So, as children develop place value understanding, there are different phases that they must go through and successfully accomplish/understand before they can move on to more complex understanding. These are the phases:

**Pre-Place-Value Understanding**– This is when children can count to a certain number as they count by one’s but may not necessarily understand what the numbers stand for. For example: a child may be able to count 23 jelly beans but doesn’t really understand that there are twenty and three jelly beans, or two sets of ten and three jelly beans. This is also known as**Pre-Base-Ten Understanding**as which required**unitary**counting.**Base-Ten Understanding**– At this level, students understand groups of 10 and can count these as a single object. Students will be able to**count by tens**and then continue counting in what is “left over” as the ones. With this understanding, it is important to give students lots of opportunity to count in different ways and express these numbers with numerals and words to make that connection. This is also when place-value understanding is specified. It is important to have students explore how to group objects for counting in this phase. So that they “know” that there are groups of ten because they created them, such as banded straws or cups of counters, or rods of cubes.**Equivalent Understanding**– At this stage of understanding, children see units of ten and hundred and do not need to singularly count each object to know that they can count with these objects as groups, but they must have had adequate time learning with groupable models first. For example, 1 hundreds-flat , 3 tens-rods, and 2 ones-cubes is equal to 132. This is also the stage at which children learn to count and work with numbers that are represented with nonproportional models.

**Learning and Modeling** – It is important that teachers provide students with adequate work and experience with each stage before they move them forward. The suggestions of counting cubes, links, crayons, shoes, students, toothpicks, buttons, beans, plastic chips, craft sticks, beans, washers, etc. was great. I wouldn’t have thought of counting anything and everything really. Also, linking number word forms with numerals is an important factor which needs to be included and continued as learning progresses. Once students have mastered equivalent understanding, the activity 11.2 in which students count various objects and record them as a number word and as tens and ones is powerful. Additionally, with that same worksheet, understanding of ten-ness is reinforced with counting “ungrouped” objects and recording the numbers, counting out a given number of objects and filling in tens-frames to represent, and “looping” objects in groups to create a certain number.

**Benchmark Numbers** – Benchmark numbers are key number of recognition such as multiples of 10, 100, and later on special numbers such as multiples of 25. This will assist students as they learn to work computations with numbers. Using a hundreds chart or a number line can assist in this understanding. For example, when students are multiplying 45 x 4 they can also see that as 45 + 45 + 45 + 45. It would be easy for students to figure out this by using benchmark numbers. So, students could say I am multiplying 50 x 4 or adding 50 together 4 times and then I am taking away those 5’s that were missing so taking away 20. 50 x 2 = 100 x 2 = 200 – 20 = 180.