education.vic.gov.au/school/teachers/teachingresources/discipline/maths/assessment/Pages/multithink.aspx
