Our first unit is a twin unit of Fractions (mostly visual) and Word problems. We begin working with both these topics very early in the year, so that we have all year to incorporate them into every unit. We find that word problems are an excellent way to teach fractions, too. Our next two units will be place value, patterns and multiplication, after which we will return to fractions in more depth.
This introduction to 5th grade word problems starts at the concrete, as it should.
- Pass out a handful of rainbow cubes (smallest base-10 blocks) or similar per child.
- Have each child count out 11 blocks, and set the others aside.
- Project or write up Problem 1.
Problem ONE: Marcia has 3 blocks more than Sam has. Together they have 11 blocks. How many blocks does each have?
- Say “Solve this with the blocks, but watch your process, and be ready to report back to the class on how you solved it.”
- This will go pretty quickly, given the blocks. When everyone is finished, discuss methods. We reward the class if they can find THREE DIFFERENT ways to solve any problem. Here are typical strategies they share:
- Guess and check (moving blocks until it works)
- “I set 3 aside because Marcia has 3 more. Then I divided the rest into 2 groups – that made four each. Then I gave Marcia back her 3, so 7 and 4.”
- “I said ‘One for Marcia, one for Sam, one for Marcia, one for Sam, one…. Until there were only 3 left, and gave them to Marcia.”
- Be sure and listen for even more strategies. This step can’t be skipped.
- This time have students count out 19 blocks.
- Project or write up Problem 2.
Problem TWO: Use 19 blocks. This time, Marcia has 1 more than twice as much as Sam. Together they have 19 blocks. How much does each have?
- Say “Again, solve this with the blocks, but watch your process, and be ready to report back to the class on how you solved it.”
- The outcomes will vary more here. Some students will solve it quickly, others will need more time. We often use this strategy to even it out: as students finish, we ask them to go to the board and write or draw out their solution. We might have 8 or 9 students at the board, so they have to share the space, by not writing too large. These ‘speedy’ students often struggle to communicate their thinking in writing, so this slows them down and helps them grow in that skill.
- The most difficult part of this problem is the “1 more than twice as much” part. Some students will use guess and check for a long time before they see the relationship of 2 to 1. This is normal, and can’t be rushed.
Marcia ¤¤¤¤¤¤¤ ¤¤¤¤¤¤ ¤
The class is doing great if everyone can get this far. Some might end up getting help from a neighbor. As your fastest students write at the board, and others are still struggling, circulate and ask students to explain their blocks to you. If someone needed help, but now has the blocks laid out correctly, ask them “Are you sure? Can you show me WHY that works? Does Marcia really have twice as many plus one?” If they can put it into words, ask how many groups of six there are IN TOTAL (3; two for M, one for S).
End with a short discussion of the explanations on the board. Ask students who are sitting to give feedback. “Mine looks like Tami’s”, “I don’t understand why Sophie wrote 18÷3 = 6.”
Have Sophie explain her thinking visually. Does color help, as in the solution above?
Reward the class for their perseverance and good thinking. (We use a marble jar that gives them extra recess when it’s full)
PROBLEMS #3 through #6, Plus Our Take on DIFFERENTIATION
- Pass out the Classwork worksheet with the problems #3 – #6.
- Tell students they can use blocks or not use blocks, but they have to record their thinking.
- Stress that students should work AT THEIR OWN SPEED. As long as they’re working, and showing work, they won’t be penalized. (we don’t take off points or send unfinished work home.) Lack of speed can be demoralizing, even traumatizing at this age.
- Remember, we’re looking for understanding, not just correct answers. We all work at our own speed. You can’t force neurons to connect! It takes the time it takes, but it’s worth it to build conceptual understanding at this age.
- Collect the worksheets. Evaluate them on work shown on the ones they did. Use a management tool other than speed assessment to keep students on task.
- Group work?
- We believe tomorrow’s world will require strong collaborative skills, so yes, we encourage group work. This activity seems to work best in pairs, with partners somewhat close in rates of working speed.
- We have a few age-appropriate stuffed animals (a dragon, a moose, a unicorn). We tell the students we’re looking for excellent collaborative work (ask students and record their answers: what will that sound like?), and when we hear it, that group gets the stuffed animals for a couple minutes. They love this, and they usually keep working, with pride. Collaborative language is something students can control, unlike working speed, which is developmental.
- Collect? We do collect the work done on these problems, but we’re looking for good form, for work shown, for evidence of thinking.
- What does this work tell us?
- Assessment in a Singapore Math classroom has to be ongoing. Collecting class work gives us a snapshot of where a student is at that moment. Since the road to learning math is CONCRETE > PICTORIAL > ABSTRACT, their work gives us insight into their journey along that road.
- The child who is still using blocks to answer questions #3 and #4 is building understanding at the concrete. They often don’t finish more than these 2 problems, which is fine. We cannot rush this; we cannot force neurons to connect. We make a note of this progress, and continue to provide concrete learning experiences. We have years of graduates from this group of students who go on to high school and do well, because they have learned how they learn. They might not take the AP Calculus test in 12th grade, but neither do they drop math or fail math. They can go on to college with a strong background in high school math, based on solid conceptual understanding.
- The child who can draw a model (based on their visualization of blocks) has mastered a pictorial tool that will serve them well all the way through algebra. They have translated abstract words to concrete blocks and then to semi-abstract drawings that have meaning. They can attack any word problem with these skills. We make a note of this progress – our grading system should reflect a student’s progress through the levels of learning, not their ranking in the class.
- The student who can find an answer mentally, or with convoluted calculations, or by guess and check shows a preference for abstract. In the past, this was considered the best – even only – path to learning math. There is nothing wrong with this approach, certainly. We DO require these ‘racers’ to draw a model occasionally. This gives them a visual underpinning for their understanding that they otherwise would not have. The inability to explain one’s own thinking can be a deficit in today’s workplace, so we help them set goals around neatness, showing work, and good form. Beware, however; there is a delicate balance to be found: enforce those requirements gradually, without killing the students’ confidence and love of math. This student enjoys struggling with problem #6.
Attachments: Power Point of Problems #1 and #2, Worksheet of Problems #3 – #6, and answers to all six problems.