Teaching students how to represent fractions on a number line can be tricky, especially when some fractions are represented on the same point.
The following scratch program helps students visualize this tricky concept – feel free to copy and modify the code if you find it useful.
One of the main benefits of mathematics is the development of abstract reasoning skills. However, fluency – or the ability to immediately process simple calculations or recognize basic math facts – is also important to develop as students progress through elementary school. Just like being able to recognize words without decoding letters (“sight reading”) aids reading comprehension, fluency with basic math facts can help students master tasks that rely on more abstract reasoning. In other words, being able to automatically add, subtract, multiply, and divide single digit numbers frees up student’s mental resources (specifically their working memory) to focus on more conceptually challenging tasks.
And building fluency is an area where online tools really shine. Unlike a human teacher, online tools can continually and carefully monitor the performance of dozens, hundreds, or thousands of students – their error patterns, response times, and growth rates – and provide each student with material that is designed to boost their performance. One of these tools is xtramath – which allows teachers to tailor specific programs – like division, or subtraction of single digit numbers – to their students, then allow students to work for 5-10 minutes a few times a week to develop fluency. The icing on the cake is that the program is free for teachers and families. You can navigate to their website by clicking the image below:
Students who are not yet “fully operational” or able to reason using abstract concepts, can still learn form mathematical concepts with support. They typically need to develop these abstract representations of concepts in three stages, using: 1) concrete representations (like a geo-board) ; 2) semi-concrete representations (like a drawing); and 3) abstract representations (like a formula). While concrete “manipulatives,” like base-ten blocks, are widely used and effective in introducing concepts like place-value, evidence suggests that “virtual manipulatives” can be similarly effective.
Toytheater has an excellent selection of “virtual manipulatives” that students can use online, including a realistic version on fraction strips or fraction blocks. The image below will take you to their website (in a new tab) where you can explore their fraction strip app, along with other virtual manipulatives.
Comments and suggestions are always welcome. Thanks for visiting!
Fractions present students in early to middle elementary school with a big challenge. For the first time, they are faced with numbers that do not represent whole numbers, and do not follow the same operational rules as whole numbers. For that reason, in some countries – like Japan – operations with fractions are not taught until students reach fifth grade. In the U.S., the common core standards introduce fractions in third grade, but emphasize developing an understanding of fractions as numbers, and developing a strong intuitive understanding of whole number fractions before diving into addition, subtraction, multiplication, and division with mixed fractions.
The set of texts below are designed to support 2-3 lessons introducing third graders to fractions. The texts focus on connecting fractions to students’ everyday experiences, providing students with concrete experiences working with fractions, and using concrete and semi-concrete experiences to introduce the symbols used to represent fractions.
“Develop understanding of fractions as numbers for fractions with denominators 2, 3, 4, 6, and 8.” (p. 40)…“Understand a fraction 1 ∕b as the quantity formed by 1 part when a whole (a single unit) is partitioned into b equal parts; understand a fraction a ∕b as the quantity formed by a parts of size 1 ∕b”
Related MA Common Core Standards
Many states require students who are not proficient in English to learn all subjects – e.g. Math, Science, Art, and Social Studies – in English along with native speakers. For teachers, many of whom already struggle to differentiate lesson plans for students with different academic backgrounds, learning styles, and interests, the requirement to teach equivalent content to students at varying levels of English proficiency poses yet another challenge.
The texts below are designed to provide equivalent content instruction (along with content-specific academic language) to students at three levels of English proficiency – early intermediate (“WIDA 2-3”) intermediate (“WIDA 4-5”) and advanced to native speaking (“WIDA 6” and native speakers). The basic template is also designed to be further differentiated in terms of complexity – for example, by using more complex assessments or examples.
In any case, check out the texts below – hope you like them, and don’t hesitate to leave comments or suggestions! [note, links will open a new tab outside of this website]
