Math mindset or mathematical thinking is a logical process that isn’t necessarily related to numbers, but basic math language is absolutely dependant on them. That’s why Happy Numbers prepared a comprehensive summary on how numbers to 1000 are taught in its curriculum. This overview unifies counting, number sequencing, notation, and place value. For Part 1, we’ll focus on numbers from 10 to 20 and from 20 to 100 through detailed examples of how two-digit numbers are presented in the curriculum. Learn more about Happy Numbers’ pedagogy, see the variety of visual models, and follow your students’ path of learning numbers through several grades!

The invention of numbers long, long ago most likely stemmed from counting. And from ancient times to the present, when learning numbers we first grasp them as a means to count. Numbers to one hundred are rich enough to provide material for the first encounter with basic concepts of our decimal system.

## 1. The Ten and Teens

### Grouping Objects || The Ten

Counting objects can be simplified by arranging them into equal groups. It’s no wonder that we most often arrange objects in groups of ten because ten is the origin of our decimal system.

Starting from Kindergarten, Happy Numbers uses two well-known types of ten-grouping:

The ten frame is used for any kind of objects while the ten rod is part of the Base-10 Blocks representation, which is also used with sets of real objects or for visualizing numbers.

Work with teen numbers starts with separating a ten group from sets of 11 to 19 objects. For example, in the following task, students arrange the objects in the *ten frame*:

Answering the question in all tasks attracts students’ attention to the fact that the box contains 10 objects

To see the full exercise, follow this link.

Students don’t count the total yet. However, they get used to this convenient representation of the total as a “box” and a “small” number of remaining objects:

Students also work on similar tasks using Base-10 Blocks:

To see the full exercise, follow this link.

There are exactly 10 places for cubes, so students here compose a ten rod and represent the total as 1 rod and some (loose) cubes. Watch the video below to see the whole exercise.

It helps students to associate the number 10 with its visual representation by a ten frame or ten rod. It’s important that these groups are also associated with the *tens place* of the numeral. Happy Numbers employs an exercise to promote this association:

When the answer is incorrect, students get full support:

To see the full exercise, follow this link.

### The “How Many” Question

The first step to mastery of counting more than ten objects is answering the “How many” question for *sets with visually separated groups of ten*. Happy Numbers provides tasks of this type using a variety of objects. For example:

To see the full exercise, follow this link.

Watch the video to see all the steps, including the animation that composes the total from two one-digit numbers of boxes and loose apples:

The next step to mastery of counting of objects is to answer the “How many” question for sets arranged less strictly than those mentioned above. At the same time, a configuration of these sets allows for separating a group of ten objects *mentally*. For example:

In case of an incorrect answer, students receive the following support:

Then comes the main question:

Notice that here the wording has become “more mathematical”: tens and ones instead of boxes, etc. The Happy Numbers support changes accordingly. For example, if a ten digit is incorrect, students see “1 ten” highlighted in their own answer to the previous question:

These exercises substantially extend the counting skills of students and prepare them for further advancement: counting a greater number of objects in various configurations.

### Names and Sequencing of Numbers to 20

In all the tasks mentioned above, students submitted their answers as numbers. One remaining skill to address is the counting sequence and connecting teen numbers to their names.

The Number Line is perfectly suited for visualization of the counting sequence, so Happy Numbers starts to employ it as early as Kindergarten. Experience with the Number Line in the range from 0 to 20 begins with two engaging exercises.

In the first exercise, students position numbers from 11 to 19 in their places on the Number Line. Numbers pop up one by one in a ball that students move along the Number Line and drop where they think the number belongs:

If placed incorrectly, the ball flies away and then returns with the same number, until students finally put it in the correct place.

In the second exercise, tasks consist of putting three numbers in their places on the Number Line:

The first number, 9, is already in its place. The current number to be placed is 18

To see the full exercise, follow this link.

To connect teen numbers to their names, Happy Numbers uses two-step tasks, like the following:

Each of the tasks focuses on one number, with all numbers 11 to 19 appearing in quasi-random order. The sequence of tasks is automatically adjusted to the needs of a student: in case of an error, the number is added again to the sequence after a few others.

It’s worth mentioning that using the Number Line here has a bonus: preparing students for comparison of numbers in Grade 1 that is effectively supported by the Number Line visual model (see Comparison of Whole Numbers within 100 || Strategy 2 in this blog article).

## 2. Two-Digit Numbers and the First Touch to Place Value

By this point in the Happy Numbers curriculum, students have extensive experience with numbers: learning the counting sequence to 100, understanding the basics of addition and subtraction, and understanding the composition of teen numbers as 1 ten and some ones. Students were also introduced to zero. With this foundation, they’re ready to master two-digit numbers. For students who need to refresh and consolidate their prior knowledge of the prerequisite subjects, there is a review in the beginning of Grade 1 in Module 1, Topic A.

Happy Numbers provides exercises on two-digit numbers in *two sets* that differ mostly in the range of the numbers involved: numbers to 40 and numbers to 100. In other respects, the exercises are very similar. For example:

Not only the task itself but also the entire interactive support are completely similar. The same is true for other pairs of сorresponding exercises. So, in what follows we show just one of the pairs.

### The Tens

The tens __10, 20, 30, 40, 50, 60 ,70, 80, 90__ are base camps on a road trip through the first hundred. Happy Numbers provides several exercises in which the tens are represented by sets of objects arranged into easily recognizable groups of 10. The exercises first employ the ten frame model. For example:

Students actually start with two auxiliary questions, asked one by one and aimed to focus them on the data needed to answer the original question. The screenshot below shows the basic questions answered correctly, the original question with an incorrect answer, and a hint provided when a non-zero digit was entered in the ones place:

In case of an incorrect tens digit, a visual hint highlights each of the boxes. To see the full exercise, follow this link.

Similar exercises employ the ten rod model:

When students successfully complete these exercises, the preparatory questions are no longer included in the tasks:

…. and support is only provided to students who need it.

Notice that *cubes play a double role* in the Happy Numbers curriculum. They can be viewed as another object to be counted, like apples, acorns, etc. On the other hand, they are often used in the Base-10 Blocks visual model to represent a number. So, tasks that involve cubes can be interpreted as “What number is shown by the Base-10 Blocks?”.

### The “How Many” Question

In the above exercises, students have already started to answer the “How many” question for sets of more than 20 objects, however, only when they are *arranged in groups of ten*. The next step is working with sets *arranged in groups of tens and some remaining loose objects*, too few of which make up another ten. For example:

Students are well-prepared for this step since they have experience in:

– working with the tens 10 to 90

– combining 1 ten and some ones to compose numbers 11 to 19

Students start with counting pre-arranged objects:

Students should first submit the correct number of rods.

Answering the question about cubes is blocked until then

Students who need to correct their answers receive visual hints:

Questions are asked here in terms of particular objects: *rods* and *cubes* or boxes of plums and loose plums. In the next step, answers about the objects are interpreted in terms of *tens* and *ones*, which leads to identifying the total:

To see how Happy Numbers provides support in case of an incorrect answer and gradually reduces the scaffolding of support, follow this link to see the full exercise.

The next step is counting objects that are *not pre-arranged* as in the above exercises. At this point of the curriculum, Happy Numbers requires students to arrange objects before counting. This simplifies the task, transforming it to the type on which students have worked recently. After solving these exercises, students work on tasks that let them choose between counting objects in the given configuration or rearranging them first:

Here, students discover on their own that rearranging objects in groups of ten simplifies counting.

### Tens and Ones

In the previous exercises, students learned to easily recognize tens and ones in the ten frame or the Base-10 Block representation of numbers.

The more advanced and important skill is *recognizing tens and ones in numerals*. Happy Numbers employs the Hundred Chart to help students develop that skill. The Hundred Chart perfectly suits the task, since it arranges numbers with the same tens in rows and with the same ones in columns.

Special introductory tasks prepare students for work with the Hundred Chart. Then students solve a number of exercises in which they fill in a missing number and identify its tens and ones:

To see the full exercise, follow this link.

The entire series of these exercises are discussed in detail in the blog Number Sense with the Hundred Chart.

### Unit and Expanded Forms of Two-Digit Numbers

With the acquired skill of recognizing tens and ones in a number, students are ready to represent two-digit numbers in *unit form*. For example, represent 28 as 2 tens + 8 ones. Students grasp the skill through two types of tasks:

The tasks are first supported by the Base-10 Blocks visual model, and then with hints highlighting the tens or the ones digit:

To see the full exercise, follow this link.

Next is representing a two-digit number as the sum of tens (their numerals end with 0) and ones, for example, 47 = 40 +7. Happy Numbers provides exercises to introduce this skill, and a typical task looks like this:

Here, the first input only accepts first a two-digit number and second one-digit number, which is of course an implicit hint. Further support is provided as explicit hints that address all possible errors:

This way, students interactively learn to represent two-digit numbers in *expanded form*.

Exercises in this topic initiate students into understanding the fundamental concepts of the decimal number system — place values and the number’s three forms of notation: standard, unit, and expanded form (the terms are not used in the Grade 1 curriculum). Students grasp on an intuitive level that:

– 1 ten is made up of 10 ones

– Tens are written as 10, 20, 30, 40, 50, 60, 70, 80, and 90

– A two-digit number is the sum of one of the tens and some (0 to 9) ones

– Digits of any two-digit number represent its tens and ones parts

At this point in the curriculum, student perception of these facts is supported by visual representation and work with objects. Happy Numbers provides corresponding activities to solidify conceptual understanding and tangible skills of counting objects and representing numbers.

### Names and Sequencing of Two-Digit Numbers

Number names go hand in hand with the counting sequence. Happy Numbers employs the Number Line representation in the corresponding exercises.

As far as two-digit numbers are concerned, the sequencing and naming of numbers is best begun with tasks using the tens: 10, 20, …, 90.

The first tasks refresh and solidify students’ knowledge of the counting sequence:

… with visual hints specifically addressing errors in each digit:

The next tasks focus on number names:

On successful completion of exercises using tens, students move to naming and sequencing all two-digit numbers. Happy Numbers maintains the style and scaffolding of the previous exercises. For example:

So far, exercises on the counting sequence to one hundred were limited to filling in isolated blanks in the sequence. In the next level, Happy Numbers requires filling several blanks in a row, counting forward and backward and including and crossing tens when counting.

The first tasks involve counting in either direction without crossing tens:

In case of an incorrect answer, students receive visual hints: if the tens digit is incorrect, then the tens digits are highlighted in the given numbers.

Similarly, if the ones digit is incorrect, then the ones digits are highlighted in the given numbers.

Next comes counting *across tens* in any direction, with the same support in case of an incorrect answer. To see the full exercise, follow the link.

Finally, there is a series of tasks that can be used for building fluency and formative assessment:

In these tasks, just a few ticks on the Number Line are labeled with numbers, and there is no scaffolding.

### Numbers: Visualization and Modeling

To ensure student success in mastering two-digit numbers and initial conceptual understanding of the decimal system, Happy Numbers employs a variety of models and means of visualization. Each of them has its own advantages in certain domains, and each, except the final one, expires when it comes to a wider range of numbers.

The ten frame is the first representation that helps students count 10 to 99 objects by arranging them in groups of 10. This effectively illustrates the main idea of the decimal system: the largest possible number of such groups is represented by a digit in the tens place, and the number of remaining objects by a digit in the ones place.

At the same time, usage of the ten frame is practically limited to numbers within one hundred or slightly beyond it. There is a good reason for this: by virtue of the decimal system, when there are 10 or more ten frames they should be grouped, too, which results in a “ten frame of ten frames.” This isn’t an effective representation for numbers beyond 100. So, the ten frame is out of the game when it comes to numbers beyond one hundred. Another example of a very helpful visual model within 100 and unsuitable beyond it is the Hundred Chart.

So, Happy Numbers extends to 1000 use of the *Base-10 Blocks* representation (within 100 it’s employed along with the ten frame and Hundred Chart). Its suitability for larger numbers is not the only reason to use this representation. Even more important is that the Base-10 Blocks model is very helpful for modeling addition and subtraction. Learn more about that in our Teacher’s Best Friend: Base-10 Blocks article. However, the Base-10 Blocks representation in its turn expires for numbers over 1000. At that point, we use the Place Value Discs representation, which is comprehensively described in the article 5 Concepts Best Explained with a Placе Value Chart and Disks.

Switching to the Place Value Discs representation is a step toward more abstraction (some curricula skip this step), while the final goal on this road is the *Number Line*. In contrast to all means of visualization mentioned above, the power of the Number Line is that it works not only for the whole numbers but for all real numbers, including negative, rational, and irrational. At the same time, this representation is so simple that Happy Numbers gradually and carefully introduces the Number Line starting in Kindergarten.

__Which strategies do you like most to use in the classroom?__

Share your experience and opinion in the comments below or via social media with mention of @happynumbers.