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BUILDING FLUENCY WITH NUMBER LINES The goal of these activities is to build understanding of attributes of number line so that it can become an effective tool for thinking, understanding, and solving problems. These attributes include:

• The number line is a length model. Quantity is represented by distance or measurement on a number line (not the tick marks themselves);

• The number line is continuous. Between any two numbers, there are more numbers. Number lines are important for developing understanding and strategies in many mathematical domains—counting, magnitude, measurement, addition, subtraction, multiplication, fractions, decimals, coordinate geometry, functions, etc. In the early grades, number lines should be used to develop number sense and understanding of number sequence and magnitude, as well as a tool for making sense of and solving addition and subtraction problems. Number lines are used in many elementary math program materials but they expect that students already understand the attributes of number lines and can use them with understanding; that is often not the case. Use these activities to embed concrete number line models into the regular use of your math program materials so that students will understand the attributes of number lines before they are expected to use them flexibly to solve problems. The activities are organized into different sections by suggested grade level; however, if you are teaching grade 1 or 2 students who have not had prior exposure to number line models, or students who are having difficulty interacting with number lines, you may need to go back and start with the beginning level activities. CONTENTS Kindergarten ................................................................................................................................................. 2

10-Bead Sticks ........................................................................................................................................... 2

20-Bead Sticks ........................................................................................................................................... 3

Number Paths ........................................................................................................................................... 3

First grade ..................................................................................................................................................... 5

Concrete Bead String ................................................................................................................................ 5

Number Path ............................................................................................................................................. 6

Number Line Showing All Numbers .......................................................................................................... 8

Transferring to an Open Number Line ...................................................................................................... 8

Empty Number Line Flips ........................................................................................................................ 11

Second grade .............................................................................................................................................. 13

Transferring from a Concrete to an Open Number Line ......................................................................... 13

Linking the Concrete Bead Rack to an Open Number Line: .................................................................... 13

Empty Number Lines Flips from 0-100: .................................................................................................. 14

Ogap Items that focus on understanding of open number lines ............................................................ 15

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KINDERGARTEN In kindergarten the focus should be on working flexibly with numbers up to 10, developing foundational number concepts like cardinality and conservation, counting to 100, and understanding teen numbers as “ten ones and some more ones.”

10-Bead Sticks Make 10-bead sticks with 5 red beads and 5 white beads on a thin wooden skewer (Use a glue gun to glue wooden beads on each end).

Have students make numbers from 0-10 by sliding the beads to one side of the stick. For example, show students 7 on the bead stick (as shown below) and ask “How many? How do you know?” Students can observe that 7 is 7 ones, 2 more than 5, or 3 less than 10.

Below are more tasks you can use as warm-ups or anytime for ongoing practice and fluency. They are listed in increasing order of difficulty, so start with one kind of question and use it regularly with different numbers until you feel students are ready for a different kind of task.

10-Bead Stick Tasks For each task, use any number between 1 and 10 for the blank. Students use their bead sticks. After every question, ask “how do you know?” Use equations to record the answers together on the board.

• Find the number ____

• Show me one more than ____

• Show me two more than ____

• Show me one less than ____

• Show me one more than _____

• Find ____. How far away from 5 is ___? How far away from 10 is ___?

• Find ____. Is ___ closer to 5 or 10?

• Find 3. What goes with ___ to make 5?

• Find 4. What goes with ___ to make 10?

• How far away is ___ from 5? from 10?

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• Start at ____ and count on to 10

• Start at ____ and count on to ____

• Start at 10 and count back to ____

• Start at ___ and count back to____

• Use bead sticks to solve addition and subtraction story problems within 10 Over time, the goal is for students to be able to find quantities without having to count every single bead.

20-Bead Sticks Later on, move to 20-sticks, made with ten black beads and then 5 red and 5 white beads.

This tool will be used to help students see teen numbers as “one ten and some more”. For example, above you can see that 17 is 10 (black) and 7 (red and white), or 17=10 +5 + 2, or 17 is 3 less than 20.

20-Bead Stick Tasks Ask the same questions listed above but with numbers between 10 and 20. Additional questions for 20-bead sticks (substitute a variety of numbers between 10 and 20):

• Find ___. What goes with the 10 to make ___?

• Find ___. What numbers do you see in ___? (10 and ___)

• Make ___(a number between 15 and 20). What numbers do see in ___? (10, 5 and ___)

• Start at 10 and count up to ____.

• Start at 20 and count back to ____.

• Show ____ and ask, how many? how many 10's? how many ones?

Number Paths After students are comfortable using bead sticks to locate numbers, introduce the Number Path, a slightly more abstract representation that still shows discrete quantities—each number is in its own square. It is important to explicitly relate the number path to the bead number line so students do not think they are separate models. Start with a 1-10 number path

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Then move to 1-20 with 10 squares shaded red. (You can also laminate a 20 strip so students can mark on them with dry-erase markers)

Number Path Tasks:

• Ask the same questions as above with Bead Sticks

• Count on and back chorally from different starting points while students follow along with their fingers on the number path.

• Start at ____. Count forward ____. Where did you land?

• Start at ____. Count backward ____. Where did you land?

• Start at ____. Where do you think you will land if you count forward ____? Check.

• Count on or back from ___ to ___. How many did you count?

• Have students use number paths as tools to solve word problems (add to, take from, put together/take apart, and compare)

There are also some OGAP Items in the Counting Patterns section of the item bank that ask students to interact with a number path: CP5 CP8

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FIRST GRADE The goals for first grade are to use number line models to gain fluency with the number sequence to 100, to count up and back by 10s on and off the decade, to see multi-digit quantities as an amount of tens and some ones, to compare numbers by magnitude, and to solve addition and subtraction problems. The sequence is to start with the concrete bead string, then move to a number path to 100, and then to a number line showing all the whole numbers.

Concrete Bead String Make a concrete bead string to 100 by alternating 10 red and 10 white beads on a 36” lacing string.

Ask the questions listed for kindergarten bead sticks but for numbers from 1-100. Students should be able to find quantities without having to count every single bead.

Additional Concrete Bead String Tasks:

• Find _____ and count by 10s

• Find ____ count back by 10's

• 10 more than _____

• 10 less than ______

• Find ___. How many tens? How many ones?

• Find 48. Is 48 closer to 40 or 50? How do you know?

• Find 89. How far is 89 from 100? How far from 0?

• Start at 7. Count by 10s. Stop at 67. How many 10s did you count?

• Start at 64. Count back three 10's. Where do you land?

• Find two numbers. Which is larger? How much larger?

• Find two numbers. How far apart are they? Which one is closer to 50? After students are familiar with the bead string, transition them to recording what they do on the bead string to the concrete number line recording sheet:

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Showing 23 and 50 on the concrete bead number line and the recording sheet

Number Path Start by having students use a number path that goes up to 20 (see First Grade). Then add another strip of ten on to make the number path go up to 30; Keep gradually adding tens, coloring them in each time, until you get to 100. If your program uses 100 chart, make an explicit link to the number path. Give a 100 chart to students and have them cut apart the rows and tape the strips together to see the relationship between the two models. (picture of 100 chart) Color every other group often red Color every other group often red

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And then tape together into one long strip:

• Use the 1-100 Number Path to answer the same questions (above) and do the same things you would have students do with the 100 grid.

• Ask students to explicitly talk about attributes of the Number path and what is the same or different from the concrete number line/ bead strings

• Move to number paths with missing numbers (increase the size of the sections you cover up).

Look for OGAP items that use number paths in the Counting Patterns section of the item bank: CP19 CP11

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Number Line Showing All Numbers The next step is to link the concrete bead number line to the number path and a number line with every number.

• Create a number line by taping a number path on the board and then drawing number line underneath. Talk about why you are making the line at the end of each square.

• Fill in all the numbers with students, emphasizing that each hash mark lines up with the end of each block on the number path.

Now students can begin using number line with all numbers showing, but if they are struggling to make sense of it, you can always go back to the number path and concrete bead string.

Transferring to an Open Number Line Once students are familiar and fluent with a number line showing all numbers, you can begin to transition to an open number line (end of first grade). Leave some numbers off the number line: cover some numbers with post-it notes and ask students to identify the numbers, then check). Next cover two or three in a row.

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Link the Concrete Bead Number Line to an Open Number Line

First mark 0 and 100:

Then mark the multiples of 10:

Have students locate numbers between 1-100 on the open number line and check with the bead rack. Next have students link their own Concrete Bead Strings to an Open Number Line:

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Linking the Number Path to an Open Number Line In the same way, use a number path to create an open number line:

Marking the end of each square on the number path with a hash mark on the number line:

Mark a spot on the open number line and ask students to identify the number:

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Add more numbers to the open number line:

Next, represent distances on the open number line:

Empty Number Line Flips https://nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk4/MM%204_12.pdf

Cut out the empty number line and fold in half horizontally on the dotted line.

Now one side shows only the numbers at the start and end. The other side shows all the numbers. Partner Activity: One student places a paper clip on the number line, then shows the empty number line side and asks their partner to guess which number the clip is on. If the students have

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difficulty starting out, their partner can help by pointing to and saying reference numbers close to the clipped number. For example, “I am pointing to 5. What number is the clip on? Empty Number Line Side: “What number is the clip on?”

Check on the reverse side :

Center/Independent Activity: Have students place a clip where they think 7 will be and then turn it over and check. Students can then transfer where they clipped to an open number line on a worksheet. Repeat for different numbers.

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SECOND GRADE In second grade, the goal is to build on work done in 1st grade to solidify the relationships between concrete number lines and open number lines. You may need to go back to first grade activities if students have not been exposed to the bead number lines or number paths yet. Below are a sample of activities.

Transferring from a Concrete to an Open Number Line Place the bead string on a piece of white paper. Mark two number and the distance between in tens and ones:

Then have students do the same thing on a bead string recording sheet:

Linking the Concrete Bead Rack to an Open Number Line:

Show addition on the

bead rack, recording it

below with an open

number line.

Here, 36 + 27 is shown.

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Similarly, the picture below shows 71-38 by jumping up or back to the nearest 10.

Empty Number Lines Flips from 0-100: https://nzmaths.co.nz/sites/default/files/Numeracy/2007matmas/Bk4/MM%204_12.pdf

Cut out a 0-100 Empty Number Line and fold horizontally on the dotted line.

One side is an empty number line from 0-100 that shows only the numbers at the start and end. The multiples of 10 are shown on the reverse side. Partner Activity: One student places a paper clip on the number line, then shows the empty number line side and asks their partner to guess which number the clip is on. If the students have difficulty, their partner can help by pointing to and saying reference numbers close to the clipped number. For example, “I am pointing to 70. What number is the clip on? Center/Independent Activity: Have students place a clip where they think 70 will be and then turn it over and check. Students can then transfer where they clipped to an open number line on a worksheet. Repeat for different numbers.

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Ogap Items that focus on understanding of open number lines You can monitor student understanding of open number lines and misconceptions by administering OGAP items from the Magnitude section of the item bank. Shown below are some examples. M1

M25

M29

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Once students have developed understanding of open number lines, students can use them as tools for solving addition and subtraction problems. Start by demonstrating how to represent student generated strategies with an open number line, for example, during a number talk. Encourage students to use the open number line as a tool to help them generate multiple strategies based on number sense. These are just a sampling of ideas for building number line understanding in K-2. Use them to create more activities to use with students.