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Section 1.3 Connecting Mathematics Through Representations

Much of mathematics involves the study of patterns. Making sense of patterns happens through interconnections among representations. In Section 1.3, we begin with a visual growing pattern of dots, connect it with a pattern of numbers, and graphs that model different interpretations of the pattern. Finally, we connect the pattern to a familiar context. All of these representations help us makes sense of one or more aspects of how the pattern is growing, how to analyze it, and how to generalize it.

Subsection 1.3.1 Video: Brain Crossing

The video shares some important new research on the power of engaging with numbers and symbols visually, which involves brain crossing. Your brain is capable of learning through visual cues, especially if you connect them to other modes of learning. The video explains that it is helpful for all students to think visually about mathematics. ‚ÄúBrain Crossing‚ÄĚ is found in Week 1 of Inspirational Math, Day 2‚ÄČ10‚ÄČ.

Subsection 1.3.2 Pattern Talk: Dot Pattern

Consider the dot arrangement in Figure 1.3.2.1. How many dots are in the arrangement? Explain how you counted the dots. Share as many ways of counting the dots as possible.

A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The triangle contains six circles.
Figure 1.3.2.1.

Show the different ways you counted the dots by drawing on each figure in Figure 1.3.2.2.

A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The first square in a group of five identical squares. Each triangle contains six circles.
A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The second square in a group of five identical squares. Each triangle contains six circles.
A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The third square in a group of five identical squares. Each triangle contains six circles.
A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The fourth square in a group of five identical squares. Each triangle contains six circles.
A square containing circles in the shape of an isoceles right triangle with its right angle in the bottom right corner. The fifth square in a group of five identical squares. Each triangle contains six circles.
Figure 1.3.2.2.

Mathematics is a creative endeavor. There is always more than one way to solve a problem. If you get stuck, try to find another route!

Student Page 1.3.3 Growing Pattern of Dots

Now consider a pattern related to Figure 1.3.2.2. See the student page, Growing Pattern of Dots.

Spend at least 5 minutes thinking about the problem on your own.

When everyone in your group is ready, share your solutions to Growing Pattern of Dots with each other. Try to resolve any differences you have in your understandings. If you and your group members are still uncertain about something after your discussion, write a question or two to ask the full class.

Consider the pattern of dot arrangements in Figure 1.3.3.1.

An image showing a series of of 5 steps. Each step is a vertical rectangle containing circles. The number of circles increases with each step creating a larger isoceles right triangle after each step.
Figure 1.3.3.1.

2.

(a)

How is each arrangement of dots related to the arrangement in the step before it?

(b)

How is each arrangement of dots related to the arrangement in the step after it?

4.

Consider the strategies you and classmates used to count the arrangement of dots in the Pattern Talk: Dot Pattern. Choose one strategy that works to count the dots in each of the arrangements.

(a)

Describe the strategy.

(b)

Use the strategy, show how you used it for each step, and record the number of dots in each arrangement in the bottom row of Figure 1.3.3.1.

5.

How many dots will there be in step 10? How do you know?

Share your responses with the full class:

Which strategy did you use to count the dots in each of these arrangements? Show how you used the same strategy to count the dots in Student Page Exercise 1.3.3.4 and Student Page Exercise 1.3.3.5.

Does the strategy you used to count each dot arrangement help you determine the number of introductions that were made in our class? Explain.

What questions do you still have? (Resolve these as a class.)

Student Page 1.3.4 Graphing a Growing Pattern of Dots

1.

Consider the data table you created for Growing Pattern of Dots. The data is plotted in the following graphs. For each graph, determine:

Figure 1.3.4.1.
Figure 1.3.4.2.
Figure 1.3.4.3.
Figure 1.3.4.4.

(a)

Is the graph a correct representation of the data?

(b)

Why or why not?

2.

What are some important characteristics of an accurate graph? List of at least 5 items.

This collection of activities ‚Äď Making Introductions, Pattern Talk: Dot Pattern, Growing Pattern of Dots, and Graphing a Growing Pattern of Dots ‚Äď shows how seemingly unrelated contexts and representations can have the same underlying mathematical structure. We see this a lot in mathematics. Making these connections requires Brain Crossing (recall the video you watched today). Remember that Brain Crossing helps you with recall and grows your brain!

When you work on a new problem, try to connect the work you're doing to problems you've worked on in the past. Connecting related mathematics problems helps reduce your reliance on memorized procedures. It also helps move your working (short-term) memory into long-term memory so that it can be retrieved to help you make sense of new problems and concepts.

Homework 1.3.5 Homework

Complete the following homework before the next class period:

3.

Critique the graph below. If the graph is accurate, say how you know. If the graph is incorrect, indicate all the ways it needs to be fixed, label the axes correctly, and redraw the graph.

\(x\) \(y\)
1 1.2
10 24
15 36
50 60

4.

The same growing number pattern is shown twice in Figure 1.3.5.1 Illustrate on the figures and explain in words how you are counting each set of dots in the pattern.

Figure 1.3.5.1.
(a)

Count the number of dots in the first set of Figure 1.3.5.1 using the same method to count all three arrangements of dots.

(b)

Count the number of dots in the second set of Figure 1.3.5.1 using a different method from the method you used in Task 1.3.5.4.a. Use this second method to count all three arrangements of dots.

(c)

What number of dots will be in the next figure? How do you know? Include a drawing of the figure in your explanation.

5.

Consider each of the stories and expressions on the student page, Interpreting Expressions. What do you notice about the expressions? Which ones name the same number? Why?

6.

Read the introduction to Section 1.4. In writing, answer each question in the reading.

Student Page 1.3.6 Interpreting Expressions

I am making apple pie and apple turnovers.
One recipe calls for 2 apples. The other
recipe calls for 3 apples. I'm making 5
recipes of each dessert. How many apples
do I need?
\(2 + ( 3 \times 5 )\)
I have 2 one-dollar bills and 3 five-dollar
bills. How much money do I have?
\(( 2 + 3 ) \times 5\)
Mom lets me keep any money I find when I
clean the house. She also quadruples any
money I find to pay me for my work. I
found $2 in the couch and $3 in my dad's
jacket. How much money will I have after
Mom pays me?
\(2 + 3 \times 5\)
I need 2 cups of almond flour to make a
batch of pancakes and 3 cups of almond
flour to make a cake. I'm going to make 5
cakes and 1 batch of pancakes. How much
flour will I need?
\(2 + 3 + ( 2 + 3 ) \times 4\)

1.

Decide which numeric expression best fits each story. Be ready to defend your choices.

2.

Decide which numeric expressions, if any, represent the same value.

youcubed.org/resources/brain-crossing-video/