Start here

Use the model to solve the first problem. Then try these quick review questions from earlier lessons.

1. 1

   Same stepsFinish what the group started. Use the model above, step for step.

   Let's do one together:

   1. The same border repeats: pink circle, purple square, blue triangle. What is shape number 20?
   2. How long is the pattern unit? (3 shapes)
   3. How many complete units fit in 20 shapes? How many shapes are left? (6 × 3 = 18 and 20 − 18 = 2. So 6 units, remainder 2.)
   4. What does a remainder of 2 tell you? (Shape 20 is the 2nd shape in the unit.)
   5. What is the 2nd shape in the unit? (The purple square.)

   My first step is ___ , because the problem asks for ___ .

   Show your work
2. 2

   Quick reviewA bicycle has a gear ratio of 3:1. For each turn of the pedals, the rear wheel turns…

   1. A1 time
   2. B1/3 of a turn
   3. C3 times
   4. D4 times
3. 3

   Quick reviewWhich ratio is equivalent to 2:3?

   1. A3:2
   2. B4:5
   3. C5:6
   4. D8:12
4. 4

   Quick reviewTwo wheels turn the same number of times, but one has a larger diameter. The larger wheel…

   1. Atravels farther
   2. Btravels the same distance
   3. Ctravels less far
   4. Ddoes not move

Try these on your own

Choose an answer. Then explain how you know.

1. 5

   Think it throughIn the circle painting, the colors, sizes, and ring counts change from circle to circle with no repeating unit. What design element does this describe?

   1. ARepetition
   2. BRhythm
   3. CA pattern unit
   4. DPredictability

   Why is that the answer?

   I chose ___ because ___ .
2. 6

   Think it throughWhy can you NOT predict what the next circle in the painting will look like?

   1. ABecause the artist used no shapes at all in the painting
   2. BBecause the painting contains too many pattern units to count
   3. CBecause circles are not allowed to be part of a geometric design
   4. DBecause rhythm varies color, size, and the number of rings instead of repeating a fixed unit

   How do you know?

   I chose ___ because ___ .
3. 7

   Think it throughA new row of circles is being added to the painting. Which addition keeps the design consistent with rhythm rather than turning it into a pattern?

   1. ANew circles with different colors, sizes, and ring counts than any existing row
   2. BAn exact copy of the very first row, repeated precisely
   3. CCircles that follow a strict, repeating 3-shape unit
   4. DReplacing all the circles with a single repeated square

   Explain your thinking.

   I chose ___ because ___ .
4. 8

   Back to the modelA border repeats pink circle, purple square, blue triangle. Using its pattern unit, how do you predict what shape number 12 is without drawing all 12 shapes?

   I predict shape 12 by dividing ___ by the ___ length.

Sort it again — on paper this time

Write the letter of the group each one belongs in.

• The same square tile repeated across a whole wall
• Circle, square, triangle repeated in the same order forever
• Circles whose colors, sizes, and thickness all vary across the canvas
• You can name exactly what the next three shapes will be
• The same shapes appear, but their size and order of appearance keep changing
• A design with a clear pattern unit that repeats in a regular arrangement
• The design has a flow, but you cannot predict what comes next
• Concentric circles where the number of rings inside each circle varies

Use the words

Fill each blank with a word from the bank.

• The same object or shape repeated multiple times is ___.
• Objects or shapes repeated in a regular order form a ___.
• The smallest chunk of a pattern that repeats is the ___ ___.
• Varying the placement, size, or order of the same shapes creates ___.

Explain the thinking

1. 9

   Find the mistakeRead the mistake. Explain what went wrong, then show the correct step.

   The mistake: Calling every repeated design a 'pattern. ' One identical shape repeated is repetition; a pattern needs a unit of shapes repeating in a regular order; and rhythm varies the shapes' size and order, so it is neither.
2. 10

   Say moreWhen you sorted design descriptions into 'predictable' and 'rhythm (varies),' what made the circle painting belong with rhythm instead of pattern?

   The circle painting is rhythm because ___.
3. 11

   Change one numberGo back to Part 1 and pick one problem you already solved. Change one number in it, then solve your new version. Show every step.

   If ___ changed to ___ , then ___ .

   Show your work

Last check

These two come from Lesson 10.4. If they are shaky, that is the lesson to revisit — not this one.

1. 12

   Show what you knowQuick check — you've got this: Which statement best matches this lesson's summary of 'Math is Boundless'?

   1. ADesigns only count as mathematical if they are perfectly predictable
   2. BMath can be used to create beautiful designs, which often have repetition, patterns, and rhythm
   3. CRhythm is the only real element of design
   4. DGeometric patterns were invented recently

   Explain your choice.

   I know it is ___ because ___ .
2. 13

   From Lesson 10.4Quick check — you've got this: Which statement best captures what this lesson means by 'Math is Ingenuity'?

   1. AMath is only useful for inventors who lived in the 1800s
   2. BIngenuity means avoiding math and just guessing
   3. CMath can help us come up with creative solutions to everyday problems
   4. DEvery bicycle must have a 1 to 1 pedal ratio
3. 14

   From Lesson 10.4A student's answer says only, 'My vehicle is powered by a battery.' Which addition would make this a complete connect answer, based on this lesson's check?

   1. ASaying the vehicle looks very futuristic
   2. BSaying the vehicle is extremely fast
   3. CRepeating that it is powered by a battery a second time
   4. DNaming a specific ratio or rate in the design, like a gear ratio between a motor and a wheel

Check your progress

Choose the box that best describes how you feel about each skill.