MATHEMATICAL CONTENT KNOWLEDGE -
There are five main concepts in fractional meaning and these include part-whole, measure, division, operator and ratio. Part-whole examples include shading a region or when there is part of a group of objects. Examples of measure includes identifying a length and using that as a measurement tool, as well as focusing on how much rather than how many parts. Division is the idea of sharing with an amount of something, it is often not thought of with fractions. Operator is used to indicated an operation. Ratio is the probability of an event.
Fractions can be difficult, so common misconceptions can include seeing the numerator and denominator as separate values, not understanding that a fraction such as ⅔ means two equal-sized parts, thinking that a fraction such as ⅓ is smaller than a fraction like 1/10, and mistaking the operation rules for whole numbers to compute with fractions.
PEDAGOGICAL CONTENT KNOWLEDGE -
There are four categories of models when working with fractions - area, length, region and set or quantity. When working with area, a whole area can be divided without necessarily having equal parts. Representing fractions with length can include using strings, paper strips or fraction mats. Using regions is a fun and practical model as this can be done with pizzas, cakes, chocolate blocks or paper folding. Lastly, using sets can include cubes, counters or blocks as fraction models. Important steps to take when teaching fractions includes helping the students identify the whole, partition this into equal parts, name the equal-sized parts which is the numerator, determine the number of parts which is the denominator, and to associate the fraction name.
Reference: (Van et al., 2015)
--------------------------------------------------------------------------------------------------------------------------------
Activity - Pizza Topping Fractions
Year Level - 3
Classroom Context -
For students who may be culturally diverse and unfamiliar with what pizzas are, use other representation models such as paper or shapes to assist them in learning the same concept as others. Keep in mind that in English, fractions such as ‘one-third’ are often expressed differently in other languages.
Content Descriptor -
Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058)
Learning Objectives -
This activity is a fun task designed to help students identify halves, thirds, quarters and fifths using pizza slices as a model for representation.
This activity is a fun task designed to help students identify halves, thirds, quarters and fifths using pizza slices as a model for representation.
Resources -
Key Mathematical Language:
Fraction, Half of, Halves, A third of, Thirds, A quarter of, Quarters, A fifth of, Fifths, Equivalent to,
Simplified to, Equal, Numerator, Denominator, Whole, Parts
Simplified to, Equal, Numerator, Denominator, Whole, Parts
Hands-on Manipulatives/materials: laminated pizza fraction cards, markers, whiteboard dusters
Using pizzas, pies, chocolates, etc. are great model representations for teaching and learning
fractions. Laminated charts and whiteboard markers allow students to erase their mistakes and
re-use them if necessary.
fractions. Laminated charts and whiteboard markers allow students to erase their mistakes and
re-use them if necessary.
Prior Knowledge -
Students should have the ability to identify halves and wholes
Students should recognise that quarters and eighths are formed by repeats of halving the length.
Instructions -
- In small groups, students will recognise and write fractions to match different pizza cards. Explain that the numerator (top) shows how many equal parts of the whole have been taken, and the denominator (bottom), shows how many equal parts there are of the whole.
- Encourage students to write both equivalent and simplified fractions
- Once the students are familiar with this activity, transition into the next activity where they draw on blank pizza cards to create their own toppings. They can draw any toppings of their choice to match the fractions already given on the cards.
Questions to ask -
How many parts are there to make this particular whole pizza?
How do you express one half?
How can you simplify this fraction?
How else can this fraction be represented?
Is one quarter smaller or bigger than one half?
How many quarters are there in this pizza?
Enabling Prompts - Students may start by working on identifying how many slices are in each pizza to make 1 whole. Identify the difference between each slice in the pizzas - e.g There are 4 slices in a whole. 1 slice has cheese.
Extending Prompts - For the fast finishers, there are extended cards they can complete as a challenge. These include simplified fractions, e.g. 1/3 instead of 2/6.
Images -
Example: Student writing in the correct fractions on the lines. The pizza is 1/4 cheese, 1/4 olives, and 1/2 pepperoni.
Further pizza fractions that students can work on.
Example: Student drawing a topping of their choice, matching the fraction that is already given. This is done on the blank pizzas.
Further pizza fractions that students can work on.
Reflection -
This week we were introduced to the topic of how children learn mathematics and developing fraction understandings. As Mou et al. (2016) discusses, children’s understanding of fractions is central to their mathematical development and is one of the first steps to extending their numerical understanding, such as working with whole numbers to other rational numbers. However as Van et al. (2015) mentions, fractions have always been a challenge for students since there are many misconceptions due to it being strongly linked to mental computation, decimals, percentages, and algebraic problems. There are a variety of ways students can develop fractional understanding and it can take years of schooling to fully become competent with the concept.
In my activity, I have chosen a part-whole concept. Pizzas were used as the representation model where students work with equal parts of whole pizzas. Circles are one of the best models for fractions as it allows students to develop mental images of the piece sizes (Van et al., 2016).
Previously when thinking about teaching fractions, I would often think of basic part-whole concepts. After research on this topic, I have developed deeper knowledge on the many other important fraction meanings (such as sets or using number lines), and the many models of representations that are effective and commonly used when teaching and learning about fractions. This research encouraged me to think deeper about the strategies I could incorporate when teaching fractions to my future students. I personally struggled with this concept in school, so I believe it is essential to use the appropriate mathematical language and learning approaches - experiencing activities that relate to everyday settings, and are catered to visual, logical, and linguistic students.
References
Mou, Y., Li, Y., Hoard, M. K., Nugent, L. D., Chu, F. W., Rouder, J. N., & Geary, D. C. (2016). Developmental Foundations of Children’s Fraction Magnitude Knowledge. Cognitive Development, 39, 141–153. http://doi.org/10.1016/j.cogdev.2016.05.002
