In math, there are usually several ways to represent any idea. For example, how could you show adding 2 and 5? You could use numbers and symbols, as in 2+5=7. You could also write it out in words, such as “The sum of two and five is seven.” Using diagrams is another possibility. Two examples of diagrams, a dot diagram and a number line, are shown below.
In the next few lessons, you will focus on finding several ways to represent a quantity(an amount of something). As you work, ask your teammates the questions that follow.
How can we represent the quantity with numbers and symbols?
How can we represent it with words?
How can we represent it with diagrams?
1-41. A HANDFUL OF PENNIES
How can you figure out how many objects are in a pile without counting each one? Are there some ways objects can be arranged so that it is easy to see how many there are?
Your task: Your teacher will bring your team a handful of pennies. As a team, organize the pennies so that anyone who looks at your arrangement can easily see how many pennies your team has. Keep working until all members of your team agree that your arrangement is the clearest and easiest to interpret.
(Note that someone looking at your pennies should know how many there are without having to believe what you tell them. For example, arranging your pennies into the shapes of the numerals of your number will not work.)
1-42. Are some arrangements easier to interpret than others? Your teacher will direct you to participate in a Gallery Walk so that you can see how other teams have arranged their pennies. You will walk to the desks or tables of the other teams in the class to see how they have arranged their pennies.
As you do, notice how easy or difficult it is for you to see how many total pennies each team has. When you see an arrangement that helps you know quickly and easily what the number of pennies is, consider what makes that particular arrangement easy to total.
1-43. How could you make your arrangement even clearer?
Work with your team to rearrange the pennies to improve how well others can understand the quantity represented. Use what you noticed on the Gallery Walk to help you do this.
On your own paper, draw a diagram that represents your new arrangement of the pennies without drawing all of the pennies themselves.
Compare your diagram with those made by your teammates. Are some diagrams clearer matches to the arrangement than others?
As a team, decide on the best way to represent your arrangement in a diagram. Consider using ideas from multiple drawings. When all team members have agreed on the best diagram, copy it onto your paper.
1-44.Work with your team to represent your arrangement of pennies using words, numbers, and symbols. Write at least three different numerical expressionsthat represent your quantity. (An expression is a combination of numbers and one or more operation symbols.) Some number-and-symbol representations may match certain diagrams more closely than others. Identify which expressions most closely match your team’s chosen arrangement.
Be sure to keep your work in a safe place as you will need to share your team’s results in the next lesson.
1-45. As a team, create a poster that shows your team’s best arrangement of your pennies along with the diagrams and numerical expressions that represent it. Show as many connections as you can among the pennies, diagrams, and numerical expressions. Use color to enhance your connections and poster as appropriate.
1-46. Match each of the following descriptions of pennies with its possible numeric expression. Then calculate the value of each expression.
1. 8(12) + 7 / 2. 6(20) + 5 / 3. 11(10) + 7 / 4. 9(12) + 5a. 11 piles of 10 pennies with 7 leftover pennies
b. A rectangular array of pennies that is 9 pennies long and 12 pennies wide with 5 leftover pennies
c. 8 stacks of 12 pennies with 7 leftover pennies
d. A rectangular array of pennies that is 6 pennies wide and 20 pennies long with 5 leftover pennies
1-47. Write one whole number or one fraction in each blank to make each statement true.
One hundred pennies equals ____ dollar(s).
Two hundred pennies equals ____ dollar(s).
Fifty pennies equals ____ dollar(s).
Ten pennies equals ____ dollar(s).
One penny equals ____ dollar(s).
1-48. Matthew’s mother asked him to go to the store for her. To get to the store, he walked seven city blocks. He caught the bus and rode 13 blocks. He got off and walked one and a half blocks to the store. He purchased the items his mother wanted and returned home the same way. How many total blocks did he travel?
1-49. Which numbers make these division problems correct? Replace each box and triangle with a single-digit number.
1-50.A line segmentis a piece of a straight line. Onyour paper, draw two line segments that are the same length and each about as long as a pen.
Draw marks on the first line segment to show how you can divide it into eight equal lengths.
Draw marks on the second line segment to show how you can divide it into five equal lengths.
Was one of these tasks easier than the other? Which one? Why?