Lesson 8

8.1: Tables of Equivalent Ratios (10 minutes)

Warm-up

The purpose of this activity is to prepare students to work with a table of equivalent ratios that appears in a later activity in this lesson.

Launch

Give students a few minutes to complete the table. Ensure that they found the correct values before they work on the second question.

Student Facing

Here is a table of equivalent ratios:

$$a$$ $$b$$
3 15
10 50
6 30
1
80
1. Complete the table with the missing values.
2. Explain what it means to say that the pairs of numbers are equivalent ratios.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Invite students to share how they reasoned about the values needed to complete the table. Keep asking, "Did anyone think about it a different way?" until a few ways of reasoning come to light. Some methods might include

• multiplying any number in the first column by 5 to find the number in the second column
• dividing any number in the second column by 5 (or multiplying by $$\frac15$$ to find the number in the first column)
• using a scale factor to move from row to row, for example, starting with 6 and 30, you can multiply each value by $$\frac16$$ to find 1 and 5 in the next row.

8.2: The Birthday Trick (15 minutes)

Optional activity

The purpose of this activity is to learn that you can click on a cell to use its contents in a new expression rather than typing the address of the cell.

Launch

Ask students to write down the day and month of their birthday. Then, give them this sequence of instructions. Allow them to use a calculator if they wish. (Alternatively, you could have one student demonstrate the procedure on their birthday for the group, and then re-run the computations with each student using her own birthday.)

• Multiply the month by 5.
• Multiply by 4.
• Multiply by 5.
• Subtract 165.

Tell students that they’ll program a spreadsheet to perform this trick, and learn a few shortcuts for using spreadsheets along the way.

Action and Expression: Internalize Executive Functions. Provide students with grid or graph paper to organize their work with the sequence of steps for their birthday calculations.
Supports accessibility for: Language; Organization

Student Facing

Use the applet to answer the questions.

1. In cell B4, we want to enter = B1 * 5 to multiply the month by 5. Enter this, but when you are about to type B1, instead, click on cell B1. This shortcut can be used any time: click on a cell instead of typing its address.
2. Practice this technique as you program each cell in B5 through B10 to perform the right computation.
3. When you are finished, does cell B10 show a number that contains the month and day of your birthday? If not, troubleshoot your computations.
4. Try changing the month and day in cells B1 and B2. The rest of the computations should automatically update. If not, troubleshoot your computations.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Why does this trick work? Try using $$m$$ for the month and $$d$$ for the day, and writing the entire computation as an algebraic expression. Can you see why the resulting number contains the month and day?

Student Response

For access, consult one of our IM Certified Partners.

Launch

Ask students to write down the day and month of their birthday. Then, give them this sequence of instructions. Allow them to use a calculator if they wish. (Alternatively, you could have one student demonstrate the procedure on their birthday for the group, and then re-run the computations with each student using her own birthday.)

• Multiply the month by 5.
• Multiply by 4.
• Multiply by 5.
• Subtract 165.

Distribute internet-enabled devices, and give students instructions to navigate to this lesson in the digital version of the materials.

Action and Expression: Internalize Executive Functions. Provide students with grid or graph paper to organize their work with the sequence of steps for their birthday calculations.
Supports accessibility for: Language; Organization

Student Facing

Navigate to this activity in the digital version of the materials or to ggbm.at/djcz6fjf.

1. In cell B4, we want to enter = B1 * 5 to multiply the month by 5. Enter this, but when you are about to type B1, instead, click on cell B1. This shortcut can be used any time: click on a cell instead of typing its address.
2. Practice this technique as you program each cell in B5 through B10 to perform the right computation.
3. When you are finished, does cell B10 show a number that contains the month and day of your birthday? If not, troubleshoot your computations.
4. Try changing the month and day in cells B1 and B2. The rest of the computations should automatically update. If not, troubleshoot your computations.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Why does this trick work? Try using $$m$$ for the month and $$d$$ for the day, and writing the entire computation as an algebraic expression. Can you see why the resulting number contains the month and day?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Some students may misunderstand the instructions and continue to refer each cell back to the number of their month. Each row after the first instruction is meant to refer back to the previous value.

Activity Synthesis

Ask students what new shortcut they learned in the spreadsheet. Namely, when students want to use the contents of the cell in a computation, they can click on a cell rather than type in its address. Ask students what other things they had to remember about using spreadsheets.

Reading, Writing, Conversing: MLR 3 Clarify, Critique, Correct. Present a first draft of an explanation for what to remember when using spreadsheets: “I found the cell and used it to get my answer.”. Prompt discussion by asking, “What were the steps that the author took?”. Ask students to clarify and correct the statement. Improved statements should include some of the following: an explanation of each step, order/time transition words (first, next, then, etc.), reasons for decisions made during steps, or the terms or phrases “formula,” “clicking on the cell,” or “dragging.” This will help students develop and understand explanations of how to use spreadsheets as computation tools. Design Principle(s): Maximize meta-awareness; Optimize output (for explanation)

8.3: Using Spreadsheet Patterns (10 minutes)

Optional activity

The mathematical purpose of this activity is for students to become more efficient at entering values into a spreadsheet. In later units, students may wish to use the techniques in this activity to quickly compute the values of sequences by continuing the pattern.

Monitor for students who use a recursive formula, for example, typing = B2 + 2 in cell B3. Also look for students who use an explicit formula, like typing = A3 / 3 in cell B3. After students complete the first question, ask students to share their various spreadsheet formulas that work. Assure them that either is okay.

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight important connections between cells and formulas.
Supports accessibility for: Visual-spatial processing

Student Facing

The spreadsheet applet contains a table of equivalent ratios.

1. Use spreadsheet calculations to continue the patterns in columns A and B, down to row 5. Pause for discussion.
2. Click on cell A5. See the tiny blue square in the bottom right corner of the cell? Click it and drag it down for several cells and let go.
3. Repeat this, starting with cell B5.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight important connections between cells and formulas.
Supports accessibility for: Visual-spatial processing

Student Facing

Navigate to this activity in the digital version of the materials or to the URL, ggbm.at/wu9t7kkd.

The spreadsheet contains a table of equivalent ratios.

1. Use spreadsheet calculations to continue the pattern in columns A and B, down to row 5. Pause for discussion.
2. Click on cell A5. See the tiny blue square in the bottom right corner of the cell? Click it and drag it down for several cells and let go.
3. Repeat this, starting with cell B5.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Tell students that the technique used in this activity extends a set of operations to additional cells in the spreadsheet. Invite students to experiment with extending some other patterns in the spreadsheet.

8.4: Cool-down - Doubling in a Spreadsheet (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

Sometimes you want to create a list of numbers based on a rule. For example, let's say that the cost of a gym membership is a \$25 sign-up fee followed by monthly dues of \$35. We may want to know how much the membership will cost over the course of 6 months. We could use a spreadsheet and set it up this way:

Which results in:

See the little square on the lower-right corner of cell B2? If we click and drag that down, it will keep adding 35 to the value above to find the value in the next row. Drag it down far enough, and we can see the total cost after 6 months.

Any time you need to repeat a mathematical operation several times, continuing a pattern by dragging in a spreadsheet might be a good choice.