The Most Versatile Tool-NCTM
As you enter . . . Solve this problem using a number line There are 18 blackberries in the bowl. Kelly eats 11 blackberries. How many are left in the bowl?
Solve this problem using a number line There are 18 blackberries in the bowl. Kelly eats 11 blackberries. How many are left in the bowl? • Find a person at your table who has a different number line than you. • Compare how your thinking / representation was similar and different.
With your group, Discuss the advantages and limitations to a ticked number line.
The Most Versatile Tool in the Elementary Mathematics Classroom: The Number Line! Connie Conroy Denise Bogart
Howard County Public Schools Ellicott City, Maryland
The Most Versatile Tool in the Elementary Mathematics Classroom: The Number Line! Thank You!
Outcomes Understand how number lines can represent mathematical thinking in the intermediate grade. Articulate ways a number line can develop understanding of whole numbers, fractions, and decimals
Mathematics Teaching Practices
NCTM (2014)
• Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.
What does Jo Boaler say. . . SEEING AS UNDERSTANDING: The Importance of Visual Mathematics for our Brain and Learning. By Jo Boaler, Professor of Mathematics Education with Lang Chen, Stanford Cognitive and Systems Neuroscience Lab, Cathy Williams & Montserrat Cordero, youcubed. Stanford University
What Does The Brain Science Say?
What Does The Brain Science Say? •
Talk with a partner.
•
Create one statement about number lines based on this article excerpt.
Number Lines and Brain Research Continues In his book How the Brain Learns Mathematics, David Sousa explores the idea of “The Mental Number Line”. • “Humans possess a mental number line, where we envision numbers as points on a line, with 1 on the left and 2 to its right . . . “ • “When we decided which of two numbers is larger, we mentally view them on our internal line and determine which one is on the right.” • He explain that over the last 40 years numerous experimenters have made discoveries about how people compare numbers.
Number Lines and Brain Research Continues • These experiments measured the time it took adults and children to decide whether a two-digit numeral was larger or smaller • Is a number larger or smaller than 65?
• The responses grew longer as the numbers got closer in value. • Is 71 larger or smaller than 65?
• The response time was shorter as the number values became more distant. • Is 43 larger or smaller than 65?
What do you think about this?
Think About It! What concepts support the understanding of this question? What does this have to do with the understanding of number lines?
Using a Number Line to Solve Addition Problems 68 + 35
• Solve using number line. • How is the number line model connected to partial sums?
68 + 35
68 + 35
68 + 35
384 + 244 Is it efficient to use a number line to solve this problem?
382 + 244 = Is it efficient to use a number line to solve this problem?
382 + 244 = Is it efficient to use a number line to solve this problem?
Adding Fractions
Use a number line to solve one of the problems below.
A Ryan is training for the Spring Fun Run and needs to run at least 1 mile per day. If Ryan runs to his school which is ⅝ of a mile away, and then to his dad’s work, which is ½ of a mile away, will he have trained enough for the day?
Adapted from Illustrative Mathematics
B Madeline and Mallory are building a castle out of wooden blocks. They need 3 ½ buckets of blocks for the castle they have in mind. Madeline has 1 ¾ buckets of blocks and Mallory has 1 ¼ buckets of blocks. If they combine their buckets, will they have enough blocks to build their castle?
How Far is it . . . From 29 to 38?
Subtraction : Constant Difference and Adjusting
For students to use this strategy they need to develop meaning by experiencing it with many numbers, testing their prediction, and making conjectures.
Constant Difference/Adjusting
What is the difference? What happens if I add more to each tower?
7 10
Constant Difference / Adjusting
What is the new expression? How did it affect the difference?
7+1=8 10+1=11
Constant Difference / Adjusting
What is the new expression? How did it affect the difference?
7+2= 9 10+2= 12
Constant Difference / Adjusting
What is the new expression? How did it affect the difference? Continue to add blocks and write the new expression. Find the difference until you have added 5 more blocks to the minuend and the subtrahend. What math rule can you generate?
Constant Difference /Adjusting
What is the difference? What happens if I take one away from each tower?
Constant Difference/Adjusting
What is the difference? What happens if I take one away?
6 =7-1
9 =10-1
Constant Difference/Adjusting
What is the difference? What happens if I take one away? Does the same math rule apply if you take the same amount away?
Constant Difference / Adjusting • Sam and Jane have 83 points all together. Jane has 56 points. How many points does Sam have? • Work with a partner to create 3 different number line representations that will have the same difference. • Choose which problem is easiest to solve? Be ready to justify your thinking.
How does this information help me? • Look at the following problem.
4000 -1725 •
Adjust each by taking one away.
3999 -1724
I can do that in my head!!!
Subtract fractions On Monday night Wendy spent ½ an hour on her math homework. On Tuesday night she spent ¾ of an hour on her math homework. How much longer did Wendy spend on her homework on Tuesday night than on Monday night? Adapated from Beyond, invert and multiply
Subtract fractions Kira’s favorite cookie recipe needs 1 ⅓ cups of sugar. Jeremy’s favorite cookie recipe needs ⅔ cup of sugar. How much more sugar does Kira’s recipe need than Jeremy’s?
Adapated from Beyond, invert and multiply
Modeling Multiplication How does solving multiplication of whole numbers support the understanding a fraction times a whole number? What misconceptions can students have solving this using the algorithm?
Fractions on the Number Line
Why is 3 x 2/6 the same as 6 x 1/6?
Reasoning On A Number Line
How can a number line help students to reason about the size of numbers?
Reason About Numbers On A Number Line
762
How could you adapt this activity for the numbers below?
44,683 4.46
Closure
Mathematical Teaching Practices?
NCTM (2014)
• Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.
In Closing. . . Number Lines help students understand the operation that they are using. Think of one way you will use number lines with your students next week.
Solve this problem using a number line There are 18 blackberries in the bowl. Kelly eats 11 blackberries. How many are left in the bowl? • Find a person at your table who has a different number line than you. • Compare how your thinking / representation was similar and different.
With your group, Discuss the advantages and limitations to a ticked number line.
The Most Versatile Tool in the Elementary Mathematics Classroom: The Number Line! Connie Conroy Denise Bogart
Howard County Public Schools Ellicott City, Maryland
The Most Versatile Tool in the Elementary Mathematics Classroom: The Number Line! Thank You!
Outcomes Understand how number lines can represent mathematical thinking in the intermediate grade. Articulate ways a number line can develop understanding of whole numbers, fractions, and decimals
Mathematics Teaching Practices
NCTM (2014)
• Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.
What does Jo Boaler say. . . SEEING AS UNDERSTANDING: The Importance of Visual Mathematics for our Brain and Learning. By Jo Boaler, Professor of Mathematics Education with Lang Chen, Stanford Cognitive and Systems Neuroscience Lab, Cathy Williams & Montserrat Cordero, youcubed. Stanford University
What Does The Brain Science Say?
What Does The Brain Science Say? •
Talk with a partner.
•
Create one statement about number lines based on this article excerpt.
Number Lines and Brain Research Continues In his book How the Brain Learns Mathematics, David Sousa explores the idea of “The Mental Number Line”. • “Humans possess a mental number line, where we envision numbers as points on a line, with 1 on the left and 2 to its right . . . “ • “When we decided which of two numbers is larger, we mentally view them on our internal line and determine which one is on the right.” • He explain that over the last 40 years numerous experimenters have made discoveries about how people compare numbers.
Number Lines and Brain Research Continues • These experiments measured the time it took adults and children to decide whether a two-digit numeral was larger or smaller • Is a number larger or smaller than 65?
• The responses grew longer as the numbers got closer in value. • Is 71 larger or smaller than 65?
• The response time was shorter as the number values became more distant. • Is 43 larger or smaller than 65?
What do you think about this?
Think About It! What concepts support the understanding of this question? What does this have to do with the understanding of number lines?
Using a Number Line to Solve Addition Problems 68 + 35
• Solve using number line. • How is the number line model connected to partial sums?
68 + 35
68 + 35
68 + 35
384 + 244 Is it efficient to use a number line to solve this problem?
382 + 244 = Is it efficient to use a number line to solve this problem?
382 + 244 = Is it efficient to use a number line to solve this problem?
Adding Fractions
Use a number line to solve one of the problems below.
A Ryan is training for the Spring Fun Run and needs to run at least 1 mile per day. If Ryan runs to his school which is ⅝ of a mile away, and then to his dad’s work, which is ½ of a mile away, will he have trained enough for the day?
Adapted from Illustrative Mathematics
B Madeline and Mallory are building a castle out of wooden blocks. They need 3 ½ buckets of blocks for the castle they have in mind. Madeline has 1 ¾ buckets of blocks and Mallory has 1 ¼ buckets of blocks. If they combine their buckets, will they have enough blocks to build their castle?
How Far is it . . . From 29 to 38?
Subtraction : Constant Difference and Adjusting
For students to use this strategy they need to develop meaning by experiencing it with many numbers, testing their prediction, and making conjectures.
Constant Difference/Adjusting
What is the difference? What happens if I add more to each tower?
7 10
Constant Difference / Adjusting
What is the new expression? How did it affect the difference?
7+1=8 10+1=11
Constant Difference / Adjusting
What is the new expression? How did it affect the difference?
7+2= 9 10+2= 12
Constant Difference / Adjusting
What is the new expression? How did it affect the difference? Continue to add blocks and write the new expression. Find the difference until you have added 5 more blocks to the minuend and the subtrahend. What math rule can you generate?
Constant Difference /Adjusting
What is the difference? What happens if I take one away from each tower?
Constant Difference/Adjusting
What is the difference? What happens if I take one away?
6 =7-1
9 =10-1
Constant Difference/Adjusting
What is the difference? What happens if I take one away? Does the same math rule apply if you take the same amount away?
Constant Difference / Adjusting • Sam and Jane have 83 points all together. Jane has 56 points. How many points does Sam have? • Work with a partner to create 3 different number line representations that will have the same difference. • Choose which problem is easiest to solve? Be ready to justify your thinking.
How does this information help me? • Look at the following problem.
4000 -1725 •
Adjust each by taking one away.
3999 -1724
I can do that in my head!!!
Subtract fractions On Monday night Wendy spent ½ an hour on her math homework. On Tuesday night she spent ¾ of an hour on her math homework. How much longer did Wendy spend on her homework on Tuesday night than on Monday night? Adapated from Beyond, invert and multiply
Subtract fractions Kira’s favorite cookie recipe needs 1 ⅓ cups of sugar. Jeremy’s favorite cookie recipe needs ⅔ cup of sugar. How much more sugar does Kira’s recipe need than Jeremy’s?
Adapated from Beyond, invert and multiply
Modeling Multiplication How does solving multiplication of whole numbers support the understanding a fraction times a whole number? What misconceptions can students have solving this using the algorithm?
Fractions on the Number Line
Why is 3 x 2/6 the same as 6 x 1/6?
Reasoning On A Number Line
How can a number line help students to reason about the size of numbers?
Reason About Numbers On A Number Line
762
How could you adapt this activity for the numbers below?
44,683 4.46
Closure
Mathematical Teaching Practices?
NCTM (2014)
• Establish mathematics goals to focus learning. • Implement tasks that promote reasoning and problem solving. • Use and connect mathematical representations. • Facilitate meaningful mathematical discourse. • Pose purposeful questions. • Build procedural fluency from conceptual understanding. • Support productive struggle in learning mathematics. • Elicit and use evidence of student thinking.
In Closing. . . Number Lines help students understand the operation that they are using. Think of one way you will use number lines with your students next week.
