October Unit: PATTERNS, especially squares. Homework will continue to review fractions and word problems visually.
What does a daily lesson look like?
- Every other day, students hand in HW, and retrieve and correct the last HW. We mark any mistakes on their HWs, but it is their job to correct them. We put an answer key on the board – it shows work – but they cannot bring a pencil if they come to look at it. They only read it, and return to their seats to correct. For each mistake, they write a “Note to Self” reflection. “Draw more carefully”, “Only draw unit boxes the same size if they’re the same”, “When adding fractions in 12ths, make all the fraction bars the same size”, for example. HW corrections take 5 minutes or so.
- We collect their binders about once a week to check corrections.
- Here are HW #6, and #7 and their keys.
CW can be an investigation, a game, or a quiz combined with something else. Other common lessons involve games, or word problems and mental math with white boards. At the moment, we’re using a lot of investigations, building and drawing patterns.
Investigations. We spend a lot of time on squares because of their importance in higher math. We’ve taught middle school and high school, and seen this too many times: “Umm… 3 squared = 6, right?”. ARRGHH! We spend as much time as needed on the visualization of what a square is – what 3 squared looks like, for example, so that won’t happen.
Activity ONE: Squares and Odd Numbers
We start this activity with this power point: CW squares 1 Note: It is VERY important to WAIT on slide 2, when students are asked “Can you use your second stick (3 blocks) to build a new square around your first square?” Their response will be confusion at not being told exactly what to do. Pretend you don’t know either? This wait time leads students into the recognition that math is about thinking and trying, not just listening and following directions. What a valuable asset they’re acquiring!
Follow up CW: CW 10_3 sq The follow-up activity takes about 20 minutes, and can be done alone or in pairs. We encourage the use of blocks, but not everyone needs them. Many students DO need them, though, and join in if we get the activity started on the floor.
We have found that IF students do NOT draw the answer to #1 – #4, then they cannot figure out the answers to #5 and #6. We walk around with a 3-squared block (we use snap cubes) and the 7 L-shaped blocks it will take to make 3 squared into 4 squared. We put the 7 cubes around 2 sides of the 3-squared cube, then take them off, on and off, on and off. Usually, the light goes on, and students can draw the answers to these problems.
Note: Not everyone finishes any given CW. It is always too long, so that a few students can get to the challenge problems at the end, and the others can capture the major concepts. This is fine, and we don’t send it home to be finished!
Activity TWO: Sum of squares
This activity takes about 30 minutes, and can be done alone or in pairs. We encourage the use of blocks.
Word doc: CW Sum of Squares
Answers: We created problem #1, but the activities 2 and 3 are from nrich.org. (answers here) :
Reminder: Not everyone finishes any given CW. We reward the class if everyone is working. Speed should not be the goal – understanding is the goal .
Activity THREE: Triangular Numbers and Squares
We start this activity with a Power Point: CW tri sq
Then we start the CW, which takes 25 – 30 minutes, and can be done in pairs. We encourage the use of drawings and blocks.
Word doc: CWtrian and squares
Key: key cw tri sq
Reminder: Not everyone will get to the challenge problems. They’re there for those who need them!
Activity FOUR: Primes and Squares
We start this activity with a Power Point:CW primes and squares
These slides introduce and clarify the expectations for the investigation. We have noticed that students know what productive group work looks like, but don’t always remember its importance. Slide 1 brainstorms the expectations the students have of themselves and others. During the following investigation, we circulate and praise or record when we hear good group work.
Investigation CW: There are 2 parts. Question 1 uses this recording sheet (key attached). The primes are listed, since students do not yet know their primes well.
Question 2: Each pair will need a set of the photocopied squares: CW fit in squares
Each pair will also need a large piece of paper (11 x 17″) to arrange the squares on. They can share the job of cutting out and then moving the squares around until they form a rectangle. This is quite difficult, and we give a clue after a few minutes: the 18 square should go in one of the corners. has to be built. Be sure to warn them that the rectangle does not fill up the WHOLE paper.
Finally, if needed, we’ll tell them the 9 square goes in the opposite corner. This activity is meant to be a little frustrating – go ahead and have that discussion about how they felt! We can TALK about mindset, but it is really a muscle that needs to be built up, day by day, challenge by challenge.