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NCTM San Diego 2010
Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra
New Teacher Workshop
Eomailani Bettercourt, University of Hawai' i, Manoa
Joe Zilliox, University of Hawai' i, Manoa
Emphasis on Process standards:
Problem solving, Communication, Connections, Reasoning, and Relationships
Fraction as:
A part-whole relationship.
A place on a number line.
An operator in computation.
Opening Question
2
7 is a representation that tells me something. What does it say to you?
Activity 1: Folding Paper Strips
Using the colored strips and making all creases along the short dimension, fold to show the
following fractions: gray is one-whole; fold the yellow into halves, blue into thirds, green
into fourths, pink into sixths, and purple into eighths
Use the portions you created to
2
Show 7 in three different ways.
On a number line 3 in exactly in the middle between 2 and 4. Is 7 exactly in the middle
between —, and 7 ?
Use the idea from above to decide where 7 would be if it made with the same strips.
Comparing fractions without common denominators. Use the strip and logical reasoning to
explain why 7 is greater or less than 7 ? (Note: what kind of language is useful here). Now
use similar reasoning without calculations to compare the following.
3
5
7
7
13
17
a. q and 7 b. 73 and 77 c. 77 and < g
What portions match — ? What names do we give to these portions.
Combiner + 7 - make the portion. Name the portion.
Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop
NCTM San Diego 2010
Activity 2: Fractions on a number line
Label 15 post-its with the following fraction numerals:
i2' 3i 3 'n4' 4i' 6 i' 6 i' 8 'i 8 'i8 'H
i i i i i
8 ' 12 ' 12 ' 12 ' 12
Suppose the left end of the log paper strip (adding machine tape) is 0 on the number line and
the right end is the number 1. Using only estimation (no folding), place each of the labeled
post-its on the number line where it belongs relative the 0 and the 1. When all are placed
stand back and reconsider the location of the post-its, moving those you think are misplaced.
Compare your marks to the other group at your table. At this point you may or may not want
to make changes. Once the placements are decided, the paper strip can be marked with the
fraction name in pencil (because the post-its tend to fall off) to record the locations.
To check their estimates, take a second paper strip and carefully fold it into 24 equal
portions. Label this strip with the same fraction names used on the first strip. If the folding is
done carefully these labels should be more accurate than the estimates. A good estimate
should be within a post-it width on either side of the fold line.
Activity 3: Patterns and Generalizations in Fraction Sums and Differences
Calculate: (a)j + 3 = ©3+4= (c)t+7 =
What patterns or connections do you see between the addends and the sums? Can you
generalize this patter in words and symbols.
Repeat for the following differences
(a)j- 3 = (b)"-4=(c)4-5 =
Do the above patterns work if the denominators are not consecutive integers (2 and 7 instead
of2and3)?
Repeat each of the above by using 2 in numerator of both fractions. How does the pattern or
generalization change?
Activity 4: How Does It Grow - (see separate handout)
Look at the "row houses" in the first pattern.
1. What is the ratio of the number of triangles to the number of squares? Squares to blocks?
Blocks to triangles?
2. If I have 15 triangles, how many squares will I need? How many houses will be in the
last set?
3. If I have 72 squares how many triangles will I need? How many houses in the last set?
4. If I have 234 blocks and they make row houses without any left over, how many of the
blocks were squares and how many were triangles? How many houses made up the last
set?
Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop
NCTM San Diego 2010
Activity 5: Ratios with Pattern Blocks
One of the vendors in the NCTM exhibit area is giving away small samples bags of their
pattern blocks. Each bag is exactly the same and contains four triangles, three trapezoids, and
five squares. Mr and Ms Freebee collected many sample bags so they can do projects with
their students.
a. If their first project requires 12 triangles, when they open the sample bags to get the
triangles how many squares will there be? How many trapezoids?
b. If the second project requires a total of 48 blocks, how many sample bags will they need
and how many of each kind of block will they have when they open the bags?
c. A final project requires 64 quadrilaterals. How many of these will be squares and how
many will be trapezoids?
Activity 6: Pattern Block Similarity
1. Begin with one square from the set of pattern blocks. Using only orange squares make
another larger square using as few small squares as possible. Still using only orange
squares make the next largest square using as few small squares as possible. There are
now three figures composed of squares. For each of the three figures record the length of
a side, perimeter, and area. What patterns can be found in these number sequences? What
will be the next number in each sequence? Why?
2. Leave the squares intact on the table. Repeat this activity but instead of orange squares
use just triangles, or just blue rhombi, or red trapezoids. So use trapezoids to make the
next larger trapezoid. How are the number patterns the same or different as they were for
the square? What does this have to do with ratios?
3. Now take a single square in your hand. Stand over the larger square made with four
small squares. Place the single square in your hand between one of your eyes and the four
squares on the table as you look down on them. At some point in your line of vision
between your eye and the four squares, the single square should appear to cover the four
squares exactly. Hold that position. Measure the distance between the eye and the small
square and then measure the distance between the eye and the larger square on the table.
How do the measures compare?
Activity 7: Function Machine
See separate handout
Fractions, Ratios and Patterns: Helping Elementary Students Get Ready for Algebra - New Teacher Workshop
NCTM San Diego 2010
Function Machines
Input
\ L
Function Rule
7 \
Output
Building Towers
The idea of a function machine is that you begin with a starting input value as the first number in
a sequence and apply the "rule" to produce an output. That output becomes the second
number in the sequence. To get additional numbers in the sequence the output is repeatedly
returned to the machine as the next input.
Example: Use 1 as the starting number and with a rule of "add 1" we would get the following
sequence:
A. 1, 2, 3,4, 5, The numbers represent the number of blocks in a tower.
Find the following sequences using the starting number and the rule. Stop after 5 towers are in
your sequence.
B. Starting number is 3, rule is "add 1".
C. Starting number is 1, rule is "add 2".
D. Starting number is 1, rule is "double the input".
E. Starting number is 1, rule is "add n, where n is 1, then 2, then 3,...".
Sketch the five on grid paper. Use the card stock to connect the towers. What does the slant in
the card stock indicate?
Name
A
AA
AAA
The pattern continues.
1. Draw the next figure.
2. When there are six triangles, how many squares are there?
3. When there are five triangles, how many shapes (squares and triangles)
are there in all?
4. Can there be seven squares in a figure?
Why?
This pattern continues.
5. Draw the next figure.
6. When there are ten squares, how many triangles are there?
7. How do you know?
8. When there are fourteen triangles, how many squares are there?
9. How do you know?
Copyright © 2001 by the National Council of Teachers of Mathematics, Inc.
www.iiclm.org. All rights reserved.
Navigating through Algebra in Prekindergarten-Grade 2
Name
Navigating through Problem Solving and Reasoning in Grade 4
Copyright © 2005 by the National Council of Teachers of
Mathematics, Inc. www'.nctm.or<r. All rights reserved.