Background
The following areas provide background information on regrouping in addition and subtraction. This section includes information on place value and the models used to present information, as well as the foundational steps that should be mastered prior to regrouping with two digit numbers. Without a mastery of these first two foundational steps (which can be addressed throughout the entire school year), students may lack the understanding of basic math concepts necessary to regroup. These steps are also included to demonstrate the connections addition and subtraction have to place value. Many curriculums are now structured as spiraling, meaning if students don't understand a concept now they will understand it sometime later. Although this can be good for review of skills it doesn't lend well to a deep lasting understanding of skills and concepts. The structure of these steps is more of a ladder: it is necessary to fully master one step before progressing to the next.
Place Value
Place value is the method for writing any whole number as a sum of other numbers in terms of its base 10 expansion. For example, the number 1234 denotes 1 thousand + 2 hundreds + 3 tens + 4 ones. The number 0 is used as a place holder to denote that there is not an amount represented in the given place, i.e. in the number 1034 there is nothing in the hundreds place. Each of the other single digit numbers gives an amount that is represented in each place while taking into account that each place in itself is an amount, as in the example above. Each number and each place combine to create their own place value.
Number Line
The number line is a horizontal line, with a designated "0" point, and a designated "1". The one is placed to the right of the 0. The distance between the 0 and 1 becomes the unit length, and dictates the placement of every other integer on the number line. For example, 2 is marked at one unit distance to the right of 1, 3 is marked at one unit distance to the right of 2, and so on. It is important to maintain equal units when working with the number line and when working with number problems to give an answer in number and word form to indentify the unit.
Concrete – pictorial – symbolic
One fundamental ideal within this unit taken from Singapore math 2 is the flow of concrete, pictorial, and symbolic models. By starting with concrete objects or manipulatives students regardless of reading level or even language skills are able to manipulate and discover patterns in numbers. This can include the use of counting bears, base ten blocks or counting tiles, and will allow students who are reading below grade level or may be English language learners to participate in the lesson and begin to build an understanding of the math concept.
After working with concrete models, the next step is to work with pictorial models. It is best to use pictures that relate back to the concrete activity. This makes this step even smother and continues to easily involve struggling readers. The second step begins the process of students transferring hands on learning to paper. While students may see some numbers, at this stage they will not work with mathematical symbols (+, -, =, for example). As students work with pictures and numbers they are exploring how numbers can be taken apart and exploring the part-part-whole aspects of numbers (i.e.13 and 4 are parts of 17 the whole) and students are continuing to deepen their knowledge and understanding of how numbers combine together to build other numbers, and how they may be taken apart.
The final model in this Singapore progression is the symbolic. At this stage, students will refine the skills from the two previous steps and begin the transition into paper and pencil work, and then onto mental math. For example students will begin using manipulatives to explore how amounts work together such as combining 4 bears with 3 bears to make 7 bears. Then students will use pictures and or number cards to continue building and breaking apart numbers, and finally use symbols with the numbers to solve problems using algorithms. See the lesson activities for an example related to each of the three models. Teaching just one or two lessons in each area is likely not enough to allow for the mastery of a concept like concrete addition or subtraction. This is essential before moving onto the next stage and integral for a deep and lasting understanding of mathematical concepts. Each of these three models will be reoccurring throughout this unit.
Addition and Subtraction Facts to 10
Addition and subtraction using the digits up to 10 is a first foundational step towards addition and subtraction of more general (larger) numbers. Prior to beginning addition and subtraction time should be given to concretely and pictorially discuss how to break apart numbers, and then build them back up. For example a student could be given 8 bears and asked to make two groups. Students should become familiar with and practice writing down all the different amounts that can make up the parts or groups of a given number or quantity. Once students have a strong understanding of this, they should be prepared to begin working with basic addition and subtractions. Along with ensuring student have a solid and deep foundation to numbers and how they work together this time will also allow for the opportunity to introduce the concrete – pictorial – symbolic model. This will allow all students to become familiar with a process we will be incorporating throughout the year and allow students who do not yet have a comfort with basic addition to develop their skills and mastery of the concept. To ensure that students who do understand their basic concepts are not distracted I will also intertwine word problem activities.
During this step students will be working with numbers for both addition and subtraction while exploring all they ways they are connected. We will be using fact families where all related numbers are explored. (i.e. 2+3=5, 3+2=5, 5-2=3, 5-3=2), as well as missing addends (i.e. 3+?=5). The concrete model will be explored through the discussion of word problems and with use of manipulatives students can physically act out the problem. An example can be seen in classroom activity 1. Moving into the pictorial model I will utilize the idea from Singapore 3 of "number bonds" which illustrate addition and subtraction without using symbols (+,-,=) It is a part-part-whole way of thinking about numbers ( for example 3 and 4 are parts of 7):
Once students have mastered all the ways to take apart and build numbers within 10 we will then move into the symbolic portion using the signs (+,-,=) and working with pencil and paper. Students will revisit all of these skills as they progress, but for a deep and lasting understanding, a good deal of time should be give to allow student to master each step. The suggested time allotment for this step is 3 to 5 weeks, with flexibility to allow for mastery.
Teen Numbers As One Ten and Some Ones
The way teen numbers are named in most languages can be confusing. As our numbers progress it becomes easier to see that twenty-three is 2 tens and 3 ones, but what is eleven? In this second step place value will be introduced and then used as a means to add and subtract. Students will be given an expended amount of time to gain a deep understanding of what at teen number is and not simply a vocabulary lesson. We will spend time discussing the names of our teen numbers and how we can "rename" them or think of them in a different way. Using concrete models with manipulatives and base 10 blocks paired with a discussion of how eleven is 1 ten and 1 one students will begin to gain an understanding of our base 10 number system and place value.
As students master these concepts and stages they will also begin to develop automaticity for the basic facts, however it is important for later steps that they continue to write out problems. This step lends itself to comparing numbers and working with greater than and less than. The integration of word problems will assist in blending all of these concept areas. The suggested time allotment for this step is 3 to 5 weeks.
Higher Addition and Subtraction Facts Within 20
After steps one and two above, students should be prepared to progress to adding and subtracting within 20 (i.e. using the numbers 1-20), using the same concepts. For example, just as we used base 10 blocks to illustrate adding within 10, 6+8 for example, we will continue to discuss word problems within 20 with base 10 blocks. As we move between the pictorial and symbolic portion of this step students will be directed on how to write out their work using the expanded form. As an example, I might ask students to solve problems such as 14 + 5 = ? , 10 + 4 + 5 = ? From here I will direct student to use the knowledge they have from the first step and their understanding of number bonds to combine 4 + 5 to make 9, then writing 10 + 9 = ? At this point, they should easily see the answer they are working toward is 19. "The Break Apart to Make a Ten" method will also be used in concrete then pictorial and finally symbolic. Within the following example of the symbolic model the Associative Property (i.e. that multiple addends can be regrouped in any ways) is also demonstrated:
6 + 8 = (4 + 2) + 8 = 4 + (2 + 8) = 4 + 10 = 14.
Due to the many layers of this step, the suggested time allotment is a flexible 4 weeks.
Two Digit Numbers As Some Tens and Some Ones
In this step we will explore how the –ty numbers (the twenties, thirties, etc) are made up of some 10's. For example, twenty is made of two tens, sixty is six tens, and just as students were taught that 13 is make of one ten and three ones, they will learn that 23 is make of two tens and three ones. These activities will again move through the concrete – pictorial – symbolic model. It is important that this step, as with the step with teen numbers, not be downgraded to a vocabulary lesson. Students will be working with larger numbers and will need to use the skills from previous steps to build or take apart these 2 digit numbers. The skills they learn now will be mirrored and built upon as they work through larger numbers and more difficult concepts. We will continue to work with the base 10 blocks to give student the concert and visual connections for arranging numbers and rearranging them. I will also emphasize writing the numbers in a variety of ways (23, twenty three, 20 + 3) and using them in more complicated word problems.
I will also introduce the use of the number line in this step allowing students to become accustomed to laying out their base 10 blocks on, drawing and writing with the number line or ray. I will make a point to discuss that we are only using part of the line and not all of it so when they are introduced later to negative numbers on the line they will not be confused by the idea that a number line must start with 0 or 1. We will also look at how numbers are related to each other on the 100's chart by mapping out patterns of 10's and 1's. I will allow for at least 4 weeks for this step.
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