## Counting Back

Pre-K-2
1

Students count back to compare plates of fish-shaped crackers, and then they record the comparison in vertical and horizontal format. They apply their skills of reasoning and problem solving during this lesson in several ways. [Because students have associated the word "more" with addition, the comparative approach to subtraction is typically more challenging for the students to understand.]

Teacher Note: As you prepare to teach this unit, you may wish to consider what management strategy you will use to distribute the fish-shaped crackers. Packaging the fish-shaped crackers in resealable plastic bags, one for each student, will keep the material sanitary.

To open this lesson, provide each student with crackers, two paper plates, crayons, and graph paper. Then read a counting book about fish (one option is Fish Eyes: A Book You Can Count On by Lois Ehlert) or some other counting book.

As the students listen to the story, have them model each number named by adding one fish-shaped cracker to a plate, writing that numeral at the beginning of a row on the graph paper, and then placing one fish cracker per square to show the number. [Encourage students to have rows of 10.] Then ask them to remove the fish, one row at a time; coloring the squares in that row that had held fish crackers. Have students color each row a different color using yellow, purple, and other colors of their choosing. If you choose not to read a counting book, you might sing a favorite counting-up song, such as "This Old Man," and have the students model and record the number featured in each verse.

 1 2 3 4 5 6 7 8 9 10

To the tune of an easily-remembered song, such as "A Hundred Bottles of Beer on the Wall," sing a more appropriate counting back song, "Ten Little Fish on the Plate." Have the students place crackers on the appropriate bar on the graph they created in the activity above as they sing each verse of “Ten Little Fish.” [This activity allows you to determine whether any students have trouble matching numerals to sets, counting, or writing numerals.] Now ask: How many more squares are colored in the yellow bar than in the purple bar? How do you know? [Possible answer: 1; I counted one more. This question introduces the new mode of subtraction, the comparative mode.]

Next hold up a number, such as eight, and ask the students to put that many fish-shaped crackers on a plate and record the number of crackers. Then ask them to make a second plate with one less cracker.

Now tell them they will compare the number of crackers on each plate. To do this, have them remove pairs of crackers from the plates by taking one cracker in each hand from each of the plates and placing the crackers in the bag until crackers are left on only one plate. Then have them count the crackers left on the plate. Ask for a volunteer to model and record the comparison using the number sentence [equation or horizontal] format. [For example, 8 – 7 = 1.] Now ask for a volunteer to record the comparison using the vertical format. Encourage the students to describe the comparison in two ways, in one instance using the word "more" and in the other instance using the word "less." [For example, there is one more cracker on the plate that had eight crackers. There is one less cracker on the plate with seven crackers.]

Ask a student to name a number from 5 to 10. Divide the spinner into that many parts. Now call on a volunteer to spin the spinner and tell which number came up. Ask the students to show the number with fish-shaped crackers. Then tell them to show a group of crackers that is one more than that number, and then a group that is one less. You may wish to have them record their responses on the One Less/One More Activity Sheet.

Repeat with other volunteers.

To end the lesson, have the students divide some crackers between two plates, then remove a cracker from each plate until all the crackers on one plate are gone. [If you wish, you may encourage them to eat each pair of crackers as they remove it. If you prefer to use the crackers another day, be sure each bag is labeled with the student's name and closed securely.]

Assessment Options

1. Students may draw plates that show "one more" and "one less" for entries for their portfolios.
2. The information that you collect about the students' understanding and skills throughout the unit allows you to focus on individual student's needs and strengths and plan additional learning opportunities. For this reason, a recording sheet, Class Notes, has been provided for you. You may also find the information that you record useful when discussing the students' progress.
Extension
Move on to the next lesson, How Many More?

Questions for Students

1. How many fish-shaped crackers are on this plate? (Show a plate with six crackers.) On this plate? (Show a plate with seven crackers.) Which plate has more? How many more? Which plate has less? How many less? How can you prove that?

[6; 7; the second plate has more; 1 more; the first plate has less; 1 less; student responses may vary.]

2. What number sentence would show that you compared a plate of six fish-shaped crackers with a plate holding five fish-shaped crackers?

[6 - 5 = 1.]

3. What is alike between the two ways that we recorded the comparisons? What is different?

[The same numbers are compared, but the way we wrote them was different (one way was horizontal, and one way was vertical.)]

4. How could you help a younger student find the answer to 7 - 6 by counting?

[Student responses may vary.]

5. What equation shows that you compared a plate of 8 fish-shaped crackers with a plate with 10 fish-shaped crackers?

[10 - 8 = 2.]

6. What does the minus sign mean?

[Subtract.]

7. What two symbols did you use to mean "equals"?

[In the equation form of recording subtraction, = is used. In the vertical form, the line symbolizes "equals."]

8. When you count backward, what comes after 10? After 7? After 1? What comes before 6? Before 4?

Teacher Reflection

• Were the students able to model the numbers as the books were read? What other books might I use?
• Which students were not yet able to record numbers of the bar graph?
• Which students met all the objectives of this lesson? What extension activities would be appropriate for those students?
• Which students did not meet the objectives of this lesson? What instructional experiences do they need next? What mathematical ideas need clarification?
• What adjustments will I make the next time that I teach this lesson?

### How Many More?

Pre-K-2
Students write subtraction problems, model them with sets of fish-shaped crackers, and communicate their findings in words and pictures. They record differences in words and in symbols. The additive identity is reviewed in the context of comparing equal sets.

### Hopping Backward to Solve Problems

Pre-K-2
In this lesson, students determine differences using the number line to compare lengths. Because this meaning is based on linear measurement, it is a distinctly different representation from the meanings presented in Lessons One and Two. At the end of the lesson, the students use reasoning and problem solving to predict differences and to answer puzzles involving subtraction.

### Balancing Equations

Pre-K-2
This lesson encourages the students to explore another meaning for operations of subtraction, the balance. This meaning leads naturally into recording with equations. The students will imitate the action of a pan balance and record the modeled subtraction facts in equation form.

### Fact Family Fun

Pre-K-2
In this lesson, the relation of addition to subtraction is explored with fish-shaped crackers. The students search for related addition and subtraction facts for a given number and also investigate fact families when one addend or the difference is 0.

### Wrapping Up the Unit

Pre-K-2
During this final lesson in the unit, the students use the mathematical knowledge and skills developed in the previous lessons as they visit five stations to review comparative subtraction.

### Learning Objectives

Students will:
• Model numbers and write numerals to 10.
• Subtract from numbers to 10 by counting back.
• Record differences in vertical and horizontal format.
• Recognize the symbols used in recording subtraction.

### NCTM Standards and Expectations

• Count with understanding and recognize "how many" in sets of objects.
• Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
• Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections.
• Use multiple models to develop initial understandings of place value and the base-ten number system
• Understand the effects of adding and subtracting whole numbers.
• Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.
• Develop and use strategies for whole-number computations, with a focus on addition and subtraction.
• Develop fluency with basic number combinations for addition and subtraction.
• Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

### Common Core State Standards – Mathematics

-Kindergarten, Algebraic Thinking

• CCSS.Math.Content.K.OA.A.1
Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

-Kindergarten, Algebraic Thinking

• CCSS.Math.Content.K.OA.A.2
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

-Kindergarten, Algebraic Thinking

• CCSS.Math.Content.K.OA.A.5
Fluently add and subtract within 5.

• CCSS.Math.Content.1.OA.B.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

• CCSS.Math.Content.1.OA.C.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

• CCSS.Math.Content.1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

• CCSS.Math.Content.2.OA.B.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.