Mathematical Background
Decomposing Numbers
Decomposing numbers is the first step in learning addition and subtraction. This step cannot be skipped! Students cannot learn to add and subtract numbers if they don't know what makes up each of those numbers. Decomposing a number is just breaking down a number by finding out what other numbers make that number. For Instance the number 7 may be decomposed into 3 and 4, or 2 and 5, or 1 and 6.
Adding and Subtracting Numbers to Ten
There are three types of problems to learn while adding and subtracting numbers to ten. The types are sum (addition), difference (subtraction) and missing addend (one of the numbers being added or subtracted in missing) and students must understand how to work each type.
Addition and subtraction are the foundational skills for teaching story problems and must be taught so they can internalize the importance of inverse operations. They must understand that 5+4 =9 is the same as 4 + 5 = 9, and the inverse operation: if you have 9 and you get rid of 4, what is left is 5. They need visual representation of these concepts and they need to be able to physically move objects to see them making the numbers. I will begin with putting unifix cubes together to make "trains" and taking them apart and then we will move to tens blocks.
The tens frame is a visual representation that helps students add and subtract because they are physically moving the counters. On addition they can see the joining happening by moving the counters and on subtraction they physically move the counters out of the tens frames.
Students need to master all of the different ways of putting numbers together and taking numbers apart within ten before moving to the next step. We will spend at least four weeks on this step alone since it is the foundational skill for all other skills in mathematics. During this step we will also discuss ways to decompose each number we are working on, this will help with fluency of facts and get them ready for doing numbers above ten later.
Breaking Teen Numbers into Tens and Ones
Teen numbers can be quite confusing for younger students. Numbers like fifty-five are easier to break apart because the name tells us there are fifty, meaning five groups of ten, and the five, meaning five ones. With numbers like fourteen students want to think that it is four tens because the word four comes first in fourteen. If the numbers with the word teen in them aren't confusing enough, we (English speakers) have the numbers like eleven and twelve that don't even have the word teen in them. The non-English speaking students have even more difficulties since they are also trying to learn a new language at the same time as learning the number words.
Teaching the vocabulary of the number words at the same time we are working on the teen numbers is ideal so they can relate to what the number words mean. We will continue to use tens frames since students should be comfortable with them by this stage. They will be breaking numbers like twelve into one ten and two ones or 10 and 2.
This will also include discussing decomposing of numbers. We will discuss different ways to break the ten apart and different ways to break the two apart. The more discussions around how numbers are composed and decomposed at the early stages of mathematics, the easier it will be for students to add and subtract the bigger numbers.
Students will not only be taking the numbers apart (decomposing), but time needs to be spent on putting them together(composing) also. For instance, they need to know that seventeen is one ten and seven ones but they also need to be fluent with one ten and seven ones is seventeen.
Adding and Subtracting Numbers within 20
Students need to have mastered decomposing numbers and breaking numbers into tens and ones for them to be ready to add and subtract numbers within 20. This is where students must understand how numbers are put together and taken apart, and not just memorizing becomes very important. The key to adding and subtracting within twenty is making groups of ten, so that the adding and subtracting is easier to manage. In subtracting six from thirteen, they need to be able to think that ten and three is thirteen. Now I need six from the ten so I can cancel out the minus six. I need to break the ten into four and six, now the sixes will cancel out. Now, we have four and three left. Since four and three is seven I know that thirteen minus six is seven.
I will address adding and subtracting numbers separately on paper but they will be taught together, just like the lower numbers. It is important that students can relate to the inverse operations and the commutative property (if you know that four plus three is seven, then you know that three plus four is also seven) for them to be successful in adding and subtracting with larger numbers.
In adding eight and seven students must be able to think: What do I need to go with eight to make ten, I need two to go with eight. Now what is left when I break the seven apart? Two and Five. Therefore I have a group of ten and five left over so I have fifteen. I now know that eight and seven is fifteen. I also know that seven and eight is fifteen, and I know my inverse operations: I know that fifteen minus eight is seven and fifteen minus seven is eight.
In subtracting five from nineteen students need to break the nineteen into tens and ones, so nineteen is one ten and nine ones. Nine minus five is four, and ten plus four is fourteen. Therefore, five minus nineteen is fourteen. So I know if nineteen minus five is fourteen, I also know that nineteen minus fourteen is five, and five plus fourteen is nineteen, and fourteen plus five is nineteen.
During the adding and subtracting of numbers the tens blocks will still be used for the visual representation and in addition this would be a good time to introduce using the one hundreds chart for mathematics. Once they are comfortable with the tens blocks, I will introduce tens rods and cubes to make trains for addition and subtraction.
My students will also learn the beginning ideas of place value: that a two-digit number has a group of tens and a group of ones. They will learn to break a two-digit number into tens and ones , and understand what ten of something means (eight groups of ten is eighty). They will also learn how to put a double digit number back together when starting with tens and ones. (nine tens and six ones is ninety-six). They will learn to add or subtract ten from any double digit number and not just multiples of ten. (twenty-two plus ten is thirty-two).
My students will also learn the beginnings of base ten arithmetic. They will learn the addition and subtraction facts, connecting them to the beginnings of place value via the approach of addition and subtraction within 20. They will also learn that a two-digit number is made up of some tens and some ones, and some ways this is related to addition and subtraction.
In addition to linear problems, story problems will be used in the section to get them used to seeing them but they will not need to know the names of the types of story problems.
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