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* *Reflection: Developing a Conceptual Understanding
Counting Up to Solve Problems - Section 3: Teaching the Lesson

As I worked today with the group of students struggling with the relative position of numbers, I began to see some progress. We looked at our blank number lines and I asked if they always had to start at 1 to count to the number they wanted. I asked what if the number they were looking for was a large number such as 68? The students looked at the number line and several students decided to start at 1, but count by 10s to get to sixty, and then go a little further to mark 68.

This shows me that they are beginning to understand that 68 is 6 tens and a little more. This is a place value concept that some of these children did not have before. They didn't understand that 68 and 86 were different numbers. By being able to show me that 68 has 6 tens, they are showing a change in their thinking.

I don't want students just to memorize that the tens place is the second digit, because this won't help them later when they begin decimals. I want them to remember that the tens place represents bundles of ten.

Teachers think this is obvious and often think a single lesson is enough for students to understand place value, but it is a deep concept, and one that is foundational to later mathematical understanding.

*Beginning to Make Sense*

*Developing a Conceptual Understanding: Beginning to Make Sense*

# Counting Up to Solve Problems

Lesson 3 of 18

## Objective: SWBAT count forward by 1's, 2's, 5's or 10's to solve an addition problem containing one and two digit numbers.

#### Warm Up

*10 min*

Articulating mathematical thinking is a critical skill in development at 2nd grade, and requires practice. Students wrote addition and subtraction word problems several days ago and some are left to share. So, I ask any child who has not shared to read his/her word problem to the class. The other students use their math journals to solve the problem.

Students are now used to this process, and I expect several students will find a solution to share. I expect others to comment on how the problem was solved or ask questions about things that don't make sense (MP3).

As students present, I particularly draw students' attentions to strategies that use counting up and number lines, because we will be using these strategies/tools today.

I also think it is important to talk about the fact that there is more than one way to solve a word problem.

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#### Practice Time

*15 min*

Today I introduce a new game to students. This game will encourage students to count up or back to solve a math sentence.

A deck of number cards is placed face down between 2 or 3 students. Students also are given a small pile of colored chips. Each student draws 2 cards and adds them up. Now they share their answers and the answer closest to 10 takes a colored chip from the pile. If 2 students are equi-distant from 10, they would both take a chip. The cards are then placed in a "discard" pile and students draw again.

I demonstrate with students how to add the 2 cards and then count up or back to see how far away from 10 they are. If a child picks up a 7 and a 4 they would add and say 7+4 =11. They would then count back 11, 10 so it is only 1 away.

Students use the number lines on their desks to help them visualize how far away from 10 they are.

Some children may use their fingers to determine how far from ten they are because the largest cards they will draw are 9+9. While I want students to move beyond fingers to other tools that will help them when they deal with larger numbers, I do not stop them from counting up or back on their fingers.

Students check each other's answers before deciding who gets the chip. They are instructed to count on the number line or number grid if they don't agree about the distance from 10.

This game requires two sets of problem solving. Both steps are building the fluency expected in the Common Core grade 2 standards.

#### Resources

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#### Teaching the Lesson

*30 min*

To start the lesson I introduce the blank number line again. I give each student a blank number line on a sentence strip. I ask them to make a mark right near the left edge. Next I tell them to put down 2 fingers and make another mark. I ask them to repeat this until they have a number line marked in even spaces.

Next I ask them to put a colored chip on the mark at the left edge. I tell them to imagine that that mark is the number 18. I ask them to use the number line to count up 10 and find 10 more than 18. We check our answers. Now I ask them to move the chip to the mark at the right end of the line. I ask them to imagine that that number is 25. I ask them to count back 10 and find 10 less than 25.

I repeat this with several more numbers. I want to reinforce the concept of ten more, ten less as a way to understand adding and subtracting tens.

Next I break the class into 3 groups based on their understanding of the number line from previous lesson informal assessments.

The group that is able to use the number line to solve problems is given a challenge paper of word problems that require adding 2 digit numbers. They are asked to work in partners, or alone to solve the problems using their number line as needed to support their thinking.

The group that has some understanding of the number line, but has difficulty with a blank line will work with a parent to locate a number on the line and then add a second number to it. The purpose here is for students to understand that they do not always have to start with the left end marking. They need to think about what number they are adding, ie is it far away from the number they are starting with, or very close. If it is far away, should they count by 2s, or 10s. The group will make some decisions about starting points together, and then solve the problems.

The group that still does not grasp how the number line works, (they always want to start at 1, even if they are looking for the number 89), will work with me to locate one number relative to another, counting by 1's or 2's and putting chips on the two numbers. If students become competent with this skill, we will try starting with one single digit number and adding on a second single digit number.

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#### Closing

*5 min*

To close, I ask student to write in their journals about how they can use a number line to help them with adding or subtracting. I encourage students to use drawings and words to show me how they are using the number line.

This may seem like a jump to abstract thinking, but I am looking an expression of mathematical thinking, at a basic 2nd grade level. According to Mathematical Practice P3, "Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades."

By asking students how they can use the number line, I encourage students to use drawings as well as words in their journals to demonstrate their thinking.

I review student journals after school to assess student understanding. One child wrote, "I count up and down on the number line." She drew a picture of a number line with hops going along the line. This shows a basic understanding of how a number line can be used.

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##### Similar Lessons

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###### The Recipe for a Great Word Problem

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- UNIT 1: What and Where is Math?
- UNIT 2: Adding and Subtracting the Basics
- UNIT 3: Sensible Numbers
- UNIT 4: Sensible Numbers
- UNIT 5: Everything In Its Place
- UNIT 6: Everything in Its Place
- UNIT 7: Place Value
- UNIT 8: Numbers Have Patterns
- UNIT 9: Fractions
- UNIT 10: Money
- UNIT 11: The Numbers Are Getting Bigger
- UNIT 12: More Complex Numbers and Operations
- UNIT 13: Area, Perimeter and More Measurement
- UNIT 14: Length
- UNIT 15: Geometry
- UNIT 16: Getting Ready to Multiply
- UNIT 17: Getting Better at Addition and Subtraction
- UNIT 18: Strategies That Work

- LESSON 1: Let Me Count The Ways to Get An Answer
- LESSON 2: Who Makes Mistakes
- LESSON 3: Counting Up to Solve Problems
- LESSON 4: Counting Backwards Works Too
- LESSON 5: Counting Bugs
- LESSON 6: Taking Apart the Problem
- LESSON 7: Getting Bigger and Smaller
- LESSON 8: Double It
- LESSON 9: Doubles Plus or Minus One
- LESSON 10: Evens and Odds
- LESSON 11: Plus Ten Minus Ten
- LESSON 12: From Tens to Nines
- LESSON 13: Equal Amounts
- LESSON 14: Understanding Subtraction
- LESSON 15: Skip Counting with 5s, 10s and 100s
- LESSON 16: Balancing Equations and Counting Backwards
- LESSON 17: Counting with Tens and Hundreds
- LESSON 18: Assessment