Algebra Across
the Grades
Tracy Hargrove, Ph.D. UNCW
NCCTM Eastern Region Conference
February 15, 2014
Developing Algebraic Reasoning and Problem
Developing Algebraic Reasoning
in Elementary School
Rather than teaching algebra procedures to elementary school children, our goal is to support them to develop ways of
thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra
Big Ideas in Algebraic
Reasoning
Patterns
Balancing
Variables
Functional Relationships
Common Core Standards
Grade 3
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
CCSS.Math.Content.3.OA.D.8 Solve two-step word
problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.3
CCSS.Math.Content.3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or
multiplication table), and explain them using properties of operations. For example, observe that 4 times a
Common Core Standards
Grade 4
Gain familiarity with factors and multiples.
CCSS.Math.Content.4.OA.B.4 Find all factor pairs for a whole
number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
Generate and analyze patterns.
CCSS.Math.Content.4.OA.C.5 Generate a number or shape
pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For
example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even
Common Core Standards
Grade 5
Write and interpret numerical expressions.
CCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. CCSS.Math.Content.5.OA.A.2 Write simple expressions that record
calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Analyze patterns and relationships.
Triangle Paths
Kiri made the following triangle paths
using 1, 2, and 3 triangles – she called
them a 1-triangle path, a 2-triangle path
and a 3-triangle path.
• Use Kiri’s method to make a 4- and then a 5-triangle path.
• How many extra toothpicks would be needed to make a 6-triangle path? A 7-6-triangle path?
• How many toothpicks would Kiri need to make a 20-triangle path?
Kiri noticed that if she rearranged the
toothpicks, she could count them quite quickly. The following picture shows how she
rearranged them.
Triangle Paths
• How does Kiri’s method work?.
• How would Kiri rearrange a 7-triangle path?
Triangle Paths
• Kiri says that using her
method, she can see a short cut way of counting the
number of toothpicks needed to make a 10-triangle path. Write down, using pictures to help you explain, what Kiri’s short cut method might be. Let’s call Kiri’s method, Kiri’s Rule.
• Using Kiri’s Rule, how many toothpicks will be needed to make a 20-triangle path?
How big a path can Kiri make with 201 toothpicks?
Triangle Paths
Kiri’s friend Jamie arranged his toothpicks
differently. His pictures looked like this:Jamie says that using his method,
he can see another short cut way of counting the number of toothpicks needed to make a 10-triangle path. Write down, using pictures to help you explain, what Jamie’s Rule is.
• How many toothpicks be needed to make a 20-triangle path?
• How big a path can Jamie make with 201 toothpicks?
Make a 4-square and 5-square path. How many extra toothpicks were added each time?
How can you develop a quick and easy way of finding the number of toothpicks needed to make a 20-square path?
Work in groups of 2. Make a picture showing how each path is made. Experiment with the toothpicks, and record your pictures.
Is there only one possible picture?
What might some of the others look like?
Some pictures will be very helpful in counting the number of toothpicks needed to make a 8-square path – some will not. Choose the picture that you think best explains how successive square paths are made up AND gives a quick and easy method for counting the toothpicks needed for an 8 -square path.
Square Paths
Use your best method’ to verify that there are 76
toothpicks needed to make a 25 -square path.
Use this method to predict the number of toothpicks
needed to make 20-, 36- and 100 -square paths.
Write down how they would use your method to count
the number of toothpicks needed to make a square path consisting of any number of squares, say 1000 squares.
How many squares are in a square path with 31, 304 and
457 toothpicks?
How many toothpicks will be left over if you make the
biggest square path that you can with 38, 100 and 1000 toothpicks?
Square Paths
Use the techniques developed in the last two activities to
explore the following problem:
A new toothpick path is being designed. It is called a
house path. Some of them are shown above. Develop a
counting rule, that is, a short-cut way of counting the
number of toothpicks needed to make a 1000-house path.
Illustrate how you developed your counting rule. You
could do this, for example, by using pictures, words or numbers (or some combination of these).
House Paths
My friend made a picture that showed how a toothpick path was made. She named it:
5 lots of 4 and add 2 (this was the counting rule used to make the path)
She sent it to me via email. However, I was
only able to read the name of the path and not see the picture! Make some possible pictures that she could have sent.
What’s My Path?
The pattern below is made up of square tiles. The 1st, 2nd, 3rd and 4th terms of the
sequence are shown
Odd Ways of Seeing Things
Two students Paul and Penina discover two different ways that they think the shapes are related.
Odd Ways of Seeing Things
Penina's way of seeing how each shape is made is given below:
This is Paul’s way of seeing how each shape is made:
Odd Ways of Seeing Things
Both of these students have written down a way of finding the number of tiles (not toothpicks!) needed to make the 10th shape in the pattern.
Which way belongs to whom?
(1)2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 –
1
(2)10 + 9
Penina notices that adding 2 all the time is the same as multiplication by 2. She refines her method
further into a short cut. Use her short cut to find the 100th shape in the sequence?
Check Penina’s result using Paul’s method.
What term of the sequence has 45 tiles?
Tiling a Patio
Alfredo Gomez is designing patios. Each patio has a rectangular garden area in the center. Alfredo uses black tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures shown below show the three smallest patios that he can design with black tiles for the garden and white tiles for the border.
a. Draw Patio 4 and Patio 5. How many white tiles are in Patio 4? Patio 5?
b. Make some observations about the patios that could help you describe larger patios.
c. Describe a method for 5 finding the total number of white tiles needed for Patio 50 (without constructing it).
d. Write a rule that could be used to determine the number of white tiles needed for any patio. Explain how your rule relates to t he visual representation of the patio.
Beth’s Approach to Patio 1
