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Use super cute fruit to introduce algebraic thinking to your young students.
Students will find factors of 161,000 and match them up with a first name.
First, students will answer some warm-up questions about 161,000.
Next, they’ll break 161,000 down into its prime factorization.
Now we begin working with those prime factors to figure out the name(s).
I reveal the answers and issue several new challenges. Can you find a smaller code than IAN?
Students will use the factors of a number to turn that number into words.
First, students encode PIE, BREAD, and BEARD.
Then they’ll find other words that would encode to 720.
Students explore the “Forbidden Letters” of 720, starting with G.
Finally, they look for the longest 720 words possible. BARED, FABLE, AIDED, and DEBAR are other 5-letter words. BALBOA and BABIED are two 6-letter words.
Students will add sets of parentheses to expressions to see how large of a change they can create.
First, I model the process and then give them a sample to try: 7 × 3 + 7 × 2
I show my best answer (see below) and then present four more samples.
I reveal my best answers, then ask students to think about what patterns or rules they spotted to help them place their parentheses. They write these out as 3 to 5 tips to create a guidebook.
In this video, students explore the relationship between multiplication and its inverse, division. They will attempt to “undo” multiplication by dividing once, twice, or even three times.
We begin with the simplest, one-step case. Multiply by a number, then divide by the same number, and you return to where you started. But then, we try a two-step version, such as: 10 × 6 ÷ 3 ÷ 2 = 10.
Now we look at cases where we divide three times to undo multiplication.
Finally, students have a chance to continue practicing this idea with a web app.
Students factor 365 in an attempt to create a better system of months and weeks than our current calendar.
Students are asked (by the Supreme Council for Calendars) to clear up the confusing mess of 28, 30, and 31-day months by factoring 365.
Students will discover that 365 has very few factors: 1, 5, 73, and 365 — not so great for even months. We introduce intercalary days: days outside of any month. Students reserve a few days for a special holiday, then create nice, even months.
Now students will divide those months into even weeks so that the year and each month have nice, whole numbers of weeks.
Finally, we do some naming: what will we call our months? What about our days of the week?
Students will double a single dollar once per day and discover how long it takes to reach $1 million. Along the way, they’ll move from repeated multiplication to using exponents.
Students will use the patterns they find in the first few steps to predict a step much further along.
Students begin by predicting how many pieces there will be at step 5.
After revealing that step 5 has 36 pieces, I ask students to predict step 12.
I reveal that step 12 has 85 squares and challenge students to predict any step. Answers: Step 49: 344, Step 93: 652, Step 728: 5,097
Students will use the patterns they find in the first few steps to predict a step much further along.
Students begin by looking for patterns and predicting the number of squares at step 5.
I reveal the solution (it’s 24) and students work on predicting step 12.
After revealing the solution (it’s 52), I reveal an algorithm to predict any step and challenge students to calculate much larger steps. Answers – Step 63: 64 × 4 = 256 – Step 97: 98 × 4 = 392 – Step 821: 822 × 4 = 3288
Students will use the patterns they find in the first few steps to predict a step much further along.
Students predict the number of squares at step 5.
I reveal the answer (it’s 29) and then ask them to predict step 12.
After confirming that step 12 has 64 squares, I ask students to predict any step! Answers – Step 47: 239 – Step 111: 559 – Step 998: 4994
Students will use the patterns they find in the first few steps to predict a step much further along.
Students predict how many squares there will be at step 5.
I reveal the solution (it’s 29) and ask students to predict step 12.
After showing the answer (it’s 64), I challenge students to predict any step! Answers – Step 50: 453 – Step 101: 912 – Step 999: 8,994
Students will use the patterns they find in the first few steps to predict a step much further along.
Students predict the number of pieces at step 5.
I reveal the solution (32) and ask students to predict step 12.
After revealing that step 12 has 74 pieces, I challenge students to predict any step. Answers – Step 50: 302 – Step 100: 602 – Step 999: 5,996
Students will use the patterns they find in the first four steps to predict the 50th step.
Students look for patterns and predict how many Xs will be at step 10.
I reveal the solution (it’s 31) and ask students to predict step 50.
After showing my solution (there are 151 Xs) I challenge students with a new formation of Xs and Os. (The secret: in this version, you start with 4 Xs and add 12 Xs at each step.)
Students will use the patterns they find in the first few steps to predict a step much further along.
Students begin by just counting and making a prediction for step 6 in the pattern.
After revealing the answer (it’s 48), students will make a prediction for step 12.
I reveal that step 12 has 168 squares and then challenge students to predict much larger steps. Answers – Step 49: 2,499 – Step 99: 9,999 – Step 999: 999,999
Students will use the patterns they find in the first three steps to predict the 50th step.
Students will look for patterns and predict how many squares there will be at step 10.
Next, they’ll predict how many squares there will be at step 50!
I reveal the answer (2,550) and propose the challenge of adding the first 50 odd numbers. (Psst, you can square 50).
Students will use the patterns they find in the first three steps to predict the 20th step.
Students will note patterns and look for how many slices there will be at step 6.
After checking the answer (it’s 64), we’ll extend the pattern and ask students to predict step 20.
I reveal the answer (1,048,576) and then propose a challenge! What if you split pieces three ways instead of two, tripling the pieces at each step?
Students will use the patterns they find in the first four steps to predict the 100th step.
Students will look for patterns and then predict how many seats there will be at step ten.
Next, they predict how many seats they’ll have at 100 desks!
I reveal the answer (402) and propose an extension involving non-rectangular desks.
Students will use the patterns they find in the first four steps to predict the 100th step.
First, students count the squares in each step, search for three patterns, and predict Step 5.
Next, they use their patterns to predict Step 10.
Finally, students try to predict Step 100.
We review the answer and I introduce two extensions: Triangular Numbers and Carl Friedrich Gauss.
Students will use the patterns they find in the first four steps to predict the 50th step.
First, students count the squares in steps 1 through 4. They’ll identify patterns and make a prediction about step 5.
We look at the patterns, unveil the truth about step 5, and students try to predict step 10.
We unveil the number of squares in step 10 and then challenge students to predict step 50!
Students see if their predictions were correct and I reveal a final pattern as well as the name of this sequence of numbers.
Students will make mathematical predictions about an infinitely repeating sequence of triangles.
First, students simply count up triangles in the first three steps (1, 3, and 9). Then they predict what the 4th step will be like.
We reveal that step 4 has 27 triangles. I ask students to predict step 6.
After revealing that step 6 has 243 triangles, we will try to predict all the way up to step 20!
I discuss the pattern of repeating 3s and show how exponents are the key to quickly finding any step. And students confirm that, yes, step 20 has over a billion triangles!