Start here

You did these with your group. Now do them on your own — the model above is yours to copy.

1. 1

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

   Try it together — then prove it: The pattern unit is still 3 shapes long. 20 ÷ 3 = 6 complete units with a remainder of 2, because 6 × 3 = 18 and 20 − 18 = 2. A remainder of 2 means shape 20 is the second shape in the unit. The second shape is the purple square. Now prove it: say why that move had to work at all — not just that it did.

   Show your work
2. 2

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

   1. A3 times
   2. B1 time
   3. C1/3 of a turn
   4. D4 times

   How do you know?
3. 3

   Warm restartWhich ratio is equivalent to 2:3?

   1. A8:12
   2. B3:2
   3. C4:5
   4. D5:6

   How do you know?
4. 4

   Warm restartTwo 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

   How do you know?

You answered these out loud. Now write them.

Answer, then say how you know — the reason is the part that counts.

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. ARhythm
   2. BRepetition
   3. CA pattern unit
   4. DPredictability

   Why is that the answer?
2. 6

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

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

   How do you know? Give a second reason as well.
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. What would have to change for a different choice to be right?
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?

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 mistakeAnother student made this exact mistake in this lesson. Explain why it is wrong, then write what they should have done.

   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?
3. 11

   Write your ownWrite your own problem about this situation, then solve it and show the answer. The situation: A photograph shows the tiled wall of an old building. The same shapes repeat across it in bands of color — geometric patterns have decorated homes and buildings like this for centuries.

   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 knowExplain your thinking — Which statement best matches this lesson's summary of 'Math is Boundless'?

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

   Explain your choice.
2. 13

   From Lesson 10.4Explain your thinking — Which statement best captures what this lesson means by 'Math is Ingenuity'?

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

   How do you know?
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. ANaming a specific ratio or rate in the design, like a gear ratio between a motor and a wheel
   2. BSaying the vehicle looks very futuristic
   3. CSaying the vehicle is extremely fast
   4. DRepeating that it is powered by a battery a second time

   How do you know?

Be honest — this tells your teacher what to do next