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http://nctm.org/pa15

http://www.nctm.org/CCSSW14

Geometry, Number, Expressions

Fred Dillon Math Coach, Ideastream

[email protected]

Amy Hillen, Jennifer Outzs, Chonda Long, Jennifer Lee

Effective Teaching with Principles to

Actions: Implementing College- and

Career-Readiness Standards

Content focus: Geometry, Number and Expressions

Effective mathematics teaching practice:

Please sit with at least one person you don’t know and work on the warm-up.

Warm-up: Look at on your calculator. Is this a repeating decimal or irrational? How do you know?

Welcome!

4

13

• Understand congruence and similarity with transformations.

• Understand and apply the Pythagorean Theorem. Explain a proof of the Pythagorean Theorem.

• Solve real-world and mathematical problems involving volume.

Geometry Key Ideas

Number Key Ideas

Progressions for the Common Core State Standards in Mathematics, 2011, p7

Know that there are numbers that are not rational, and approximate them by rational numbers.

Work with square roots and cube roots as solutions to equations, such as p=x2 and q=x3. Use only small perfect squares or cubes.

Expression Key Ideas


Work with square roots and cube roots as solutions to equations, such as p=x2 and q=x3. Use only small perfect squares or cubes.

Using line segments and number lines

Using cash register tape, make a segment. Mark the distance x on the number line.

Label the endpoints so you have .

AB


Now mark another distance, y, tape . Label this segment

Are the segments congruent? How can you tell?

How could you make a segment that is congruent to ?

CD

CD


Use the tape to make a segment that is equal in length to

CD + AB

What is the length of the new segment?


What physical actions did you use to perform these tasks?

Describe your actions as transformations.


What questions could you pose to your students to have them visualize x + x = 2x?

How could you model 2x - 3y?


How could you model x2 on a single number line?

Do you think this is the best model for thinking about x2 ? Why?

Connecting to Previous Learning

Standard Algorithms in the Common Core State Standards, Fuson, K. and Beckmann, S. NCSM Journal

, Fall/Winter 2012-2013, pp. 14-30.

Thinking about x2

How can you connect the area model for multiplication to have students consider the meaning of x2?

Visual Proof

Where is c?

Visual Proof

Trace the

shape of the

large square,

and label with

a,b abd c

Visual Proof

Cut apart your

pieces to make

four triangles

and two

rectangles.

Visual Proof

Use your models to prove a2+b2=c2

Visual Proof

MoMath

Illuminations Lesson

• http://illuminations.nctm.org/uploadedFiles/Content/Lessons/Resources/9-12/LawCosinesGeo-AS-PythReview.pdf

Finding Distance

• On grid paper make a segment from A(0,0) to B(3,0). Then make a segment from A(0,0) to C(0,4). Connect B and C .

• How can we determine the distance from B to C?

More with Distance

• On the coordinate grid, make an isosceles right triangle whose legs are 5 units long and in quadrant. Have one leg extend from (0,0) to (5,0) and the other leg vertical, but not on the y-axis.

Finding Distance

• What happens when you try to count how many “squares” long the hypotenuse is?

• How long is the hypotenuse? How do you know?

• What two perfect squares is 50 between?

Finding Distance

Since 50 is between 49 and 64, what

two whole numbers is 50 between?

Since 50 is between 7 and 8, what are

two numbers that 50 is between that

are closer together?

Finding Distance

We can keep making smaller and smaller intervals, but we never get exactly to one answer. This leads us to the concept of irrational numbers – decimal representations that do not repeat and that do not have a pattern.

Finding Distance

How can you generalize what we’ve done to find the distance between any two points on our coordinate grid?

Pythagorean Theorem

How can you make segments of length

on your number line?

Try these at your table, and be ready to share your results.

2, 3, 5

Pythagorean Theorem

Did you use transformations in your work?

If so, how?

If not, what methods did you use?

Modeling Geometrically

• How can you model x2 = 16 geometrically?

• How can you model x2 = 20 geometrically?

Modeling Geometrically

• How can you model x3 geometrically?

• What applications from geometry can you think of that would give students a need to know about cube roots?

Making the connections

Thank You!

Please drop off any questions, things you’ve noticed, and/or wonderings to The Parking Lot.

Please put any awesome amazing resources (web sites, books, articles, etc.) on the Ideas page.

Disclaimer

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, official professional development offerings of the National Council of Teachers of Mathematics, support the improvement of pre-K-12 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. Each Institute is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

Contact Info

Fred Dillon

[email protected]


• Presentation slides • Handouts • Contents of the Course Binders



Grade 8 Statistics


[email protected] Gail Burrill, Jane Porath

Effective Teaching with Principles to Actions: Implementing College- and Career-Readiness

Standards

Content focus: Statistics

Mathematics Teaching Practice: Facilitate meaningful mathematical discourse

Please sit with at least one person you don’t know and work on the warm-up.

Warm-up: A newspaper article reports on a study that says if someone exercises daily, doesn’t smoke, eats fruits and vegetables, and drinks moderately, the person will live an average of 7 years longer. What questions do you have?

Welcome!

Expectations

• Compare data by constructing a scatter plot and investigate patterns of association.

• Informally find lines of best fit and use these to make predictions and determine the meaning of the slope and intercept in the context of the problem.

• Your understanding of mathematical discourse will evolve today

Women’s Salaries Are Catching up to Men’s

How would you determine the validity of this statement?

a.Google it. b.Read an article c.Do a statistical analysis d.Ask some acquaintances what their salaries are

What should 8th graders do?

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association


Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.


Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Earnings

As you look at the data. . . • What are some things you notice? • What are some things you are wondering

about?

Median Earning of Full Time Workers Men Women Year Men Women Year

48,202 37,118 2011 27,331 18,769 1989 47,715 36,931 2010 26,656 17,606 1988 45,485 35,549 2009 25,946 16,911 1987 45,161 35,104 2008 25,256 16,252 1986 45,113 35,102 2007 24,195 15,624 1985 43,460 33,437 2006 23,218 14,780 1984 41,386 31,858 2005 21,881 13,915 1983 40,798 31,223 2004 21,077 13,014 1982 40,668 30,724 2003 20,260 12,001 1981 39,429 30,203 2002 18,612 11,197 1980 38,275 29,215 2001 17,014 10,151 1979 37,252 27,462 2000 15,730 9,350 1978 36,476 26,324 1999 14,626 8,618 1977 35,345 25,862 1998 13,455 8,099 1976 33,674 24,973 1997 12,758 7,504 1975 32,144 23,710 1996 11,863 6,970 1974 31,496 22,497 1995 11,186 6,335 1973 30,854 22,205 1994 10,202 5,903 1972 30,407 21,747 1993 9,399 5,593 1971 30,197 21,375 1992 8,966 5,323 1970 29,421 20,553 1991 27,678 19,822 1990

Median Earning of Full Time Workers

When looking at this data, what key points would you like your students to bring up when making comparisons between the men’s and women’s salaries?


• Do you think that women will ever earn more than men, and if yes, when?

• How can we examine the data represented in the table to answer the question?


• Is the gap between men and women closing?

• Create a table of the ratio of women’s salaries to men’s salaries as a function of time.

Ratio of Women to Men’s Salaries Men Women Year Men Women Year

48,202 37,118 2011 27,331 18,769 1989

47,715 36,931 2010 26,656 17,606 1988

45,485 35,549 2009 25,946 16,911 1987

45,161 35,104 2008 25,256 16,252 1986

45,113 35,102 2007 24,195 15,624 1985

43,460 33,437 2006 23,218 14,780 1984

41,386 31,858 2005 21,881 13,915 1983

40,798 31,223 2004 21,077 13,014 1982

40,668 30,724 2003 20,260 12,001 1981

39,429 30,203 2002 18,612 11,197 1980

38,275 29,215 2001 17,014 10,151 1979

37,252 27,462 2000 15,730 9,350 1978

36,476 26,324 1999 14,626 8,618 1977

35,345 25,862 1998 13,455 8,099 1976

33,674 24,973 1997 12,758 7,504 1975

32,144 23,710 1996 11,863 6,970 1974

31,496 22,497 1995 11,186 6,335 1973

30,854 22,205 1994 10,202 5,903 1972

30,407 21,747 1993 9,399 5,593 1971

30,197 21,375 1992 8,966 5,323 1970 29,421 20,553 1991 27,678 19,822 1990

Analyzing Data

• Looking at the table, what predictions or conclusions can we make?

• What answers would you expect from your students?

• Now graph the data in your table to see what type of relationship exists.

Table of Ratios

Men Women Ratio of Women

to Men Year Men Women Ratio of Women

to Men Year 48,202 37,118 0.7701 2011 27,331 18,769 0.6867 1989 47,715 36,931 0.7740 2010 26,656 17,606 0.6605 1988 45,485 35,549 0.7816 2009 25,946 16,911 0.6518 1987 45,161 35,104 0.7773 2008 25,256 16,252 0.6435 1986 45,113 35,102 0.7781 2007 24,195 15,624 0.6458 1985 43,460 33,437 0.7694 2006 23,218 14,780 0.6366 1984 41,386 31,858 0.7698 2005 21,881 13,915 0.6359 1983 40,798 31,223 0.7653 2004 21,077 13,014 0.6175 1982 40,668 30,724 0.7555 2003 20,260 12,001 0.5923 1981 39,429 30,203 0.7660 2002 18,612 11,197 0.6016 1980 38,275 29,215 0.7633 2001 17,014 10,151 0.5966 1979 37,252 27,462 0.7372 2000 15,730 9,350 0.5944 1978 36,476 26,324 0.7217 1999 14,626 8,618 0.5892 1977 35,345 25,862 0.7317 1998 13,455 8,099 0.6019 1976 33,674 24,973 0.7416 1997 12,758 7,504 0.5882 1975 32,144 23,710 0.7376 1996 11,863 6,970 0.5875 1974 31,496 22,497 0.7143 1995 11,186 6,335 0.5663 1973 30,854 22,205 0.7197 1994 10,202 5,903 0.5786 1972 30,407 21,747 0.7152 1993 9,399 5,593 0.5951 1971 30,197 21,375 0.7079 1992 8,966 5,323 0.5937 1970 29,421 20,553 0.6986 1991 27,678 19,822 0.7162 1990

Graph of Ratios

Analyzing the Graph

Looking at the graph, what predictions or conclusions can we make? Are your surprised by anything? What information does the graph give you that the table does not?

Creating a Function

• What is meant by the phrase “line of fit”?

• How can you compare lines to determine if one is a better “fit” than others?

Creating a Function

Analyzing the Graph

• Is there a positive, negative or no relationship between the year and the ratio of earnings?

• Explain your answer.

• What would no relationship look like?

Analyzing the Graph

At this point, it might be best to remind students of the GOAL:

“Women’s Salaries Are Catching up to Men’s” How would you determine the validity of this

statement?

Analyzing the Graph

• Now that we have a line of best fit, how can we write an equation to represent this line?

• What questions can we ask students to help them write an equation of the line?

• What answers would we expect from students?

Analyzing the Line of Fit

• What does the slope mean in the context of this problem? (What is another name for slope?)

• What is the Independent variable? • What is the Dependent variable? • How do you know which variable is which?


• At this point, students may start to notice that the equations of their lines are not the same as the equations of their partner’s lines.

• This is an excellent time for discussion as to why that is the case.

• Write down two questions you would ask your students to start a discussion on why the equations may look different.


• “Why are the equations different?” • “What did you use to write your equation?” • “Looking at the graph, what would cause

your line to be placed differently than Joey’s line?”

• “Is it ok for our equations to be different? Explain.”

What do you anticipate students to ask or say?


• How can we use our line to make predictions for the future?

• How confident are you about your predictions? • What other questions can we ask about the median

earnings for men and women and how can we use our line to answer those questions?


• When will men and women have the same salaries?

• How can we use the graph, table, or equation to answer these questions?

• Which method do you prefer? • How confident are you in your response?


• We used the method of graphing the ratio of women’s salaries to men’s salaries as a function of time to make future predictions.

• How else could we have used the data to answer our question? To make other predictions?


• If we plotted the men’s salaries and the women’s salaries as functions of time on the same graph, how would we know when they had the same salary?

• What would the x-intercepts represent in that situation?

Two Sets of Data

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

90,000

1970 1975 1980 1985 1990 1995 2000 2005 2010

SALA

RY

DATE

SALARIES BY GENDER


• In what ways could this task promote discourse?

• What would you have to know in advance to be prepared for encouraging student discourse?

• How did the tasks connect to previous learning?

Promoting Discourse

• Listen for what can be learned about students' thinking rather than for correct answers

• Identify & check a “hinge point” in the lesson where student understanding is critical for moving on

• “no hands up, except to ask a question” Leahy et al, 2005

• Be relentless in asking what does it mean/why it work

Ten Strategies for Making Questions Central to Teaching , Burrill, Gail, CMC-South 2103

Promoting Discourse

• Maintain neutral stance with respect to answers • Record responses so everyone can think about them • Wait time before responses/after response • Deflect questions to students • Offer examples/counterexamples to test understanding • Plan questions in advance

Thank You!


Please put any awesome amazing resources

(web sites, books, articles, etc.) on the Ideas page.

Contact Info

Fred Dillon [email protected]

Disclaimer The National Council of Teachers of Mathematics is a public voice

of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, official professional development offerings of the National Council of Teachers of Mathematics, support the improvement of pre-K-12 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. Each Institute is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.



• Handouts




Function


[email protected]

Amy Hillen, Jennifer Outzs, Erik Tillema, Jane Porath, Alyssa Hoslar, Kim Morrow-

Leong




Content focus: Function Mathematics Teaching Practice:

Connecting multiple representations

Please sit with at least one person you don’t

know and work on the warm-up.

Warm-up: A graphing program takes one

vertex of a square and moves it to a new

location. Try this with sketches. Is the

relationship of image to pre-image a

function? Why?

Welcome!

Introductions

Please introduce yourself to your tablemates

–Name

–Where you teach

–Grades/classes you teach

–How long you have been teaching

–Two “fun facts” about yourself

Introductions

“In too many schools, professional isolation severely undermines attempts to increase collaboration among colleagues, both between teaching peers internally in the school and among teachers, mathematicians, and mathematics educators externally (Scholastic and the Bill & Melinda Gates Foundation, 2012). Such isolation stands as an obstacle to ensuring mathematical success for all students as well as teachers’ continual growth.” (NCTM, 2014, p. 100)

Challenge for today and tomorrow: Keep your eyes and ears open for people that you could build professional relationships with!

Principles to Actions – Professionalism.

Mathematical Teaching Practices

1. Establish mathematics goals to focus learning.

2. Implement tasks that promote reasoning and problem solving.

3. Use and connect mathematical representations.

4. Facilitate meaningful discourse.

5. Pose purposeful questions.

6. Build procedural fluency from conceptual understanding.

7. Support productive struggle in learning mathematics.

8. Elicit and and use evidence of student

thinking.

Goals

• Define “function”

• Use multiple representations when working with functions

• Understand “rate of change”

• Work with linear functions

Change quantities


If two pounds of beans cost $5, how much will 15 pounds cost?

Solve at least two ways without using cross multiplication.

Tables



Unit Rates



Factors



Algebraic Equations



2

515

p

p2

5

p

15

p

52p

5

5(15)

2p 75

p 37.5

How does

this compare

to method 3?

Which would

you prefer to

see?

Cross Multiplication

Cross-multiplying should not be the first step in teaching about proportional reasoning because it obscures the proportional relationship between quantities when it is not developed with understanding.

Briars, et al, Common Core Mathematics in a PLC at Work™, (Solution Tree Press, 2012,) 94.

Then the method becomes an algorithm,

easily confused with the division of

fractions algorithm.

Multiple Representations

Mathematics Teaching Practice

Use and connect mathematical representations.

In what ways did these examples use different representations?

How can a concept of function create another representation?

College and Career Ready

http://www.ncee.org/college-and-work-ready/

What Does It Really Mean to Be College and Career Ready?

“A very high priority should be given to the improvement of the teaching of proportional relationships…”

Executive Summary, National Center on Education and the Economy, 2013, page 12

What Does It Really Mean to Be College and Career Ready?

“Percent, graphical representations, functions, and expressions and equations, including their application to concrete practical problems” are a key skill set.

Executive Summary, National Center on Education and the Economy, 2013, page 12

What have 6th and 7th graders learned?

• Define a ratio

• Write ratios

• Use multiple representations with ratios

• Make equivalent ratios

• Use unit rates

• Solve problems with ratios

Rate of change

What is the “rate of change”?

What does that mean to you?

What may rate of change mean to your

students when they first hear it?

Cost of coffee beans

The steepness of the ramp is the ratio. Which quantities affect the steepness?

How can this picture show the effect of changing one attribute at a time?

Essential Understandings of Ratios, Proportions and Proportional Reasoning Grades 6-8, p 25.

Rate of Change

Are these points all on one line?

(0, 0) (9,11) (5/4, 1) (8, 10) (1, 5/4)

How do you know?

Essential Understanding

Look at similar triangles

Look at the ordered pair (1,r)


When thinking about coffee beans and cost:

What do the “parts” of each equation mean?

y 20

x 8

5

2

y 30

x 12

5

2

Coffee Beans

Cost of coffee beans

What does the graph of this information look like?


Consider the graph of the coffee bean data

Can we generalize what we’ve done here to help make a “formula”?

y 30

x 12

5

2

If (?, *) is on this line, what proportion can you write?

Building functions

Chinua is measuring the drips of water coming from a faucet. He puts a container under the faucet and notes the heights counting drips.

After five drips, there were 10 mm; after 10 drips there were 20 mm, and after 15 drips there were 30 mm.

Is this a linear relation? How do you know? If so, make an equation that fits this data. If not, state why.

Recognizing Direct Variation

Mathangi is 18 years and her sister, Mathura is 22. How old will Mathangi be when Mathura is 50?

Lars and Olga can paint a long section of fence in three hours. How long will it take them to paint four similar sections of fence?

Mathematics Assessment Resource Service, 2012b, Interpreting Distance-Time Graphs

Keisha’s Bike Ride

Margaret Schwan Smith

Tony’s Walk

Margaret Schwan Smith

What Is a Function?

A function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output

Common Core State Standards for Mathematics

Linear Functions

Consider the eight points

on the grid,

Which set of four make a

linear function?

How can you describe

that function?

Copyright © 2008 by Mathematics Assessment Resource Service. All rights reserved

Linear Functions

Describe different

domains that could

occur with different

contexts.


Linear Functions

Consider the eight

points on the grid.

Are these four

points a function?

Why?


Are these four

points a linear

function? How do

you know.

Linear Functions


Are these four points a function?

How do you know?


• In what ways did we use multiple representations?

• How did the tasks we used promote the use of multiple approaches?

• How did the tasks connect to previous learning?

Common Core Functions grade 8

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.


Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.


Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.


Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.


Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Thank You!



Disclaimer


Contact Info

Fred Dillon

[email protected]



• Handouts




Number and Operation grade 6


[email protected]

Sara Moore, Gail Englert, Arlene Mitchell,

Kim Morrow-Leong




Content focus: Number and Operation Mathematics Teaching Practice: Fluency

Please sit with at least one person you don’t

know and work on the warm-up.

Warm-up: Longest kangaroo jump is 45 ft. This is 3 ¾ as long as the longest human jump. How long is the longest human jump? Create a visual for this.

Welcome!

What Students Have Done

• Place value, fluency with whole number addition, subtraction, multiplication

• Almost all of the intensive work for the concept of fractions and decimals and for computation with both

Expectations

Students can solve and create problems that require rational numbers.

6.NS.1 Interpret and compute quotients of

fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient

Fluency

FAST

ACCURATE

WITH UNDERSTANDING

Computing with Decimals

I have one whole dollar. I need to share it with five people. How much does each get?

Partitive model – split into equal size parts.

Computing with Decimals

I have one dollar. How many sets of size .2 are in the dollar?

Quotative – take a set and form groups of a given size from that group.

Money may not be the best context here. Why?

Compare

• How are the two previous problems alike?

• How are the two previous problems are different?

Looking at Division of Fractions

Partitioning – or fair share – for division

Fractions: How to Fair Share, Wilson, et al, Mathematics Teaching in the Middle School, Vol. 17, No. 4, November 2011

Using Fractions

Kristin has four pizzas. She wants to share it equally among her five friends and herself. How much pizza will each one get?

How does this context help set the stage for our lesson? Would you start this way?

Why?

Picture strategies


You try one

How many sets of size 1

5 are in

1

2 of a whole unit?

Make the picture. What does the “left

over” mean ?

Sharing and Dividing

Chonda has 3 sheet pizzas. She is separating them into sections that are 2/3 of a pizza each. How many sections will she have?

Create a picture representation of this prompt


Partitive Division: Diagram Model

There are 4 sets of slices that are 2/3 of

the original pieces.

How do we describe the blue part in the context of the problem?


Partitive Division: Diagram Model

What part of the “section” is the remaining blue rectangle?

Unit to

Begin

The new unit is the

8 of the 12 pieces from the original unit.

3 of unit

4

39 pieces in

4

2 of unit

3

28 pieces in

3

2" unit"3

Sharing and Dividing – another pictorial example

Numerical Problem

Rearrange the 9 pieces from to cover the 8

pieces from

3

4

2

3How many

2

3s are there in

3

4?

3 of unit

42

of unit3

ANSWER = 1

1

8

3

4

2

3

This covers one

2

3-unit with one piece left over.

Different Models

Ms. Hillen has 7 pizzas she wants to share among three students. How much does each student get?

For 7 shared among 3, create a “split all” representation, a “benchmarking” representation and a “deal-then-split” representation.

Reflecting

What are some questions you can ask to help students see the relationship among the representations.

How can the different approaches help students understand what division of fractions is doing?

Using division

A piece of twine that is 6/7 feet long is going to be split into pieces that are 2/7 feet long. How many pieces will be made?

A typical student method

A piece of twine that is 6/7 feet long is going to be split into pieces that are 2/7 feet long. How many pieces will be made?

6 2 33

7 7 1

Question

Does this always work?

a c a c

b d b d ?

Consider dividing across

A race that is ¾ of a furlong is split into sections that are ½ furlong each. How many sections are there?

3 1

4 2

Make a verbal description explaining our answer in terms of fractions.

More dividing across

This is called a “naked problem.”

What advantages are there to a naked problem?

What disadvantages?

8 2

21 3

Considering dividing across

What strategies can you think to try?

3 1 3

5 4 1 1R

One solution

What other strategies can you think to try?

3 1 12 1 12

5 4 20 4 5

An Unusual Fraction

How might your students interpret this?

12

3

2

Is this slide needed?

Reconsidering our fraction

12

3

2

3

15

2

Transition problems

How can you move students to think of these steps?

4 1 20 1 20

9 5 45 5 9

4

9

1

5

4

1 4

5

4

1 4

5

5

5

49

5

5

5

20

9

Transition problems

How can you move students to think of these steps?

Make a picture for 4 divided by 1/5.

Do this problem using reasoning?

Try our divide across method.

Lemma

Always true?

24 6 3 144 3 48

24 6 3 24 6 3 24 2 48

A possible route

For you to consider -

4

9

1

5

A possible route


4

9

1

5

5

5

4

9

1

5

4

9

5

5

1

5

4

9

5

5

1

5

4

9

5

5

1

5

4

9

5

1

20

9

A possible route

Why do we use 5 for the form of one?

2

3

4

5

5

5

2

3

4

5

A possible route

Why do we use 5 for the form of one?

2

3

4

5

5

5

2

3

4

5

2

3

5

5

4

5

2

3

5

4

1

2

3

5

4

10

12

5

6

A possible route


1 2

1

5

1 2

1

5 1

11

5

2

1

5

1

5 of the 11 blocks are colored – so 5

11 of 2 1

5 is in 1.

A possible route


A possible route

For you to consider

A possible route


2 11 2

3 5

5

3

11

5

5

3

11

5

5

5

25

3

11

25

3

11

3

3

25

33

A possible route


2 11 2

3 5

5

3

11

5

5

3

11

5

5

11

5

11

25

33

1

25

33

A possible route


4 2

1

5

4 2

1

5 4

11

5 4

5

11

If one unit contains

5/11 of 2 and 1/5,

then four units

contain 20/11

Real math!

Ross is building a stair rail. The center piece on the midsection is 3 ¼ inches long. The kerf from his saw is one-eighth inch long. The piece of oak he is using is 4 ½ feet long. How many center pieces can he cut from the oak?

Fluency

What do you define as fluency when dividing fractions?

How will your students demonstrate fluency?

Thank You!



Disclaimer


Contact Info

Fred Dillon

[email protected]

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