Make 10 to add

Base 10 Lesson 1: Materials for teaching base 10, and early work with base 10.

A list of some things that your should know about teaching and learning the base 10 number system:

Decomposable and/or Groupable materials have a lot of advantages for introducing the base 10 number system.

Craft sticks, grouped into 10s using rubber bands are a groupable, decomposable, student grouped material:

3 groups of 10 and 2 single craft sticks

Beans, grouped into 10 in a cup are a groupable, decomposable, student grouped material

3 cups of 10 beans and 5 extra beans

Decomposable/Groupable materials are materials that consist of objects that represent ones, and a system for grouping those objects into tens. A common example is craft sticks and rubber bands to make bundles of 10.
Decomposable and/or Groupable materials are particularly good for kindergarten and first grade.
Base 10 work is explored somewhat in kindergarten, and it is a major focus of math in first and second grade.
Children need lots of experience with grouping and ungrouping objects in tens as they make the transition from thinking of numbers as ones to thinking of numbers as tens and ones (some discussion of this can be found in Van de Walle pgs 123-124)
The added difficulty of navigating numbers when thinking of them as tens and ones rather than as just a large collection of ones makes solving problems more difficult for a while, but then makes it easier as children get used to thinking in terms of tens and ones.
A base 10 task that's tricky at first is counting using tens and ones, in particular switching from counting by tens to counting by ones is tricky. For example, if a child has 3 sets of tens and 4 ones, they may be able to count 10, 20, 30, just fine, but then they may continue counting :10, 20 , 30, 40, 50, 60, 70 instead of switching to ones and counting 10, 20, 30, 31, 32, 33, 34.

Counting a group fo 34 by tens and ones

Groupable/decomposable materials can be grouped to add and decomposed to subtract

Proportional Pregrouped materials are somewhat more abstract than decomposable proportional materials

The most common pregrouped proportional materials are Diens blocks, which are often called base 10 blocks (invented/popularized by Zoltan Diens)
Proportional means that a ten is exactly the same size as 10 ones, and hundred is exactly the same size as 10 tens (and 100 ones). Decomposable materials are also proportional (a ten is made of 10 ones so it is the same size as 10 ones)

Cuisenaire rods which represent the numbers 1-10 have the same size unit as Diens blocks (which is 1cm x 1cm x 1cm)
Pregrouped materials can't be put together and taken apart, so moving from 10 ones to a ten is a trading action rather than a grouping action.
Pregrouped materials are more abstract than groupable materials.
Pregrouped materials encourage seeing amounts in groups rather than as ones.
Pregrouped proportional materials are sometimes more obviously proprortional than student grouped materials (the equality of size is a main property of the material)
Work with pregrouped materials should come after work with groupable materials.
Instead of grouping and decomposing, with pre-grouped materials, we have to trade or exchange amounts to make or break down tens.
Children can productively repeat the same sorts of counting and number making exercises with pregrouped materials that they did earlier with groupable materials.
Pregrouped materials are ideal for second grade and third grade, and when available, can also be productively used in first grade.

Non-Proportional materials are more abstract than proportional materials

Non-proportional materials include number discs, Montessori stamp game tiles and money (particularly coins).

Non-proportional materials means that the value of the material is represented in a way that is different from size. It can be amounts of money for coins or a numerical value written on the chips or tiles.
Non-proportional materials rely on childrens knowledge of trading and exchanging place value materials when representing numbers.
When using coins to solve problems and represent numbers, it's can be a good idea to begin with only pennies, nickels and dimes, and add quarters later (when children have mastered pennies, nickels and dimes) because quarters don't match the base 10 structure of our numbers as well.
Non-proportional materials are more abstract than proportional materials, and should follow work done with proportional materials:
- Children can productively repeat the same sorts of counting and number making exercises with non-proportional materials that they did earlier with proportional materials.
- Work with non-proportional materials moves children further towards the abstraction of working with only numbers, rather than numbers and materials both.
Children as young as second grade can work with non proportional materials effectively, but it should follow work with proportional materials. Non-proportional materials shouldn't replace proportional materials until (ideally) late third grade or fourth grade.
Non-proportional materials are more convenient for work with larger numbers (especially thousands) because the problems require less space.

Activities for learning about base 10 with materials

There are two main kinds of work for learning about the base 10 structure of numbers: counting or adding; and making or naming a number
Counting or adding tasks consist of repeatedly adding on to an amount of counters, and grouping when appropriate. This can be done as a task where the child repeatedly adds on 1 more each time--usually while recording the total number of counters at each step. This can also be done as a game (race to 50 or 100 or 200 etc), where children repeatedly add on a number of counters given by a die roll--grouping when appropriate, and telling the total amount they have at the end of each turn (the game ends when someone reaches the target number).
Making and naming a number tasks usually involve a teacher's participation in naming numbers for children to build using the base 10 materials or providing bags of materials for students to count and name.

Children can also do counting back or subtracting tasks with base 10 materials. This would consist of starting with some amount of counters, and repeatedly taking some away from the start number. As with the adding on work, this can be done as a task where children repeatedly take away one from a start number (ungrouping where necessary), or it can be done as a game where children take away an amount determined by a die roll (in this case the game ends when the first person gets down to 0).

Standards

The Common Core standards for kindergarten call for students to count and write numbers to 120, but do not emphasize base 10 concepts
The first grade standards emphasize place value understanding to 100 (tens and ones)
The second grade standards empasize place value understanding to 1000 (hundreds tens and ones)
Building base 10 place value understanding is especially an area of emphasis in grades 1 and 2.
Base 10 place value concepts remain an important part of the curriculum with additional sophistication and with more of a focus on operations (addition, subtraction, multiplication and division).