Using a ten frame and counters helps children develop a sense of quantity and number sense relative to the base of ten. It also provides a visual and mental image of numbers.

To model a number such as 4, children should be asked to place one counter in the ten frame and then add one more at a time. With each step, ask which group has more and which has fewer to emphasize number relationships.

Invite children to fill the ten frame across the first row, starting from the left just as they read a book. When they fill the first row, anchor the number 5 so that children can use 5 as a reference when they model numbers 6–9. After the first row is full, have children begin placing counters in the bottom row, again starting on the left.

Have children use a blue counter for the last counter in the model. This change introduces part-part-whole relationships—that there are different combinations that make the same number. Children who can represent a number in a ten frame with different color combinations will be able to translate this skill to addition concepts.

Children will need to place counters outside the ten frame for modeling numbers 11 and 12.

Children can extend this pattern for other numbers to 20 and beyond, and when they model sums greater than ten.

Base ten blocks facilitate children’s understanding of place value and help them recognize the values of numbers in larger sets. As they use the blocks, children often find patterns that help them discover the meaning of larger numbers.²

To model 2-digit numbers, children can use ten rods and ones cubes. Be sure children recognize that a ones cube is a single unit that represents the number 1 and belongs in the ones place in the base-ten system. A ten rod is comprised of ten connected ones cubes and represents 10 and the tens place.

The number of ten rods 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 a number.

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

To identify a number represented by a model, children should count the number of ten rods to find the digit in the tens place. The number of ones cubes tells 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.

Children learning to subtract 2-digit numbers benefit from using manipulatives to develop a firm understanding of place value. The use of base ten blocks helps to reinforce placevalue concepts and build visual representations of regrouping routines.

A concrete activity, such as the following, will help children as they move to the abstract algorithm. Have children work in pairs to practice building 2-digit numbers using rods and cubes. Encourage them to build each number in a variety of different ways and then challenge them to build the number in the most efficient way. The number 56, for example, can be built with 3 rods and 26 cubes, but is most efficiently built with 5 tens and 6 ones.

Then, have children model subtraction problems, such as 42 – 18. Have children build 42 (4 rods and 2 cubes), and invite them to generate strategies for “taking away” 18. Children should see that they can “trade” 1 rod for 10 cubes allowing them to take away 18 (1 rod and 8 cubes). The remaining rods and cubes show the difference. As children gain confidence in their concrete work, encourage them to record their subtraction on a place value mat.

You may wish to challenge children to solve “missing number” problems. Give them a starting number, such as 51, and challenge them to find the number that is subtracted in order to end up with a given difference, such as 35 (16).

Coin manipulatives enable children to examine each coin’s attributes closely and to estimate a quantity of money based on a visual scan of a handful of coins. Making amounts in different ways develops flexible thinking, practices the value of coins, reinforces the understanding of place value, and is a practical skill that motivates children to learn the value of different coins.

Invite children to play a game of Coins Three Ways. Begin by giving partners a selection of coins. Then explain that you are going to name an amount and ask them to show that amount in three different ways. You may wish to demonstrate how the game is played by counting out 27 cents as five nickels and two pennies. Then, show the same amount as a quarter and two pennies. Finally, show 27 cents a third way as two dimes and seven pennies. Count each grouping with the class.

Ask children to show you three ways that they can show an amount. Begin by asking partners to show 33 cents three ways. Use children’s coin combinations to build a chart that shows the various combinations. Invite children to identify which combination requires the fewest coins and which combination requires the most. Encourage children to use the data in the chart to describe any trends, patterns, or similarities that they see.

Some possible combinations that make 27 cents:

Connecting cubes are useful for exploring part-whole relationships.

Give children a different color cube for each addend. The colors will help children visualize the different parts they are joining. For example, children can see that 1 blue cube and 6 red cubes make 7 cubes, as do 2 blue cubes and 5 red cubes, 3 blue cubes and 4 red cubes, and so on.

Have children begin by building the parts completely and then joining the parts to make a whole.

Reading or writing the combinations encourages reflective thought on the “part-partwhole” relationship. You may wish to have children record their work through drawings, numbers written in blanks, or by writing addition sentences.

Dominoes are useful for modeling related facts because they can represent the addition and subtraction facts through 6 + 6. The center divider keeps the parts distinct, yet the dots are clearly part of a single whole. In subtraction situations, children sometimes have a tendency to focus only on the number left or the answer, and return the number taken away to a supply pile. This maneuver prevents children from reflecting on the situation and reversing it. With dominoes, since both the whole and parts are shown, children connect the ideas of addition and subtraction.

Children typically begin by noting the addition facts associated with the domino, such as 5 + 4 = 9 and 4 + 5 = 9.

Have them cover one side of the domino with a finger or index card and write the related subtraction fact. Repeat for the other subtraction fact by covering the other side of the domino. Ask: “How much is left when you take away 5? How much is left when you take away 4?” Have children say and record the related subtraction facts: 9 – 5 = 4, 9 – 4 = 5.

After organizing the four written number sentences, have children use different colors of highlighters to mark each number (all of the 5’s, for example, could be yellow). Help them see how the placement of the number changes, and connect the relationship between the whole and the parts in the addition and subtraction sentences.

Relate the different written forms of notation (vertical and horizontal formats) by having children rewrite their number sentences in the other format.

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. They should master length measurements with both standard and nonstandard units. An activity such as the following will help children develop measurement sense.

Separate the class into teams of two and give each team an inch ruler and a pile of paper clips. Ask each team to choose 3 classroom objects to measure. (Objects should be no larger than a desk.)

Ask teams to estimate the length of each object and record their estimate. Then have children measure the length of each object in paper clips. Show children how to record their data in a chart such as the one shown here. You may wish to have children find the difference between their estimates and actual measurements by subtracting the lesser number from the greater.

Have children repeat the activity by estimating and measuring in inches and completing a second chart.

Allow time for the teams to discuss how they made their estimates and how close their estimates came to the actual measurements.

Ask children to consider why the activity would be more difficult with larger objects, such as the distance of a room. Ask for a nonstandard unit (such as a jump rope) and a standard unit (foot) that might be more effective for measuring greater distances.

Children in the primary grades are still developing their understanding of numbers, counting, and number patterns. For this reason, it is helpful for children to work with number lines as they build their addition skills. In particular, children benefit from working with number lines as they round numbers and estimate sums.

Create a number line on the floor with the tens labeled. Ask them to count aloud by tens as they walk along the number line, stopping at each “ten.”

Then have children practice locating individual numbers on the number line and establishing which “10” is closest to that number. Given the number 52, for example, children should identify 50 as the closest number on the number line. You can teach children the convention that numbers ending in 5 round “up.”

Finally, show children how to use the number line to estimate sums. The estimate of 52 + 25 is 80, for example, because 52 is closer to 50 than to 60, and so it rounds down to 50. On the other hand, 25 is midway between 20 and 30 and ends in 5, so 25 rounds up to 30. 50 + 30 = 80.

To reinforce the concept of rounding, children can work together to create numbers to round and then add together. Have each partner take turns tossing a number cube twice to make and record a 2-digit number. Then, have partners use their number line to round their number to the nearest ten and record it beside the original 2-digit number. Finally, have children add their rounded numbers.

Pattern blocks help children identify shapes by attributes, reinforce shape names, and compare shapes. As children become familiar with basic geometric shapes, it is helpful to offer problem-solving activities that require children to explore and compare attributes. For example, first graders should be able to explain how shapes are similar or different based on a shape’s specific properties.

A game such as What Shape Am I? gives children a problem-solving experience with shape identification. To play the game, give each child a selection of pattern blocks. Then, put a shape behind your hand. Tell children you will give them clues to help them guess what shape you are hiding. Explain that after each clue, they are to put aside all of their shapes that do not match the clue. Then, offer clues about the “mystery shape” that you are holding. For example, you might say, “I have four sides.” At that point, children would remove the triangles and hexagons from their piles. Then you might say, “I have square corners.” Children would remove the rhombus, the small rhombus, and the trapezoid, leaving the square. As children eliminate possible shapes, encourage them to verbalize their problem-solving processes.

As you progress through each lesson in this chapter, you can use the pattern blocks and the What Shape Am I? game to assess children’s comprehension of concepts. You may wish to challenge advanced learners by offering longer descriptive clues that include terms such as: and, but, and not. For example, you might say. “What Shape Am I? I have no square corners and I do not have four sides.”

First graders should experience probability in an informal, pictorial, and practical way. Probability concepts should relate to the likelihood of real-world events, using terms such as more likely and less likely. Spinners are particularly useful for helping children visualize probability concepts. Children will benefit from an activity such as the following.

Separate the class into groups of two or three. Give each team a spinner divided into three unequal parts. Ask children to describe their spinners and compare the amount of space devoted to each color.

Have each group predict which color will “win” when they spin the spinner 20 times. As children take turns spinning the spinners, have them record their group’s results on a chart such as the one below.

After children have completed their charts, have them discuss the results and report which color “won.” Encourage children to explain why they think a particular color was landed on more often than another color. Guide children to understand that the more space a color takes up on the spinner, the greater the likelihood that the arrow will land on it, and the greater the likelihood that it will “win.”

Have children repeat the activity with different spinners. When children use a spinner with equal-size sections, help them understand that the probability of landing on a particular color is the same for any other color though the results may not be exact for each color.

Children were introduced to the ten frame in kindergarten and reviewed its use in number identification early in Grade 1. Besides helping children identify numbers, the ten frame also assists children in learning addition facts. For numbers greater than ten, children can use a second frame or include additional counters outside of the frame. The ten frame anchors the value of ten and gives children a benchmark reference of 5 (the top or bottom of a row) for quickly finding a sum or difference.

The addition strategy “make ten” relies on place-value concepts and children’s ability to quickly recognize the number 10. This strategy requires children to break the second addend in a addition sentence into parts, so that the first number and part of the second number make ten. Then, the second part of the second number is added to ten to find the total. This strategy is difficult to explain abstractly, but is quickly understood when demonstrated and experienced with a ten frames and counters as below:

Children can also use two ten frames to add. Give children two numbers, such as 9 + 8, to add. Ask them to model each number in a separate ten frame, and then invite them to suggest ways to use the ten frames to add the two numbers by joining the ten frames. Guide children to understand that they should move counters from one ten frame to fill the other, then add or count on from ten to find the sum. Have children explain what they did, and focus especially on the idea that counters can be taken from a frame that represents one number and put together with the other frame to make 10. Then, there is 10 and some more.

Allow time for children to practice both methods with a variety of problems, including doubles (7 + 7) and near doubles (7 + 8).

Children at this level have had some experience sorting objects in a set by attribute. Sorting objects by attribute using a Venn diagram helps children to sort by two or more attributes and to gain a visual sense of the groups and their intersection. You can use attribute blocks and circles drawn on an overhead projector to introduce children to sorting objects in a Venn Diagram. Focus the task by providing a limited number of attributes; too many choices may result in a Venn Diagram with no intersection.

Real graphs, such as concrete graphs, use the actual objects being sorted and graphed. Most of the graphs that children work with in Chapter 4 are symbolic graphs, which use something like squares, blocks, tallies, or X’s to represent the things being counted. Sorting actual objects into Venn Diagrams formed with yarn ovals is an excellent activity building a real graph, and will prepare children for the symbolic activity in later lessons.

Have children sort a variety of red shapes and different color triangles into two overlapping yarn circles. They can work with partners or small groups. Allow time for children to discuss different ways to sort the shapes. Challenge them to find a shape that belongs in the overlap area, and discuss why this shape belongs there.

Then, have children sort a collection of coins and challenge them to come up with their own attributes to sort by. Help children label their circles and have them share their results with the class as a method of checking student work.