Maths Portfolio Misconceptions
Decimals are a part of our everyday life in some way, when we put fuel in our cars to buying meat from the butcher. Mastering this critical mathematical concept is a necessity (Stephanie Welch, 2010). A decimal is a proper fraction, which is a number less than 1. It is a part of a whole number. Since our numbering system is based on the powers of 10, it is called a decimal system. Decem in Latin means ten (The Maths Page, 2012, Lesson 3). Decimal fractions are represented as the numbers found between two whole numbers.
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The decimal fraction shows part of a whole number and is written after the decimal place. Some key understandings in learning about decimals would be- * the idea that there are numbers between two consecutive whole numbers, for example between 6 and 7 is 6. 54. * the place value system can be extended to the right to show the numbers between two whole numbers * to record a number you write the whole number followed by a decimal point then the part of the number * the numbers to the right of the decimal point have decreasing values in powers of ten ie. /10, 1/100, 1/1000 and so on. * decimal numbers can be partitioned just like whole numbers (0. 84 = 8/10+4/100 or 84/100 or 840/1000)
Prior to learning about decimal numbers students must have a clear understanding of place value, ordering and rounding whole numbers. Without this secure understanding and ability to work with whole numbers, students will not have the prerequisite skills and understanding to move into decimal numbers. Many of the misconceptions students have with decimals arise from the lack of confidence and skills with whole numbers.
When representing whole numbers and parts of whole numbers the decimal place is a separator between the whole number and the smaller part of the whole number. A major misconception students have with decimals is the idea that the decimal place separates two different whole numbers. This is demonstrated when students read 29. 15 as, “twenty-nine decimal fifteen”. This misconception is compounded by the fact that the first experience of decimals for most students is working with money. Instead of seeing $28. 5 as twenty-eight dollars and thirty-five hundredths of a dollar, students are taught that all the numbers to the left of the decimal point represent dollars and everything to the right represents cents.
This leads to further difficulties because with money we only ever use 2 decimals places to the right to represent the hundredths of a dollar. When confronted with three decimal places ie. $5. 362 students will read this as $5 and 362 cents instead of 362 thousandths of a dollar. Another misunderstanding is that the number’s length determines its greatness. With whole umbers, the longer the number the bigger its value (247 397 is larger than 45 673).
We determine this by assessing the value of the number systematically, beginning with the left hand place column. When comparing decimal numbers, students commonly misunderstand that the longer the number the greater its value ie. When comparing 3. 45 and 3. 12345 students may rely on their whole number understanding and see 3. 45 as 3 and 45, and 3. 12345 as 3 and 12345. Therefore they perceive 3. 12345 as the larger number. The introduction of a place value chart to include decimal numbers to the right of the ones place can also be confusing for children.
The first column to the right is called tenths, if we were to work symmetrically from the decimal point the first column would be the oneths. Problem solving using addition and subtraction algorithms involving decimal numbers is an area where students commonly make mistakes. Students list the numbers down the page, lining up the digits according to their length not their place value. 3. 25 + 12. 6 Section 2 Objective| Misconception| Teaching Activity| SWBAT correctly read decimal numbers| Students read 3. 25 as “three decimal twenty-five” instead of “three and twenty-five hundredths”.
This incorrectly suggests that the number contains 25 ones. | Use the extended place value chart to reinforce placement and value of each digit. | SWBAT order a mixed set of numbers with up to 3 decimal places| Numbers with more digits are larger. | Refer to place value charts to prompt students to always order numbers reading the value from the far left hand column first. | SWBAT plot a number on a number line demonstrating that to the left of the decimal we have ones, tens, hundreds and to the right is tenths, hundredths, thousandths.
The first place to the right of the decimal place is ‘oneths’| Concrete materials demonstrating that whole number is shared into 10 parts (tenths) or into 100 parts (hundredths). | SWBAT work with money representing dollars and parts of a dollar (cents) after the decimal place. SWBAT calculate money up to 3 decimals places| The decimal place separates two different mediums. Students read $7. 125 as $7 and 125 cents or $8. 25| Use play money kits with notes and coins and hundreds boards to allow students to manipulate and record money to two decimal places.
Students also need exposure to financial maths problems where the answer contains more than two decimal places, and be guided to consider the reasonableness of the answer.
Record algorithms on extended place value charts to ensure correct values and alignment of digits. Double check with calculator. | | Section 3 This extended place value chart will assist with reading and placement of decimal numbers (First Steps in Mathematics: Number, 2004, p. 72) undreds| tens| ones| hundreds| tens| ones| Tenths| Hundredths| thousandths| thousands| ones| ? fractions| 3 4 6 4 2 7 ? 1 2 5 This resource is invaluable for understanding decimals, reading and placement of digits. This shows students that the decimal does not have its own place but is a separator between whole and decimal numbers. It simplifies reading large numbers by showing grouping of three places. You can see clearly the ones column is our base unit of number and that’s our axis of symmetry. This resource is very useful and concise.
It clearly shows columns, grouping of digits and is also useable when performing addition or subtraction of whole and decimal numbers. It doesn’t lack anything in regards to being a useful resource, as it can be expanded or reduced depending upon what grade level is being taught.