Decimals are tricky things. They don’t seem to follow the same rules as whole numbers at first glance. However, if you look deeper, the rules are similar and it’s a matter of knowing and understanding the difference to be successful while using them. The article Models for Initial Decimal Ideas by Kathleen A. Cramer, Debra S. Monson, Terry Wyberg, Seth Leavitt and Stephanie B. Whitney details how to implement the use of a 10x10 and 10x10 +/- frames in constructing decimal concepts. Additionally, this article explores the use of 10x10 grids to help deepen students’ understanding of decimals. This helps to understand the concept of place value, comparisons (and subsequently addition or subtraction) between decimals, and demonstrating the link between fractions and decimals. 10x10 frames provide an easy way for students to create their own understanding of decimals and the rules they follow. They should be used to help scaffold students’ success with building initial decimal ideas.

The first step in using 10x10 frames in the conceptual understanding of decimals is to have the students create them for themselves. This is a very important step as it empowers students who are not proficient with using them a way to make them should they not be provided.
    Colour coding is essential. By coding single units, and groups of ten units (commonly rows) different colours, it cements the idea that tenths and hundreths are separate place values by visually disniguishing 0.17 as one tenth, and seven hundreths as opposed to seventeen hundredths. By visually representing decimals, students will be able to arrange them in order of largest to smallest, and from there they can add and subtract them. It’s much easier for students to comprehend 0.17 – 0.13 as one tenth, seven hundreths minus one tenth, three hundredths is zero tenths (1 tenth minus 1 tenth is zero tenths) and four hundreths (seven hundreths minus three hundreths is four hundreths) and subsequently harder equations.
    One point that I found interesting was the introduction of 10x10 +/- frames. Students demonstrated more success in addition or subtraction of decimals when they were given two boards to represent their decimals on, and a third one to display their answer. Some students required them for the entire process, others to represent the two decimals being manipulated and they did the operation in their heads, and still others simply as a visual reminder of how to visualize the decimals and came up with the answer without marking the frames at all. This is a great example of how students can use these frames as a scaffold for their learning of initial decimal concepts.
    By using the 10x10 frames, students were able to overcome some of the most common mistakes associated with decimal concepts on their own. They were able to use appropriate language, compare decimals correctly, and add and subtract them without losing place value in the process.
    The first ‘place’ to the right of the ‘dot’ is the tenths place, followed by the hundredths, whereas the first ‘place’ to the left of the ‘dot’ is the ones, followed by the tens. It’s easy to see a link between tens and tenths, but not as easy to understand that they are offset, and why this is. Decimals provide a key link to place value. In fact, one of the most common problems students face when adding decimals is they line up the numbers from the left most digit, not at the decimal point. (Cramer, 2009) Another is their lack of understanding that there are an infinite number of zeros behind the leftmost digit, thus 0.120 is the same as 0.12. However, representing 0.12 on a 10x10 grid will show students that it is the same as 0.120 on a 100x100 grid, helping them to link these two facts together.
    Traditionally, when you compare whole positive numbers, the one with the more digits is larger (1 < 10). While the trick of counting the number of digits still applies, it functions in reverse when it comes to decimals, so 0.1 > 0.01. The easiest way for students to learn to understand this is on a 10x10 grid, as they can see the visual comparison and discover this concept for themselves.
    Fractions can be thought of as a simplified decimal. ¬1/10 is the same as 0.1, and 2/5 is the same as 0.4. But, how can this be? Students who are given fraction circles may understand the concept of 2/5, represent it pictorially, and may even be able to divide 2 by 5 to get the result of 0.4. However, this doesn’t tie 0.4 and 2/5 together completely. By using 10x10 frames, students can compare 2/5, 4/10 and 40/100 to come up with their own conclusions regarding how they are associated visually and conceptually.
    Based on this research, I will use 10x10 and 10x10 +/- frames when introducing decimals and place value concepts. The results showed their use was extremely effective by scaffolding previous mathematics knowledge and organizing it in such a way that students are able to formulate their own ideas about decimals. With a strong foundational knowledge, students will be able to move on to more complex equations with less problems than if they had a weak understanding of decimals.
Decimals are not an easy thing to understand naturally: the language is different, comparisons are backwards, and relationship to fractions is obscure. However, decimals are an integral part of mathematics, and not understanding them leads to frustration and problems with higher level mathematics. Students need to learn them, so we need to teach them so they are understood rather than simply known.

    Cramer, Kathleen A. ; Monson, Debra S. ; Wyberg, Terry ; Leavitt, Seth ; Whitney, Stephanie B.
    Teaching Children Mathematics, v16 n2 p106-117 Sep 2009. 12 pp


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