Today’s warmup refers to a longterm problem solving “brainstretcher” homework consecutive sums we have been working on for a while in previous lessons (Number System Assessment & Garden Design ) called consecutive sums. They have been trying to figure out which type of numbers can not be made by adding positive consecutive whole numbers. In their warmup they are reminded that in a previous warmup (break it down) they described the number 16 in several ways:
I now ask my students what 16 has in common with 8 and 32. I expect them to conclude that 8 and 32 are both even like 16 and they should automatically start factoring and trying to use exponents to represent both numbers. They should conclude fairly quickly that 8 and 32 are both powers of 2. If they don’t notice this I would ask students to describe the numbers 8 and 32 like they did for 16. Someone may see the connection to the consecutive sums work we have done, because the numbers on the “can’t make” list include 8, 16, & 32, in which case we will work now on the problem solving section instead of the white board section. If not, we will proceed to white board practice of simplifying multistep expressions.
Students are also told to double check their homework simplifying multistep expressions with their math families and to make sure they explain why they think a particular way is correct or why they think a mistake was made. I circulate to check two answers and give specific feedback. See: Common mistakes multistep simplifying
I may tailor these practice problems so that students are practicing the ones that were hard for them. Sometimes I may give them a choice of challenge levels. Before seeing what they struggle with I would predict that the following might be good practice problems. I may not do all of them or I may give choices depending on challenge level students want.
Some of the common mistakes I look for and hope to work on are following the correct order of operations and distributing the multiplication before combining like terms (adding). I also look to make sure they are distributing throughout the parentheses and not combining constant numbers with variable terms.
5(2x + 1) + 2x 5x + 3(3 + x) 2 + 4(2n + 3) 3(2x + 1) + 4
2 + 4(x + 3) + 3x 2(3x + 1) + 4(x + 2) 6(2n + 1) + 3(2+ c)
As we go over each one I would be sure to reference order of operations and continue to remind them that a number outside parentheses must be multiplied by all terms inside. It may help to tell them that the only reason we are doing the multiplication this way is because we can't do the addition inside the parentheses because they are not like terms.
This is a long term problem we have been working on together in previous lessons (Number System Assessment & Garden Design) to help them with perseverance and also because it engages their natural curiosity in math. Today is the day they will figure out what kinds of numbers cannot be made by adding positive consecutive whole numbers.
So far we have list of all the numbers students have been able to make and all the numbers they have not been able to make. 1,2,4,8,16, & 32 are the only ones not possible so far. I ask what similarities they find between the numbers. If they don’t notice right away that three of the numbers were in our warmup I will say “these numbers seem super familiar to me….I feel like we have seen these before recently…how have we described some of these numbers?”
I want them to notice that, except for 1, all the others are powers of 2. They may first point out that except for 1 they are all even. I would ask if they think it is even numbers that cannot be made, but they will notice all the even numbers that we were able to make. This may give rise to some additional questions from students that may warrant exploration like:
If they don’t come up with the last one I would ask so they could check it and see if anyone can make 64. Some students might want to make sure we can make all the ones in between.