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NC Mathematics Standard(s):

Operations and Algebraic Thinking

Represent and solve problems.

NC.2.OA.1 Represent and solve addition and subtraction word problems, within 100, with unknowns in all positions, by using representations and equations with a symbol for the unknown number to represent the problem, when solving:

o   One-Step problems:

§  Add to/Take from-Start Unknown

§  Compare-Bigger Unknown

§  Compare-Smaller Unknown

o   Two-Step problems involving single digits:

§  Add to/Take from- Change Unknown

§  Add to/Take From- Result Unknown

Number and Operations in Base Ten

Use place value understanding and properties of operations.

NC.2.NBT.8 Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Standards for Mathematical Practice:

1.       Make sense of problems and persevere in solving them.

2.       Reason abstractly and quantitatively

3.       Construct viable arguments and critique the reasoning of others

4.       Model with mathematics

5.       Use appropriate tools strategically

Student Outcomes:

·         I can use a number line to represent addition and subtraction word problems.

·         I can solve addition and subtraction word problems using strategies related to place value.

·         I can communicate how I solved problems to my teacher and classmates.

Math Language:

What words or phrases do I expect students to talk about during this lesson?

      addition, count, count on, group, hundreds, ones, subtraction, tens

Materials:

·         ten strips, activity sheet, base ten blocks

Advance Preparation:

●     Gather materials

Launch:

Patterns in Two-Digit Numbers (8-10 minutes)

Use ten strips, ten frames or base ten blocks to have students count by tens.

For example:

Start by displaying 4 ones. Ask students to count with you (4). Add a ten, students should say 14.

Continue to add tens and have students count on (24, 34, 44, 54).

Do this for 3-4 different start numbers. A student can record the numbers on the board. Ask students, “What do you notice about the numbers that we counted?”

Emphasize that when we add 10, the digit in the ones place stays constant, but the digit in the tens changes by 1.

Repeat with a few different start numbers.

Subtracting Backwards

Start with a large two-digit number such as 91.

Remove tens and have students count backwards by tens (81, 71, 61, etc.).

Explore

Exploring the Open Number Line (20-25 minutes)

Part 1:  Explain to students, “We are going to use counting by tens to help us solve story problems.”

Display an open number line on the board. Start at 4 and make hops of 10. Relate this to the counting they did with the ten strips (or ten frames.) The teacher can model this for several different start numbers.

Note: An open number line is just an empty line used to record children’s addition and subtraction strategies. Only the numbers children use are recorded and the addition (or subtraction) is recorded as leaps or jumps.

Display this story problem and read it aloud. 

Tim had 18 baseball cards.  For his birthday he got 29 new cards.  How many cards does Tim have?

You can change this problem to include names of students in the class. Feel free to change the numbers if the students in class need smaller/larger numbers. Easier problems have high numbers in the ones place (7, 8 or 9) so that it is easier to jump to a multiple of 10.

For example, if a child’s strategy for adding 18 + 29 is to keep 29 whole and decompose the 18 into smaller pieces, the jumps on the open number line could be to start at 29, jump 10 to 39 and then jump 8 more to 47. Another strategy is moving to a landmark or friendly number of 30. Since the jump from 29 to 30 was a jump of 1, the student needs to still jump 17 more from 30, which gets them to 47. Those jumps on the number line can be written as: 29+1+10+7= 47.

Part 2: Another Story Problem

Tom and his mom are driving to the zoo. It is 75 miles away. They have already driven 36 miles. How many more miles do they have to drive? (This is an Add to, Change Unknown problem.

Have students pair-share (talk with a partner) about what the problem is asking and how they would solve it. After about 1 minute ask students to share their thoughts with the class.

Ask, “What is the problem asking?” and “How would you solve it?”

Ask, “How far away is the zoo?” Students should say 75 miles. On a number line mark 0 and mark 75. Ask, “How far have they gone already?” Students should say 36. On the number line mark 36.

Ask, “What do we need to find?” Students should talk about finding how far 36 is from 75.

If students struggle, guide them with the following questions, “Should our answer be more than or less than 75? Why?”

You could also have a student act the problem out by walking in front of the classroom.

Have students share how to solve the problem. Examples of strategies:

Start with 36 and count up until you get to 75. If a number line is posted in the class the teacher could have a student start at 36 and have the class count up by ones to 75. Keep track of the count with tally marks. In this lesson, you want students to realize that this is not a very efficient method for solving the problem.

Draw an empty number line (horizontal line). Explain that is a tool for solving problems. Include an arrow on either end to show that the number line continues indefinitely in both directions. Place a point on the number line labeled 36.  Remind them about how they are counting by tens and how this is a way to count to 75 without saying all the ones. Record the jumps of ten saying, “36, 46, 56, 66, 76— oops that too far. I’ll go back to 66. How should I hop to 75?” Some students may suggest going by ones.  Say, “OK 67, 68, 69, 70, 71, 72, 73, 74, 75.”

Record each number beyond 66 as individual hops.

After recording on the number line ask, “How will this (referring to the number line) help us know how far they have to drive to the zoo?” Have students come up to the number line and show the hops and how to determine the answer.

Another approach could be start at 36 and jump to 40, then jump to 50, 60 70 and then to 75.  The equation would be 4 + 10 + 10 + 10 + 5 = 39

After recording the number line ask, “How will this help us know how far they have to drive to the zoo?” Have students come up to this number line and show how to use it to determine the answer.

Questions to ask?

How much farther did Tom and his mom have to drive to get to the zoo?  How do you know?     Show us on the number line.

There are multiple ways to jump on the number line from 36 to 75. Other suggestions may be: 36 to 46 to 56 to 66 to 76 and then subtract 1 (36 + 10 +10 +10 +10 -1); 36 to 66 to 70 to 75 (+30 +4 + 5).

Ask, “How are these strategies similar or different?” Possible responses on how they are alike: They all jump by tens. They start at 36 and end at 75.

Possible responses on how they are different:

One starts at 36 and jumps to a “friendly” number 40 and then jumps by tens. One makes bigger jumps (40).

After a student has shared what the problem is asking, the teacher asks students to think of an equation that they could write for the problem that they just solved.

36 + _____= 75.

Ask, “What does the blank mean in this equation?” Answers might include: “It’s the part you figure out. It’s the answer.  You have to solve 36 plus what equals 75.”

Part 3: One More Story Problem

Maria and John are going to the beach. It is 68 miles away. They have already driven 31 miles. How many more miles do they have to drive?

Ask, “What equation would represent this story.” 31 + ____ = 68.

Some students may know that you can subtract to solve this problem 68-31 = ___.

Draw an open number line on the board.

Ask, “How can we use the number line to solve the problem?  Think-pair-share for a minute.” After pairs have discussed how to solve it. Have them work the problem, using a number line, on a white board or notebook paper.

As they are solving the problem, observe students. Look for students who:

·         know to start at 31.

·         know how to jump by tens and label the number line.

·         know how to jump by tens but do not label the number line.

·         “hop” up the number line by ones.

·         are not making the connection of how to use the number line to solve the problem.

·         see this as a subtraction problem.  Can they start at 68 and hop backwards to 31?

  As you observe, choose the strategies that you want shared.                                                                              

Discuss

Discussion of Strategies Used to Solve Story Problems (8-10 minutes)

After students have had a few minutes to solve the problem ask students to share their strategies: Show the students’ strategies on the board. Let students draw the number lines or have them use the document camera to show their work.

After 2-3 different ways of using the number line are given ask:

How are these two ways alike?

How are they different?

How are we using what we know about counting by tens to work on the number line?

Note: The class discussion is critical to helping students build an understanding of how place value can be used to solve addition and subtraction problems. The open number line is a tool for students to use their knowledge of adding multiples of 10 and 100 to solve a problem. Sharing strategies and having students compare them helps students become more fluent in using place value understanding and properties of operations to add and subtract.

Additional Activities (20-30 minutes)

More Story Problems

After sharing strategies have students complete the activity sheet Solving Problems Using a Number Line.

The teacher can have students work independently on the worksheet or work with their Think-Pair-Share partner to solve the tasks.

As the students are working look to see:

·         Do students know where to start on the number line?

·         Do students accurately jump by tens and label the number line correctly?

·         Do students know how to decompose a one-digit number to make jumps of 1 that land on a landmark (number that ends in a zero)?

·         Can students tell you how to use the number line to find the answer?

·         Do students see tasks as subtraction tasks?

There are additional story problem sheets attached to this lesson.  These problems give students choice on the numbers to use.  They are designed so that if the first number is chosen in the parentheses, then the first number should be chosen in the second set of parentheses.

For example: I saw (15, 67, 145) butterflies in the garden. (10, 20, 100) joined them. How many butterflies are now in the garden?  The numbers would be 15 + 10, or 67 + 20 or 145 + 100. 

Evaluation of Student Understanding

Informal: Checked through questioning during the lesson. Also formative assessment is done  while students are working on the worksheet.  As students are working questions to ask are;

·    Why did you start here?—pointing to the number line.

·    Where will you stop on the number line?

·    What is the problem asking?

·    How can you use the number line to find the answer to the question?

Formal: The student worksheet will be used to evaluate their initial understanding of jumping on the number line to solve the problems.

The activity sheet provides you with data on students’ understanding about using the open number line. It is normal for students to struggle with the strategy the first few times they use it. Additional lessons and tasks should be given to help students further develop understanding of this method.

Use data from worksheets and observations to plan future lessons.

Do students need to work with smaller numbers to get use to using the number line?

Is it clear that some students understand this strategy and others are struggling? If so, the lesson tomorrow could be a brief overview of this method and then divide the class into groups.

Meeting the Needs of the Range of Learners

Intervention: Students who do not understand how to use the number line may use a 100 board to solve the problem. Have them start at the beginning number and move to the ending number. Observe if they move by ones or by tens? This can be related to the game, “Plus-Minus Stay the Same and The Game of Tens and Ones.”

If the numbers seem too large, change the numbers in the problem so they only have to move one ten and a few ones. If they then move on the number line by ones show them a jump of ten for the ten ones.

Students could use ten sticks and ones to solve the problem. The teacher could help them see the relationship between the ten sticks/ones and the open number line.

Extension: Some students will be able to make jumps larger than ten—larger multiples of ten. Ask, “How would you record your moves?

Other students will understand that they can move in tens beyond the targeted number and then subtract. For example, when determining how far 56 is from 92 a student may make 4 jumps of 10 or a move of 40 and then subtract 4. Ask, “How would you record your moves on the number line?”   

Possible Misconceptions/Suggestions: