Groups of objects are useful for a variety of real-world math applications, especially for recognizing attributes, and sorting, classifying, and ordering by attributes.

Attribute materials are sets of objects that lend themselves to being sorted and classified in different ways. Natural or unstructured attribute materials include things such as seashells, a set of children's shoes, or even the children themselves. The attributes will determine the ways in which the materials can be sorted.

A structured set of attribute pieces has exactly one piece for every possible combination of values per attribute. For example, attribute links have multiple attributes, such as color, shape, and size. The specific number of attributes that a set may have is not important.

Children should begin by exploring the group of objects. You can help them put their observations into words by asking questions such as:

• How would you describe the objects to someone who can’t see them?
• In what ways are the objects alike?
• What do you notice different about the objects?

Allow time for children to sort the links into different groups by the attributes they choose. Children should show each other their groups and discuss the attributes they sorted.

You can also help children draw connections between the sorting they do at home, such as sorting toys or laundry, and the sorting they do in class. Have children practice their sorting skills on everyday objects in the classroom, such as coins or shirts in a variety of colors.

Base ten blocks facilitate children’s understanding of 2-digit numbers and help them recognize the value of these greater sets. As they use the blocks, children often find patterns that help them discover solutions.²

To model 2-digit numbers, children can use ten rods and ones cubes. It is important that children recognize that a ones cube is a single cube that represents both the number 1 and the ones place in the base-ten system. This understanding will help children appreciate that a ten rod represents ten connected ones cubes as well as 10 and the tens place.

The number of ten rods used corresponds to the digit in the tens place of a number. The number of ones cubes corresponds to the digit in the ones place of the number.

To model a number, children should identify the digit in the tens place and lay out that many ten rods. They identify the digit in the ones place and lay out that many ones cubes.

To identify a number represented by a model, children count the number of ten rods to find the digit in the tens place. The number of ones cubes shows the digit in the ones place. Children can also count the total number of ones cubes in the entire model, but this brute force method is time-consuming and ignores the merits of place-value concepts.

Clock manipulatives can help children strengthen basic time concepts in hands-on activities. These clocks allow children to examine a clock’s attributes closely, to manipulate the hands, to see how time wraps back on itself, and to develop a sense of how clocks track the passage of time.

Give each child a clock. Encourage children to manipulate the hands and watch what happens when the hour hand travels around the clock. Together count aloud, “one o’clock, two o’clock, three o’clock,” and so on, until you reach 12:00. Help children to see that the next hour is not “thirteen o’clock,” but 1:00.

Use the clocks in a class activity to address sequencing concepts. Ask children to show what the clock’s hands might look like in the morning when they wake up (seven o’clock, for example), what the clock’s hands look like at lunch time (twelve o’clock), and what the clock’s hands look like at bedtime (eight o’clock). Use this as an opportunity to have children make and show specific times to the hour.

You can extend the activity by conducting elapsed time activities. Ask children to show two o’clock, for example, and then ask them to show what the clock would look like one hour later (3:00).

Since money is a concrete part of children’s everyday experiences, it is an easy way to engage students. Coin manipulatives enable children to examine each coin’s attributes closely, reinforce counting concepts, and develop a sense of how to estimate a quantity of money based on a visual scan of a handful of coins.

Give each child a selection of coins. Tell the children you are going to play a game called “Count It Up”; here you will challenge them to “build” a certain amount of money using their available coins. For example, you might say, “Build 27 cents.” At that point, each child will create a combination of coins equaling 27 cents, such as two dimes and seven pennies.

Invite children to share their results and to compare and contrast the combinations. Ask them to identify their peer’s combinations and notice who used the most and the fewest numbers of coins. Then, challenge each student to make a different combination equaling 27 cents. Encourage them to see that the number 27 can be built in different ways.

Connecting cubes can be used to show both the taking away and the part-part-whole models of subtraction.

To model “taking away,” have children start with a number of connecting cubes less than ten. Have them name the starting number of connecting cubes.

Have children break off 3 of the cubes, and tell how many they took away.

Finally, have children tell how many are left.

The taking away activity can be repeated with any variety of numbers less than 10.

The next activity encourages children to use a related addition fact to solve a subtraction problem. Each pair of children will need 10 connecting cubes and an index card. Have partners work together to make a train of 10 cubes or less. One partner breaks off part of the train and hides it under the card while the other partner looks away.

The second partner then uses the visible counters and logical reasoning to determine how many cubes are under the card. He or she should consider what number goes with the part they see to make the whole. Children should then remove the card to check the answer.

Partners take turns and repeat the activity.

Connecting cubes can be used to help children visualize tens and ones in a two-digit number because they allow singles to be grouped as tens. This grouping most clearly reflects the relationshop between ones and tens.

To model a number, children should identify the digit in the tens place and lay out that many groups of ten. Then, they should identify the digit in the ones place and lay out that many individual cubes. Have children practice by writing the number 43 and modeling this number. Check the models for accuracy and repeat with other numbers.

When children have demonstrated fluency with the skill, have them work in pairs. One student tosses a number cube two times to create a 2-digit number. The other student uses connecting cubes to model it. Children work together to check the model. They alternate roles and repeat the activity.

To develop a deeper understanding of how and why to measure, children should be given many opportunities to measure and compare real objects with both nonstandard and standard units. Children at this level of development should become familiar with the use of rulers, scales, liquid cup measures, and thermometers, as well as their related units of measure. It is important to introduce these tools gradually as a complement to the nonstandard units of measure more commonly used in kindergarten.

Have children work in teams of two to trace each other’s feet onto a sheet of paper.

Have children estimate and measure the length of their feet in paperclips. Introduce the ruler. Have children estimate and measure the length of their feet in inches. Have them record the data in a class chart like the one shown below. Invite children to note any patterns they see. Allow time for them to discuss how they measured, and to compare and contrast measurements made with paperclips and with a ruler.

You can extend the activity by asking children to describe other measurements they might find on their bodies. Ask them to discuss why it might be difficult to find the distance around one’s head, for example, and challenge them to generate strategies for finding that measurement.

Number lines help children visualize the magnitude of a number and the relationships among numbers. They assist children in counting up or counting down from one number to another, and reinforce the concepts of “one more” and “one less.” Number lines also help with proximity—that 2, for example, is closer to 0 than to 10.

Relative magnitude refers to the size relationship one number has with another—“Is it much larger, much smaller, or about the same?”

This Mystery Number Game develops children’s understanding. Draw a number line labeled with 0, 5, and 10.

Have children count the hatches between 0 and 10 so they can visualize each mark as having its own value, one more than the number before it, and one less than the number after it. Then label one of the unmarked hatches with a question mark, and have children identify the mystery number. Ask them to explain their thinking so that the class benefits from hearing and seeing different strategies.

Children can also play the game in pairs—taking turns pointing to a hatch mark, and then identifying the missing number.

Children developing spatial awareness and learning geometric concepts progress through increasingly complex levels of understanding. “Level 0” learners—for the most part, children at the kindergarten level—benefit from activities that engage them in sorting, identifying, and describing of shapes as well as the use of physical models that they can hold and manipulate.

Pattern blocks help children explore relationships of direction and position. Pattern blocks are colorful figures available in common shapes such as squares and triangles. The physicality of the blocks, as well as the colors and sizes, maintain student attention and reinforce the learning of shape names and their properties.

Give each child a selection of pattern blocks to work with. Then, direct children to build pattern block pictures based on clues that you provide. For example, you might say, “The rectangle is above the square,” or “The circle is below the triangle.” As children become more adept at using the shapes, offer more complex arrangements such as, “The triangle is above the rectangle,” and “The square is to the right of the triangle.”

As you progress through each lesson, use the pattern blocks to assess children’s comprehension of concepts. Children who can work independently can work in pairs. Direct one child to offer the clue while the partner builds with pattern blocks, and then have them switch roles. Independent workers can also build more open-ended patterns. You can assess comprehension by asking them to describe their work using position words.

Real graphs, such as concrete graphs, use the actual objects being sorted and graphed. Most of the graphs that children work with in Chapter 6 are symbolic graphs, which use something like squares, blocks, tallies, or X’s to represent the things being counted. Tally marks give children a visual model of number quantities. The advantage to young math learners is that tallies do not require a great deal of hand-eye coordination or sophisticated graphing skills to be able to accurately record data.

Explain to children that the word tally means “to count.” Tally marks are a symbol where each mark represents one element in a set. Tell children that as they count, or tally, they should make a single mark for each object.

Give each pair of children blank paper, a pencil, and a handful of red and blue counters. Have each pair sort the counters into a red pile and a blue pile. Have one child arrange the counters into rows and count the counters aloud. As they do this, have the other child make a tally mark for each counter counted.

Explain that five tally marks are grouped with a diagonal line across the first four marks. Have children erase the fifth mark and draw it diagonally across the first four.

Set up small centers of a variety of objects with between five and 10 elements in each set. Have partners count each group together and represent the number in each group with tally marks.

Children can use a ten frame to model numbers and the addition of numbers. The models provide a visual image, reference the benchmark number 5 (a top or bottom row), and develop a sense of quantity relative to the base of ten.

To model a number such as 4, children should be asked to place one counter in the ten frame starting in the upper left corner and then add one more at a time. Children should fill the ten frame across the first row, starting from the left just as they read a book.

After the first row is full, children begin placing counters in the bottom row—again starting on the left. Children use 5 as a reference when they model numbers 6–9.

As children begin to work with sums, they can use counters in two colors to fill the tenframe. The counters represent the two addends.

After children can model sums with counters, give them 10 two-color counters. Ask them to model the sum 10 in at least three different ways. Challenge them to find all the different ways to model the sum.